boundary element methods in heat transfer

Boundary Element Methods in Heat Transfer

International Series on Computational Engineering

Aims:Computational Engineering has grown in power and diversity in recent years, and for the engineeringcommunity the advances are matched by their wider accessibility through modern workstations.The aim of this series is to provide a clear account of computational methods in engineering analysisand design, dealing with both established methods as well as those currently in a state of rapiddevelopment.The series will cover books on the state-of-the-art development in computational engineering and assuch will comprise several volumes every year covering the latest developments in the applicationof the methods to different engineering topics. Each volume will consist of authored work or editedvolumes of several chapters written by the leading researchers in the field. The aim will be to providethe fundamental concepts of advances in computational methods as well as outlining the algorithmsrequired to implement the techniques in practical engineering analysis.The scope of the series covers almo>t the entire spectrum of engineering analysis. As such, it willcover Stress Analysis, Inelastic Problems, Contact Problems, Fracture Mechanics, Optimization andDesign Sensitivity Analysis, Plate and Shell Analysis, Composite Materials, Probabilistic Mechanics, Fluid Mechanics, Groundwater Flow, Hydraulics, Heat Transfer, Geomechanics, Soil Mechanics,Wave Propagation, Acoustics, Electromagnetics, Electrical Problems, Bioengineering, KnowledgeBased Systems and Environmental Modelling.

Series Editor:

Dr C.A. BrebbiaWessex Institute of TechnologyComputational Mechanics InstituteAshurst LodgeAshurstSouthampton S04 2AAUK

Editorial Board:

Professor H. AntesInstitut fur Angewandte MechanikTechnische Universitii.t BraunschweigPostfach 3329D-3300 BraunschweigGermany

Professor H.D. BuiLaboratoire de Mecanique des SolidesEcole Polytechnique91128 Palaiseau CedexFrance

Professor A.H-D. ChengUniversity of DelawareCollege of EngineeringDepartment of Civil Engineering137 Dupont HallNewark, Delaware 19716USA

Associate Editor:

Dr M.H. AliabadiWessex Institute of TechnologyComputational Mechanics InstituteAshurst LodgeAshurstSouthampton S04 2AAUK

Professor D. BeskosCivil Engineering DepartmentSchool of EngineeringUniversity of PatrasGR-261l0 PatrasGreece

Professor D. CartwrightDepartment of Mechanical EngineeringBucknell UniversityLewisburg UniversityPensylvania 17837USA

Professor J.J. ConnorDepartment of Civil EngineeringMassachusetts Institute of TechnologyCambridgeMA 02139USA

Professor J. DominguezEscuela Superior de Ingenieros IndustrialesAv. Reina Mercedes41012 SevillaSpain

Professor G.S. GipsonSchool of Civil EngineeringEngineering South 207Oklahoma State UniversityStillwater, OK 74078-0327USA

Professor S. GrilliThe University of Rhode IslandDepartment of Ocean EngineeringKingston, RI 02881-0814USA

Professor D.B. InghamDepartment of Applied Mathematical StudiesSchool of MathematicsThe University of LeedsLeeds LS2 9JTUK

Professor P. MolinaroEnte Nazionale per l'Energia ElettricaDirezione Degli Studi e RicercheCentro di Ricerca Idraulica e StrutturaleVia Ornato 90/1420162 MilanoItaly

Professor Dr. K. OnishiDepartment of Mathematics IIScience University of TokyoWakamiya-cho 26Shinjuku-kuTokyo 162Japan

Professor H. PinaInstituto Superior TecnicoAv. Rovisco Pais1096 Lisboa CodexPortugal

Dr. A.P.S. SelvaduraiDepartment of Civil EngineeringRoom 277, C.J. Mackenzie BuildingCarleton UniversityOttawaCanada K1S 5B6

Professor A. GiorginiPurdue UniversitySchool of Civil EngineeringWest Lafayette, IN 47907USA

Professor W.G. GrayDepartment of Civil Engineering andGeological SciencesUniversity of Notre DameNotre Dame, IN 46556USA

Dr. S. HernandezDepartment of Mechanical EngineeringUniversity of ZaragozaMaria de Luna50015 ZaragozaSpain

Professor G.D. ManolisAristotle University of ThessalonikiSchool of EngineeringDepartment of Civil EngineeringGR-54006, ThessalonikiGreece

Dr. A.J. NowakSilesian Technical UniversityInstitute of Thermal Technology44-101 GliwiceKonarskiego 22Poland

Professor P. ParreiraDepartamento de Engenharia CivilAvenida Rovisco Pais1096 Lisboa CodexPortugal

Professor D.P. RookeDRA (Aerospace Division)Materials and Structures DepartmentR50 BuildingRAE FarnboroughHampshire GU14 GTDUK

Professor R.P. ShawS.U.N.Y. at BuffaloDepartment of Civil EngineeringSchool of Engineering and Applied Sciences212 Ketter HallBuffalo, New York 14260USA

Professor P. SkergetUniversity of MaiiborFaculty of Technical SciencesYU-62000 MariborSmetanova 17P.O. Box 224Yugoslavia

Professor M.D. TrifunacDepartment of Civil Engineering, KAP 216DUniversity of Southern CaliforniaLos Angeles, CA 90089-2531USA

Dr P.P. StronaCentro Ricerche Fiat S.C.p.A.Strada Torino, 5010043 Orbassano (TO)Italy

Professor N.G. ZamaniUniversity of WindsorDepartment of Mathematics and Statistics401 SunsetWindsorOntarioCanada N9B 3P4

Acknowledgement is made to Professor N. Tosaka for the use of figure 8.18 (isotherms) on page 262,which appears on the front cover of this book.

Boundary Element Methods inHeat Transfer

Editors:L.C. Wrobel and C.A. Brebbia

Computational Mechanics PublicationsSouthampton Boston

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Elsevier Applied ScienceLondon New York

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L.C. WrobelWessex Institute of TechnologyAshurst Lodge, AshurstSouthampton S04 2AAUK

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A Catalogue record for this book is availablefrom the British Library

C.A. BrebbiaWessex Institute of TechnologyAshurst Lodge, AshurstSouthampton S04 2AAUK

ISBN 1-85166-726-1 Elsevier Applied Science, London, New YorkISBN 1-85312-103-7 Computational Mechanics Publications, SouthamptonISBN 0-945824-86-6 Computational Mechanics Publications, Boston, USA

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@Computational Mechanics Publications 1992

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CONTENTS

Preface xi

Chapter 1 - Solving Heat Transfer Problems by the DualReciprocity BEMC.A. Brebbia, A.J. Nowak

1.1 Introduction 11.2 Steady-State Problems with Heat Sources 71.3 Transient Heat Conduction 121.4 Numerical Examples and Conclusions 18Acknowledgement 25References 26

Chapter 2 - Transient Problems using Time-DependentFundamental SolutionsR. Pasquetti, A. Caruso, L.C. Wrobel

2.1 Introduction 332.2 Boundary Integral Equation 342.3 Space and Time Discretization 362.4 Evaluation of the Coefficients of Matrices HI, H2, Gl and G2 " 372.5 Boundary Conditions 432.6 Initial Conditions 462.7 Treatment of Heat Sources 482.8 Applications 50References 60

Chapter 3 - Solving Linear Heat Conduction Problemsby the Multiple Reciprocity MethodA.J. Nowak

3.1 Introduction 633.2 Fundamentals of the Multiple Reciprocity Method 643.3 Heat Conduction with Heat Sources 683.4 Linear Transient Problems 703.5 Numerical Examples 77Acknowledgements 82References 82

Chapter 4 - Solving Nonlinear Heat Transfer ProblemsUsing the Boundary Element MethodR. Bialecki

4.1 Introduction 874.2 Applying BEM to Nonlinear Problems. General Remarks 954.3 Nonlinear Boundary Conditions 964.4 Nonlinear Material (Nonlinear Differential Operator) 1014.5 Nonlinear Source Term 1094.6 Moving Boundaries 1104.7 Conclusions 112Acknowledgements 113References 114

Chapter 5 - Coupled Conduction-Convection ProblemsL.C. Wrobel, D.B. DeFigueiredo

5.1 Introduction 1235.2 BEM Formulation for Steady-State Problems 1245.3 BEM Formulation for Transient Problems 1295.4 BEM Formulation for Variable Velocity Fields 1335.5 Conclusions 142Acknowledgements 142References 142

Chapter 6 - Solving Coupled Problems InvolvingConduction, Convection and Thermal RadiationA.J. Nowak

6.1 Introduction 1456.2 Coupled Thermal Problems with Non-Participating Medium 1476.3 Coupled Thermal Problems with Participating Medium 1586.4 Concluding Remarks 168Acknowledgement 169References 169

Chapter 7 - Advanced Thermoelastic AnalysisV. Sladek, J. Sladek

7.1 Introduction 1757.2 Governing Equations 1797.3 Fundamental Solutions 1817.4 Integral Representations of the Temperature and the Displacement

Fields. Boundary Integral Equations 1857.5 Integral Representations of the Temperature Gradients and Stresses 1907.6 Stress Tensor and Temperature Gradient on Boundary 1957.7 Numerical Solution 196

7.8 Stationary Problems in Media with Temperature DependentYoung's Modulus and Coefficient of Thermal Expansion 215

Appendix A 223Appendix B 227Appendix C 228Appendix D 229References 232

Chapter 8 - Integral Equation Analyses of NaturalConvection Problems in Fluid FlowN. Tosaka, N. Fukushima

8.1 Introduction '" 2358.2 Natural Convection Problems 2368.3 Steady Analysis 2388.4 Unsteady Analysis 2428.5 Numerical Examples 2468.6 Conclusions 256Acknowledgements 266References 266

Chapter 9 - Improperly Posed Problems in Heat TransferD.B. Ingham

9.1 Introduction 2699.2 Formulation 2729.3 Non-Linear Formulation 2749.4 Existence of Solution of Problem I 2759.5 Mathematical Models for the Solution of Problem I 2789.6 Mathematical Model for the Solution of Problem II 2829.7 Solutions of Some Test Examples for Problem I 2859.8 Solution of Some Test Examples for Problem II 2899.9 Conclusions 290Acknowledgements 293References 293

PREFACE

Heat transfer problems in industry are usually of a very complex nature, simultaneously involving different transfer modes such as conduction, convection, radiationand others. Because of that, very few problems can be solved analytically and onegenerally has to resort to numerical analysis.

The boundary element method is a numerical technique which has been receivinggrowing attention for solving heat transfer problems because of its unique abilityto confine the discretization process to the boundaries of the problem region. Thisallows major reductions in the data preparation and computer effort necessary to solvecomplex industrial problems.

The purpose of this book is to present efficient algorithms used in conjunctionwith the boundary element method for the solution of steady and transient, linearand nonlinear heat transfer problems. It also aims to reflect research being carriedout by several active groups around the world, and its chapters have accordingly beenwritten by scientists working in renowned centres of excellence.

The first three chapters all deal with transient heat conduction using alternativeboundary element formulations which require boundary discretization only. Chapter1 presents the dual reciprocity technique which is attracting considerable interestbecause of its ability to transform domain integrals, resulting from effects such asinternal heat generation, into equivalent boundary integrals. The technique is generaland is applied in this chapter to steady and transient, linear and nonlinear problems.A more traditional approach using time-dependent fundamental solutions is describedin chapter 2. Also included is a discussion on the treatment of some types of initialconditions and internal loadings by equivalent boundary integrals, and an efficientconvolution-type time-marching scheme. The multiple reciprocity method is describedin chapter 3. This method may be seen as an extension of Galerkin-vector techniquesfor non-harmonic loads, and can also be applied to transient problems.

Chapter 4 deals with nonlinear heat transfer problems. The types of nonlinearitydiscussed include those of material, boundary conditions, heat sources and movingboundaries. Several practical examples of application are presented, and areas pointedout where further research is still necessary.

Boundary element solutions to the convection-diffusion equation are the subjectof chapter 5. The fundamental solution to the steady-state equation with constantcoefficients is employed, and features such as transient effects and variable parametersare accounted for by using dual reciprocity approximations. Coupled problems are alsodiscussed in chapter 6 which deals with heat transfer involving conduction, convectionand radiation in enclosures. The formulation developed can be implemented intostandard boundary element codes, and is equivalent to introducing a new fundamentalsolution. Special consideration is given to the resulting set of nonlinear equationswhich is solved by an efficient pre-elimination technique employing the Gauss-Jordanalgorithm.

Chapter 7, on thermoelasticity, starts with a brief classification of thermoelasticproblems; next, it defines the fundamental solution for the Laplace transforms in general coupled thermoelasticity, and the time-dependent fundamental solutions when

these are available. A pure boundary formulation is then given for both the Laplacetransform and the time-dependent fields, and boundary integral equations writtenin an advanced regularized form without any singular integral. Finally, a BEM formulation for solution of stationary problems in media with temperature-dependentYoung's modulus and coefficient of thermal expansion is presented.

Natural convection in fluid flow is the subject of chapter 8. This chapter is acollection of recent results obtained by the authors using an integral equation methodbased on boundary-domain discretization for solving two-dimensional thermal convection problems. The formulation uses the primitive variables, i.e. velocity and pressure,and constructs fundamental solution tensors for the differential operators corresponding to a linearized set of equations. Approximate solution procedures of the nonlinearsystem of integral equations are derived based on Newton-Raphson techniques.

The last chapter deals with inverse heat conduction problems. Three differentmathematical models, namely direct, least squares and minimum energy methods,are presented for two Laplace-type problems. It is found that the minimum energymethod always gives a good, stable approximation to the solution, whereas the directand least squares methods do not.

We are indebted to all the authors for their contribution, patience and continuous support during the production stages of this book. Special thanks are due toMs. Christine Seward for the excellent work in the preparation of the final manuscript.

Luiz C. WrobelCarlos A. Brebbia

February 1992

Chapter 1

Solving Heat Transfer Problems by the DualReciprocity HEM

C.A. Brebbia (*), A.J. Nowak (**)(*) Computational Mechanics Institute, Ashurst Lodge, Ashurst,Southampton S04 2AA, U.K.(**) Institute of Thermal Technology, Silesian Technical University,44-101 Gliwice, Konarskiego 22, Poland

1.1 Introduction

Background

The impressive success of the Boundary Element Method [1],[2] in the last decade hasbeen mainly due to the fact that the technique became applicable to a wide rangeof engineering problems e.g. [3-8]. The boundary value problems of heat transfer areamong those which can be effectively solved by BEM. For many practical cases, e.g.linear and non-linear steady-state heat conduction, some problems of thermal radiation and many others, BEM can result in a simple and elegant formulation. However,some heat transfer problems lead to an integral equation with domain integrals. Thepresence of these integrals diminishes the efficiency of the method and makes the datapreparation needed for the computer codes much more difficult.

The development of methods for transforming these integrals into boundary oneswas a milestone in the rapid advance of BEM [9-12]. The most effective approaches arebased on applying the reciprocity theorem. Two different ways exist for accomplishingthis transfcrmation.

The integrand of domain integrals is generally a product of two functions. Thefirst function, say ~, depends on the problem under consideration whereas the secondone is the fundamental solution u·. The first approach consists in approximating thefunction ~ by a set of functions to which the reciprocity theorem is then applied. Thisapproach proposed by Nardini and Brebbia [13-15] and then considerably developed byBrebbia and Wrobel [16],[17] is called the DUAL RECIPROCITY METHOD (DRM).Its efficiency and accuracy has already been proved in many numerical experiments.

In the second approach the fundamental solution u· instead of the function ~, istransformed. As a consequence, this method introduces a sequence of higher orderfundamental solutions. Since the reciprocity theorem is applied to each term of this

2 Boundary Element Methods in Heat Transfer

sequence the method has been named the MULTIPLE RECIPROCITY METHOD(MRM). The exact solution of the problem is expressed as a series of boundary integrals only [18],[19] (cf. chapter 3).

In this chapter, only the Dual Reciprocity Method will be discussed in detail.First the fundamentals of the method are presented. The basic formulation is thenparticularized to the type of heat transfer problem under consideration. Special casessuch as steady-state problems with heat sources as well as transient heat conductionare discussed in the following sections. It should be stressed however, that the methodis not limited to solving only these type of problems and can easily be extended tomany other types of heat transfer problems which lead to integral equations containing domain integrals.

Fundamentals of the Dual Reciprocity Method

As stated in the previous section, the Dual Reciprocity Method is a general methodof transforming all kinds of domain integrals into equivalent boundary ones. In theBEM formulation of heat transfer problems the type of resulting domain integralsdepends on the class of fundamental solution used when second Green's theorem (orreciprocity theorem) is applied to transform the boundary value problem into anintegral equation. In this chapter the fundamental solution of Laplace's equation willbe used

(1.1 )

where k is a thermal conductivity, u" is the fundamental solution and ~i is Dirac'sfunction acting at point i. Thus the resulting integral equation has the following form

Ci Ui +t u q" df =t q u" df +D (1.2)

where u is a potential (temperature), q is the heat flux density defined as q =- k au/an, q" is the heat flux density analog q" = - k au" /an and Ci is a constant dependent on the geometry of the boundary at the point i [1], [2]. D is thedomain integral having the most general form

D = in k <P u" dfJ (1.3)

where <p is a problem dependent function. For steady-state heat transfer problems <p

represents distribution of heat sources within the body and varies only with positionwhen the problem is linear. For non-linear problems this function also becomes solution dependent. For transient heat conduction, <p represents the derivative of thesolution (temperature) with respect to time.

The Dual Reciprocity Method is based on approximating the function <p by a setof coordinate functions jJ(x) as follows [15],[16],[17]

NP

<p = 'L,jJ(x) oJj=l

(1.4)

(1.5)

Boundary Element Methods in Heat Transfer 3

N P represents the number of functions considered in the set. This number is equal toN P = N +P, where N stands for the number of boundary nodes and P is the numberof selected internal nodes (in particular P can be equal to zero). The coefficients a i

which are still undetermined, depend on the problem. For steady-state problemscoefficients ai are constants whereas for transient heat conduction they are functionsof time.

Introducing Eq. (1.4) into (1.3) one obtains

NP

D = E ai i k Ii u* dO.i=l fl

In order to apply the reciprocity theorem in the domain integral D, the functionsIi have to be replaced by the Laplacian of new function, say iJ,i

V 2iJ,i = IiOne can then substitute the above into Eq. (1.5) which yields

(1.6)

NP

D =E ai 1k V 2Ui u* dO. =i=l fl

= I:ai { [ (k u*aui

_ k ui au* )dr +i=l ir an an

+k k V 2u* ui dO.} (1.7)

where aO/an is an outward normal derivative. Taking into account the definitionof the fundamental solution (1.1) one obtains for the domain integral the followingexpressIOn

(1.8)

(1.9)

. auiwhere q) = - k an'

Introducing Eq. (1.8) into Eq. (1.2) one arrives at the following boundary integralequation

CiUi + lr (q*u - u*q) dr =

NP=Eai{Ciu1+ [(q*ui-u*qi)dr}

)=1 i r

This equation can now be discretized. In order to accomplish this, the boundaryr is subdivided as usual into a number of elements. The problem variables u and q arethen approximated within each boundary element using the appropriate interpolationfunctions tP [1], [2]


and q = t/JT qe (1.10)

Since functions uand qare known functions of space the integrals in the summationshown in Eq. (1.9) can be calculated directly. However, the same type of approximation as used for the unknown functions u and q can also be applied to the functions uand qand this saves considerable computing time. Thus, Eq. (1.9) when applied to allboundary nodes and internal poles can be expressed in terms of standard boundaryelement influence matrices Hand G

H U - G Q = [H iJ - G QJ 0: (1.11)

where vector 0: contains the values of (Xi for j = 1,2, ... , NP. The values of functionsui and qi at nodal points form vectors iii and qi. These vectors are subsequentcolumns of two-dimensional matrices iJ and Qin Eq. (1.11).

The last step of the analysis is to express the vector 0: in terms of the originalvariables of the physical problem. To accomplish this, Eq. (1.4) is written for allconsidered points

~ = Fo: (1.12)

where vector ~ contains the values of function <p at these points, and F is a matrixformed by the values of functions Ii at nodal points.

Assuming that the matrix F is nonsingular, Eq. (1.12) gives after inversion

0: = F-1 ~ = E~ (1.13)

Substituting Eq. (1.13) into (1.11), after some simple algebra manipulation onearrives at the following boundary only formulation

with matrix C defined as

HU-GQ=C~

C = [H iJ - G QJ E

(1.14)

(1.15)

and called the diffusion (or capacitance) matrix for the case of transient problems.

Internal polesIn many practical situations like transient problems with a small number of degreesof freedom, problems governed by Helmholtz equation with a large wave number,accuracy of the formulation based only on boundary points (P in Eq. (1.4) equalsto zero) is not satisfactory. To overcome this difficulty additional selected internalpoints (called internal poles) are incorporated into analysis [16]'[17J. The values ofu function at these internal poles are calculated simultaneously with the functions uand q determined at the boundary nodes. Thus, approximation (1.4) represents theinternal effects much better.

The number of internal poles is recommended to be small. Usually one or twopoles guarantee high accuracy of the formulation. Internal poles should be chosen as

(1.16)


central as possible and should not coincide with any other point arising in approximation formula (1.4).

Solution at internal pointsOnce Eq. (1.14) is solved for the unknowns at boundary nodes and internal poles, values of potential u at any internal point (if needed) can be computed from relationship

Ui +1(q* u - u* q) dr =

NP

='L ai {u1+ r (q* ui - u* qi) dr})=1 i r

Heat fluxes at internal points are computed by differentiation of the above equation with respect to the coordinates of the internal points.

Approximation Functions

The efficiency and accuracy of the Dual Reciprocity Method depends on the choiceof approximation functions Ii. They should satisfy the following requirements:

i) the summation in Eq. (1.4) should represent the original function <P accurately

ii) each function Ii of the sequence has to be such that Eq. (1.6) can be solvedanalytically to obtain the set of functions ui

iii) matrix F in Eq. (1.12) has to be nonsingular

Many sets of functions fi satisfy the above conditions. The one usually recommended[15],[16],[17] contains functions which are in line with the behaviour of fundamentalsolution of Laplace's equation. These functions are the geometrical distance' R' between the considered point 'j' (called pole or source point) and any other point in n(called field point), i. e.

(1.17)

(1.18)

Functions (1.17) constitute a sequence of linearly independent functions (providedthat no boundary or internal pole coincide with any other) and hence matrix F in Eq.(1.12) is not singular. The functions fi produce the following sets for ui and qJ

A)· 1 R3U =-

d

qi = _ ~ R2 oR (1.19)d an

where d = 9 for 2 - D problems and d = 12 for 3 - D problems.For axisymmetric problems the fundamental solution u* depends not only on the

distance between points but also on the distance from the source and field points tothe axis of revolution. This implies that the form of fi function should be as follows[20]


. rjjl = (1 - -) R(x, x j )

4r(1.20)

where ri stands for distance from the pole to the axis of revolution whereas r is adistance from any field point in n to the same axis. Functions iJ) and if are given byformulae (1.18) and (1.19) assuming d = 12.

For infinite regions the function Ii must vanish on roo in order to eliminate theinfluence of integrals along this surface. Loeffler and Mansur [21] proposed for 2 - Dtransient potential problems functions fi defined by the expression

jl(x) = 2C - R(x,xi)[R(x, Xi) +C]4

C is an arbitrary constant satisfying the relationship

C ~ 50 3Va t1l t

(1.21)

(1.22)

where a is thermal diffusivity, t1l is the length of the smallest element and t is thetotal time in analysis.

In order to fulfil the requirement regarding the accurate representation of function</> by formula (1.4) the following power representation can be used [11],[22]

(1.23)

where 13m are arbitrary coefficients.This generates functions ui and if given by expressions

where

d = (m +3)(m +3)

and

d= (m+4)(m+3)

for 2 - D problems

for 3 - D problems

(1.24)

(1.25)

The approximation functions fj for internal poles are generally the same as thoseproposed for boundary nodes. However, it was found that function Ii = constant,better simulating heating up of the whole body by a constant heat source, should beincluded [17]. Corresponding iJ) and Iji functions, in this case, are

. 1iJ,J = - R2 (x x·) constantd 'J (1.26)


. 2 oRi/ = d R(x, Xj) constant an (1.27)

in which d = 4 for 2 - D problems and d = 6 for 3- D problems. For axisymmetricalproblems these functions have the forms

.1 2 1iJ,J = (6 R (x,Xj) + 3r rj) constant (1.28)

. 1? = 3(r nr + r rj nz ) constant (1.29)

where n r and n z are direction cosines of the normal to the boundary at point x.To avoid ill-conditioning of matrix F the constant employed above should have ageometrical dimension. Usually the maximum distance between boundary nodes isassumed.

Although the above sequences of fj, iJ) and if functions are the most popular,others can be postulated and some of them can produce better results for particularproblems. An idea about the influence of Ii functions on the accuracy can be deducedfrom results published by Aral and Tang [23].

Some remarks on the functions Ii required for transient problems as well as forthe Helmholtz equation are given in the relevant sections of this chapter.

1.2 Steady-State Problems with Heat Sources

Formulation

In this section, details regarding application of the Dual Reciprocity Method for solving steady-state heat transfer problems with heat sources are discussed. Analysisstarts with linear boundary value problems which are governed by Poisson's equation

inO (1.30)

where b represents heat source generation rate and k is heat conductivity. In manypractical situations heat conductivity k is a constant within the domain 0 and b is asimple function of coordinates (e.g. b is a constant or linear function). Thus domainintegral D (Eq. (1.3)) having the form

D = in b u* dO (1.31)

can be transformed into boundary directly using reciprocity theorem. In general, whenb is position dependent, solution of the problem (1.30) results in Eq. (1.2) providedfunction ¢> in domain integral D is replaced by quantity bj k

D = r k b(x) u* dO1n k

Thus approximation formula (1.4) can be written as

(1.32)

(1.33)


b(x) =~ F(x) (Xi

k i=l

where coefficients (Xi are unknown constants.Following all steps of analysis described in the section on Fundamentals of the

Dual Reciprocity Method, one arrives at the final matrix formulation of the problem

HU - GQ = [HU - GQ] F-1 B = - CB (1.34)

where vector B contains values of expression b(x) / k calculated at nodal points.Since all matrices on the right hand side of Eq. (1.33) are known, one can perform

appropriate multiplication to obtain the following set of equations

HU-GQ=R (1.35)

The last step of the analysis is to build up the final set of equations. To accomplishthis, boundary conditions prescribed on the boundary r have to be considered [1],[21.After simple algebra manipulation one can obtain

AX=P (1.36)

where matrix A contains appropriate columns of matrices H or G (depending onboundary conditions) whereas vector P comes from vector Rand nonhomogenousterms of boundary conditions [1],[2]. Eq. (1.36) can be solved using any linear equation solver such as the Gaussian scheme.

Non-Linear Problems

In steady-state non-linear heat transfer problems with heat sources, the followingtypes of non-linearities can be distinguished

i) non-linear boundary conditions, e.g. free convection, thermal radiation on theboundary

ii) non-linear material behaviour, i. e. material properties are temperature dependent

iii) non-linear heat source term, i.e. temperature dependent heat source generationrate b

iv) problems with moving boundaries, mainly due to phase change

Obviously, often more than one source of non-linearities can occur.The first type of non-linear problem leads to Eq. (1.36). Although the set of

equations becomes non-linear, the whole nonlinearity affects only vector P. Thus, theset of equations can easily be solved iteratively, e.g. [24],[25],[26].

When nonlinearity of the problem is caused by the behaviour of material, the veryfirst step of the analysis is to apply Kirchhoff's transformation [24],[1]. This techniquetransfers non-linearity from differential operator to the boundary conditions (namely


Figure 1.1: Geometrical definitions.

to the boundary condition of the third kind). In other words, this technique changesthe type of non-linearity from type ii) to the type i).

When heat source generation rate depends on the temperature u the Dual Reciprocity Method can be applied directly. However, it should be noticed that the valuesof b function at nodal points are not known as they depend on the temperature u.As a consequence, the solution has to be obtained iteratively. To begin the iterationloop, an initial guess of the temperature field is required. When distribution of temperature is known, the vector B in Eq. (1.35) and then the vector R in Eq. (1.36) canbe determined and the next iteration can be found as a solution of Eq. (1.36). OnceEq. (1.36) is solved, vector B is updated and calculations are repeated until requiredaccuracy is achieved.

When both sources of nonlinearities, i.e. non-linear material behaviour and nonlinear heat source take place simultaneously, Kirchhoff's transformation as the firststep of the analysis is recommended. After this transformation the problem becomesof the previously discussed type and can be solved as described above.

Methods of solution of the boundary value problems with moving boundaries aswell as comprehensive survey of possible approaches to deal with other non-linearproblems are described in chapter 4 of this volume [27].

Partially Distributed Heat Sources

In many engineering problems heat sources are present only in the part of the domain.For example electric current causes heat sources but only in that part of the bodywhich is an electric conductor. Such a situation is schematically shown in Fig. 1.1where heat sources act only within the shadowed region nb. In the remaining partn - nb heat source generation rate is equal to zero.

(1.38)


The most straightforward approach to solve these kind of problems is to treat themas problems with two subregions. Both boundaries rand r b are divided into elements,influence matrices Hand G are computed for both subregions and after taking intoaccount the boundary condition of the fourth kind (continuity requirement of u andq) on the boundary r b the final set of equations is built up. The main disadvantage ofsuch an approach is that unnecessary unknowns on the boundary r b are introducedinto analysis. That is why alternative approaches were proposed by Niku and Brebbia[22] and by Azevedo and Brebbia [12]. They consist in considering the whole domaino as one region only. The resulting integral equation has the form of Eq. (1.2) inwhich domain integral D can be written as

D = 10 k ~ u* dO = lob k ~ u* dO b (1.37)

Using the DRM this integral is transformed into an equivalent boundary integral. It should be noticed however, that this time the right hand side of Eq. (1.9) iscalculated as an integral over r b only

Ci Ui +1(q* u - u* q) dr =

NP= La

j{Cib it{ + r (q* it j - u* qj) dr}

j=1 Jrb

Hence, if temperature is calculated at a point which lies outside the region Ob , constant Cib in the right hand side of Eq. (1.38) is equal to zero.

Solving the Helmholtz Equation by the DRM

Solution of the boundary value problems of heat conduction by the method of separation of variables or by integral transforms results in the Helmholtz equation. Thisequation also governs other types of heat transfer problems such as those involving thermal waves, heat transfer in fins and others. Hence, it is of great practicalimportance to develop an efficient approach to solve the Helmholtz equation. In astraightforward technique proposed mainly by Kobayashi, Niva and co-workers [28]solution of the Helmholtz equation is expressed in terms of complex fundamentalsolutions, i.e. Hankel's functions. Notice that the Helmholtz equation

(1.39)

can also be treated as a steady-state heat conduction equation with generation rateof the heat source being proportional to the solution u. Hence, the DRM can be usedto transform the resulting domain integral D into the boundary, as described in thesection on Non-Linear Problems. In order to do this, one is required to assume thefollowing approximation

NP/12 u(x) = LP(x) ai

i=1

(1.40)

(1.41)


Then, the domain integral D is evaluated as usual which leads to the boundary onlyformulation.

However, the assumption (1.40) also allows the calculation of the Laplacian of uas

1 NPV7 2u(x) = 2 LV72fi(x) oi

11 i=l

The introduction of Eqs (1.40) and (1.41) into the Helmholtz equation yields thefollowing expression

NPLoi [V7 2p(x) + Jl2 fi(x)] =0i=l

(1.42)

Since relationship (1.42) holds for any point x, each term of the summation hasto satisfy the condition

(1.43)

This equation enables the sequence of the functions fi, which gives the best approximation of the solution of the Helmholtz equation, to be found. Eq. (1.43), being againof Helmholtz type, can be solved easier than the original Eq. (1.39) as no boundaryconditions are prescribed for p. If Laplacian in Eq. (1.43) is written in cylindricalcoordinate system with origin at pole 'j', this equation (for axisymmetrical problem)becomes the well known Bessel's equation satisfied by any Bessel's function, e.g. byBessel's function of the first kind and zero order

(1.44)

If spherical coordinate system with origin at pole 'j' is considered, for symmetricalproblems Eq. (1.43) is satisfied by trigonometric functions divided by the distancebetween point x and pole xi' i. e.

f i(x) __ sin[1l. R(x, Xi)](1.45)

Jl R(x, Xi)

Approximation functions given by formulae (1.44) and (1.45) are different fromthose usually proposed and listed in the section on Approximation Functions. Noticethat they are obtained under a constraint that considered differential Eq. (1.39) hasto be satisfied by the functions p in (1.40). Thus, this approximation seems to bemuch more adequate when using functions (1.44) for 2 - D problems or (1.45) for3 - D problems, even if there is no symmetry in the problem at all. Unsatisfactoryaccuracy of the DRM for solving the Helmholtz equation reported by Loeffler andMansur [29] was probably caused by using formulation with standard approximationfunctions. These authors have overcome the problem by incorporating a large numberof internal poles in the analysis as discussed in the section on Internal poles.

(1.48)


1.3 Transient Heat Conduction

Linear Problems

Linear heat conduction without heat sources is governed by the following differentialequation

V 2u(X, t) = ~ aU~, t) = ~ U(X, t) (1.46)

in which p is density, c is specific heat and thermal diffusivity a = kJpc. Any fieldpoint with coordinates (Xl, X2, X3) is represented by X whereas t stands for time. Onthe boundary f, boundary conditions of arbitrary kind are prescribed. To get a uniquesolution of the problem, initial condition, i.e. distribution of temperature u at initialmoment to, should also be specified.

Applying Green's theorem to the temperature field u and time independent fundamental solution u* one obtains Eq. (1.2) in which the domain integral is associatedwith temporal derivative of the temperature

D = ~ ruu* dO (1.47)a 1n

In order to transform this integral to the boundary f by the DRM, approximation(1.4) has to be slightly modified as follows

NPu(x,t) = Lfi(x) ai(t)

i=l

Notice that this approximation is a sort of separation of space variable X and timevariable t [30J.

The remaining portion of the analysis is very much the same as presented in thesection on Fundamentals of the Dual Reciprocity Method, provided ai is replaced byai . Thus, the resulting equation has the following form

CU+HU=GQ

with diffusion matrix defined as

1 A A

C = - -[H U - G QJ Ea

(1.49)

(1.50)

System (1.49) is similar in form to that obtained using the FEM. However, dueto the presence of the heat fluxes vector Q the DRM formulation is of mixed type, asopposed to 'displacement' only FE formulation.

System (1.49) can be solved analytically or numerically. To get numerical solutionof the problem, functions u and q have to be interpolated between two time-levelsmarked by superscripts m and m+l. The simplest interpolation is the linear one

(1.51)


(1.52)

where e = (t - trn)/(t rn+! - trn ) is a parameter (0 ~ e ~ 1) which positions actualtime t in the current time step.

Differentiation of Eq. (1.51) wi th respect to time yields

. du de urn+! - urn urn+! - urnU = de "dt = trn+! _ trn = ~trn (1.53)

Substituting approximations (1.51-1.53) into system (1.49) one obtains after simplealgebra manipulation

(~~mC+e H) urn+! - e GQrn+! =

= C:i~rn C - (1- 0)H) urn +(1- 0)GQrn (1.54)

The right hand side of Eq. (1.54) is known from the previous time step or fromthe initial condition. Hence, upon introducing boundary conditions at current timestep ~trn one can rearrange the system to obtain the standard set of linear equations

AX=P (1.55)

and solve it by any method such as Gauss elimination.Notice that matrices H, G and C depend solely on geometry and they are calcu

lated only once. If the time step is kept constant throughout the analysis, all matricesin formula (1.54) do not depend on time and system (1.55) can be inverted only once.Thus, the final solution is obtained recursively with only simple algebraic operationsinvolved.

Some Remarks on Approximation FUnctions

General suggestions as to what kind of approximation functions can be used in theDRM are given in the section on Approximate Functions. It should be noticed howeverthat for transient heat conduction, formula (1.48) simultaneously postulates followingapproximation of temperature u

NPu(x,t) = 'Lfi(x) ai(t)

i=l

Introducing Eqs (1.48) and (1.56) into (1.46) one arrives at

(1.56)

(1.57)~ (V 2fi(x) ai(t) - ~fi(x) ai(t)) = 0

Since this relationship holds for any point x and any time t, the following equationmust be satisfied for each term of the summation

(1.58)

(1.59)


After simple algebra manipulation one obtains

\72Ji(X) _ 1 ai(t)Ji(x) a ai(t)

The left hand siGe of Eq. (1.58) is a function of space only, whereas its right handside depends on time only. Thus, both sides have to be constant and equal, say to_p2

1

a(1.60)

Hence, approximation functions for transient heat conduction should be chosen assatisfying the equation

(1.61 )

Types of possible functions j1(x) being solutions of Eq. (1.61) are discussed in thesection on Solving the Helmholtz Equation by the DRM. An apt choice of these typeof approximation functions is confirmed by results published by Aral and Tang [23].As reported, for 2 - D problems, approximation functions, being Bessel's functions ofthe first kind, first order J1 generally produced the best results.

Transient Problems with Heat Sources

A term representing any kind of heat source can also be present in the differentialequation of transient heat conduction. If the problem is linear the equation has thefollowing form

\72u(x, t) + ~ = ~ u(x, t) (1.62)

Solving the problem (1.62) one needs to deal with an integral equation in whichtwo domain integrals appear. The first integral is associated with the heat sourceand being of type (1.32) is evaluated as described in the section on Formulation. Thesecond integral is given by expression (1.47) and is transformed according to detailsdiscussed in the section on Linear Problems. Thus, final formulation results in theequation being the combination of Eq. (1.34) and (1.49)

c U +H U - G Q= -a C B = R (1.63)

(1.64)

with matrix C given by Eq. (1.50).The solution procedure is similar to the one described for the transient problem

without heat sources. The only difference consists in the right hand side vector whichin this case is not zero. This implies the following iteration formula

(A~mC+8 H) urn+1 - 8 G Qrn+1 =

= C~~rn C - (1 - 8)H) urn + (1 - 8)GQrn +R


which is solved within each time step ~tm. Vector R is calculated as given in thesection on Formulation.

Non-Linear Conduction

In this section non-linear heat conduction problems without heat sources are considered. If the nonlinearity is caused by the dependence of material properties onthe temperature u, the differential equation governing this kind of problem has thefollowing form

auV[k(u) Vu] = pc at (1.65)

The most efficient and elegant approach to solve Eq. (1.65) is based on Kirchhoff'stransformation. This is equivalent to converting Eq. (1.65) into simpler form byintroducing a new variable U(u) defined as

or, in the integral form

dU- =k(u)du

U = Uo+1: k(u) du

(1.66)

(1.67)

(1.68)

Notice that U o and Uo are arbitrary reference values which do not affect the results.Upon the above transformation Eq. (1.65) can be written as

V 2U = ~ aua at

Although Eq. (1.68) looks like a differential equation for linear conduction (cf Eq.(1.46)), it is still a non-linear one, as the thermal diffusivity a is temperature (orsolution) dependent

k(u)a = a(u) = p(u) c(u)

Hence the volume integral D has a more complicated form, i. e.

(1.69)

(1.71)

D = f ~ au u· dO. (1.70)in a at

In order to transform this integral into a boundary using the DRM, the followingtransformation is employed (notice l/a term included this time into approximationformula)

1 au NP. .-- = EF(x)<Y1(t)a at ;=1

The inverse transformation expressing the unknowns a; (t) in terms of the originalvariables can be written in matrix form as


a(t) =F-1 DU (1.72)

Matrix D has all coefficients equal to zero except those on the main diagonal whichare defined as

1Djj =

a·J(1.73)

where aj is understood as thermal diffusivity calculated for the temperature at pointJ.

Using the above formulae the domain integral is evaluated as usual (see section onFundamentals of the Dual Reciprocity Method) and the DRM results in the formulation (1.54). It should be noticed however, that this time matrix C is not constant, asit contains (in matrix D) solution dependent thermal diffusivity, i.e.

C = -[HU - GQl ED (1.74)

(1.75)

Thus, the solution can only be obtained in the double-looped iteration process.External loop is associated with time steps, whereas within the internal, updating ofmatrix C is carried out. In order to proceed iterations in the internal loop, residualvector for considered time step ~tm is defined as [31],[32]

tJ1(Xm+I) = C~~m C + eH) um+1- eGQm+I

- (~~mC - (1- e)H) Um- (1- e)GQm

This vector should be zero if Xm+I is the exact solution of the problem, i. e.

(1.76)

System (1.76) can be solved by the Newton-Raphson algorithm which results inthe following expression

J~!ll ~X~+I = - tJ1(X~n (1.77)

where subscripts nand n-l represent iteration numbers in the internal loop and increment ~Xn = Xn - X n- 1 . J stands for the jacobian matrix, coefficients of whichare calculated as

fJ-/.m+lJm+l _ --.;..'f'..:.,.i~

ij - aX~+IJ

If the unknown Xm+I is the heat flux Qj+I these coefficients have the form

(1.78)

r!'+l = -ea·· (1.79)IJ IJ

When the unknown Xr+1 is the temperature Ur+I they can be obtained as

(1.80)

(1.82)


Jm+l - _1_ C.. +eB.. _ eG.. aQi+1 +

ij - ~tm IJ IJ IJ 8Um+1 •J

C·· Um+! - U'!' 0 (!).IJ J J a J

+~ ~tm oU'!'+!J J

The derivative of (l/a)j with respect to Uj+! is computed as follows

o(~) __~~ ou']'+!oUm+1 - a2 ou'!l+! oU,:,,+! (81a)

J J J J

~ __l_(~_kj~_kj~) ()ou'!l+! - p'C' dum+! c· dU'!l+1 p' dU'!l+1 81b

J JJ J J J J J

oum+! 1orim+! = Y; (81c)J J

If the dependence on temperature of the conductivity k, density p and specificheat c is modelled as a piecewise linear representation, their derivatives with respectto temperature u, at any point j, are given by the slope of the segment to which lineUj belongs.

The derivative of Qj with respect to Uj in Eq. (1.80) depends on the kind ofboundary condition at node j. For the boundary condition of the second kind, thisderivative is equal to zero, as Qj is prescribed value and does not depend on Uj • Forthe boundary condition of the third kind the following relationship can be obtainedas

oQm+! oQm+! Oum+! h.J _ J J_...l.

oUm+1 - Ou'!l+1 oU,:,,+! - k·J J J J

where hi is the heat transfer coefficient at point j.To begin iteration loops an initial condition is used to determine all matrices

needed in Eq. (1.75). Then, Eq. (1.77) is solved and the values of U are obtainedat all boundary points (and internal poles). As a next step, an inverse Kirchhoff'stransformation is performed to find the temperature U and to update the matrix C.Subsequent iterations are calculated from Eq. (1.77). The internal loop is completedwhen the residual vector (1.76) becomes sufficiently small. Finally, all the abovedescribed operations are repeated for the subsequent time steps.

It should be noticed that, if the domain under consideration is made up of severalsubregions of different materials, function U is not continuous on the interface. Thus,the final set of equations has to have such .form that continuity is satisfied in theprimitive variable u. The Newton-Raphson scheme can deal with these type of nonlinear problems without any difficulties [26J.

The procedure of solving non-linear transient heat conduction described in thissection can also be interpreted as a technique to transform problem (1.68) into newtime space T defined as [31]'[32J

OT = a ot (1.83)

Although, this transformation enables Eq. (1.68) to be written in another form,


Table 1.1: Convergence with refining discretization.

Time Analyt. 7 elem. 14 elem. 28 elem.2 0.025 0.016 0.012 0.0124 0.154 0.167 0.158 0.1586 0.298 0.304 0.290 0.2928 0.422 0.420 0.405 0.40710 0.526 0.518 0.502 0.50415 0.710 0.696 0.681 0.68320 0.823 0.809 0.796 0.79830 0.934 0.924 0.917 0.917

\l2U = au (1.84)aTit does not linearize the problem. Thus, the approach explained in [31] and [32] offersno advantages over that one presented in this section and both formulations are fullyequivalent.

1.4 Numerical Examples and Conclusions

In order to demonstrate convergence and accuracy of the Dual Reciprocity Methodsome numerical examples are being studied in this section. Although results discussedhere are mainly obtained with the boundary element system BEASY [33], others havealso been published, e.g. [23]. Whenever possible, results are compared with analytical solution or some established benchmarks.

Example 1The first problem studied is the transient heat conduction within the infinite slabsubjected to a thermal shock. Comparison of the results obtained by the Dual Reciprocity Method with analytical solution gives the idea about the rate of convergenceof the method when refining the discretization and also when decreasing the valueof the time step. Although the physical problem is a 1 - D problem, it has beenmodelled as a 2 - D problem with the following boundary conditions

u = 1

q=O

along faces Xl =±L

along faces X2 = ±l

Since the homogeneous initial condition was assumed (constant temperature andequal zero within the region), the above boundary conditions cause thermal shock onthe faces Xl = ± L. The numerical values chosen when carrying out the calculationsare as follows L = 5, I = 4, k = 1. Results for central point Xl = X2 = 0 are presentedin Table 1.1.

Due to symmetry of the problem, only one quarter of the region was considered.Faces were divided into 7, 14 and 28 constant boundary elements. It can be seen that


Table 1.2: Convergence with decreasing time step.

Time Analyt. ~ t=1 ~ t = 0.5 ~ t = 0.12 0.025 0.012 0.012 0.0114 0.154 0.158 0.162 0.1656 0.298 0.290 0.298 0.3048 0.422 0.405 0.414 0.42210 0.526 0.502 0.512 0.52015 0.710 0.681 0.691 0.69820 0.823 0.796 0.804 0.81130 0.934 0.917 0.921 0.925

Table 1.3: Influence of the number of internal poles.

Time Analyt. No pole 1 pole 2 poles 4 poles2 0.025 1.006 0.012 0.033 -0.0184 0.154 0.142 0.162 0.141 0.1356 0.298 0.154 0.298 0.275 0.2828 0.422 0.307 0.414 0.396 0.40510 0.526 0.450 0.512 0.498 0.50715 0.710 0.692 0.691 0.684 0.69120 0.823 0.826 0.804 0.801 0.80630 0.934 0.942 0.921 0.921 0.924

results converge to values which are, approximately, within 3% of the appropriateanalytical solutions. The reason for this is a linear approximation of temperaturesand heat fluxes over the first time step whereas temperatures should have had a jump.The results can be improved by smoothing the thermal shock according to algorithmknown from the Finite Element Method [34].

To study the influence of the length of the time step onto the accuracy, the timestep has been changed from ~t = 1 to ~t = 0.1. The number of constant boundaryelements was kept equal to 14. Since the thermal shock is better represented withsmaller time steps, results obtained are closer to analytical ones (cf Table 1.2). Itshould be stressed however that the existence of the lower limit of time step is reportedin many works, e.g. [35],[36]. Although the problem still requires more research, theminimum value of time step (for 2 - D problem) usually proposed is of order

where ~l is a mesh size and a stands for thermal diffusivity.Table 1.3 presents the influence of the number of internal poles selected within the

rectangle in order to represent accurately the internal effects. Notice that one internalpole (associated with a function fi = constant) guarantees considerably accuracy of


Table 1.4: Results for different values of coefficient 0.

Time Analyt. 8 = 1 8 = 2/3 8 = 1/22 0.025 0.012 -0.002 -0.0094 0.154 0.162 0.151 0.1466 0.298 0.298 0.292 0.2898 0.422 0.414 0.411 0.41010 0.526 0.512 0.511 0.51015 0.710 0.691 0.692 0.69320 0.823 0.804 0.807 0.80730 0.934 0.921 0.923 0.924

Table 1.5: Influence of the type of boundary element.

Time Analyt. Const. Linear Quadrat.2 0.025 0.012 0.016 0.0174 0.154 0.158 0.166 0.1666 0.298 0.290 0.302 0.3028 0.422 0.405 0.418 0.41810 0.526 0.502 0.515 0.51615 0.710 0.681 0.694 0.69420 0.823 0.796 0.807 0.80730 0.934 0.917 0.923 0.923

the DRM.Coefficient 8 used in interpolation formulae (1.51) and (1.52) also affects the

results. From data given in Table 1.4 one can learn that the implicit scheme (backwarddifferences), equivalent to 8 = 1, presents some advantages over others.

Results listed in Tables 1.3 and 1.4 were obtained for time step fJ.t = 0.5 and 14constant elements.

The influence of the type of element was examined in calculations, the results ofwhich are given in Table 1.5. Other numerical data selected are as follows: number ofboundary elements = 14, number of internal poles = 1, time step fJ.t = 1, coefficient8=1.

Example 2In the current example, the axisymmetric problem is being studied. Solid cylinder,initially at unit temperature, was subjected to the following boundary conditions

u=o forr=l

q = 2u for z = ±1

Taking into account symmetry with respect to the r-axis, only half of the cross-


T

1.000.75

uX

0.8t- 0.026

)(

~·O.{)5

0.6

x't=O.1

0.4t·O.~5

0.2 t-O.25

t-O.40

0 0.25 0.50

Ano.lyhcuL

• W..,-obel [:)7Jx Present/coarseo P..,-e,sent, fLne

Figure 1.2: Temperature distribution along the face z = ±1.

section was discretized into 8 constant boundary elements. Time step was chosen asf:!:.t = 0.025s, exactly the same as employed by Wrobel when using a time-dependentfundamental solution [37]. Results of both formulations are shown in Fig. 1.2 andcompared with analytical solution of the problem.

Since the above mentioned discretization was rather coarse, some oscillations atthe beginning of the process can be observed. They are damped out however, whenfine mesh with 16 boundary elements is used and the time step is reduced to the valuef:!:.t = 0.0125s (cfFig.1.2).

Example 3Transient heat conduction within a prolate spheroid is being examined. A parametricrepresentation of the surface depicted in Fig. 1.3 can be written as

r = £1 cos </>;

where values £1 and £2 are chosen as £1 = 1, £2 = 2.The initial temperature is zero everywhere, and then a unit thermal shock is

applied.


u

1.0

AnCl.Lljt~calI!

0.8 I0 e>EM I

!x FEM

0.6 ,z

~0.4

Lz. I

~0.2-

¢ .,.L1

0.2.. 0.4 0.6 0.8 1.0 t.

Figure 1.3: Temperature at the center of a prolate spheroid.

Since the problem has no degrees of freedom some internal poles have to be introduced in order to properly represent internal effects. Results presented in Fig. 1.3 wereobtained with 5 internal poles. They are compared with analytical solutions and withresults obtained by FEM with the same time step tit = 0.025s. Accuracy is very good.

Example 4In this example the response of a mercury-in-glass thermometer immersed into a gaswhose temperature varies with time t as a sine function

U g = 37.78 +18.89 sin (21l't)

is being investigated.The thermometer is idealized as a cylinder of length I" and of diameter 0.25".

The internal resistance has been neglected and initial temperature was assumed to be16.56°C. On the external surface, the boundary condition of the third kind has beenprescribed with heat transfer coefficient h = 28.4 Wj m2 K. The physical properties ofmercury are k = 8.3 Wjm K, p = 13.6.103 kgjm3

, c = 136.1 Jjkg K.


Table 1.6: Temperatures for selected times.

Time [min] Analytical 2-D Axisym. 3-D6 48.89 47.22 47.22 47.2216 64.44 64.44 64.44 64.4432 36.67 35.56 35.56 35.56

I

IIL--------

Figure 1.4: Two-dimensional and three-dimensional idealization of thermometer.

The above problem can be modelled as two-dimensional, axisymmetric or threedimensional. All meshes with quadratic boundary elements are shown in Fig. 1.4 and1.5. Results obtained with all the three formulations are practically the same andconverge to analytical solution very well, ef Table 1.6.

Example 5The example discussed here concerns heat transfer problem in rectangle being thecross-section of a wall 20em long and lem high. Initial temperature is constant andequal to 100°C. Boundary conditions vary with time. Namely, the temperature ofthe left edge of the cross-section is suddenly raised up to 200°C, kept at this value forlOs and then it is decreased to 100°C again. The temperature of the right edge of thecross-section has constant value 100°C, whereas the two remaining edges are insulated.Heat capacity of the material is assumed to be equal ep = 8.103 kJ1m3 K. Since heatconductivity varies with temperature according to relationship k = 20+0.1 u WlmK,the problem becomes non-linear with non-linear material. The boundary of the crosssection was divided into 44 constant or 22 quadratic elements. The results obtainedby the DRM are compared (Tables 1.7 and 1.8) with the finite element solutionscalculated when the region was discretized into 20 equal linear finite elements [38].The time step chosen in the DR analysis was t::.t = Is and the average number ofiterations in internal loop was 4. The analogous value in FE calculations was 3. Itshould be noticed that there is very good agreement of both results.


.2.

7 4 1

----- -

Figure 1.5: Axisymmetric idealization of thermometer.

Table 1.7: Temperatures along the thickness of wall at time t = lOs.

x FEM BEM-Q BEM-C0 200.00 200.00 200.001 176.16 174.86 175.292 153.21 151.03 151.483 133.47 131.33 131.744 118.60 117.32 117.635 108.98 108.74 108.946 103.72 104.14 104.277 101.29 101.91 102.018 100.37 100.87 100.979 100.08 100.39 100.5010 100.01 100.14 100.27


Table 1.8: Temperatures along the thickness of wall at time t = 13s.

x FEM BEM-Q BEM-C0 100.00 100.00 100.001 128.53 130.46 130.152 139.97 138.46 138.953 136.95 132.01 132.474 124.72 121.29 121.715 114.40 112.37 112.646 107.18 106.56 106.717 103.24 103.27 103.368 101.29 101.56 101.629 100.45 100.72 100.7710 100.13 100.33 100.36

Looking through the discussed examples and listed results of calculations it canbe stated that the Dual Reciprocity Method is a general tool for solving heat transferproblems leading to integral equations with domain integrals. These integrals aretransformed into equivalent boundary integrals. As a consequence the DRM resultsin elegant boundary-only formulation of the problem. The method enables one todeal with a variety of linear and non-linear thermal problems. The consistency andaccuracy of this approach should be pointed out. The implicit scheme with quadraticboundary elements and with one internal pole results in a good accuracy of calculations. This accuracy can also be improved by proper choice of interpolation functionsfi. Although some sequences of these functions are recommended in this chapter aswell as in other references, e.g. [23], more research needs to be carried out in thisimportant topic.

In presence of the thermal shock, employing the smoothing algorithm can alsoconsiderably improve the accuracy of the method.

Acknowledgement

The present chapter was written while the second author was a visiting research fellowat the Computational Mechanics Institute, Ashurst Lodge, UK. He wishes to thankthe financial support received from the Ministry of National Education within theCentral Plan for Fundamental Research - direction 4.4 coordinated by Technical University of Warsaw.


References

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2. C.A. Brebbia and J. Dominguez, Boundary Elements - An Introductory Course,Compo Mech. Publications, McGraw-Hill Book Co., 1988.

3. C.A. Brebbia and L.C. Wrobel, The boundary element method for steady-stateand transient heat conduction, in R.W. Lewis and K. Morgan, eds., NumericalMethods in Thermal Problems, Pineridge, Swansea, U.K., 1979.

4. H.L.G. Pina and J.L.M. Fernandes, Three-dimensional transient heat conductionby the boundary element method, in C.A. Brebbia, T. Futagami and M. Tanaka,eds., Boundary Elements V, Springer - Verlag, Berlin, 1983.

5. L.C. Wrobel, A boundary element solution to Stefan's problem, in C.A. Brebbia,T. Futagami and M. Tanaka, eds., Boundary Elements V, Springer - Verlag,Berlin, 1983.

6. C.A. Brebbia and P. Skerget, Diffusion-convection problems using boundary elements, in J.P. Laible, C.A. Brebbia, W. Gray and G. Pinder, eds., Finite Elements in Water Resources V, Springer - Verlag, Berlin, 1984.

7. P. Skerget, A. Alujevic and C.A. Brebbia, The solution of Navier-Stokes equationin terms of vorticity-velocity variables by boundary elements, in C.A. Brebbia,ed.,Boundary Elements VI, Springer - Verlag, Berlin, 1984.

8. W.J. Mansur and C.A. Brebbia, Transient elastodynamics, in Topics in Boundary Elements Research 2, Springer - Verlag, Berlin, 1985.

9. D. Nardini and C.A. Brebbia, A new approach to free vibration analysis using boundary elements, in C.A. Brebbia, ed., Boundary Element Methods inEngineering, Springer - Verlag, Berlin, 1982.

10. W. Tang, C.A. Brebbia and J.C.F. Telles, A generalized approach to transformthe domain integrals onto boundary ones for potential problems in BEM, in C.A.Brebbia, ed., Boundary Element Methods in Engineering VII, Springer - Verlag,Berlin, 1985.

11. W. Tang, Transforming Domain into Boundary Integrals in BEM - A GeneralizedApproach, Lecture Notes in Engineering, C.A. Brebbia and S.A. Orszag, eds.,Springer -Verlag, Berlin, 1988.

12. J.P.S. Azevedo and C.A. Brebbia, An efficient technique for reducing domainintegrals to the boundary, in C.A. Brebbia, ed., Boundary Element Methods inEngineering X, Springer - Verlag, Berlin, 1988.


13. D. Nardini and C.A. Brebbia, The solution of parabolic ~nd hyperbolic problems using an alternative boundary element formulation, in C.A. Brebbia, ed.,Boundary Element Methods in Engineering VII, Springer - Verlag, Berlin, 1985.

14. D. Nardini and C.A. Brebbia, Boundary integral formulation of mass matricesfor dynamic analysis, in Topics in Boundary Elements Research 2, Springer Verlag, Berlin, 1985.

15. D. Nardini and C.A. Brebbia, The solution of parabolic problems using an alternative Boundary Element formulation, in C.A. Brebbia, ed., Boundary ElementsVII, Springer - Verlag, Berlin, 1985.

16. L.C. Wrobel, C.A. Brebbia and D. Nardini, Analysis of transient thermal problems in the BEASY system, in J.J. Connor and C.A. Brebbia, eds., BETECH86, Compo Mech. Publications, Southampton, 1986.

17. C.A. Brebbia and L.C. Wrobel, The solution of parabolic problems using theDual Reciprocity Boundary Element, T.A. Cruse, ed., Advanced Boundary Element Method, San Antonio, Springer - Verlag, 1988.

18. A.J. Nowak and C.A. Brebbia, The Multiple Reciprocity Method. A new approach for transforming BEM domain integrals to the boundary, EngineeringAnalysis with Boundary Elements, vol. 6, No 3, 1989, pp. 164-167.

19. A.J. Nowak, The Multiple Reciprocity Method of solving transient heat conduction, C.A. Brebbia, ed., Boundary Elements XI, Springer - Verlag, Berlin,1989.

20. L.C. Wrobel, J.C.F. Telles and C.A. Brebbia, A dual reciprocity boundary element formulation for axisymmetric diffusion problems, in C.A. Brebbia and M.Tanaka, eds., Boundary Elements VIII, Springer - Verlag, Berlin, 1986.

21. C.F. Loeffler and W.J. Mansur, Dual Reciprocity Boundary Element formulation for potential problems in infinite domains, C.A. Brebbia , ed., BoundaryElements X, Springer - Verlag, Berlin, 1988.

22. S.M. Niku and C.A. Brebbia, Dual reciprocity boundary element formulation forpotential problems with arbitrarily distributed sources, Engineering Analysis,vol. 5, No 1, 1988, pp. 46-48.

23. M.M. Aral and Y. Tang, A boundary only procedure for time - dependent diffusion problem, Appl. Math. Modelling, vol. 12, 1988, pp. 610-618.

24. R. Bialecki and A.J. Nowak, Boundary value problems with non - linear materialand non - linear boundary conditions, Appl. Math. Modelling, vol. 5, 1981, pp.417-421.

25. L.C. Wrobel and J.P.S. Azevedo, A boundary element analysis of nonlinear heatconduction, in R.W. Lewis and K. Morgan, eds., Numerical Methods in ThermalProblems, Pineridge, Swansea, U.K., 1985, pp. 87-97.


26. J.P.S. Azevedo and L.C. Wrobel, Nonlinear heat conduction in composite bodies:A boundary element formulation, Internat. J. Numer. Methods Eng., 26, 1988,19-38.

27. R. Bialecki, Solving nonlinear heat transfer problems using the Boundary Element Method, Chapter 4 in the present book.

28. U. Niwa, S. Kobayashi and M. Kitahara, Determination of Eigenvalues by Boundary Element Methods, Chapter 7, in Developments in Boundary Element Methods, vol. 2, Applied Science Publishers, 1982.

29. C.F. Loeffler and W.J. Mansur, Analysis of time integration schemes for boundary element applications to transient wave propagation problems, in C.A. Brebbia and W.S. Venturini, eds., BETECH 87, Compo Mech. Publications, Southampton, 1987.

30. M.N. Ozi§ik, Boundary Value Problems of Heat Conduction, International Textbook Company, Scranton, 1968.

31. L.C. Wrobel and C.A. Brebbia, The dual reciprocity boundary element formulation for nonlinear diffusion problems, Computer Methods Appl. Mech. Eng.,65, 1987, 147-164.

32. L.C. Wrobel and C.A. Brebbia, Boundary elements for non-linear heat conduction, Communications in Applied Numerical Methods, vol. 4,1988, pp. 617622.

33. C.A. Brebbia, D. Danson and J. Bayham, BEASY Boundary Element AnalysisSystem, in C.A. Brebbia ed., Finite Element System, A Handbook, Springer Verlag, Berlin, 1985.

34. O.C. Zienkiewicz, The Finite Element Method, McGraw-Hill, New York, 1977.

35. D. Morvan, Application of the BEM to the resolution of an unsteady diffusionproblem, C.A. Brebbia, ed., Boundary Elements X, Springer - Verlag, Berlin,1988.

36. A. Chaudouet, Three-dimensional transient elastic analysis by the BEM method,Int.J.Numer.Meth.Eng., Vol. 24,1987, pp. 25-45.

37. L.C. Wrobel, Potential and viscous flow problems using the Boundary ElementMethod, Ph.D. Thesis, Southampton University, 1981.

38. S. Orivuori, Efficient method for solution of nonlinear heat conduction problems,Int.J.Numer.Meth.Eng., Vol. 14,1979, pp. 1461-1476.


Notation

a thermal diffusivity; m 2I s

A matrix of linear BEM equations, Eq. (1.36)

b heat generation rate due to internal heat sources; WI m3

B vector containing the values of blk at nodal points

c specific heat; JIkgK

Cj function of the internal angle the boundary r makes at a given point.For a smooth boundary Cj = 0.5

C diffusion (capacitance) matrix

d coefficient in the approximation functions depending on theproblem dimensionality

D domain integral

D diagonal matrix defined by Eq. (1.73)

Ii coordinate functions

F matrix formed by the values of functions Ii at nodal points

G heat flux influence matrix in BEM formulation

h heat transfer coefficient; Wlm 2K

H temperature influence matrix in BEM formulation

J Jacobian matrix

Jo Bessel's function of first kind zero order

J1 Bessel's function of first kind first order

k thermal conductivity; WlmK

n outward normal to the boundary

N number of boundary nodes

NP number of functions considered in the DRM

P number of internal poles

P right hand side vector in the final set of linear equations

R geometrical distance between two points

R product of matrix C and vector B

q heat flux; Wlm 2

q* heat flux analog associated with fundamental solution

qi auxiliary heat flux associated with field iJ,i


Q

Qt

u

u*

uuiJu

uX

vector containing the values of function qJ at nodal points

vector containing the values of heat fluxes at nodal points of e-thboundary element

vector of nodal heat fluxes in matrix BEM formulation

matrix whose columns are the vectors qitime; s

temperature; K or °C

fundamental solution satisfying Eq. (1.1)

auxiliary field defined by Eq. (1.6)

vector containing the values of function iJ,i at nodal points

vector containing the values of temperatures at nodal points of e-thboundary element

Kirchhoff's transform of temperature u

vector of nodal temperatures in matrix BEM formulation

matrix whose columns are the vectors fIi

temporal derivative of temperature

vector containing the values of temporal derivatives of temperatures

vector of nodal unknowns, Eq. (1.36)

region under consideration

density; kgjm 3

part of integrand in domain integral D, Eq. (1.3)

vector containing the values of function </> at nodal points

wave number in Helmholtz equation

residual vector

Jl

PO

p

Greek lettersa) unknown coefficients in the approximation formula (1.4)

Q. vector containing the values of a i

13m arbitrary coefficients used in approximation functions P~i Dirac's function acting at point i

r boundary of the region 0

~l minimum length of boundary element

~t time step

~X increment of solution

</>

~

T

e


time; s

parameter which positions actual time in the current time step

Subscriptsn iteration number

o reference values of the Kirchhoff's transformation or initial

condition

Superscripts* fundamental solution

auxiliary functions

temporal derivative

m time step number

T transformed matrix

Other symbols\7 gradient

\72 Laplace's operator

oO/on differentiation along outward normal to the boundary

bold face designates matrices and vectors

Chapter 2

Transient Problems using Time-DependentFundamental SolutionsR. Pasquetti (*), A. Caruso (**), L.C. Wrobel (***)(*) Laboratoire de Mathematiques, URA, CNRS 168, Universite deNice, Parc Valrose, 06034 Nice, France(**) EDF - Electricite de France, Direction des Etudes et Recherches,beft. Lab. National d'Hydraulique, BP49, 78401 Chatou Cedex, France(* *) Computational Mechanics Institute, Wessex Institute ofTechnology, Ashurst Lodge, Ashurst, Southampton S04 2AA,England

2.1 Introduction

The present chapter discusses a boundary element formulation for transient heat conduction using time-dependent fundamental solutions. This formulation can be viewedas a direct extension of potential theory since the proper fundamental solution of thediffusion equation is used to obtain an equivalent boundary integral equation. Numerical techniques are then employed to solve the integral equation in discrete formthrough a time-marching procedure.

The fundamental solution adopted here is a free-space Green's function which hasbeen used by Morse and Feshbach [1], Carslaw and Jaeger [2], among others, to obtainanalytical solutions to some simple problems. Chang et al. [3] and Shaw [4] were thefirst to apply this fundamental solution in the context of the direct BEM, but theiremphasis was on the analytical rather than numerical aspects of the method. Theformulation was later extended by Wrobel and Brebbia [5] to allow higher-order spaceand time interpolation functions to be included, thus making possible the analysis ofpractical engineering problems.

The technique found widespread use, and a large number of papers dealing withit have been published in the 80's. We notice, in particular, the papers by Onishi[6] where the second-order convergence of the technique was mathematically proved,Wrobel and Brebbia [7] who developed a ring-type fundamental solution and appliedthe formulation to axisymmetric problems, and Skerget and Brebbia [8] who treatednon-linear materials and boundary conditions.

This chapter starts by reviewing the formulation as applied to linear and non-


linear problems. The Kirchhoff transformation is employed together with linearizationto treat the material non-linearities. Space and time discretization algorithms arediscussed in detail for two- and three-dimensional problems, and expressions resultingfrom the analytical integration of the fundamental solutions with respect to time areincluded.

The next section describes how linear and non-linear boundary conditions areimplemented. A 'local linearization' technique developed by Pasquetti and Caruso [9]is employed for the latter. The following two sections describe original work carriedout by Pasquetti and Caruso [10] to transform the domain integrals arising frominitial conditions and internal heat sources, into equivalent boundary integrals. Thisproduces a final system of equations which can be solved with boundary discretizationonly, thus retaining the main advantage of the BEM.

Finally, results of several examples of application are presented to demonstratethe efficiency of this technique, including some practical engineering problems.

2.2 Boundary Integral Equation

This chapter discusses boundary element solutions to the transient heat conductionequation

\72T +b = ~ ~~ (2.1)

where t is time, T is temperature, k is the thermal diffusivity (k = AIpc), Ais thethermal conductivity, p density, c specific heat, and the term b accounts for internal heat generation. The problem definition is completed with the specification ofboundary and initial conditions.

The above equation can be recast as an integral equation over space and time,with the help of the corresponding fundamental solution; between the initial time toand the final time tF, the integral equation for a source point M on the boundary fof the domain 0 is written as [11]

CM TM,tF +ltF

f k T q* df dt = ltF

f k qT* df dt +to ir to ir

l tF

f k bT* dO dt + f To T* dO (2.2)to in in

in which eM is a coefficient depending on the geometry of f at point M, q = aTIan,q* = fJT*Ian, and ii is the unit outward normal vector.

The fundamental solution T* is a free-space Green's function, describing the temperature field generated by a unit heat source applied at point M at time to [1],[2],z.e.

(2.3)

(2.4)


with d = -n' r, r being the distance vector connecting source and field points,T = tF - t and s is the number of spatial dimensions of the problem.

This formulation can also be applied to non-linear problems in which the thermalconductivity .A is a function of temperature. In this case, it is convenient to employthe Kirchhoff transformation [12]

VJ = (T .A(T') dT'IT. (2.5)

where Tr is a reference temperature. For steady-state problems, the above transformation is sufficient to linearize the governing equation (although boundary conditionsmay still be non-linear). For transient problems, the integration by parts of equation(2.1) weighted by T* gives rise to extra domain integrals involving the derivatives ofk. Assuming, however, that the space and time derivatives of k are small quantitiesand can be neglected, i.e.

an integral equation similar to (2.2) is obtained for the non-linear problem

eM VJM,tF + rtF { k VJ q* dr dt = rtF { k p T* dr dt +ho k ho k

rtF { kg T* dfl dt + { VJo T* dfl (2.6)lt~ In In

in which p = .A q is the heat flux density and 9 = .A b is the heat source intensity.The numerical solution of the boundary integral Eqs (2.2) and (2.6) requires space

and time discretization. Two different time-marching schemes can be used in thesolution [11]:

1. For each time step, we consider as initial time the time level tF-l previous to theresolution time tF . This approach minimizes the time integrations but requiresthe evaluation of a domain integral associated to the temperature field TF - 1 ateach time step;

2. For each time step, the solution is restarted from the initial time to; in this way,the domain integral associated to the initial conditions can be avoided for themajority of practical situations.

The second approach was adopted in this work. Recently, truncation algorithmswere developed to improve the computer efficiency of such an approach, by computingonly approximately the influence of initial steps after some time had elapsed. Onesuch algorithm, named 'study temporal domain'[9], is herein presented in detail. Forother techniques, the interested reader is referred to [13-15].

(2.7)

(2.8)


2.3 Space and Time Discretization

Equations (2.2) and (2.6) present two domain integrals due to initial conditions andinternal sources. For simplicity, it will initially be assumed that there is no internalheat generation and that the initial temperature is constant. In this case, it is possible to rewrite the problem into an equivalent one with zero initial condition for thetemperature difference T - To. For the non-linear case, it is sufficient to take To asthe reference temperature Tr in the Kirchhoff transformation (2.5) to obtain 7/Jo = o.

With the above simplifications, Eq. (2.2) can be rewritten for a point M in asmooth portion of r as

1 l. tF J l tF J-2 TMh + k T q* dr dt = k qT* dr dt~ f ~ f

Dividing the boundary r into N boundary elements and the time span tF - to into Ftime steps, the following discretized equation is obtained

~ Ti,F +t -t [1 JkTq*drdt = t -t [1 JkqT* drdt2 f=I j=I tl-1 f) f=I j=I tl-1 f)

Ti,F being the temperature at node i at time tF.Assuming that the boundary elements are constant in space and linear in time,

the temperature variation on element j between time levels f - 1 and f is given by

T = Tj,J-I (if - t) +Tj,J (t - tf-I)tit f

with tit f = t f - t f-I· A similar expression can be written for q. Calling

1 it! 1Hli,j,F,J = ~ . k (t f - t) q* dr dtutf tl-1 f)

A 1 it! [H2i,j,F,J =~ ir k(t - tf-dq* drdt

utf t!-1 f)

1 it! JGl i,j,F,J =~ .k (tf - t) T* dr dtutf tl-1 f)

1 1t1 JG2i,j,F,J =~ k (t - tf-I) T* dr dtutf t!_1 fj

A 1H2· .Ff - H2· .Ff +- fJ· fJFfI,)" - I,)" 2 I,) ,

in which fJ is the Kronecker delta, Eq. (2.8) becomes

(2.9)

(2.10)

(2.11)

(2.12)

(2.13)

F N F N

L L (Hl i,j,F,J Tj,J-I + H2i,j,F,J Tj,J) = L L (G1i,j,F,J qj,J-I + G2i,j,F,J qj,J)f=Ij=I f=Ij=I

(2.14)


Writing the above equation at all boundary nodes i(1 ~ i ~ N) using a collocationtechnique, the following system of equations is obtained

F FE (HIF,J T /- 1 +H2F,J T/ ) = E (GIF,J Q/-l +G2F,J Q/) (2.15)1=1 1=1

The temperature and flux at each node are known for all time levels previous to tF,so that the above equation can be rewritten in the form

(2.16)

with

F-l

SF = - E (HIF,J T/ - 1 +H2 F,J T/ ) - HIF,F TF-l +1=1

F-1

E (GIF,J Q/-l +G2F,J Q/) +GIF,F QF-l1=1

(2.17)

2.4 Evaluation of the Coefficients of Matrices HI, H2, GIand G2

It is important to notice that the fundamental solution T* and its derivative q* arefunctions of r; thus, in the linear case, we have that Hl i,j,F,1 = Hl i,j,F',J' if F - f =F' - f', and the same for H2, Gl and G2. This means that these matrices can bestored and reused whenever needed. Thus, for computer efficiency, it is possible tocompute and store all these matrices, which depend only on geometry and time stepvalues, and perform several simulations, for instance for different types of boundaryconditions.

For non-linear problems, such preliminary calculation is not possible since thediffusivity k is temperature-dependent. On the other hand, one should avoid recomputing all coefficients for each new time step due to efficiency reasons. In what follows,we shall comment upon how this problem can be overcome, thanks to the introductionof 4N2 functions, namely J(l i ,j, J(2 i,j, J(3 i ,j and J(4i ,j, closely related to the coefficients Hl i,j,F,J, H2 i ,j,F,J, Gl i,j,F,J and G2i ,j,F,J but depending only on the product kT.

Time Integration

Expressions (2.3) and (2.4) can be written in the condensed form

with

1 (r2

)X* = C ex--(4kT )'/2+( P 4kT (2.18)

c = _1_ and t: = 07r'/2

if X* = T*


2dC = - and f- = 1

1r./2if X· = q.

After inverting the order of integration in expressions (2.10) to (2.13), the followingtime integrals have to be evaluated

itl/2 = kt X· dt

t'_1

For the linear case, k is a constant; for the non-linear one, it is assumed that the timestep !:it is sufficiently small for the approximation k =constant between t f-I and t fto be valid, e.g. k(t) = k(tf-d over !:itf.

Evaluation of 11

Substituting the expression for X·, i. e.

i tl k (r2)

I1 = C ( k )'/2+< exp --k dtt'_1 4 r 4 r

and performing the change of variables

withr 2 x

dx = --dt = -dt4kr2 r

the integral I1 is rewritten as

I C l x I kr e-x d1= - X

XI-I (4kr)·/2+< x

Eliminating the product kr, according to expression (2.20), gives

I1 = C LXI x·/2+<-2 e-Xdx4r·+2<-2 XI_I

the result of which is

C [(8 )]XI-II1 = 4r.+2<-2 r 2"+f--l,x XI

where r is the incomplete Gamma function [16],

(2.19)

(2.20)

(2.21 )

(2.22)

(2.23)

(2.24)


Evaluation of 12

Substituting the expression for X*, i.e.

itf kt (r2)

I2 = C tf-I (4kT )'/2+< exp - 4kT dt

and taking into account that r = tF - t,

(2.25)

(2.26)

Introducing again the change of variables given by (2.20), the integral J2 becomes

lXf kr2 e-X

J2 = C -dxXf-I (4kT)·/2+< x

or, eliminating the product kT,

C 1 lxfJ2 = - X'/2+<-3 e-x dx16r·+2<-4 k Xf-I

which gives

J2 = C ! [r (-28 +f - 2,X)] Xxf

f-

1

16rs+2<-4 k

Evaluation of r(n, x)

(2.27)

(2.28)

(2.29)

The following recurrence relation can be used for evaluating the incomplete Gammafunction [16],

f(n + l,x) = nr(n,x) + xne-X

The following particular expressions should also be noted [16]:

f(1, x) = e-X

in which E1 is the exponential-integral and erf the error function.


Space Integration

The final step in the calculation of the coefficients of matrices HI, H2, GI and G2is the spatial integration along the boundary elements. Calling

it is possible to introduce four functions of y, namely I<1 i ,j, I<2 i ,j, I<3 i ,j and I<4 i ,j

such that

r I1(Tt) df = I<1 i,AYI-l) - Kl i ,j(YI) = [I<li,j]~~-1 (2.30)ir}

j I1 (q;) df = [I<2i,jj~~-1 (2.31)}

r J2(T*) df = ~ [I<3· .]Yf-1 (2.32)ir 1 k I,) Yf}

j J2(q;)df = l [I<4i,j]~~-1 (2.33)}

The preliminary calculation of the above 4N2 functions (N: number of boundarynodes), for points distributed between 0 and Ymax, makes it possible to determine byinterpolation the values of I<l i,j to I<4i,j at YI, for all f and all pairs i,j, during theresolution procedure.

Thus, taking into account expressions (2.10) to (2.13), we have from the definitionof I1 and J2:

HI 'FI = _1_ {(tl - tF) [I<2 .jYf-1 + ~ [I<4· .]Yf-1 } (2.34)l,), , l::i.t I l,) Yf k I,) Yf

i12· FI = _1_ {(tF - tl- 1 ) [I<2· .]Yf-1

- ~ [I<4· .]Yf-1 } (2.35)I,), , ~t I I,) Y/ k I,) Y/

G1 .FI = _1_ {(t l - tF) [I<I· .]Yf-1 + ~ [I<3· .]Yf-1 } (2.36)I,)" l::i.t I I,) Yf k I,) Yf

G2 FI = _1_ {(tF - tl _1 ) [I<1 .]Yf-1 - ~ [/(3· .]Yf-1 } (2.37)I,)" ~t I I,) Yf k I,) Yf

The boundary elements employed in the present work are superparametric, i.e. theshape functions used to model the geometry are of higher order than the interpolationfunctions used to represent the functional variation. The shape functions are derivedby using Lagrangian polynomials in non-dimensional, natural coordinates.

For two-dimensional problems, the expressions for /(1 to /(4 are (omitting thesubscripts i,j for simplicity):

1 1+1

I< 1 = - £1 (x) I J I dry471" -1

(2.38)

(2.41)

(2.40)

(2.39)


1 1+1 dK2 = - - e-x I J I d1]211" -1 r 2

1 J+1 [e- X

]K3=- r 2 --E1(x) IJld1]1611" -1 X

1 1+1K4 = -8 dE1(x) I J I d1]11" -1

where 1] is the natural coordinate and 1 J I the Jacobian of the transformation fromr to 1], i.e.

1J 1=1 dr 1d1]

The calculation of the off-diagonal coefficients of matrices HI, H2, GI and G2involves only regular integrals, for all time levels tJ up to and including the actualtime tF. In this case, it is noted that T = 0 and x --t 00 according to (2.20); thus, allintegrands in expressions (2.38) to (2.41) become nul.

For the diagonal coefficients (i.e. i = j in expressions (2.34) to (2.37)) we havethat, for any t i= tF, there is a singularity in the calculation of K1 at the source pointi where r = 0 (and x = 0). It is possible to remove this logarithmic singularity by achange of variables of the form [9]

1] = 1 ~ 13

/2 sgn(O

in which sgn(O takes the sign of ~ and the integration limits are not changed. TheJacobian of this transformation is

d1] 3 1 1/2d~ = 2 ~ I sgn(0

It can be seen that this Jacobian is zero at the singular point 1] = ~ = 0, thus thenew integral is regular and can be evaluated using standard quadrature schemes. Ageneralization of this idea has been developed by Telles [17] and applied to singularand nearly-singular integrals.

The expressions of f{1 to f{ 4 for three-dimensional problems can be written as

1 11)2 11'2(1)) 1f{1 = - - [1- erf(JX)] 1 J 1 dJid1]

411" 1)1 1'1(1)) r

11'1211'2(1)) [ 1]f{3 = -8 r erf(JX) -1 + ~ e- x I J I dJid1]11" 1)1 1'1(1)) y1l"X

1 11)2 11'2(1)) df{4 = - - [1- erf(JX)] I J I dJid1]

811" 1)1 I'd1)) r

(2.42)

(2.43)

(2.44)

(2.45)


~

~

+1

+t.p

1

- 1

- 1 +2

11

- 1

Figure 2.1: Integration domain for three-dimensional boundary elements.

The elements used for three-dimensional analysis are quadrilateral or triangular, flator curved. With the node placed at position "l = 0, J1. = 0, the domain of integrationfor each type of element is shown in Fig. 2.1.

The singular integrals that appear in the evaluation of ](l i,j and ](2 i,j at r =°for i = j can be more easily dealt with using cylindrical polar coordinates. To thisend, these integrals are split in the form

](1 = ](1' + fOil

](2 = ](2' + ](2"

where

and rewritten as

1 1'12 11"2(1)) 1](1' = - - I J I dJ1. d"l

41l' 1)1 I"J(I)) r

1 1211" 1P

(8) P](1' = - - I J I dp dB41l' 0 0 r

1 1211" 1P(8) d p](2' = - - - I J I dp dB

41l' 0 0 r 2 r

(2.46)

(2.47)

(2.48)

(2.49)


When TJ and Jl simultaneously tend to zero, the ratios p/r and d/r2 tend to constants.Thus, the integrals Kl' and K2' become regular.

2.5 Boundary Conditions

~inear Boundary Conditions

The most common linear boundary conditions in heat transfer problems are of thefollowing types:

• Dirichlet condition (prescribed temperature):

• Neumann condition (prescribed flux):

aT _q= -=qan

• Fourier condition (convection):

p(T) = >.. q = -h (T - Ta ) + 4> = aT + JJ

in which h is the heat transfer coefficient, Ta the ambient temperature and 4> isa known value. The adopted convention implies that q is positive if the surfaceflux p = >.. q is inwards the region. Moreover, a and JJ may vary in space andtime.

With the above, the equation relative to a boundary point i in the matrix system(2.16) can be decomposed as follows:

" H2" "FFT F +" H2 "FFT F +"(H2" "FF - G2" "FFa -)T F -L...J I,), I 3, L...J 1.,), J J, L...J t,), 1 1.,1" J j, -

j(l) j(2) j(3)

L G2i ,j,F,Fqj,F +L G2i ,j,F,FQj,F +L G2i ,j,F,FJJj +Si,F (2.50)j(l) j(2) j(3)

where j (k) means element j is in part r k of the boundary.Writing the above equation for all boundary points produces the system

H2~,F T F = G2F,F QF +S~ (2.51)

similar to (2.16), in which:

H2'*J" F F = H2i J" FF" , I I ,

H2'~J"FF = H2iJFF - G2iJ"FF a J"" I " , " ,


S;F = Si,F +L G2 i ,j,F,F (3jj(3)

Non-Linear Boundary Conditions

The third kind of boundary condition (convection) can be more generally expressedas

aTan = q(T)

in which q is a non-linear function of T. A typical case is the mixed convectionradiation condition

p(T) = Aq(T) = -a ((T4- T;) - h (T - Ta ) +~

in which a is the Stefan-Boltzmann constant and ( the emissivity.A simple and efficient approach to consider the above consists in linearizing q(T);

in the neighbourhood of a point P with temperature Tp and flux q(Tp) = qp it ispossible to write a truncated Taylor series in the form

(2.52)

j3 = qp - aTp

which gives:

a = [dq]

dT Tp

The determination of a(Tp) and j3(Tp) requires an iteration process, which can bestarted by assuming e.g. Tp = (TF + TF - 1 )/2; if sufficiently small time steps areused, iteration can be avoided by using the cruder approximation Tp = TF - 1 . Thisscheme was derived by Pasquetti and Caruso [9] and named 'local linearization' since,after the boundary integral equation is discretized, the above linearization is appliedindependently to each element. Note that, for radiation boundary conditions,

a = -(h +4a(T~)/A ;(3 = qp/A - aTp

A more refined 'global linearization' technique based on a Newton-Raphson procedure has been developed for steady-state problems by Azevedo and Wrobel [18] andemployed for transient analysis by Wrobel and Brebbia [19] using a dual reciprocityboundary element formulation.

Non-Linear Diffusion

The boundary conditions in this case are written as follows:

• Dirichlet condition:


• Neumann condition:

aT at/JP = ,,\ - = - = p on r2an On

• Fourier condition:

ot/Jp=-=at/J+f3onThe terms a and f3 are function of t/J since the Kirchhoff transformation transfers thematerial non-linearity to the boundary condition in this case. Linearizing the field inthe vicinity of point P, i.e.

we obtain:

(2.53)

[dP] [dP dT]

a( t/Jp) = dt/J ,pp = dT dt/J ,pp [1 dP],,\ dT ,pp

Another type of non-linearity may appear if the region under consideration ismade up of piecewise homogeneous sub-regions of different materials. Enforcement ofthe compatibility condition along the interface between sub-regions

where the subscripts represent different regions, produces a discontinuity in the integral of conductivity t/J, since ¢I =I- ¢II'

Pasquetti and Caruso [9] suggested a local linearization around a point P on theinterface in the form:

From this expression, ¢ II can be eliminated by writing it as a function of t/JI, i. e.

Such an approach is also valid for mixed conditions between sub-regions, as in thecase of heat transfer resistance [10].

An alternative global linearization using a Newton-Raphson technique applied tothe final system of equations was presented by Azevedo and Wrobel [8].


2.6 Initial Conditions

If the initial temperature field is uniform, it is possible to eliminate its domain integralin Eq. (2.2) by solving for the temperature difference T - To. For non-linear problems,it suffices to choose To as the reference temperature Tr in the Kirchhoff transformation (2.5) to obtain 1/10 = 0, thus eliminating the corresponding domain integral inEq. (2.6). Two other cases which also permit the suppression of this domain integralwill be discussed in what follows for linear materials, with extension to non-linearproblems being straightforward.

Stationary Initial Temperature Field

Consider that the initial temperature field To satisfies the equation

(2.54)

in which be, is the value of b at time to. Subtracting the above equation from Eq. (2.1),we obtain

'V 2(T - To) +b- bo = ~ aT = ~~(T - To)k at kat

with the initial condition T - To = 0 at time to.The boundary integral equation equivalent to (2.55) is

(2.55)

itF rk (q - qo) T* df dt +i tF rk (b - bo) T* dO dt (2.56)

to ir to illSo, with simple changes of variables, a boundary integral equation is obtained in whichthe domain integral due to initial conditions no longer appears.

'Study Temporal Domain'

If the span of time elapsed between the initial time to and the resolution time tF islarge enough, it is not necessary to carry out the integrations up to the initial timeto. Thus, all computations can be kept within the time range TO (the 'study temporaldomain'), implying substantial savings in the evaluation of boundary integrals. Tocalculate the domain integral due to initial conditions, associated with the new timetF-TO, a recurrent approximation is developed which uses previously calculated valuesto avoid the evaluation of new domain integrals.

Considering two successive instants to and t1 and the final time tF' and callingT1 = tF- t1 and To = tF- to, it is possible to write an equation similar to (2.2) withtime limits tF- TO and tF and another with time limits tF- T1 and tF' The differencebetween the two yields the integral equation


1 1 l tl -TIlTtl_TIT*(TI)dn= Tt'-TOT*(To)dn- F kTq*dfdH

o F 0 F t~-TO r

1tF-

TI[ k qT* dr dt +1tF

-TI

[ k bT* dO dt (2.57)tF-To 1r tF-TO 10

With tF = tF+ tit, consider now the boundary integral equation from tF - TO totF; then, the problem is to calculate the domain integral 10 such that

10 = [ TtF-TO T*( TO) dO = [ Tt' -TI T*(TO) dO (2.58)10 10 F

A solution can be found by differentiating the function T* with respect to T,

(2.59)

so that, if

(2.60)

one gets

aT* ~ -T* ~ (2.61)aT 2T

A second-order central finite difference approximation to the above expressiongIves

or

Consequently,

Thus, if

T*( TO) - T*( TI) T*( TO) +T*(Td 8-"--'------'--'-~- --

tit 2 TO +TI

T*(TO) ~ T*{rt) TO +T} - 8t1t/2TO + T} +8 t1t /2

1 ~ TO+TI-8tit/2 [ T" T*(T )dOo TO +TI +8 tit/2 10 tF-TI 1

(2.62)

(2.63)

(2.64)

max(r2)

TO» 28 min(k)

which is the most severe possible case from condition (2.60), Eq. (2.57) and approximation (2.64) of 10 yield the recurrent algorithm

[ Tt' -TO T*(TO) dO => [ Tt' -TI T*(Td dO => [ TtF-TO T*(TO) dO (2.65)10 F 10 F 10

(2.67)

(2.68)


The above algorithm permits an estimation of 10 at each time step without anycalculation of a domain integral if this integral is zero at time to.

2.7 Treatment of Heat Sources

The contribution of internal heat sources to the temperature field IS gIven in theboundary integral Eq. (2.2) by an integral over space and time

h = l tF

f k bT* dO dt (2.66)to in

The above integral involves only known terms but, for a general function b, it requiresdomain discretization for its numerical evaluation. Some special cases for which domain discretization can be avoided are discussed in what follows.

Point Sources

If a certain number L of point sources exists within the domain 0, the source term bcan be written as:

L

b = Lblhl1=1

where 81 is a Dirac delta function applied at point 1 and bl is the time-dependentsource intensity at this point.

Substituting the above into expression (2.66) for I b, we have

L l tF

Ib = L k bl Tt dt1=1 to

in which 1/* is the value of the fundamental solution at point 1 (for a source point Mas in equation (2.2)).

Harmonic Sources

If a harmonic source is considered, i.e. a source function b such that V'2b = 0, twodifferent approaches can be used to eliminate the domain integral h. The first derivesfrom an application of Green's second identity in the form

(2.69)

If a function w* can be found such that V'2W* = T*, the above equation reduces to

r bT* dO = f (b ow* _ w* Ob) dfin ir on on

Thus, the domain integral Ib in expression (2.66) becomes

(2.70)

(2.71)


h= l: F

kkbT*dOdt= l: F

Jk (ba

a:* -w* ;~) drdt

An expression for w* for two-dimensional problems is given in [l1J in the form

A different approach proposed by Pasquetti [20J involves a simple change of variables. Introducing a function G1 such that

~ aG1 =-bk at

we can integrate the above to obtain:

Taking the Laplacian of both sides, assuming that k is a constant,

2 2ltF ltF2\7 G1 = - \7 k bdt = - k \7 bdt = 0

to to

The next step is to write the original governing Eq. (2.1) in the form

\72T _ ~ aG1 = ~ aTk at k at

or, taking Eq. (2.73) into account,

\72(T G) = ~ a(T +Cd+ 1 k at

(2.72)

(2.73)

(2.74)

(2.75)

Since from the definition of G1 its initial value at time to is zero, the boundaryintegral equation equivalent to the partial differential Eq. (2.75) is of the form

(2.76)

with

aG1 ltF ab-= k-dtan to an

(2.77)


Use of Particular Solutions

Consider now a space-dependent heat source bfor which a corresponding function G2

can be found which is a particular solution of the Poisson equation

'V2G2 = b

Rewriting the original Eq. (2.1) in the form

'V2T + 'V2G2= ~ fJT (2.78)kfJt

and considering that G2 is not time-dependent, Eq. (2.78) can be further manipulatedto produce

'V2(T G) = ~ fJ(T +G2)+ 2 k fJt (2.79)

With the above transformation, the governing differential equation of the problembecomes a homogeneous diffusion equation; thus, the corresponding boundary integralequation presents no domain integrals and is again of the form

eM TM,tF +l tFf k T q* df dt = l tF

f k qT* df dt + f To T* dn-~ k ~ k ~

(2.80)

2.8 Applications

A computer program incorporating the previously described features was developedby the first and second authors [9,10]. Results of some applications of the programare presented in this section.

Solid Cylinder

A solid cylinder with zero initial temperature is heated by an internal source whose intensity is constant in space and variable in time, in the form 9 = 105(1+0.5sin21l'tj400)Wm-3 • The boundary conditions represent convection into a medium at ooe, withheat transfer coefficient h = 200 Wm- 21<-1. The cylinder radius is 0.1 m, the thermalconductivity A= 50 Wm- 11<-1 and diffusivity k = 5 X 10-4 m 2s-1 .

Figures 2.2 and 2.3 compare the BEM results obtained with 8 quadratic superparametric elements with a reference solution for this one-dimensional problem obtainedby the Control Volume Method (CVM). A 'study temporal domain' equal to 400 swas employed. In this linear problem the BEM and CVM results are nearly identical,attesting the efficiency of the 'study temporal domain' technique.


{\ 1\ /\ (\ 1\j\ 1\ j \ J\ 1\j \ / \ : \ j' ! \: \ /' :\ / : i \ J' \

/' :, i \ J . : l: \ i : / : i \ ' .

, , \ i \ J : ! ,\ j \ Jl :\ i 'i \\/ \: :\Ji \j \\: :j .' ,: : ..j \: ....j ....1 \!

20.

15.

10. i

25.

30.

45.

40.

35.-g,wa::::>tcta:wc.:ewttl)

><ct

5.

2000.1 500.1000.

TIME (s)500.

0. '--_~_~_----L._----Jl.--_~_-l.-_--L._---J0.

Figure 2.2: Cylinder axis temperature variation (HEM - dashed line; CVM - dotted line),


45.

- 4"'.0-W 35.a:::J... 3"'.«a:w 25.c..:iw 2"'•...W0 15.«

iLLa: 1 "'.::J ien5.

121.121. 5121121. 1121121121. 1 5121121.

TIME (5)2121121121.

Figure 2.3: Cylinder surface temperature variation (BEM - dashed line; CVM - dotted line).


Two Concentric Cylinders

This analysis concerns two concentric cylinders of different, non-linear materials withthe following properties:

• radius of inner cylinder: rI = 0.05m

• radius of outer cylinder: rll = O.lm

• thermal conductivity:

Al = 10[1 + (T - 273)/3000] Wm- 1K-1

All = 4[1 + (T - 273)/2000] Wm- 1K- 1

• density:

PI = 7500 kg m-3

Pll = 1000 kg m-3

• specific heat:

CI = 500 J kg- 1 K- 1

Cll = 2000 J kg- 1 K- 1

The initial temperature distribution is given by a stationary field, as shown in Fig.2.4. The boundary conditions along the surface of the outer cylinder are non-linear,of the radiative type, with unit emissivity and ambient temperature of 300K. Theheat source intensity is g = 105 Wm- 3 at t = 0, and g = 106 Wm- 3 for t > O.

Results are presented in Fig. 2.5 for a discretization of 16 superparametric elements, with 8 located along the interface between cylinders and 8 along the externalsurface of the outer cylinder. A variable time step (varying from 50s to 400s) wasadopted. It can be seen that the agreement between the BEM results and a CVMsolution is reasonable, with a difference that is only due to the BEM formulation itself,which introduces some approximations in the case of non-linear materials.

Sphere

This example considers a sphere at initial temperature 300K of a non-linear material with conductivity A = 1.023[1 + 1.25 x 1O-3 (T - 300)] Wm- 1 /(-1, densityP = 3500 kg m-3 and specific heat c = 841.3 Jkg- 1 /(-1. The boundary conditionsare also of the radiative type, with emissivity i = 0.8 and ambient temperature 300/(.The sphere radius is O.lm, and a heat source exists with intensity g = 105 Wm- 3 .

Figures 2.6 and 2.7 depict the BEM results obtained by using 12 quadrilateralboundary elements with 9 nodes and 12 triangular boundary elements with 6 nodes.Also shown in the figures is the reference one-dimensional solution obtained with theCVM. To show the effect of the non-linearity, linear results for A = 1.023 are alsodepicted.

Nuclear Reactor Core

The present application concerns a simplified model of a nuclear reactor core for


"'-"'-"'-"'-"-"'--""-""

'''.8[21[21.-~-w

a: 7 fa 121.:)~«a:w 6121121.Q.

:EUJ~

"-""

5121121.

'·'-"'-"'-"'-"'-"'-"-"'-"0

"'-"-"'-... - ... - ...-4121121.

01.121121 • 1214 • 1216 • 1218 • 1121

RADIUS (m)

Figure 2.4: Temperature variation along radial direction for two concentric cylinders system,for the initial and final stationary states (BEM - dotted line; CVM - dashed line).


1 5!21!21 [21.

...................................................

1 !21 !2112I12I.5 !21 !21 !21.

.............................................

.... .::::.:;::..:;~~.=.===..:.:..:.:..:..:.:..:.:..:..:.:..:.:..:"""""' ....................

: :'

4 !2112I.121.

9121121.

8121121.-~-wa: 7 f2l12l.~to-e:{a:w 6£21 £21.a.~wto-

S 121121.

TIME (5)

Figure 2.5: Temperature variation at the centre, interface and external surface of the twoconcentric cylinders system (BEM - dotted line; CYM - full line).


30000.10000. 20000.

TIME (s)

3121(21. IL-----JL...----J_--I._--L_---L._-.L_--J-_-L-_--J--_....L-_..L-.~

0.

550.

-~- 500.Wa::JI-cta: 450.WDo:EwI- 400.W0ctLLa: 350.:Jen

Figure 2.6: Sphere surface temperature variation (BEM - dashed line; CVM - dotted line;CVM with constant conductivity - dotted and dashed line).

7121121.

65121.

SZ-W 6121121.a::JI- 55121.<a:w 5121121.C.:5wI- 45121.a:WI- 4121121.ZW0 35121.

3121121.121.


----------------------_.,..........

/.,/.-;.... ::::: .•. ;:7::••.•~••••= ...=...="...~

.<:;, ~ ./' ..-P//.....

j ...j"

IIf

/TIME (s)

Figure 2.7: Sphere centre temperature variation (BEM - dashed line; CYM - dotted line;CYM with constant conductivity - dotted and dashed line).


Figure 2.8: Schematic cross-section of the nuclear reactor core.

which the temperature distribution is calculated following a step variation of the heattransfer fluid velocity.

The core is simulated as a homogeneous region traversed by cylindrical conduits(Fig. 2.8). In this region, an internal uniform heat source is generated; the energy istransferred, initially by diffusion and later by convection, to the heat transfer fluidin the conduits. The longitudinal heat transfer can be neglected, and the problemanalysed as two-dimensional.

Away from the geometrical boundaries of the core, the symmetry of the problemleads to consideration of the diffusion domain as shown in Fig. 2.9. Along the symmetry lines the flux density is zero; along the interface with the fluid elements, theboundary condition is of the mixed type. This last condition is non-linear becausethe Nusselt number Nu is calculated, knowing the Reynolds and Prandtl numbers Reand Pr, by using the Hausen formula [21]:

( )

0.14

Nu = 0.116 (Re2/ 3 _ 125) Pr 1/ 3 PfvfPffvff

in which P is density and v kinematic viscosity; the subscripts f and f f denote fluidand film fluid, respectively.

On the domain, the diffusion problem is weakly non-linear because of the variationof the thermal properties of the core with temperature (the following data correspondto stainless steel):


.'

p • Ojt:ifl~~~~i4

p=p(T)

2

p=o

3

p=O

Figur 2.9: R ion und r stud and di r tization.

720

- 700~

illa: 680 ~ TEMP(6):::> ... TEMP(S)f- .. TEMP(3)<{ ... TEMP(4)a:ill 660 .. TEMP(l)a.. ~ TEMP(2)~illf- 640

6200 10 20 30 40 50 60

TIME (8)

Figure 2.10: Temperature variation at four boundary points and two internal points (locations defined in Fig. 2.9) after step variation of the fluid velocity.


c = 14.59 +0.139 x 10-1 (T - 273) J kg- 1K- 1

P = 7700 kg m-3

A = 269.96 +0.7172T - 0.40625 x 10-3 T2 W m-1K- 1

The fluid is assumed to be at a temperature of 300°C; at this temperature, thefluid parameter values are (the following data are those of a synthetic oil whichwould permit operation at low pressure): c = 2587 J kg- 1K- 1

, P = 805.1 kg m-3, K =

0.1076 Wm- 1K- 1 , v = 0.502 X 10-6 m2s-1 •

For the liquid film:

p = 1016 - 0.703 (T - 273) kgm-3

v = 0.80656 X 10-5 exp( -0.48455 x 1O-2T) m2 S-1

The distance between the centre of the conduits is 30 x 1O-3m and their diameteris 20 x 1O-3m; the boundary is discretized by using 35 elements (Fig. 9), of which30 are straight linear and 5 circular quadratic. The heat generation is equal to 9 =30 MW m-2. The step variation of the velocity occurs at time t = 0, with a changefrom 4 to 6 ms- l . The transient study starts from an initial steady state.

Figure 2.10 presents the temperature variation versus time for 4 boundary nodes(points numbered 1 to 4 in Fig. 2.9) and two internal points (numbered 5 and 6).These results have been compared with those obtained using the finite element method(FEM); the maximum deviation of the temperature difference between the two resultsis about 1%.

References

1. Morse, P.M. and Feshbach, H., Methods of Theoretical Physics, McGraw-Hill,New York, 1953.

2. Carslaw, H.S. and Jaeger, J.C., Conduction of Heat in Solids, 2nd edn, ClarendonPress, Oxford, 1959.

3. Chang, Y.P., Kang, C.S. and Chen, D.J., The use of fundamental Green's functions for the solution of problems of heat conduction in anisotropic media, Int.J. Heat Mass Transf., Vol. 16, pp 1905-1918, 1973.

4. Shaw, R.P., An integral equation approach to diffusion, Int. J. Heat MassTransf., Vol. 17, pp 693-699, 1974.

5. Wrobel, L.C. and Brebbia, C.A., The boundary element method for steady-stateand transient heat conduction, in Numerical Methods in Thermal Problems, Vol.1, Ed. R.W. Lewis and K. Morgan, Pineridge Press, Swansea, 1979.


6. Onishi, K., Convergence in the boundary element method for heat equation,TRU Mathematics, Vol. 17, pp 213-225, 1981.

7. Wrobel, L.C. and Brebbia, C.A., A formulation of the boundary element methodfor axisymmetric transient heat conduction, Int. J. Heat Mass Transj., Vol. 24,pp 843-850, 1981.

8. Skerget, P. and Brebbia, C.A., Time-dependent non-linear potential problems,in Topics in Boundary Element Research, Vol. 1, Ed. C.A. Brebbia, SpringerVerlag, Berlin, 1984.

9. Pasquetti, R. and Caruso, A., A new software for the modelization of transientand non-linear thermal diffusion, in BEM X, Ed. C.A. Brebbia, ComputationalMechanics Publications, Southampton, and Springer-Verlag, Berlin, 1988.

10. Pasquetti, R. and Caruso, A., Boundary element approach for transient andnonlinear thermal diffusion, Num. Heat Transj., Part B, Vol. 17, pp 83-99,1990.

11. Brebbia, C.A., Telles, J.C.F. and Wrobel, L.C., Boundary Element Techniques,Springer-Verlag, Berlin and New York, 1984.

12. Ames, W.F., Nonlinear Partial Differential Equations in Engineering, AcademicPress, New York, 1965.

13. Demirel, V. and Wang, S., An efficient boundary element method for twodimensional transient wave propagation problems, Applied Mathematical Modelling, Vol. 11, pp 411-416, 1987.

14. Davey, K. and Hinduja, S., An improved procedure for solving transient heatconduction problems using the boundary element method, Int. J. Num. Meth.Engng, Vol. 28, pp 2293-2306, 1989.

15. Silva, W.L. and Mansur, W.J., A truncation scheme for BEM analysis of twodimensional transient wave propagation problems, in BEM XII, Ed. C.A. Brebbia and M. Tanaka, Computational Mechanics Publications, Southampton, andSpringer-Verlag, Berlin, 1990.

16. Abramowitz, M. and Stegun, LA., Handbook of Mathematical Functions, DoverPublications, New York, 1965.

17. Telles, J.C.F., A self-adaptive co-ordinate transformation for efficient numericalevaluation of general boundary element integrals, Int. J. Num. Meth. Engng,Vol. 24, pp 959-973, 1987.

18. Azevedo, J.P.S. and Wrobel, L.C., Non-linear heat conduction in composite bodies: a boundary element formulation, Int. J. Num. Meth. Engng, Vol. 26, pp19-38, 1988.

19. Wrobel, L.C. and Brebbia, C.A., The dual reciprocity boundary element formulation for nonlinear diffusion problems, Compo Meth. Appl. Mech. Engng, Vol.65, pp 147-164, 1987.


20. Pasquetti, R., A boundary element software for the numerical integration of theheat conduction equation, in Fifth Int. Symp. on Numerical Methods in Engineering, Ed. R. Gruber, J. Periaux and R.P. Shaw, Computational MechanicsPublications, Southampton, and Springer-Verlag, Berlin, 1989.

21. Eckert, E.R.G. and Drake, M., Heat and Mass Transfer, McGraw-Hill, NewYork,1959.

Chapter 3

Solving Linear Heat Conduction Problemsby the Multiple Reciprocity Method

A.J. NowakInstitute of Thermal Technology, Silesian Technical University,

44-101 Gliwice, Konarskiego 22, Poland

3.1 Introduction

Background

Heat transfer processes, which always take place when differences in temperatureoccur, play an important role in many industrial plants. Apart from the fact that calculation of the temperature field is needed in order to determine a suitable heating orcooling system of industrial object, the temperature field also frequently substantiallymodifies mechanical properties of the body and causes additional stresses. Thereforeefficient methods of solving heat transfer problems are of great interest for all theseengineers who are actively involved in mathematical modelling.

In this chapter, linear heat conduction will be discussed with emphasis on transient problems. When using the Boundary Element Method for these type of problemseither a time dependent or a time independent fundamental solution is applied. Itshould be stressed however, that whatever approach is used, the BEM generally leadsto an integral equation with domain integrals, e.g. Brebbia, Telles and Wrobel [1],Brebbia and Dominguez [2]. When a time dependent fundamental solution is employed, the final formulation contains a domain integral of the initial conditions (cfchapter 2). Time independent fundamental solutions produce a domain integral oftemporal derivative of temperature. These integrals spoil the elegance of the formulation and affect its efficiency. Hence, a substantial amount of research has beencarried out to find a general and efficient method of transforming domain integralsinto equivalent boundary ones.

In formulations with time dependent fundamental solutions, domain integrationcan be avoided by transforming the primary boundary value problem into the formwith homogeneous initial condition and applying special integration technique withrespect to time, e.g. Brebbia, Telles and Wrobel [1]. However, although this formu-


lation is the most natural one, it requires substantial storage and is recognized to betime consuming.

The first approach employing the time independent fundamental solution, calledthe Dual Reciprocity Method (DRM), was proposed by Nardini and Brebbia [3],[4] in1982 and was afterwards extended for a variety of thermal problems by Wrobel andBrebbia [5],[6]. Many calculations carried out so far have proved the accuracy andefficiency of the method (cf chapter 1).

More recently, an alternative approach called the Multiple Reciprocity Method(MRM) [7],[8],[9] was proposed by Nowak and Brebbia. This technique is also basedon employing the reciprocity theorem and it can be thought of as an extension of theDRM idea. It should be pointed out however that instead of approximating the partof integrand by the set of coordinate functions multiplied by constants or functionsof time, a sequence of functions associated with the fundamental solution is built up.This set can be understood as a sequence of higher order fundamental solutions andit enables one to use the reciprocity theorem as many times as required. As a result,the method leads in the limit to the exact boundary only formulation. Solution ofthe problem is expressed in terms of series. It is worth pointing out that the DRMapproximation can also be applied in order to estimate the remainder of the truncatedseries. In that sense, this chapter can be treated as a sort of generalization of bothMRM and DRM techniques.

Since MRM has been developed quite recently the present formulation is still faraway from the final stage and more research has to be carried out in this promisingtopic.

3.2 Fundamentals of the Multiple Reciprocity Method

The Multiple Reciprocity Method is a general method of transforming domain integrals arising in BEM for thermal problems. These integrals have the following form

Do = in bo u~ dO (3.1)

where n stands for a region under consideration, Uois a fundamental solution and borepresents arbitrary function of coordinates (e.g. heat source generation rate, temporalderivative of temperature). The fundamental solution and analog of a heat flux usedin this chapter are defined as

k \72 u~ = ~i (3.2)

(3.3)

where L\; is Dirac's function acting at point i, k is the thermal conductivity, andoO/on stands for the outward normal derivative.

In order to transform integral (3.1) into boundary r one can introduce functionui and its normal derivative associated with fundamental solution Uoby formula

(3.4)

(3.5)


• k ouiq -- -I - an

Thus, applying the reciprocity theorem the domain integral Do can be evaluated asfollows

Do =~ bo u~ dO =~ bo \J2ui dO =

= tj (ui Wo - qi bo) df +D I

where

aboWo = - kan

Domain integral DI has the form

DI = ~ ui \J2bo dO

Calling

(3.6)

(3.7)

(3.8)

ObIWI = - kan (3.9)

(3.11)

one immediately arrives at

DI =~ ui bI dO (3.10)

Since the final form of the domain integral DI is very much the same as that ofintegral Do it can be integrated by parts as before, i.e.

DI = ~ r(u; WI - q; bI ) df + D2k irwhere

(3.12)

and

D2 =~ u; b2 dO (3.13)

The procedure described above can be generalized in a natural way. Introducinga sequence of higher order fundamental solutions defined by the recurrence formula

forj=O,1,2, ... (3.14a)

au·q. = _ k-J

J an (3.14b)


and a sequence of functions bj and Wj

abw=-k-J

J an

(3.15a)

(3.15b)

the integral Do can be transformed into the following series of boundary integrals only

Do = f ~ l(uj+l Wj - qj+l bj ) dr (3.16)J=O r

It should be noted that Eq. (3.16) is the exact form of the primary domain integral(3.1) (no simplifications were made).

Higher Order Fundamental Solutions

Higher order fundamental solutions are defined by the recurrence formula (3.14). Thisequation can be easily solved analytically when Laplace's operator is written in termsof cylindrical (for 2- D problems) or spherical (for 3- D problems) coordinate system.For example, it can be proved by the mathematical induction, that for 2 - D problemsthe general form of function uj is given by the following expression

uj=2~kr2j(Ajlnr-Bj) forj=0,1,2,... (3.17)

where r represents distance between points, and coefficients A j and B j are obtainedrecurrently from the relationship

(3.18a)

for j = 0,1,2, ...

(3.18b)

(3.19)

Usually recommended values Ao = 1 and Bo = 0 [8],[9] result from the form of classicalfundamental solution u~

• 1 Iu = - nro 21rk

However, any value of Bo can be selected. It was found that this value affects theconvergence of series (3.16). The problem seems to be related to the non-uniquenessof the solution as reported in [10],[11] and is being investigated at this moment [12J.

Heat flux analog qj is calculated from relationship

q; = - 2~ [(2 j In r + 1) A j - 2 j B j )] r 2j-

1 ~: (3.20)

Notice that formula (3.18) introduces the factorials into the denominators of the coefficients A j and B j and this guarantees convergence of series (3.16). It should also


be stressed that the functions u; and q; have no singularities (for j = 1,2, ... ) thus,the integration does not require any special technique.

Error Estimation for Truncated Series

In practical applications series, (3.16) has to be truncated. If the number of termstaken into account is equal to J - 1 one can write

J-l 1Do = L -k 1(Uj+l Wj - qj+l bj ) df +RJ

j=O r(3.21 )

where RJ is the remainder of J-th order. Since this term is the following domainintegral

RJ = in uj bJ dfl (3.22)

it may be estimated applying the DRM idea of approximation of function bJ by a setof coordinate functions r (x)

NPbJ = L r(x) an

n=l

(3.23)

NP represents the !"lumber of functions considered in the set (cfchapter 1). For steadystate problems coefficients an are constants whereas for transient heat conduction theyare functions of time.

Introducing Eq. (3.23) into (3.22) one obtains

NP

RJ = Lan1r uj dfln=l n

Following the MRM algorithm the above integral is evaluated as follows

NPRJ = Lan [ r V'2u j+l dfl =

n=l 1n

where integral R9 is defined as

and

'lj;n = _ k aran

If function r in approximation (3.23) fulfils the following condition

(3.24)

(3.25)

(3.26)

(3.27)


(3.28)

one can convert integral (3.26) into

NP NP

R9 = Lan1uj+I dO, = Lan1\72Uj+2 dO, =n=l n n=l n

NP 1 [= - L k an ir qj+2 dr (3.29)

n=l r

The approach presented in this section combines both DR and MR techniques.MRM is used to obtain the series (3.16) whereas DRM is applied to estimate theerror made when this series is truncated. In this sense DRM can be thought of as atechnique in which transformation into the series is avoided and only the remainderis taken into consideration.

Approximation functionsFunctions r satisfying Eq. (3.28) can easily be found by analytical integration whichfor 2 - D problems gives

r = ~ r2 (3.30)

4

Although such quadratic approximation is also used in the DRM, the most popularapproximation is the linear one in which r is equal to r. Coefficients an are calculated as usual from Eq. (3.23) written for all considered nodes.

3.3 Heat Conduction with Heat Sources

Governing Equations

Linear heat conduction with heat sources in any region 0, surrounded by boundary ris governed by the Poisson equation

inn (3.31)

(3.32)on r

where u(x) stands for temperature at point x and bo represents heat source generationrate. Since the problem is linear, function bo is a known function of space. On theboundary r any kind of boundary condition can be prescribed but in this section,boundary condition of the third kind is considered

auq = - k an = h (u - uf)

where q is a heat flux, h represents heat transfer coefficient, u f is a known temperatureof fluid which exchanges heat with the boundary r by convection.

(3.34)


MRM Formulation

Applying the second Green's theorem to Eqs (3.2) and (3.31) one obtains the followingintegral equation

Cj Uj +!r qo U df = !r Uoqdr +~ bo UodO (3.33)

The domain integral in Eq. (3.33) can now be evaluated as described in section 3.2.Introducing Eq. (3.16) into Eq. (3.33) one arrives at

Cj Uj +!r qo U df - !r u~ qdf =

OOlr oolr=L k ir Uj+l wjdf - L k ir qj+l bjdf

)=0 r )=0 r

The integrals can now be evaluated numerically by subdividing the boundary f asusual into a number of boundary elements. The problem variables are approximatedwithin each boundary element using the appropriate boundary elements interpolationfunctions (e.g. [1],[2])

(3.35)

Since heat source generation rate bo is a known function of space the sequence offunctions bj and Wj can be obtained by analytical differentiation. Thus, integrals inthe series (3.34) can be calculated directly. However, the same type of approximationsused for the functions U or q can also be applied to the bj and Wj terms in Eq. (3.34)and this saves considerable computing time.

Hence, Eq. (3.34) can be expressed in terms of standard influence matrices Hoand Go plus those influence matrices H j and G j (j = 1,2, ... ) which result from theuse of higher order fundamental solutions

1 00

Ho U - Go Q = k 2)G j +1 W j - H j +1 B j )

j=O(3.36)

The vectors B j and W j contain the values of functions bj and Wj respectively at theboundary nodes.

The terms of the series in the right hand side of Eq. (3.36) decrease rapidlyprovided the problem has been scaled (i.e. all dimensions are divided by the maximumdimension in the problem). A rough idea about the convergence of the series can bededuced from the calculation of coefficients of influence matrices presented in [8].Furthermore, the convergence can easily be controlled by calculating the contributionof subsequent terms. It is worth pointing out that in practical linear cases the sourceterm is usually represented by a fairly simple function. Thus, instead of series in Eq.(3.36) one frequently obtains finite summation.

In those cases when convergence of the series is not satisfactory, series in Eq.(3.36) can be truncated. Taking into account remainder (3.29) one ends up with thefollowing final formulation


1 J-1

Ho U -Go Q = k L(Gjt1 W j -Hjtt Bj )+j=O

+ ~ [(G Jt1 \II - HJt1 F) - HJt2JF-1 BJ (3.37)

where F-1 is an inversion of matrix F built of functions r and \II is a matrix built offunctions 'lj;n.

Upon introducing the boundary condition (3.32), Eq. (3.36) or (3.37) is transformed into an algebraic equation. Then standard boundary element routines [1J,[2Jare applied in order to obtain temperature and heat flux distribution along the boundary.

3.4 Linear Transient Problems

Equations Governing Transient Heat Conduction

The proposed technique will be studied with reference to the transient heat conductionwithin the region 0 in which no heat sources exist. Since thermal properties areassumed to be constant, the problem considered hereby is linear and is governed bythe following differential equation

V'2U (x, t) = ~ ou~x, t) = ~ u(x, t) in 0a t a

(3.38)

where u(x,t) is a temperature at point x for time t, u represents temporal derivativeof temperature and a = k/pc stands for the thermal diffusivity. Heat conductivity isrepresented by k, p stands for density and c is the specific heat.

On the boundary r an arbitrary kind of boundary condition is prescribed. As themost general one (similarly to section 3.3) the boundary condition of Robin's type isconsidered in this section

auq= -k an =h(u-uf) onr (3.39)

To obtain unique solution of the problem (3.38), (3.39) an initial condition has tobe specified

¢>(x) = U (x, t = 0)

where ¢>(x) is a known function of space.

The MRM Formulation

inO (3.40)

Applying the second Green's theorem to Eqs (3.2) and (3.38) one obtains the followingwell known integral equation, i.e.

Ci Ui + [ U q~ dr = [ q u~ dr + ~ [ uu~ dOir ir a in (3.41)


The domain integral in Eq. (3.40) is now transformed into a series of equivalentboundary integrals following the technique described in section 3.2. Thus, the domainintegral appears

1J(· f)q • f)u) df D=- u--q- + 1ar 1at 1f)t

where the domain integral D1 has the following form

(3.42)

(3.44)

D1 = ~ f u~ \72udO (3.43)a in

Taking into account Eq. (3.38) and changing an order of operations one obtains aftersimple algebra manipulation

D1= ~ f u· !!-. (\72u) dO =a in 1 f)t

k 1 . f)2u k 1 ...= 2 U1 f) 2 dO = 2 U1 U dOant a n

where the double dot index indicates second derivative with respect to time.Integral D1 is evaluated analogously, i.e.

where

k 1 f)3uD2 = 3 ui f) 3 dOant

In the limit, the procedure leads to the following formula for integral Do

(3.45)

(3.46)

(3.47)

(3.48)

00 1 lr ( . f)jq • f)ju)Do = L ---.,. Uj -f). - qj -f)' df

. ~ r V V)=1

Introducing Eq. (3.47) into Eq. (3.41) one can obtain the exact boundary onlyformulation of the problem (3.38-3.40)

00 1 i .f)ju 00 1 i . f)jqCj Uj +L ---.,. qj -f). df = L ---.,. Uj -f). df

j~ ~ r V j~ ~ r V

where derivative of zero order is the function itself.Notice that Eq. (3.48) is the integral equation with respect to space and the differential one with respect to time.


Discretization in Space

Integration in Eq. (3.48) is performed numerically, i.e. boundary r is subdivided intoa number of boundary elements and appropriate boundary elements interpolationfunctions (3.35) are used. As a result Eq. (3.48) is expressed in terms of influencematrices

1 ' 1 "H o U +- HI U + 2 H 2 U +... =

a a

1 ' 1 "= Go Q +- GI Q + 2 G2 Q +...a a

(3.49)

Equation (3.49) constitutes the system of ordinary differential equations of infinite order. Since derivatives of both functions u and q are present in Eq. (3.49) theobtained formulation is essentially different from the DRM formulation. Even whenconstant temperature is prescribed on the whole boundary r, transient process canstill proceed in time, as opposed to DRM.

Introducing Boundary Conditions

Now the discretized boundary condition (3.39) can be introduced into Eq. (3.49).Differentiating, one can write (assuming that the heat transfer coefficient does notdepend on time)

Q = h CU - Uf) etc.

(3.50)

(3.51 )

(3.52)

where h is a square matrix containing, on its main diagonal, values of heat transfercoefficient associated with the boundary elements and zeros elsewhere.

Substitution of Eqs (3.50-3.52) into Eq. (3.49) yields

1 ' 1 ,.Ao U +- Al U +2 A 2 U +... =

a a

1 ' 1 .,= Ro U J +- R I U f +2 R 2 U f +...

a a

where entries of matrices A j and R j are calculated from formulae

(3.53)

(3.54)

R j = -h G j (3.55)

It should be noticed that when fluid temperature is constant the right hand side ofEq. (3.53) consists of one term only.


If any other than the 3rd boundary condition is prescribed on the boundary r,the procedure of it being introduced into Eq. (3.49) is very similar and therefore willnot be discussed here.

Discretization in Time

Although the system (3.52) can be solved either analytically or numerically, onlythe numerical solution is discussed in this section. To obtain numerical solutiontemperature u has to be interpolated between values associated with different timelevels. If linear approximation is applied between values marked by m and m+l onecan write

(3.56)

where parameter e (0 :::; e :::; 1) positions the actual time t within current time stepf'::.t m = tm+l _ tm

t - tm

e =-- (3.57)f'::.t m

Knowing how fluid temperature varies with time, the right hand side of Eq. (3.53)may be calculated directly. However, when approximation (3.56) is also applied forfluid temperature, considerable computing time can be saved. Hence

V f = (1 - e) Vj +e Vj+1

Differentiation of Eqs (3.56) and (3.58) with respect to time yields

(3.58)

(3.59)

iTf = (Vj+1 - UJ)/b.tm (3.60)

Notice that higher order derivatives vanish and system (3.53) can be converted intothe following equation

AVm +! = amwhere appropriate matrices are calculated as follows

A = AtI(af'::.tm) +Ao e

VI = -AtI(af'::.tm) +Ao (1 - e)

V 2 = -RtI(af'::.tm) +Ro (1 - e)

V3 = RtI(af'::.tm) +Ro e

(3.61 )

(3.62)

(3.63a)

(3.63b)

(3.63c)

B m = VI Um + V 2 Vj + V3 Uj+1 (3.64)

Since approximation (3.56) is a linear one, the above described approach is called firstorder approach.


Instead of linear approximation (3.56), the well known three point recurrence timestepping scheme can be proposed

U = 0.5 (8 - 1) 8 um-l_

- (1 - 8 2) U m +0.5 (8 +1) 8 U rn+1 (3.65)

where 8 is an arbitrary value from the interval < -1,1 > and is defined by the Eq.(3.57). Obviously function (3.65) better represents changes of temperature within thetime than function (3.56), especially when parameter 8 is equal to 0 (at the middleof the interval).

Differentiation of Eq. (3.65) gives

u= [Urn - 1- 2 urn + Um +1] j(b.t rn )2 (3.67)

Formulae (3.65-3.67) lead to the second order approach described by Eq. (3.61)with matrices defined as

B rn = VI urn +V2 Urn - 1 +V3 Uj+l +V4 Uj +Vs Uj-l (3.74)

Another quadratic approximation based on two time levels can be found in [13].Obviously approximation of order higher than 2 can also be applied.

Iteration Process

Solution of the problem is given by the expression (3.61). This equation has to besolved within each time step. To determine the unknown temperatures at time trn +1

one needs to know temperatures urn at time t rn and for the second order approachadditionally temperatures urn-l at time t rn - 1

•


These values are known (for each time step except the first one) from the previousiteration. If the first time step for the first order approach is considered, to obtainvector UO the initial condition (3.40), usually associated with stationary state, hasto be taken into account. As it is difficult to initiate iterations in the higher orderapproaches it is recommended to perform a sufficient number of time steps using firstorder approach and then apply the higher order approach for subsequent time steps.

Having the vector Bm determined, the Eq. (3.61) is ready to be solved by any linearsolver, e.g. Gauss elimination. Notice that if time step is kept constant throughout theanalysis matrix A does not depend on time and system (3.61) needs to be factorizedonly once.

Once vector vm+l being the solution of Eq. (3.61) is found, remaining unknowns,i.e. vector Qm+1 is obtained from the boundary condition (3.39)

Qm+l = h (Vm +1- Vj+l) (3.75)

Knowing all unknowns the calculations for the next time step can be carried out.

Internal Poles

As presented in the previous sections MRM leads to the boundary only formulation.It means that internal effects are represented in the resulting Eq. (3.49) by the certainboundary quantities. In the obtained formulation they are higher order derivatives oftemperature and heat flux on the boundary. Knowing these functions one can exactlydescribe internal effects.

In practice however, series (3.53) is truncated. This is done by assuming theapproximation (3.56) or (3.65). As a consequence, in the final Eq. (3.61) internaleffects are not represented exactly. Therefore the idea of including selected internalpoints called poles (cf. chapter 1) is valuable also for MRM. Introducing influencematrices H~ and Gj for internal poles and substituting Ci = 1 in Eg. (3.48) one canwrite

. . 1· 1 ..V· +Hb V +- H~ V + 2" H~ V +... =

a a

. 1· 1 ..= G~ Q +- G~ Q + 2" G~ Q +...

a a(3.76)

where vector Vi contains temperatures at internal poles.Since boundary conditions are introduced in exactly the same way as discussed

in the section on Introducing Boundary Conditions one obtains for internal poles ananalog of Eq. (3.53)

. 1·· 1 ...A~ V +- A~ U + 2" A~ V +... =

a a

. 1· 1 ... .= R~ V f +- R~ V f + 2" R~ V f +... - V·

a a(3.77)

In order to transform Eg. (3.77) into an algebraic equation the approximation oftype (3.56) or (3.65) is applied. Assuming parameter 0 = 0 one avoids introducingnew unknowns. Thus


(3.78)

where vector Ui,rn contains temperatures at internal poles calculated for time t rn.

Remaining matrices are calculated from relationships (3.62-3.64) or (3.68-3.74), provided influence matrices for boundary nodes are replaced by appropriate influencematrices for internal poles and e is equal to O.

Equations (3.61) and (3.78) considered simultaneously, form the overdeterminedsystem of type

A urn+} = C (3.79)

The least squares solution of this system can be obtained from the following expression

A T A urn+} = AT C (3.80)

It has been found that taking into account one or two internal poles locatedcentrally within the domain leads to the accurate and stable formulation.

Once Eq. (3.80) is solved, temperatures at internal poles are determined fromdiscretized Eq. (3.77) assuming e = 1.

Some Remarks on Convergence

Accuracy of the described technique depends on convergence of series (3.49). Thisconvergence is guaranteed by the presence of factorials in the denominators of influencematrices coefficients (cf. section on Higher Order Fundamental Solutions). It shouldbe stressed, however, that convergence of series (3.49) depends strongly on time and isaffected by thermal shock, if it occurs. When temperature distribution is calculated atlarge time values the series converges very rapidly. Coming closer to initial conditionconvergence of the series becomes unsatisfactory. It means that for a short time (orstrictly speaking for a small Fourier number, e.g. Fo < 0.1) one can realize that inorder to achieve satisfactory accuracy it is necessary to consider many more than onlytwo terms of the series (3.49). In other words for small Fourier numbers, changes oftemperature and heat flux are so irregular that approximations (3.56) or (3.65) cannotrepresent them accurately enough. This results in the existence of a lower limit forthe time step 6t. A similar phenomenon for DRM is also reported in literature, e.g.[14],[151·

One way to overcome this difficulty is to estimate the remainder of the series(3.49) according to material in the section on Error Estimation for Truncated Series.It should be however stressed that although such an approach makes the formulationmore accurate it does not remove all problems. Therefore developing for small Fouriernumbers quite a different formulation, perhaps based on applying time dependentfundamental solution, seems to be the best solution. It is worth pointing out that asimilar situation takes place when using analytical methods. Widely employed seriesof eigenfunctions are numerically inefficient for small Fourier numbers. Thus anotherform of the solution employing error function erfcO has been constructed, e.g. [16], inorder to perform calculations for small Fourier numbers.


ReS"Ulls for h =1.0 and be) =0.5

6.0 8.0 10.0 12.0iime

-'--'---

Dr,

~\I~~.

\~,

~*'"~

~O(~~

-_. ---- -0.40.0 2.0 4.0

1.2

f.O~::J.....dI...Cl)

0.8QE~

-41

0.6

o z=O.O 0 x~f.O

Figure 3.1: Temperature of surfaces of the slab as a function of time.

Research on developing a boundary only formulation which is efficient for determining temperature at early times is now being carried out.

3.5 Numerical Examples

In order to demonstrate main features of the described technique, simple numericalexamples have been studied. They were selected to check accuracy by comparisonwith analytical solution.

Example 1Heat is conducted through a 1 - D slab insulated at x = 0 and subjected to theboundary condition of the third kind (with fluid temperature equal to 0) at x = l.Source term bo is uniform within the slab. Initial temperature of the slab is alsouniform and equal to 1. Slab has thermal conductivity k = 1W /mK and thermaldiffusivity a = 1m2 /s.

The problem was modelled as a 1-D problem [17] and some representative resultsare shown in Fig. 3.1 and 3.2. The solid lines correspond to the analytical solutions


Results for h = f.O and bo=f. 01.6

f.4

Q)L-J-.J f.2.";.JUCL(.

C 1.0(,)

-;-.

0.8

D.C

r~'

IIf

0;' A A

~V vvv vvv

/L

I I I I I ,0.0 2.0 4.0 6.0 8.0 fO.O f2.0

time

o x=O.O 0 x=f.O

Figure 3.2: Temperature of surfaces of the slab as a function of time.

whereas the symbols stand for the results obtained by MRM.

Example 2The problem described in Example 1, assuming bo = 0, was modelled as a 2 - Dproblem. The geometry and numerical mesh is displayed in Fig. 3.3.

Calculations were carried out applying constant boundary elements. Results arecompared with the analytical solution (solid line) and with results obtained by DRM;see Fig. 3.4.

Example 3In this example, a 2 - D transient problem in rectangle has been studied. Geometryof the region having thermal conductivity k = 1 WJmK and thermal diffusivitya = 1m2Js is shown in Fig. 3.5.

Faces x = 0 and y = 0 are insulated, whereas faces x = 0.5 and y = 1 aresubjected to the boundary condition of Robin's type with the heat transfer coefficienth = 0.5 WJm 2K and fluid temperature Uf = O.

Since the region was initially at temperature 1 the above boundary conditions


I' 1 • 2

12.

5

6

1-1 • 10 I 9 • 8 ~ 7 >(.

Figure 3.3: Geometry of the region in example 2.

o DVClI Rec,proc-lly

o Mulh Rec.qo"'ocL\~

1.20

1.00

~:J .80-vV\...(\)Q .60E\lJ.,..,

.40

.20

.00.0 1.0 2.0

/:lrne

3.0 4.0 5.0

Figure 3.4: Temperature of element 6 as a function of time.


8

23

-~""-"""'-l-"--_x

Figure 3.5: Geometry of the region in example 3.

cause thermal shock on the faces x = 0.5 and y = 1. Hence, calculations were carriedout applying smoothing algorithm in the first time step, e.g. Zienkiewicz [18]. Theabsolute error defined as the difference between numerical results and analytical solution for two selected boundary nodes is plotted in Fig. 3.6 and 3.7. Results marked ascurve 1 are obtained using the DRM formulation with one internal pole. The curve 2is associated with the first order approach of the MRM with no internal poles whereascurve 3 refers to the same approach but with one central internal pole. The secondorder approach of the MRM, based on a two point quadratic approximation withinthe time [13], produces results marked as curve 4 - no internal poles, and curve 5 one internal pole.

Calculations were carried out using constant elements and constant time stept:i.t = 0.2. Parameter e was assumed 1. Although very good accuracy of the methodcan be observed, especially of the second order approach, it should be mentioned thatfor a smaller time step some instabilities appeared.

Summarizing, it is worth stressing that the MRM is an alternative to the DRMtechnique of solving linear heat transfer problems. It consists in employing time independent fundamental solution and transforming domain integrals to the boundary.Although the approach can be seen as 'adjoint' of the Dual Reciprocity Method, itis essentially different from this technique. While the DRM applies the same fundamental solution during the process of transformation domain integrals into boundaryones, the MRM uses increasingly higher order fundamental solutions. Since thesefunctions can be obtained recursively the method presented here requires only smallchanges within the standard steady-state boundary element computer code.

The present formulation has been found to be very efficient when temperature iscalculated at a moderate or large Fourier number. However, at the very beginning ofthe transient process the technique still needs to be improved.


2

.80.40

tlme+----+---+------jl----If-----+I------.j

1.20 1.60 200 2.40

.070

.060

.050

.040i---0i--- .030i---~

.020

.010

.000

-.01000

Figure 3.6: Absolute error of the MRM and the DRM for boundary element no 8.

time200 2.401.601.20

5/

.80

-,--------.------,------,---..,.------,.__._--.100

.080

.060

i---0i--- .040i---~

.020

000

- .02000 .40

Figure 3.7: Absolute error of the MRM and the DRM for boundary element no 23.


Acknowledgements

This work was partially carried out when author was the visiting research fellow atthe Computational Mechanics Institute.

The financial assistance of CMI and the Ministry of National Education withinthe Central Plan for Fundamental Research - direction 02.21 coordinated by the Institute of Fundamental Technological Research in Warsaw is gratefully acknowledgedherewith.

References


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5. L.C. Wrobel and C.A. Brebbia, The dual reciprocity boundary element formulation for nonlinear diffusion problems, Computer Methods Appl. Mech. Eng.,65,1987, 147-164.

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11. M.A. Jaswon and G.T. Symm: Integral equation method in potential theory andelastostatics. Academic Press, London, 1977.

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15. C.F. Loeffler and W.J. Mansur: Dual Reciprocity Boundary Element formulationfor transient wave propagation analysis in infinite domains. Proc. 11th BEMConference, Cambridge, Massachusetts, USA, (ed. C.A. Brebbia & J.J. Connor)Springer-Verlag, 1989, vol. 2.

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Notation

a thermal diffusivity; m2/ s

A matrix in linear BEM equation, Eq.{3.61)

Aj coefficient of higher order fundamental solution, Eq. (3.18a)

A j matrix defined by Eq. (3.54)

bo heat source generation rate; W/m3

bj sequence of functions defined by Eq. (3.15a)

Bj coefficient of higher order fundamental solution, Eq. (3.18b)

B j vector containing the values of function bj at nodal points

Bm right hand side vector in Eq. (3.61)

C right hand side vector in Eq. (3.79)

C specific heat; J / kg1<

Ci function of the internal angle the boundary r makes at a given point.For a smooth boundary Ci = 0.5

Dj domain integral (j = 0,1,2, ... )

r coordinate function used in approximation (3.23)

F matrix formed by the values of functions r at nodal points

G j heat flux influence matrix in BEM formulation

h heat transfer coefficient; W/m 21<

h square matrix containing the values of heat transfer coefficient

H j temperature influence matrix in BEM formulation

k thermal conductivity; W/m1<

n outward normal to the boundary rN P number of functions considered in the approximation (3.23)

r geometrical distance between two points

R9 domain integral defined by Eq. (3.26)

RJ remainder in series (3.21)

R j matrix defined by Eq. (3.55)

q heat flux; W/m 2

q~ heat flux analog associated with fundamental solution, Eq. (3.3)

qj auxiliary heat flux analog defined by Eq. (3.14b)


qe vector containing the values of heat flux at nodal points of e-thboundary element

Q vector of nodal heat fluxes in matrix BEM formulation

time; s

u temperature; K or °C

Uo fundamental solution satisfying Eq. (3.2)

uj higher order fundamental solution defined by Eq. (3.14a)

U e vector containing the values of temperature at nodal points of e-thboundary element

U vector of nodal temperatures in matrix BEM formulation

V j auxiliary matrices defined by Eqs (3.63),(3.69-3.73)

Wj sequence of functions defined by Eq. (3.l5b)

W j vector containing the values of function Wj at nodal points

Greek lettersan coefficients in the approximation formula (3.23)

.6i Dirac's function acting at point i

r boundary of the region n.6t time step

¢> initial condition in transient problem, Eq. (3.40)

'l/Jn function defined by Eq. (3.27)

'!l!... vector containing boundary elements interpolation functions, Eq. (3.35)

\lI matrix containing functions 'l/Jnn region under consideration

p density; kg/m3

o parameter which positions actual time in the current time step

Subscriptsn number of iteration

o initial condition


temporal derivative

second derivative with respect to time

internal pole


m number of time step

T transformed matrix

Other symbolsV gradient

V2 Laplace's operator

aO/an differentiation along outward normal to the boundary


Chapter 4

Solving Nonlinear Heat Transfer ProblemsUsing the Boundary Element Method

R. BialeckiInstitute of Thermal Technology, Silesian Technical University,

44·101 Gliwice, Konarskiego 22, Poland

4.1 Introduction

General

Nonlinear boundary value problems are frequently encountered in heat transfer problems of practical interest. In heat conduction problems, the nonlinear analysis is evenmore common than in structural analysis. It is obvious that for an engineer the latter problems are of primary interest. The computational complexity of determiningstress and strain distribution is much greater than the effort associated with computing temperature fields in solids. The reason for this is that the governing equationsof stress distribution are of coupled vector type, whereas the differential equation ofheat conduction is a scalar one.

On the other hand mathematical models of heat conduction are often more complex than those encountered in structural analysis. Heat transfer problems of practicalimportance are usually formulated as nonlinear boundary value problems. This is dueto at least two features:

- the exact knowledge of temperature fields is important only in cases of largetemperature differences, thus e.g. material properties cannot be treated as constant throughout the whole body. Small temperature differences usually do notdemand heat transfer calculations to be carried out (the body can be treated asisothermal).

- the prescribed boundary conditions are frequently complex and contain nonlinearterms resulting from radiation, free convection or moving boundaries.

These features are in contrast with structural analysis where the bulk of engineeringproblems can be treated as linear with very simple boundary conditions prescribed(homogeneous Dirichlet or Neuman type).


It should be stressed here that temperature fields are often an initial step in thermoelastic (thermoplastic) computations. The accuracy of heat transfer calculationsin these cases strongly influences the eventual stress values. This is another reasonto treat the thermal problems as nonlinear since only such formulation guaranteesappropriate accuracy of the sought for stresses.

All these features of heat conduction problems meant that a lot of work was devoted to solve nonlinear problems. The literature on general nonlinear heat conductionis vast (a monograph on this problem [1J cites over 900 references)

The BEM literature on nonlinear heat conduction is growing rapidly. Review ofearlier works can be found in [2]'[3J. The aim of this study is to give the state ofart in applying BEM to nonlinear conduction in solids. Therefore a large number ofreferences will not be quoted here if the methods described therein have been latelyimproved or generalized.

The chapter is an expanded and updated version of the author's paper [4].

Nonlinear Boundary Value Problems of Heat Conduction

An exhaustive description of nonlinear boundary value problems can be found instandard monographs on heat conduction [1]'[5],[6J. Some aspects of both physicalsituations leading to nonlinear problems and mathematical models corresponding tothese phenomena will be now discussed briefly.

Nonlinear differential equationThe derivation of the differential equation of heat conduction is based on the law ofenergy conservation. For a stationary solid the equation has a general form

- \7 q(r, t) + b(r, t) = a~:) rEV (4.1 )

where: r vector coordinate; m

t time; s

q heat flux; Wlm 2

b rate of internal heat generation due to heat sources; WI m3

u - specific internal energy; JI kg

p density; kglm3

V region occupied by the solid

Heat flux q for a stationary, isotropic solid (i. e. material in which thermal conductivity is independent of direction) linearly depends on temperature gradient dueto Fourier's Law. From basic thermodynamics it is known that internal energy isproportional to the temperature

(note the minus sign!)

q

u

- k \7 T(r,t)

cT(r,t)

(4.2)

(4.3)

(4.5)


where: T temperature; K

k thermal conductivity; W/mK

c - specific heat; J/kgK

Taking into account Eqs (4.2) and (4.3) the final form of the differential equationof heat conduction in solids takes the form

\7 [k V' T(r,t)] + b(r,t) = cp 8T1:,t) (4.4)

If the material properties, i.e. heat conductivity, specific heat, and density dependon temperature, the differential Eq. (4.4) becomes nonlinear. This type of nonlinearityis referred to as material nonlinearity.

Another source of nonlinearity associated with the differential equation is the nonlinear source term. There are physical situations when the internal heat generationrate depends on the temperature. Examples of this kind of nonlinearity are chemicalreactions taking place within the solid medium (self-heating of fossil fuels, contactcatalyst bed). Also joulean heating with temperature dependent electric conductivitycan be a source of that nonlinearity.

Nonlinear boundary conditionsNonlinear boundary conditions encountered in heat conduction can be caused by

i) temperature dependent heat flux

ii) radiation in cavities and enclosures

iii) temperature dependent thermal conductivity

The case of temperature dependent heat flux is the most common situation in heattransfer calculations. It can be, without loss of generality, written as a nonlinearNeuman's condition

q = q(T, r, t)

where: q - known functionBoth q and T are space dependent (intransient states they depend also ontime).

Care should be exercised when using these kind of boundary conditions in steadystate problems. In case Neuman's conditions are the only boundary conditions presentin the formulation of the boundary value problem the heat equation has a (nonunique)solution only if some additional condition is satisfied. This condition is that theintegration of the prescribed heat flux over the whole boundary should be zero. Thisis an analog of the rigid body movement condition encountered in structural analysis.

To avoid this difficulty and also for computational reasons it may be advantageousto rewrite Eq. (4.5) as a (pseudo) convective boundary condition upon extracting alinear term from the function q.


q h(r,t) (T - Tf ) + f::1qn (T,r,t) (4.6)

heat transfer coefficient; W/m 2 K

temperature of a fluid exchanging heat with the boundary

nonlinear term, given function of temperature

In the above, the heat transfer coefficient h can have a certain physical interpretation but it can be also treated as a purely computational quantity.

Here are some examples of Eq. (4.6) often encountered in practice:

- temperature dependent heat transfer coefficient, e.g. due to natural convection

(4.7)

where: f::1hn (T) - additive nonlinear part of the heat transfer coefficient

h - linear part of the heat transfer coefficient

- heat convection plus radiation (convex boundaries, radiation into environment)

where: Rt - r

Tr

q = h (T - Tf ) + Rt - r (T4- Tr

4)

- radiative heat exchange factor- temperature of the radiating environment

(4.8)

Solving heat conduction problems in bodies having concave radiating boundariespresents severe difficulties, as in such cases heat radiation produces a strong interactionbetween all points lying on those boundaries. The heat flux at a given positiondepends not only on the temperature at this very point, but also on temperaturesat each point that can be seen from the considered point. The couplings are verystrong and nonlinear. On the other hand the mutual radiative heat exchange isgoverned by an integral equation. Thus, applying BEM to these problems is evenmore straightforward than in the case of heat conduction, as radiative problems do notrequire the transformation of the boundary value problem into an integral equation.Appropriate integral equations and methods of solving them will be discussed in thesection concerning Heat Radiation on Concave Boundaries.

Temperature dependent heat conductivity makes the boundary conditions of thefourth kind nonlinear i.e. the continuity requirement of both temperature and heatfluxes when passing the interface of two different materials. This kind of nonlinearityis always associated with nonlinear differential equations, thus, this case will be discussed in connection with nonlinear material.

Unknown location of a portion of the boundaryThe unknown location of a portion of the boundary often referred to as moving boundaries are an inherent part of phase change phenomena such as melting, solidification,


ablation, welding, electrochemical machining etc. The physics of these phenomena isso complex that in fact no consistent theory of these problems exists. The reason forthis seems to be the lack of thermodynamic equilibrium often encountered in phasechange phenomena. This feature means that laws of classical thermodynamics do notapply here thus, more sophisticated rules should be employed when formulating themathematical model.

The literature on computational methods to deal with moving boundaries is vast(a recent monograph [7] contains over 600 references). As an exhaustive descriptionof physical and mathematical models is not possible in a chapter of moderate length,only a brief review of commonly employed formulations will be given here.

The simplest possible model is change of phase of a pure substance with constant temperature of phase change (Stefan's problem). For this model the boundaryconditions on the interface of solid and liquid read

q. - q/ = V P L (heat balance) (4.9)

Tc = T/ = T. (temperature continuity) (4.10)

where: 1, s - subscripts referring to solid and liquid phaseTc - temperature of phase change; 1<v - component of the velocity of the moving boundary in a direction

normal to this boundary; m/sL - latent heat; J/ kg

Even such a simple model, in the presence of internal heat generation, can lead tosingular solutions containing isothermal zones corresponding physically to the presence of mushy regions i.e. transition zones filled with a mixture of crystals and liquid[7],[8].

The case of multicomponent phase change is far more complex as both the meltingtemperature and the latent heat depend on the concentration of components at theinterface. This demands the transient diffusion equations for each component to besolved simultaneously with transient heat equation. The boundary conditions on theinterface consist of a mass balance requirement for each component, both temperatureand chemical potential continuity conditions (provided thermodynamic equilibrium isattained) and energy conservation (Eq. (4.9)).

The next step of complexity is introduced by the dependence of the phase changetemperature on the curvature of the interface and surface tension of the liquid. Thisrelationship has been reported in the presence of subcooling and superheating. Thephenomenon is known in the literature [8] as the Mullin Sekerka instability.

Another severe complexity is introduced when taking into account the buoyancyeffects within the liquid phase, and thermal properties dependence on both temperature and composition.

BEM Formulation of Linear Heat Conduction Problems

To make the chapter self contained and introduce appropriate terminology and notation, some basic knowledge from the linear BEM formulations, necessary to discuss


nonlinear problems, will be recalled.

Steady stateLinear boundary value problems of heat conduction can be transformed into an equivalent integral equation by several methods e.g. by the reciprocity theorem. The detailsof these techniques can be found in monographs and textbooks on boundary elements[2J,[9J. The direct formulation of the integral equation corresponding to steady stateconduction (temporal derivative in Eq. (4.4) vanishes) can be written as

C T =1[q" T - T* qJ dS + Iv bT* dV (4.11)

where: S - surface bounding region VT" - fundamental solution satisfying, in the whole space, the differential

equation

k \J2T" = -h(r - p)

with h - Dirac's distributionp - vector coordinate of a source point

q" - heat flux analog defined as

q" = -kaT"an

(4.12)

(4.13)

(note the minus sign!)

with ~~ - denoting differentiation along the outward normal to the boundary

C - function of the internal angle the boundary S makes at thepoint p. For smooth boundary C=O.5

Discretization of (4.11) is accomplished upon dividing the boundary S into boundary elements and the domain V (if necessary) into cells. Within each element and cell,locally based interpolation shape functions are employed to represent the distributionof temperature, heat flux and heat generation rate. Details of this procedure can befound in standard BEM literature [2]'[9J.

The result of the discretization is a set of linear algebraic equations of a form

b+HT=GQ (4.14)

where: H,G - square matrices depending on the geometry of the boundary. Entriesof these matrices are computed upon performing integration alongsubsequent boundary elements

b - vector depending on internal heat generation. The entries are computed by domain integration.

Some nodal values of the temperature and heat flux are known from the boundaryconditions. Upon introducing these known quantities on the right hand side and the


unknowns on the left hand side one arrives at a set of linear equations having the

appearance

where: A

fox -

A X= fo

known, square matrix

known vector

vector of unknown nodal temperatures and heat fluxes

(4.15)

TransientClassical BEM formulation of linear transient parabolic equations employs the timedependent fundamental solution i.e. a function being within the whole space a solutionof a differential equation

oT*D \J2T +- = -<5(r - p)<5(t - T)at

where: D thermal diffusivity D = k/pc; m2/ s

T - time;s

(4.16)

The integral equation equivalent to the boundary value problems of transient heatconduction can be obtained by employing the reciprocity theorem or integrating byparts the weighted residual formulation [2]. The final form of this equation is

C T = ~ faT is [Tq* - T*qJ dS dt + ~ /: Iv bT* dVdt +

+ IT;T*lt=o dV

where: T; - initial temperature distribution within V

(4.17)

Discretization of the space in Eq. (4.17) is accomplished using the very samemethod as in the case of steady state fields. The time domain can be discretized intwo different ways.

One technique (referred to as scheme 1 [2]) consists in treating each time stepas a new problem with initial condition within the region computed as the resultingtemperature distribution at the end of the previous time step. This approach alwaysdemands domain integration to take account of the 'new' initial conditions.

Alternatively, (scheme 2) one can start time integration at each time step alwaysfrom the very beginning of the process. Then, the time integration is carried out asa convolution of fundamental solutions and previously obtained distributions of bothtemperature and heat flux along the boundary all the way back to the initial conditions. If the initial condition satisfies the Laplace equation, the domain integrationcan easily be avoided by employing the superposition principle and introducing a newunknown defined as an excess over the initial temperature distribution.


Both schemes result in a set of linear equations to be solved at each time step.The general form of these equations is identical as in steady states

b+HT=GQ (4.18)

where: H, G matrices depending on geometry of the boundary, current time stepand material properties

T, Q vectors containing nodal temperature and heat flux values respectively, at the end of a given time step. The nodes are placed on theboundary.

b vector depending on the history of the process and/or internal heatgeneration

Eq. (4.18) can be transformed into a set of linear equations of a general formgiven by Eq. (4.15) characteristic of steady state problems. As in steady states thetransformation consists in grouping known nodal quantities on the right and theunknowns on the left hand side.

The time dependent fundamental solution approach demands in general performing domain integration in order to compute the b vector, unless scheme 2 is employedand no internal heat generation occurs.

To avoid domain integration a technique termed Dual Reciprocity [10],[11],[12]has been developed. The key idea is to employ time independent i.e. steady statefundamental solution (Eq. (4.12)). The weighted residuals produce in the absence ofinternal heat sources an integral equation

C T = r [Tq* - T*q] dB - r~ aT*T dVis iv D at (4.19)

The domain integral in Eq. (4.19) is converted into a boundary one assuming thetemporal derivative of the temperature can be expressed as a sum of some specifically chosen functions Ji(r) depending solely on coordinates, multiplied by unknownfunctions of time a j (t) i. e.

Taking t j to be a solution of

aT N . ..- ~ L F(r)laJ(t)at j=l(4.20)

(4.21 )

enables one to transform the domain integral in Eq. (4.19) into a sum of boundaryintegrals

C T = is [Tq* - T*q] dB +

(4.22)


. afiwhere if = - k an (4.23)

The integral Eq. (4.22) contains only boundary integrals. For each nodal point rithe unknown function (Xi can be expressed as a linear combination of nodal values ofthe temporal derivative of the temperature. This can be accomplished upon solvingdiscretized version of Eq. (4.18) for (Xi(t). The result reads

(4.24)

The entries of the matrix fijI are coefficients of an inverse matrix F-1 with theoriginal matrix F defined as [11]

{F}ji = p(ri) (4.25)

Standard boundary discretization of Eq. (4.22) and elimination of unknown timedependent functions (Xi employing Eq. (4.24) results in a set of ordinary differentialequations of a form

CT+HT=GQ

where T - vector containing time derivatives of nodal temperaturesC - capacitance matrix defined as

C = - [H T - G Q] F-1 D-1

with

(4.26)

(4.27)

(4.28)

{QLj = (ji(ri) (4.29)

Eq. (4.27) is solved employing standard finite differences schemes. Details of thisoperation can be found in the original paper [11J and in a chapter of the present book[12].

4.2 Applying BEM to Nonlinear Problems. General Remarks

BEM, as with any numerical technique, is well suited to deal with some kinds ofproblems while others demand special treatment. Thus, it can happen that somespecific problems can be solved by BEM less effectively than by means of methodsbased on domain discretization.

As a method based on the boundary discretization BEM can be used withoutany difficulties to solve nonlinear problems with nonlinearity concentrated only inboundary conditions.

Nonlinearities in the source term i.e. temperature dependent internal heat generation rate 'b' in Eq. (4.2), do not cause difficulties but these type of problems require


domain integration, a feature disliked by the BEM community. Using BEM for thesekind of problems generally leads to long computation times. Some new possibilities of avoiding domain integration in nonlinear cases are connected with using DualReciprocity [12] and Multiple Reciprocity methods [13]

Nonlinear material problems demand linearization of the nonlinear differentialoperator before BEM can be used. The reason for this is the reciprocity theorememployed when deriving the boundary integral Eqs (4.11),(4.17),(4.22). The theorem as a generalized superposition principle is applicable only to linear differentialoperators. This type of nonlinearity can be often very effectively treated within theBEM framework provided the linearization of the entire differential equation can beperformed without additional heat sources being introduced.

4.3 Nonlinear Boundary Conditions

As mentioned above no serious difficulties arise when solving these type of problemsby BEM. Moreover, problems with nonlinearities concentrated only in boundary conditions can be attacked using BEM more effectively than by means of other numericaltechniques. Therefore nonlinear boundary conditions were one of the first areas ofnonlinear analysis applied in BEM.

Temperature Dependent Heat Flux

The discretized version of the nonlinear boundary condition (4.5) can be written as

Q = Q (T) (4.30)

Substituting Eq. (4.30) into matrix BEM formulation (Eqs (4.14), (4.18)) one arrivesat a set of nonlinear equations for the unknown nodal temperatures

b + H T - G Q(T) = 0 (4.31 )

Eq. (4.31) has been solved using different variants of the Newton-Raphson solver[14],[15],[16]. The procedure demands a solution of a set of linear equations at eachiteration step. The general form of this equation can be written as

(4.32)

where:

(4.33)

(4.34)

(4.35)

with


(4.36)

in the above superscript (k) refers to the number of the iteration step.The jacobian matrix J can be calculated at each step of the iteration but it saves

a lot of computing time when this matrix is recomputed after several iteration stepshave been performed.

Other iterative approaches are also reported in the literature [3],[17],[18], for bothsteady state and transient problems. In Reference [3] a brief discussion of the convergence of an iterative scheme is given. This is of importance as some simple iterationsschemes of Gauss Seidel type can diverge [19] in the presence of severe nonlinearities caused by thermal radiation. In [20] the important case of multizoned media istackled but no details of the employed solver are given there. In Reference [18] aninterpolation scheme of third order to approximate the time dependence of the heattransfer coefficient within a time step is described. Unfortunately neither details ofthe solver nor numerical results for nonlinear problems are given in this paper.

Employing general purpose nonlinear solvers such as Newton Raphson algorithmproved to be efficient in most cases. There is, however, another possibility of solvingthe nonlinear boundary condition problem. This approach can be computationallyadvantageous and it is based on the pseudo convective formulation of the nonlinearboundary conditions (Eq. 4.6). The discretized version of this equation has a form

Q = h (T - T f) +LiQn (T)

where: h - square, diagonal matrix defined as

{h} .. _ {h(rj,t) j = iI) - 0 j f. i

~Qn(T) - vector{~Qn(T)}j = ~qn[T(rj, t)]

(4.37)

(4.38)

Upon substituting Eq. (4.38) into matrix BEM formulation (Eqs (4.14),(4.18))one obtains

(4.39)

Generally, in the mixed boundary condition case, some boundary conditions maybe linear. Then, putting all known nodal quantities and the nonlinear vector on theright hand side and all unknowns on the left hand side one arrives at a set of nonlinearequation of a form

Ax = f o+B AQn (T) (4.40)


where: A, B, fo - matrices depending on the geometry of the boundary, materialproperties and heat transfer coefficients. These matrices do notdepend on the unknown, which is of importance when solving Eq.(4.40) iteratively. Matrices A and fare calculated exactly as in thelinear case (cfEq. (4.15)) i.e. with nonlinear contribution in Eq.(4.6) equal to zero (~qn = 0). Matrix B contains columns of theinfluence matrices Hand G (Eqs (4.14), (4.18)) corresponding tonodal points where nonlinear boundary conditions are prescribed.

x - sought-for nodal values of unknowns, temperatures or heat fluxesdepending on the kind of boundary condition prescribed at a givenpoint

To get rid of matrix B (this step may not be performed but it often saves somecomputation time) a Gaussian preelimination is carried out so that the final form ofEq. (4.40) becomes

(4.41)

with AP, f~ - matrices with constant entries. Superscript p designates theresult of Gaussian preelimination

As can easily be seen, all the nonlinearities of the problem have been shifted tothe right hand side vector. At this stage the standard Newton-Raphson algorithmcould be advisable as the jacobian can be calculated very cheaply (only diagonalterms should be recomputed at a given iteration step). However, another approachseems to be more efficient. It is a version of the known incremental technique (cf [21D.The solver consists in increasing in a stepwise manner the value of the nonlinear term~Qn(T) starting from zero up to its actual value. The contribution of the nonlinearterm is calculated for the values of unknowns determined in the previous iterationstep. The algorithm can be written in a compact form as

AP x(k)= f~ + A(k) ~Qn (T(k-l))

where: subscript (k) designates the current number of iteration and

A(k)=k/M k=0,1,2, ... MM - chosen number of increments

(4.42)

As can be seen for k = 0 one gets the linear case with linearized boundary conditions, whereas for k = M Eq. (4.42) constitutes the original set of equations. Thereare three major advantages of the incremental approach:

- procedure does not require any starting point as it starts from the solution ofthe linear problem

- the left hand side matrix is not solution dependent thus solvers employing thefactorization (Crout algorithm) can be very effectively used. The factorization


T=1773K 1/11;) 'IIIIII/III/,~j!!j//;1W1/linSulntion 11/!0;t~II/IIIIIII/ Jill

1600KA1700K

ILI

----B -1300K900K

lr----600K---

1 500K-C

~ 400K----

Dconvection ~ fe-diG- tion

hb=30YJ / fY)2 K Tf =308K [=0,8 Tr =320K

Figure 4.1: Temperature field within a hearth of a blast furnace.

is to be performed only once. Thus on each step of iteration only the resolutionshould be carried out

in the case of multizoned media a linear block solver [22],[23],[24] can be employedas the block structure of the matrix is preserved.

The above described solver can be viewed as a simplified, specific technique appliedto solve problems with both nonlinear material and nonlinear boundary conditions[25],[261. Computations proved the method is also valid and efficient for problemswith linear material and nonlinear boundary conditions. In all examined problemsonly 3 - 5 incremental steps were required to achieve the solution with reasonableaccuracy.

Figure 4.1 shows the temperature field in a blast furnace hearth. The temperature field was assumed steady state and axisymmetric. The materials were treatedas linear. The boundary conditions at the bottom part of the hearth were assumednonlinear due to radiative heat exchange with environment having constant, known


temperature Tr = 320K. The emissivity was taken t = 0.8. The cooling air exchangingheat with the bottom of the hearth had temperature Tf = 308K. The heat transfercoefficient was hb = 30W/m2K. The frustum of the hearth was cooled with waterhaving temperature Tf = 308K. The heat transfer coefficient was h = 152W/m2K.The boundary conditions over the upper surface of the hearth were: insulation (q = 0within the lining) and constant temperature (T = 1773K within the molten pig ironzone). Numerical values of the heat conductivities were:

zone A - molten pig ironzone B - mullitezone C - carbonzone D - graphite

ks = 29.1 W/mKks = 1.34 W/mKkc = 7.25 W/mKko = 120. W/mK

Five incremental steps were performed. The relative accuracy of the global heatbalance in each zone was better than 0.003.

In nonlinear boundary conditions considered so far, the heat flux at a given boundary point depended solely on the temperature at that very boundary point. In thecase when the boundaries are concave, the heat flux at a chosen position depends ontemperatures of all points which can be seen from that position (provided the fluidfilling the enclosure is transparent to thermal radiation). This case will be discussedbelow.

Heat Radiation on Concave Boundaries

The problem of coupling BEM radiative analysis with the conductive one in case ofbodies having concave boundaries has been discussed in [27],[28]'[29],[30]. As a chapterof the present book [31] gives a review of applying BEM to these kind of problems,only the main ideas lying behind application of BEM to heat radiation analysis willbe discussed here.

Ref. [27] deals with the problem of a nonparticipating medium filling the cavitywhereas Ref. [28] addresses a more sophisticated participating, grey fluid case. Theintegral equation of radiative heat exchange has for the latter case the form

qr(r) - t(r) r qr(p)[l - t(p)]t(pt1K(r, p) exp( -a 1r - p I) dS =is.=at(r)T4 (r) - t(r) r aT4 (p )K(r, p) exp(-a 1r - p I) dSis.

- r aTe4(p)K(r,p)[1-exp(-alr-pl)]dS (4.43)is.

where: r, p - vector coordinates of points lying on the boundaryI r - p 1 - distance between these two points

q - radiative heat fluxar

- Stefan-Boltzmann constantt - emissivity of the radiating surfacea - absorptivity of the fluid filling the cavity

Te - temperature of that fluidSe- surface of the enclosure


cos </>r cos </>pin 3D

1r I r - p 12

K(r,p) = (4.44 )cos </>r cos </>p

in 2D21 r- pi

with </>r being the angle between the outer normal at the point r and aline connecting rand p. </>p is defined analogously

Discretization of Eq. (4.43) can be performed exactly as in standard BEM i.e.employing the concept of locally defined shape functions (in the references quoted, thesimplest possible step function approach has been employed). Moreover, incorporatingthe radiative analysis option into an existing BEM code is fairly simple as it is roughlyequivalent to a definition of a new fundamental solution. Discretization of Eq. (4.43)yields a set of algebraic equations relating the radiative heat fluxes and the fourthpowers of the temperatures. This set has an appearance

(4.45)

where: R, 5, k - depend on geometry, material properties( E and a) k depends also on Te

T 4 - vector containing fourth powers of nodal temperatures{T4}; = T4(r;,t)

Q - vector containing nodal radiative heat fluxes, {Qr}; =qr(r;, t)

Eq. (4.45) is to be coupled with that describing the heat conduction within solidwalls forming the cavity (Eqs (4.14), (4.18)). The latter set contains as boundaryconditions the unknown radiative heat fluxes. Eliminating the radiative fluxes fromthe equations of conduction and radiation one arrives at a set of algebraic equationsof a general form

LT4 +M T = n

where: L, M, n - known matrices of constant coefficients

(4.46)

The Newton-Raphson solver has been successfully used to solve this set of equations [27],[28].

The analysis described above has been restricted to grey surfaces and grey fluidseparating the radiating walls (E and 'a' not wavelength dependent). This method hasbeen generalized to a grey band model of both walls and fluid [29]. A recent papergives some numerical results of 3D radiation problems [30].

4.4 Nonlinear Material (Nonlinear Differential Operator)

Linearization of the Differential Operator

As already stated, BEM applied to this kind of nonlinearity demands linearization of


the differential operator. There are two ways of accomplishing such a linearization.The first approach, though computationally less effective has the advantage of beinggeneral. It consists in extracting the linear term from the differential operator andtreating the remaining part as a fictitious source term. Let the differential equationbe of a form

N [T(r,t)] = -b(r,t,T)

where: N - nonlinear differential operatorT - unknown (temperature)b - actual source term (generally nonlinear)

(4.47)

The differential operator is split into two terms. The linear one denoted by Noand a nonlinear reminder ~Nn . The resulting equation can be written in a formsuitable for BEM applications

No[T(r, t)] = g(r, t, T)

with obvious relationship defining the fictitious source term

g(r, t, T) = -b (r, t, T) - ~Nn [T(r, t)]

(4.48)

(4.49)

The integral equation corresponding to (4.48) will always contain a domain integral (even in problems with no actual source term). This feature is the main disadvantage of the pseudo-source approach. However, in the presence of internal heatsources in the original differential equations this disadvantage is not very strong.

The resulting nonlinear integral equation should be solved iteratively. The nextquestion is what is the convergence rate of such a process. Reasoning in quite anintuitive way one can observe that the nonlinear operator ~Nn is treated here as acorrecting, additive term to the linear operator. Thus, the procedure described abovecan be computationally effective when the nonlinear operator ~Nn is small comparedwith linear operator No i. e. the norms of the operators satisfy in appropriate spacethe inequality

II No 11»11 ~Nn II (4.50)

The second linearization technique that can be employed to linearize Eq. (4.47) isan ad hoc approach. This method consists in introducing a new variable so that in thespace of this variable the differential equation becomes linear. If such a transformation is possible one should use it before converting the boundary value problem into aboundary integral. The Kirchhoff's transformation, widely used in BEM context anddiscussed in the next section, is a good example of this kind of linearizing transformation. Unfortunately only for very few nonlinear operators is such a transformationknown so that this technique cannot be by any means regarded as a general method.

It is also possible to use some hybrid linearization methods. In such approachesa new variable is introduced to make the differential equation 'almost' linear i.e. withsome nonlinear terms remaining after transformation. One can work further in thespace of the new variable putting the remaining nonlinear term into a pseudo sourceform.


As elliptic (steady state) and parabolic (transient) equations require differenttreatment these two cases will now be dealt with in two separate subsections.

Steady State

The first attempts to solve nonlinear material problems in steady states [32] employedthe pseudo source approach. The governing equation in a steady state for an isotropicsolid with no internal heat generation can be written as (cf Eq. (4.4))

\7 [k(T) \7 T] = 0

The differentiation of Eq. (4.51) yields

\72 T + dk(T) _ k(Tt l (\7T)2 = 0dT

(4.51 )

(4.52)

The second term in Eq. (4.52) can be treated as a fictitious, temperature dependent heat source. This formulation requires domain integration with the integranditeratively corrected. The rate of convergence of such formulations has been studiedtheoretically [33]. A similar method has been described in Ref. [34] where productk(T)T has been treated as a primary unknown. The resulting integral equation has(with some notation changes) a form

C k(T) T =~ T k (T) q* dS - ~ T* q dS+

+ r T dk \7 T * dV (4.53)Jv dT q

in the above the value of k = 1 has been chosen in the definition of the fundamentalsolution (cfEq. 4.12). Thus, in Eq. (4.53)

aT*q*=-an

As already mentioned the unknown in Eq. (4.53) is the product k(T)T rather thanthe temperature T. However, knowing the values of this product one can easily calculate the temperature. A rapid convergence of a simple iterative procedure employedto solve the discretized version of Eq. (4.53) has been observed. The nonlinearity ofthe pseudo source term (domain integral in Eq. (4.53) is milder than those present instandard formulation (Eq. (4.52)).

Domain integration due to nonlinear material behaviour can be simply avoided byusing Kirchhoff's transformation as pointed out in Refs [19]'[20]. The transformationis defined as

iT k(x)1/' = K [T] = -k- dx +1/'0

To 0

where: To, 1/'0 arbitrary reference values

ko = k (To)

(4.54)


In the space of Kirchhoff's transform Eq. (4.51) becomes a linear Laplace equation.Boundary conditions of both Dirichlet and Neuman types are linear in the Kirchhoff'stransform space and the whole nonlinearity is concentrated in boundary conditions ofRobin's type even if these conditions are originally linear.

Linear boundary conditions of Robin's type also referred to as the boundary conditions of the third kind can be in the space of the original variables written as

q = h(r,t) (T - Tf )

The transformed version of Eq. (4.55) reads

in the above

(4.55)

(4.56)

q = -k(T) aT = -koalj;an anJ - stands for inverse Kirchhoff's transformation (a nonlinear opera

tion). A very efficient way of inverting Kirchhoff's transformationby approximating the k(T) dependence using a linear spline hasbeen proposed independently in Ref. [35] and [26]

The important case of multizoned media with nonlinear material behaviour wasexamined by using Kirchhoff's transformation in Refs [25],[26],[35]. The new difficultyhere when compared with homogeneous nonlinear materials is the lack of function(transform) continuity on the interface. This is circumvented by writing the continuityrequirement on the interface as Ref. [35]

(4.57)

or [25J,[261

(4.58)

In both equations indices i and j refer to the adjoining subregions and Tij is thetemperature on the interface coupling subregions 'i' and 'j'. Functions f3 and !llj;nare known.

The Newton-Raphson algorithm has been used to solve the resulting set of nonlinear equations in Ref. [35] whereas in Refs [25],[26] the incremental technique described in the previous section has been employed. In both approaches the case whenboundary conditions are also nonlinear is discussed and appropriate test problems aresolved.

Temperature field in an industrial kiln is depicted in Fig. 4.2 [26]. Both boundaryconditions and material properties have been treated as nonlinear. The upper surfaceof the kiln was cooled both by radiation (Tr = 290K, t = 0.7) and convection (Tf =300K h = 15Wjm2K). Similar data for the right boundary line read Tr = 290K, t =0.2, Te = 300K, h = 10Wjm2I<. The left and lower surfaces were insulated. Thesurface lying within the kiln was held at temperature T = 1700K. Materials were


7001300

1j =300K £=0.7 Tr =2~K

310 ~C\lII

E-t-C\lC)

II~

~C)C)t")

II

~

~t\l

~~C).,...II~

A

B

h=15W/m2 K

c

Figure 4.2: Temperature field within a lining of an industrial kiln.

treated as nonlinear. The dependence of the heat conductivities on temperature isshown in Fig. 4.3. The materials used were:

zone A - silica brickzone B - mullitezone C - chamottezone D - kaolinzone E - brick

Some materials of crystalline or fibrous structure (e.g. wood, graphite) exhibitstrong anisotropy. Unfortunately nonlinear anisotropic material cannot be handled inan elegant way using Kirchhoff's transformation. For such substances one can onlyuse the above mentioned pseudo-source approach either in primitive variable space orin the Kirchhoff's variable domain.

Transient

Nonlinear transient problems are much more difficult to handle than their elliptic(steady state) analogs. The difficulties inherent in parabolic equations are due to the


conductivity of materials in Fig. 2

o KIIC#I - - C 0.- - • E Eiick ••••• A ~ _. 8-M.#r.mott. trlc*

260,..------.------.----...-------;,..---/-,-0_

,B /

200 / .~:":-:-:-:":":-:-~ :":-:-:-:":":-:-:-:" :-:-:-:":":-:-:-:":":"~:"~. . . . . A

1.401------+------+------...,1------+---~

- - ---------;.. - -c--------

0.20 I--'-------'-------l_-'---'------L_J---'------!-.,.-:-'_-'----:'--:--,,--.._=O~300 600 900 1200 1500 1800

E0.801------+------160-..::.--.::::::---1--.::::::-----:...--+--=.::-----1---

Tempera ture KFigure 4.3: Temperature dependence of the heat conductivity for materials constituting thelining of the kiln shown in Fig. 4.2.

dependence of their fundamental solutions on material properties (thermal diffusivity).A linearization method that does not demand much ingenuity is to subdivide the

region under consideration into a number of small subregions where mean values ofdiffusivities at each time step can be used with sufficient accuracy.

Also the fictitious source approach can be used to cope with transient nonlinear problems. This approach is not numerically attractive and thus to the authorsknowledge, no paper using such a technique has been published yet.

The first papers dealing with transient heat conduction problems with materialnonlinearities seems to be Refs [14] and [15]. The approach described therein wasthat of Kirchhoff's transformation. Transformation of the original differential Eq.(4.4) with no internal heat sources (b = 0) yields

(4.59)

It should be stressed here that Eq. (4.59) is still nonlinear as the thermal diffusivityis solution dependent. However, the assumption of constant diffusivity is often morerealistic than that of constant values of heat capacity and heat conductivity. Thereforein Refs [14],[15] diffusivity has been assumed constant (not temperature dependent).Under this assumption the problem becomes linear in the space of Kirchhoff's transform and standard weighted residuals procedure can be used to obtain the boundary


integral equation. The result of this procedure is an integral equation of a form analogous to the linear case (Eq. (4.17)). Both of the mentioned above references concernnonlinearity caused by nonlinear material and boundary conditions.

A similar approach has been reported in Ref. [36]. The technique developed therecan be regarded as a mixed Kirchhoff's transformation pseudo-source approach. Theresult of the transformation i.e. Eq. (4.59) has been rearranged upon extracting linearpart of the diffusivity written as

(4.60)

The linear part of diffusivity Do was used to define the fundamental solution.The remaining part of the diffusivity has been put into a fictitious source term. Inthe notation used here the integral equation for homogeneous initial conditions andwithout internal heat generation has an appearance (notice a misprint in the originalpaper)

C 1/J = ~:1T l [q* 1/J - T* q] dS dt+

+ r f !:lDn ( 1/J) T* iN dV dtJo Jv Do + !:lDn (1/J) at (4.61 )

As no additional simplifications have been introduced when deriving the integralEq. (4.61) from the original boundary value problem, these two formulations areexactly equivalent. However, the domain integral does not vanish even for constanttime elements as erroneously suggested in Ref. [36]. It is true, as stated therein, thatfor constant time elements the temporal derivative vanishes within the time step, butat the end of each time increment the derivative exhibits a Dirac's delta behaviour(there is a jump of the function 1/J there). Neglecting the domain term would be thenequivalent to the assumption of constant diffusivity discussed previously. Anyhow, theimportance of formulation (4.61) lies in the fact that it is the first integral formulationtaking into account the real nonlinear material behaviour in transient states withoutintroducing any additional simplifications.

It would be interesting to get the proper solution of Eq. (4.61) also for the multizoned media. Again the incremental technique could be used to solve the resultingnonlinear algebraic equations as the nonlinear material behaviour will influence onlythe right hand side vector.

The difficulties associated with the conversion of the nonlinear differential equation of transient conduction are as already mentioned due to the dependence of thefundamental solution on the diffusivity. Thus, using the Dual Reciprocity where (notdepending on material properties) the fundamental solution of the Laplace equationis employed offers new possibilities of handling material nonlinearities. This approachhas been recently developed in Ref. [11]. The technique consists in performing twointegral transformations. The first one is the Kirchhoff's transformation whereas thesecond transformation is a modification of the time variable. The time domain transformation is defined as

T = iT D(r, t) dt (4.62)


After making use of the above definition of new time variable Eq. (4.59) takes aform:

(4.63)

Eq. (4.63) can be readily converted into a Dual Reciprocity boundary integralequation. It has been recognized recently that the trick of introducing the timetransformation is not necessary [12], moreover, it is not recommended. The reasonfor this is that the transformation (4.62) only apparently linearizes the differentialequation only apparently [37]. As shown in Ref. [37] the results obtained via DualReciprocity (Ref. [11]) are correct but transformation (4.62) leads to erroneous resultswhen using time dependent fundamental solutions.

Dual Reciprocity in the Kirchhoff's transform space produces an integral equationhaving a form

(4.64)

where: ko reference value of conductivity used in the definition of Kirchhoff's transform (Eq. (4.54)). In the definition of the fundamental solution (Eq. (4.12))ko has been taken as the thermal conductivity k.

In linear problems the temporal derivative of the solution has been approximatedby a sum of appropriately chosen functions (cf Eq. (4.20)). As shown in Ref. [12] itis possible to employ similar approximation to the nonlinear term

1 a1/; ~. .- - S:! LJ P (r) o:J (t)D(1/;) at j=1

(4.65)

After space discretization has been performed the unknown functions o:j (t) can beexpressed as a linear combination of nodal values of temporal derivatives divided bydiffusivities

j( ) _ ~ f- 1 1 a1/;(ri, t)o:t-LJ ..i=1 J' D[1/;(ri, t)] at

(4.66)

with fi"/ defined as in the linear case (cfEqs (4.24), (4.25))The matrix formulation of the Dual Reciprocity has the same form as for the linear

case but the capacitance matrix is no longer constant as it depends on the solution

c = - [H T - G Q] F-1 D- 1 (4.67)

where - D-1 a diagonal matrix with entries computed as

(4.68)for i =I- j

{D- 1 };j = { 01 for i = J'

D[tI>(r..t)]

As can be observed the capacitance matrix should be recomputed within a loopat each time step to account for the nonlinearity, but the whole procedure of time


stepping is very much the same as in the linear formulation. The reader should consult Ref. [11] and [12] for more details of the numerical implementation of the method.

4.5 Nonlinear Source Term

The importance of these problems arises not only in the presence of actual nonlinearinternal heat sources in the original boundary value problems but also when usingthe pseudo source approach of differential operator linearization. Generally speakingnonlinear source problems can be solved by BEM in a quite straightforward manner.However, in its classical formulation BEM demands domain integration to solve thesekind of problems. Thus, the main advantage of using BEM which is the reduction ofdimensionality, is lost.

Nonlinear problems discussed in this section demand evaluation of a domain integral (cfEqs (4.11) (4.17))

I =1Iv b(T) T* dV dr (4.69)

(obviously time integration is to be performed only in transient problems).Integral (4.69) contributes only to the right hand side vector. The value of this

term is to be corrected iteratively using temperatures determined in the previous stepof iteration.

The literature on this subject in heat transfer context is surprisingly scant. Assimilar problems are encountered in nonlinear stress analysis e.g. elastoplasicity [38]and fluid flow [39],[40] one can employ the methods described therein to deal withnonlinear problems in scalar fields.

As shown in Ref. [12] Dual Reciprocity offers an elegant way of handling this typeof nonlinearity and avoiding domain integration. In the presence of heat generationthe Dual Reciprocity equation has a form

C T = is [Tq* - T*q] dS-

- r~ aT T* dV + r b(T) T* dVlv D at lv (4.70)

The additional domain integral to be converted into a sum of boundary ones isthen (cfEq. (4.19))

J = Iv b(T) T* dV (4.71 )

The heat generation rate b is approximated by a similar expression as in the caseof transient states (cf Eq. (4.20))

N

b(T) ~ L fj(r)a j

j=l

(4.72)

In the above a j are constants and the functions Ii are chosen so as to satisfy Eq.(4.21). This enables one to transform the domain integral into a sum of boundaryintegrals upon employing the reciprocity theorem. Unknown constants a j can be


taken outside the integral. They are eliminated after the discretization of Eq. (4.70)by making use of the inverse transformation (cf Eqs (4.66) and (4.24))

N

al = L fT;) b[T(rj, t)];=)

(4.73)

For the case of nonlinear heat generation being the only nonlinearity present in thedifferential equation, the final form of the matrix Dual Reciprocity BEM formulationcan be wri t ten as

. DC (T - kg) +H T = G Q

where: g - contains values of the heat generation rate at nodal points

{g}; = b [T(ri, t)]

C - is defined as in linear case (cfEq. (4.27))

(4.74)

(4.75)

Eq. (4.74) constitutes a set of ordinary differential equations that degenerates toa set of nonlinear algebraic equations for steady states. Solution of Eq. (4.74) can beaccomplished using standard nonlinear equations solvers (within each time incrementfor time dependent problems). It should be stressed that applying Dual Reciprocityto these kind of nonlinear problems has been proposed very recently. To the authorsknowledge no numerical results concerning this field of BEM application has beenpublished yet.

4.6 Moving Boundaries

The primary nonlinearity of phase change problems is concentrated on the boundary,thus the phenomenon is well suited to be solved by BEM. The importance of phasechange problems and the relative ease of applying BEM to solve moving boundaryproblems attracted the attention of the BEM community from a very early stage ofdevelopment of this technique.

The first papers on moving boundaries date back to the pre-BEM era [41 ]'[42]. Theproblems solved in Refs [41 ],[42] were one dimensional Stefan's problems in Cartesianand cylindrical geometries. The authors derived appropriate integral equations anddiscretized them i.e., employed the BEM technique, though the name of this methodhas been given much later. Melting of iron-carbon alloys with simplified diffusionequations has been dealt with in the aforementioned papers. The integral equationgoverning the melting process was derived there in a form generalized later [43] tomultidimensional problems

C T = Dk

r r (T* q - T q*) dS dt + Dk

r r b T* dV dt+Jo J5 (T) Jo JV(T)

+ r T T* !T=O dV + r T T* I~=~+ dV (4.76)JV(T=O) JC.V(T) p=p+

where: ~V( T)- region swept by the moving boundaryT* - fundamental solution of the parabolic equation z. e. solution of

(4.77)


Eq.(4.16)r+ - time when the position of the moving boundary was p+

The first integral in Eq. (4.76) gives a contribution of the boundary conditions tothe solution. The second integral describes the influence of the internal heat sources.The third integral is due to initial conditions whereas the last integral gives the contribution caused by the movement of the boundary.

Equations derived in Ref. [41] were solved using a mature BEM formulation [44]and applied to a one dimensional ablation [45].

The paper [46] seems to be the only BEM formulation with complete interactionbetween diffusion and heat transport present. The authors solved a 2 - D problemof melting of a steel bar with concentration dependent melting temperature in conjunction with 2 - D transient diffusion of alloy components. Also ablation problemswith convective-radiative boundary conditions on the ablating surface were solved inthat Reference. The time marching scheme employed there was scheme 1 (cf section'Transient') whereas in other earlier quoted papers scheme 2 was used.

In the Refs [47],[48],[49] another complexity was taken into account, namely therealistic case of solidification of metals (alloys) in steel and sand moulds. The imperfect thermal contact on the casting-mould interface has been dealt with. Theauthors suggested the possibility of employing Kirchhoff's transformation to accountfor nonlinear thermal conductivities (Skerget's approach [14],[15]) but no computations using this concept were discussed. Reference [48] deals with a hybrid FDM-BEMformulation and points out a strong interaction between time steps of both numericalmethods.

In the paper [50] another integral equation describing the moving boundary problems has been derived. In this formulation the domain integral over the part of thebody swept by the moving boundary has been converted into a surface integral overthat boundary. This formulation reads:

C T = Dk

r r [T q* - T* q] dS dt + Dk

r r bT* dV dHJo JS(T) Jo JV(T)

+ r T T* !T=O dV + r r T T* v dS dtJV(T=O) Jo JS(T)

In the case of uniform initial condition and no internal heat sources, no domainintegration is to be performed when using this formulation provided marching timescheme 2 is employed (cf section 'Transient')

In Ref. [51] integral Eq. (4.77) has been discretized. The resulting algebraic equations have been solved on each time step iteratively to get the precise location of themoving boundary (front tracking method). The numerical examples are in 2 - D andconcern solidification of pure substances in square mould. Comparison with analytical'corner solutions' showed good accuracy of the proposed scheme. This neat numerical behaviour of the scheme was achieved upon adopting the double node conceptto handle corner singularity and approximate improvement of the solution for shorttimes.

Reference [52] deals with subcooling and interface instabilities of Mullin Sekerkatype. The author considered a 2 - D model in infinite region with temperature field


exhibiting cylindrical symmetry. The aim of this study was to find a stable shape of aneedle crystal advancing steadily into an undercooled melt. The temperature on theinterface was assumed to satisfy a thermodynamic condition

T(r, t) = ~u - L[1 +d(1 - cos 48)] I«8) (4.78)

where: ~u

1/L8

I«8){)

thermal undercooling of the liquid phase

capillarity length

angle formed by the normal to the interface and the z axis

interfacial curvature

measure of anisotropy

The 2 - D solidification problem of a pure substance has been solved in Ref. [53]using the enthalpy concept (fixed domain approach) and finite differences in time domain. The fixed domain consists in introducing a new unknown (enthalpy) being anintegral of the heat capacity [7]. This approach is a very useful tool of solving movingboundary problems. When using this technique it is possible under some additionalassumptions (equal conductivities of both phases) to treat the whole domain as an homogeneous one. However, because the location of the moving boundary is determinedas a secondary unknown the technique is not very accurate [51].

The moving boundary problems demand further research. First of all the physicsof the problem should be clearly stated. Then, numerical methods should be developedto deal with realistic engineering problems with nonlinearities caused simultaneouslyby the material behaviour in multizoned media as well as the boundary conditions.

An interesting phenomenon often referred to as free boundary location is also encountered in the thermal science. It consists in determining the location of a phaseinterface in steady states. An example of this kind of problem is the blast furnacelining dissolution in the molten pig iron. The process of dissolution takes place on theinterface of carbon blocks forming the lining and the molten pig iron until the temperature of the interface is higher than the solidification temperature of the moltenpig iron. Models of water seepage in porous media well established in BEM practicecan be used to solve these kind of problems [54]. First results in this area has beenobtained by Nahlik [55].

4.7 Conclusions

After one decade of its development BEM proved to be be a powerful tool for solvingnonlinear boundary problems of heat conduction. Practically all nonlinear phenomenaoccurring in situations of engineering interest can now be solved by BEM using integralformulation strictly equivalent to the the original boundary problems.

Some kinds of problems can be handled by BEM in a very effective way offeringmany advantages over domain integration techniques. Such phenomena as all kindsof nonlinear boundary conditions, nonlinear isotropic material in steady states andmoving boundaries, belong to this group of problems. A characteristic feature of this


group is a boundary-only integral equation. Applying BEM retains here the mainadvantage of this technique i.e. the reduction of the dimensionality of the problem.

The second group of nonlinear problems is that with domain integration present inthe integral equation. This term can result from internal heat sources acting within thedomain of interest. It can also contain the nonlinear part of the governing differentialoperator. A situation of this kind is characteristic of nonlinear material in transientstates. When dealing with this group of problems it is now common, though not veryelegant practice, to introduce additional simplifications (constant diffusivity) to makethe domain integral vanish.

The Dual Reciprocity and the recently developed Multiple Reciprocity technique[56],[57] open a new possibility of converting domain integrals into boundary onesalso in nonlinear cases. This direction of research is a very promising one as employing these techniques greatly improves the efficiency of using BEM by relievingthe headaches of generating internal mesh and integration over the whole domain.However, there is little experience in applying these techniques to nonlinear analysis.

In the author's opinion there are four main directions of further BEM research:

- nonlinear material in transient states (also multizoned media), by Dual or Mul-tiple Reciprocity, pseudo-source approach or some hybrid formulations.

- nonlinear heat sources by Reciprocity techniques

- moving boundaries with all inherent physical complexness

- coupling different modes of heat transfer e.g. conduction, radiation and convec-tion modelled by complete Navier-Stokes equations (to avoid the usage of theexperimentally determined heat transfer coefficients)

Acknowledgements

The work has been financially supported by the Ministry of National Education withinthe Central Plan for Fundamental Research- direction 4.4 coordinated by the WarsawTechnical University. The financial help is gratefully acknowledged herewith.


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29. Bialecki R. (1988). Radiative heat transfer in cavities. BEM solution, in BEM10th, vol 2, pp 246 - 256 Proceedings of the 10th Boundary Element Conference,(Ed. Brebbia C. A.) Southampton, Springer Verlag, Berlin and New York.

30. Bialecki R. (1989). Modelling 3D spectral band heat radiation using BEM, Tobe presented at 11th BEM Conference, Cambridge, USA.

31. Nowak A. J. (1990). Solving coupled problems involving conduction convectionand thermal radiation. Chapter 7 of the present book.

32. Banerjee P.K. (1979). Non-linear problems of potential flow. Chapter 2, Developments in Boundary Element Methods, (Ed. Banerjee P.K. and ButterfieldR.), Vol. 1, pp 21-30, Applied Science Publishers, London.

33. Sakahikara M. (1985). An iterative boundary integral equation method formildly nonlinear elliptic partial differential equations, in 7th BEM, Vol. 2, pp 1349 to 13-58, Proceedings of the 7th Boundary Element Conference (Ed. BrebbiaC.A. and G. Maier), Como, Italy, Springer-Verlag, Berlin and New York.

34. Onishi K. and Kuroki T. (1982). Boundary Element Method in singular and nonlinear heat transfer, in 4th BEM, pp 141-155, Proceedings of the 4th BoundaryElement Conference (Ed. Brebbia C.A.), Southampton, UK, Springer - Verlag,Berlin and New York.

35. Azevedo J.P.S. and Wrobel L.C, (1988). Nonlinear heat conduction in compositebodies. A Boundary Element formulation, Intern. j. numer. method. eng., Vol.26., No.1, pp 19-38.

36. Kikuta M., Togoh H., Tanaka M. (1987). Boundary Element analysis of nonlinear transient heat conduction problems, Compo Meth. in Appl. Mech. andEngn, Vol 62, pp 321-329.

37. Bialecki R. and Nowak A.J. (1990). Some remarks on transformation techniquesfor transient nonlinear problems, Engineering Analysis, Vol 7, No 3, pp 145-146.


38. Telles J.C.F. (1983). The Boundary Element Method Applied to Inelastic Problems. Lecture Notes in Engineering, Vol. 1, (Ed. Brebbia C.A. and OrszagS.A.), Springer-Verlag, Berlin and New York.

39. Wu J.C. (1985). Boundary Element Method and inhomogenous elliptic differential equations, in 7th BEM, Vol 1, pp 9-95 to 9-104, Proceedings of the 7thBoundary Element Conference (Ed. Brebbia C.A. and G. Maier), Como, Italy,Springer-Verlag, Berlin and New York.

40. Skerget P., Alujevic A., Brebbia C.A. (1985). Analysis of laminar flow withseparation using BEM, in 7th BEM, Vol. 1, pp 9-23 to 9-36, Proceedings of the7th Boundary Element Conference (Ed. Brebbia C.A. and G. Maier), Como,Italy, Springer-Verlag, Berlin and New York.

41. Chuang K. and Szekely J. (1971). On the use of Green's function for solvingmelting and solidification problems, Int. J. Heat and Mass Transfer, Vol. 14,No 9, pp 1285-1294.

42. Chuang K. and Szekely J. (1972). The use of Green's function for solving meltingand solidification problems in cylindrical geometry, Vol. 15, No.5, pp 1171-1174.

43. Banerjee P.K. and Shaw R.P. (1982). Boundary Element formulation for meltingand ablation problems. Chapter 1, Developments in Boundary Eements Method,(Ed. Banerjee P.K. and Shaw R.P.), Vol. 2, Applied Science Publishers, London.

44. Wrobel L.C. (1983). A Boundary Element solution to Stefan's problem, in 5thBEM, (Ed. Brebbia C.A. Futagami T. and Tanaka M.) pp 173-182, Proceedingsof the 5th Boundary Element Conference, Hiroshima, Japan, Springer-Verlag,Berlin and New York.

45. Shaw R.P. (1982). A boundary integral approach to the one dimensional ablationproblem, in 4th BEM, (Ed. Brebbia C.A.), pp 127-140, Proceedings of 4thBoundary Element Conference Southampton, UK, Springer-Verlag, Berlin andNew York.

46. Takahashi S., Onishi K,. Kuroki T., Hayashi T. (1983). Boundary Elementsto phase change problems, in 5th BEM, (Ed. Brebbia C.A. Futagami T. andTanaka M.) pp 163-172, Proceedings of the 5th Boundary Element Conference,Hiroshima, Japan, Springer-Verlag, Berlin and New York.

47. Hong C.P., Umeda T., Kimura Y. (1983). Application of the Boundary ElementMethod in two and three dimensional unsteady heat transfer involving phasechange; Solidification problem. in 5th BEM, (Ed. Brebbia C.A. Futagami T. andTanaka M.) pp 153-162, Proceedings of the 5th Boundary Element Conference,Hiroshima, Japan, Springer-Verlag, Berlin and New York.

48. Hong C.P., Umeda T., Kimura Y. (1984). Numerical methods for casting solidification problems. Part 1. The coupling of the Boundary Element and FiniteDifference methods for solidification problems. Metallurgical Transactions B,Vol. 15B, pp 1984-1991.


49. Hong C.P., Umeda T., Kimura Y. (1984). Numerical methods for casting solidification problems. Part 2. Application of the Boundary Element method tosolidification problems. Metallurgical Transactions B., Vol. 15B, pp 101-107.

50. O'Neill K. (1983). Boundary integral equation solution for moving boundaryphase change problems, Intern. j. numer. method eng., Vol 19, pp 1825-1859.

51. Zabaras N. and Mukherjee S. (1987). An analysis of solidification problem bythe Boundary Element Method, Intern. j. numer. method eng., Vol 24, pp1879-1900.

52. Meiron D. I. (1987). Boundary integral formulation of the two dimensionalsymmetric model of dendritic growth, Physica D, Vol. 23D No 1-3, pp 329-339.

53. Tiba M. (1987). the Boundary Element Method in two phase Stefan's problem,Engineering Analysis. Vol. 4, No 1, pp 46-48.

54. Ouazar D., Namli A., Saidi A., Brebbia C.A. (1987). An efficient Zoned PorousMedia Problem with free surfaces., in 9th BEM (Brebbia C.A. Wendland W.L.and Kuhn G.) Vol 3pp369 - 385 Proceedings of the 9th Boundary Element Conference, Stuttgart, FRG, Springer-Verlag, Berlin and New York.

55. Nahlik R. (1989). Algorithm of free surface location in heat conduction problemsusing BEM, PhD Thesis, Silesian Technical University (in Polish).

56. Nowak A.J.and Brebbia C.A. (1989). The Multiple Reciprocity Method. A newapproach for transforming BEM domain integrals to the boundary. Eng. Anal.with Bound. Elem. vol 6. No 3, pp 164-167.

57. Nowak A.J. (1990), Solving linear heat conduction problems by the MultipleReciprocity Method, Chapter 3 of the present book.


Notation

a

A

b

b

B

c

c

C

D

F

g

G

h

h

H

J

k

k

1<

1«0)

K

L

absorptivity of the fluid; m-1

matrix of linear BEM equations, Eq. (4.15)

heat generation rate due to internal heat sources; W/ m 3

vector in matrix BEM formulation, depending on the history of theprocess and/or on internal heat generation

matrix in incremental BEM formulation, Eq. (3.40)

specific heat; J/ kg1<

function of the internal angle the boundary S makes at a given point.For a smooth boundary C = 0.5

capacitance matrix in the Dual Reciprocity technique

thermal diffusivity; m 2/ s

linear part of thermal diffusivity, Eq. (4.60)

diagonal matrix defined by Eq. (4.68)

set of known functions employed in the Dual Reciprocity technique

right hand side vector in linear BEM equations set, Eq. (4.15)

matrix defined by Eq. (4.25)

vector containing values of the heat generation rate, Eq. (4.75)

heat flux influence matrix in matrix BEM formulation, Eqs (4.14),(4.15)

heat transfer coefficient; W/m 21<

diagonal matrix of heat transfer coefficients defined by Eq. (4.38)

temperature influence matrix in matrix BEM formulation, Eqs (4.14),(4.15)

inverse Kirchhoff's transformation

thermal conductivity; W/m1<

known vector arising in BEM radiative heat exchange formulation inthe presence of participating medium filling the enclosure, Eq. (4.45)

kernel function arising in heat radiation problems and defined byEq. (4.44)

interfacial curvature, Eq. (4.78)

Kirchhoff's transformation defined by Eq. (4.54)

latent heat; J / kg


n outward normal to the boundary

N number of nodal points

N nonlinear differential operator

No linear part of the differential operator

M number of incremental steps

p vector coordinate of the source point; m

p+ location of the moving boundary at the time r+, Eq. (4.76); m

q heat flux; WJm 2

qr radiative heat flux; WJm 2

ij prescribed heat flux on the boundary; WJm 2

q* heat flux analog defined by Eq. (4.13)

qJ auxiliary functions depending on coordinates and defined by Eq. (4.23).Used in the Dual Reciprocity technique

Q matrix arising in the Dual Reciprocity method and defined by Eq. (4.29)

Q vector of nodal heat fluxes in matrix BEM formulation

Q' vector of derivatives of the nodal heat fluxes with respect to nodaltemperatures used by the Newton Raphson solver, Eq. (4.36)

Qr vector containing nodal radiative heat fluxes

r vector coordinate; m

R radiative heat flux influence matrix, Eq. (4.45)

R l - r radiative heat exchange factor

S boundary surface

Se surface of an enclosure

S temperature influence matrix in radiative problems, Eq. (4.45)

t time; s

T temperature; K

Te temperature of a participating fluid filling an enclosure; K

Tf temperature of a fluid exchanging heat with the boundary; K

Tj initial temperature distribution; K

Tr temperature of the radiating environment; K

T* fundamental solution satisfying Eq. (4.12) or (4.16)

fi auxiliary functions depending on coordinates and defined by Eq. (4.21).Used in the Dual Reciprocity technique


Tij temperature of the interface of subregions 'i' and 'j'

T vector of nodal temperatures in matrix BEM formulation, Eqs (4.14),(4.15)

T 4 vector arising in radiative BEM analysis and containing fourth powers ofnodal temperatures, Eq. (4.45)

t matrix arising in the Dual Reciprocity method and defined by Eq. (4.28)

u specific internal energy; J j kg

v component of the moving boundary velocity in a direction normal tothis boundary; mls

V region occupied by the solid

x vector of nodal unknowns, Eq. (4.15)

Greek letterso:i unknown function depending on time employed in the Dual Reciprocity

technique

f3 known function depending on the interfacial temperature, Eq. (4.57)

8 Dirac's distribution

f).Dn additive, nonlinear part of the heat diffusivity, Eq. (4.60)

f).h n additive, nonlinear part of the heat transfer coefficient, Eq. (4.7)

f).N n additive, nonlinear part of the differential operator N

f).qn additive, nonlinear part of the boundary heat flux, Eq. (4.6)

f).1/Jn known function of the interfacial temperature, Eq. (4.58)

f)... thermal undercooling of the liquid phase, Eq. (4.78)

f).Qn vector containing additive nonlinear part of the boundary heat flux,Eq. (4.37)

<:: emissivity of the radiating surface

, IL capillarity length, Eq. (4.78)

</>p angle between a normal at a point p lying on a boundary and a lineconnecting two points rand p, Eq. (4.44)

</>r angle between a normal at a point r lying on a boundary and a lineconnecting two points rand p, Eq. (4.44)

1/J Kirchhoff's transform, Eq. (4.54)

,\ incremental coefficient, Eq. (4.42)

p density; kgjm3

(J Stefan - Boltzmann constant (J = 5.66710-8Wjm 21<4


T auxiliary time; s

T+ time when the location of the moving boundary was p+, Eq. (4.76); s

{} measure of anisotropy, Eq. (4.78)

e angle formed by the normal to the interface and the z axis, Eq. (4.78)

Subscriptsc phase change in moving boundary problems

initial conditions

e

n

o

r

s

o

enclosure

nonlinear

liquid phase in moving boundary problems

linear part of a function or operator

radiative heat exchange

solid phase in moving boundary problems

reference values of the Kirchhoff's transformation, Eq. (4.54)


auxiliary functions in the Dual Reciprocity formulation

temporal derivative

+ location of the moving boundary, Eq. (4.76)

(k) current number of iteration

p result of Gaussian preelirnination, Eq. (41)

Other symbolsV7 gradient


r - p distance between points rand p

{A}ii

aan

element of a matrix A lying in the i-th row and j-th column

differentiation along outward normal to the boundary

Chapter 5

Coupled Conduction-Convection Problems

L.C. Wrobel, D.B. DeFigueiredo (*)Computational Mechanics Institute, Wessex Institute of Technology,Ashurst Lodge, Ashurst, Southampton S04 2AA, England

(*) On leave from CNPq, National Council for Scientific and TechnologicalDevelopment, Brazil

5.1 Introduction

Many physical processes of engineering interest can be modelled by the convectiondiffusion equation, such as the transport and dispersion of pollutants in ground andsurface water, diffusion in semiconductor devices, travelling magnetic fields and magnetohydrodynamics, etc. In the field of heat transfer, we can mention the problems ofcrystal growth, laser-assisted surface hardening, metal cutting, casting and forming,and others. A special case is the telegraph equation, used to model heat conductionfor very short times, taking into account a finite thermal propagation speed [I].

A substantial number of numerical models for the convection-diffusion equationhas been presented in the literature. Most of these models employ either the finitedifference or the finite element methods of solution, and give emphasis on algorithms tosuppress the well-known phenomenum of 'artificial diffusion' intrinsic to these methods[2]'[3].

Applications of the Boundary Element Method for steady-state convection-diffusion have shown that the BEM seems to be relatively free from this problem [4-6].This was also the case for some transient applications using formulations with timedependent fundamental solutions [7],[8]. The main restriction of these formulations,however, is the fact that fundamental solutions are only available for equations withconstant coefficients, or coefficients with very simple variations in space [9]. Alternative formulations for treating problems with variable velocity fields have employedthe fundamental solution of Laplace's equation and treated the convective terms aspseudo-sources [4],[10],[11]. Extensions to transient analysis adopted the same idea,i.e. the fundamental solution of the diffusion equation [10J or even that of Laplace'sequation [12],[13]. The disadvantage of such an approach is that domain discretizationis required to account for the pseudo-sources.


This chapter presents a boundary element formulation for the solution of generalconvection-diffusion problems based on a dual reciprocity scheme [14]. For transientproblems, the domain integral resulting from the dual reciprocity approximation istransformed into equivalent boundary integrals by expanding the time-derivative termas a summation of global interpolation functions and introducing particular solutionswhich satisfy an associated non-homogeneous steady-state convection-diffusion equation [15]. For problems with variable velocity, the velocity field is decomposed into anaverage and a perturbation, and the fundamental solution of the convection-diffusionequation with a constant velocity (the average) is employed. This feature already represents an advantage over previous formulations in that the domain approximationsare then confined to the perturbation field only, which in many problems is smallerthan the average. Although domain discretization and integration can then be performed for the perturbation terms, it is shown that the domain integral can also betransformed into equivalent boundary integrals through a dual reciprocity approach.Thus, the problem in all cases is ultimately described in terms of boundary valuesonly, consequently reducing its dimensionality by one.

Results of several analyses are presented and compared to analytical solutions.They show that the boundary element formulation described in this chapter does notdisplay any artificial diffusion or oscillatory behaviour, thus precluding the need for'upwind' or other algorithms common to finite element analysis.

5.2 BEM Formulation for Steady-State Problems

(5.1 )2 of ofD\l f - V x - - v - = aox y oy

where f is temperature, V x and V y are the components of the velocity vector v and Dis the thermal diffusivity (assuming the medium is homogeneous and isotropic). Themathematical description of the problem is complemented by boundary conditions ofthe Dirichlet, Neumann or Robin (mixed) types.

The above differential equation can be transformed into an equivalent integralequation by applying a weighted residual technique. Starting with the weighted residual statement

The two-dimensional steady-state convection-diffusion equation can be written in theform

(5.4 )

(5.2)

(5.3)

J ( 2 of Of),D\l f - V x - - v - f dO = a12 ox y oy

and integrating by parts twice the Laplacian and once the first-order derivatives, thefollowing equation is obtained

f(O = D [ f' Ofdr - D [ f of' dr - [ ff'vndrir On ir On ir

where V n = v . n, n is the unit outward normal vector and the dot stands for scalarproduct.

In the above equation, f' is the fundamental solution of equation (5.1), i. e. thesolution of

(5.5)


in which ~ and X are the source and field points, respectively. It can be noticed thatthe sign of the first-order derivatives is reversed in (5.1) and (5.4), since this operatoris not self-adjoint. For two-dimensional problems, ¢. is of the form

1 IT¢·(CX) = 27rD e- 2D I<o(p,r)

whereIv I

p, = 2D

and r is the modulus of r, the distance vector between the source and field points. Thederivative of the fundamental solution with respect to the outward normal directionis given by

(5.6)

In the above, I<o and /(1 are Bessel functions of second kind, of orders zero and one,respectively.

Equation (5.3) permits calculating the value of ¢ at any internal point once theboundary values of ¢ and a¢/an are all known. In order to obtain a boundary integralequation, the source point ~ is taken to the boundary and a limit analysis carried outdue to the jump of a¢·/an. The result is the equation

(5.7)

in which c(O is a function of the internal angle the boundary f makes at point ~ [16].For the numerical solution of the problem, Eq. (5.7) is written in a discretized

form in which the integrals over the boundary are approximated by a summation ofintegrals over individual boundary elements, i. e.

N a¢ N (a¢. v )Ci¢i = DL, l ¢·-df - DL, l - + ....!3:.¢. ¢dfj=1 if} an j=l if} an D (5.8)

where the index i stands for values at the source point ~ and N elements have beenemployed. In the above equation, it can be seen that

(5.9)

Next, the variation of functions ¢ and aNan within each element are approximated by interpolating from the values at the element nodes. Herein, linear elementsare used, for which the expressions are

(5.10)

(5.11)


where <PI and <P2 are linear interpolation functions. Substituting the above into Eq.(5.8), the following expression is obtained

NC.),.. = '"' (gl. q l + g2. q 2 _ h1.),.1 _ h2.),.2)

I 'f'1 L.J I)) I)) I) 'f') I) 'f')

j=1

Note that the indexes 1 and 2 refer to the nodal (extreme) points of each element,and

lj = D l <Pk¢>*dfJ

Adding up the contributions of adjoining elements to each nodal point, Eq. (5.10)can be rewritten as

N

Ci¢>i = L (Gijqj - Hij¢>j)j=1

The above equation involves N values of ¢> and N values of q, half of which areprescribed as boundary conditions. In order to calculate the remaining unknowns, itis necessary to generate N equations. This can be done by using a simple collocationtechnique, i.e. by making the equation be satisfied at the N nodal points. The resultis a system of equations of the form

H<j> = Gq (5.12)

(5.13)

where the Ci values have been incorporated into the diagonal coefficients of matrix H.After introducing the boundary conditions, the system is reordered and solved by adirect method, e.g. Gauss elimination.

Evaluation of the coefficients of matrices Hand G is carried out numerically. Forthe off-diagonal terms, a selective Gaussian integration with number of integrationpoints as a function of the distance between source point and field element is employed,as described in [16]. The diagonal coefficients of matrix G have a weak singularityof the logarithmic type, and are calculated using the self-adaptive scheme of Telles[17]. The coefficients Hii can be calculated by noting that a consistent solution for aprescribed uniform temperature along the boundary can only be obtained if matrixH is singular, i.e.

N

Hii = - LHij (i ::/=j)j=1

As an example of application, the BEM formulation was used to study the problemof a bar moving parallel to the x-axis with constant velocity V x and with specifiedtemperature at the edges, i.e. ¢> = 300K at x = 0 and ¢> = 0 at x = L. The problemwas analyzed as two-dimensional with cross-section 6.0m x 0.7m, with the boundarycondition a¢>Ian = 0 specified along the faces parallel to the x-axis. Symmetry was


q=O

I-: -: -: - : - : - : - : - : - : - : - : - : -: - : - : - : -1</> = 300K </>=0

Figure 5.1: Moving bar: geometry, discretization and boundary conditions.

taken into account by reflection and condensation as described in [16], thus only theupper half of the region needed to be considered. A sketch of the problem is shownin Fig. 5.1.

The discretization employed 17 elements on the longer side and 1 element on eachof the smaller ones, making up a total of 19 linear elements and 22 nodes, for doublenodes were used in the corners to allow for the discontinuity of the normal at thesepoints [161.

Results are plotted in Figs 5.2 and 5.3 for several velocity values, compared withthe analytical solution (assuming D = 1m2 /s):

~ sinh [T(L-x)]</> = 300e 2 x -----'--=----,---.---""-

sinh [TL]It can be seen from the figures that the agreement between the two solutions is

excellent.Another problem studied was that of a rectangular plate with cross-section 6m x

8m, with uni-directional velocity in the x-direction. Symmetry was again consideredand only the upper half of the plate discretized with 41 linear boundary elements.The geometry, discretization and boundary conditions of the problem are sketched inFig. 5.4, in which a mixed-type boundary condition along the face x = 6m can benoticed. The temperature variation for V x = 0.3m/sand V x = 1m/s are compared inFigs 5.5 and 5.6 with the analytical solution (D = 1m2 /s):

A-. ( ) 400 '!.;Lx ~ 1 sinh .8nx . mr(y +4)'f' X Y = - e 2 i...J -- Sill --'-'-----'-

, 7r n=l n.8n cosh 6.8n 8

whereand 2 _ (vx ) 2 n

27r

2

n = 1,3,5,... .8n - 2 + 64

Again, excellent agreement between numerical and analytical solutions can be observed.


200(])L

=:i-+--'

oL(])

u 100E(])

f-

••••• v = 0.2 m/s~ ~ ~ ~ ~ v = 1.0 m/s00000 v = 6.0 m/s__ Analytical

6.02.0 4.0

X (m)

0-+--r---,-,,---,--,,.-;-r-,--,--..,,--,-,---,-,,---,--,,.-;-r-,.,..--,,---,-,---,-,400

Figure 5.2: Results for positive velocity.

300

200(])L

=:i-+--'

oL(])

u 100E(])

f-

••••• v -0.2 m/s~ ~ ~ ~ 0 v -1.0 m/sa a a a a V -1.6 m/s00000 v -6.0 m/s__ Analytical

6.02.0 4.0

X (m)

o-h....~.,..,.--r<t---r<>r-,e:;:=J=r~~>ril~=rwr-...,.,..~00

Figure 5.3: Results for negative velocity.


¢(x,4) = 0

¢(O,y) = 0

~(6,y) - ~¢(6,y) = 100 e6~

Figure 5.4: Two-dimensional plate with mixed-type boundary conditions.

5.3 BEM Formulation for Transient Problems

The two-dimensional transient convection-diffusion equation can be written in theform

(5.14)

where the variables are defined as for the steady-state case. Since the problem is nowtransient, initial values of ¢ at time to should also be provided.

Applying a weighted residual technique to the above equation, using as weighting function ¢* the fundamental solution of the corresponding steady-state equation(expression 5.5), we obtain

(5.15)

Integrating by parts twice the Laplacian and once the first-order space derivativesgives

(5.16)


600••••• y 0''''~~~ Y 1 moooooy 2 m

~~""~y 3m__ Analytical

Q) 400'--::J

-+--J

o

~200~Q) .

f- ~

602.0 4.0

X (m)

o~~~~--r-r-T---,-,-,rr-r-rrT""""""""'''-'-'00

Figure 5.5: Temperature variation for Vx =O.3rn/ s.

3000

2000ill'-::J

-+---'o'--ill

E- 1000

illf-

•.••• y 0~~~~oy 1 m~ ... lot ... 101' Y 2 moooooy 3m__ Analytical

6.02.0 4.0

X (m)

o~~~~~~~-.--r-r...,...-,...-r-r---"00

Figure 5.6: Temperature variation for Vx = l.Orn/s.


The corresponding boundary integral equation, for a source point ~ on the boundary,takes into account the jump of o¢* /an, in the form

In order to obtain a boundary integral which is equivalent to the domain integralin Eqs (5.16) and (5.17), a dual reciprocity approximation is introduced [14]. Thebasic idea is to expand the time-derivative o<p!at in the form,

p

~ = L !k(X, Y)Ok(t) (5.18)k=l

where the dot denotes temporal derivative. The above series involves a set of knownfunctions !k which are dependent only on geometry, and a set of unknown coefficientsOk which are time-dependent only. With this approximation, the domain integralbecomes

(5.19)

The next step is to consider that, for each function !k, there exists a relatedfunction 'l/Jk which is a particular solution of the equation

2 o'l/J o'l/JD\l 'l/J - Vx - - v - = !ax yayThus, the domain integral can be recast in the form

(5.20)

r ~¢*dO = f. Ok r (D\l2'l/Jk - Vx O'l/Jk - Vy O'l/Jk) ¢*dO (5.21)ill k=1 ill ax oy

Substituting expansion (5.21) into Eq. (5.17), and applying integration by partsto the right side of the resulting equation, one finally arrives at a boundary integralequation of the form

c(O¢(O - D lr ¢* ~~dr + D lr ¢~~* dr + lr ¢¢*vndr =

t, Ok [C(O'l/Jk(O - D lr ¢*7nkdr +D lr 1}Jk:*dr + lr'l/Jk¢*Vndr] (5.22)

For the numerical solution of the problem, Eq. (5.22) is discretized in a similarway to Eq. (5.8), i.e.


Next, the variation of functions ¢, q = 8¢/8n, 'l/J and "l = 8'l/J/8n within eachboundary element is approximated by interpolating from the values at the elementnodes. It should be noted that functions 'l/J and "l need not be approximated as theyare known functions for a specified set f. However, doing so will greatly improve thecomputer efficiency of the technique with only a minor sacrifice in accuracy.

Applying Eq. (5.23) to all boundary nodes, taking into account the previous interpolations, results in the following system of equations (see expression 5.12):

H4> - Gq = (H1P - G1])a (5.24)

In the above system, the same matrices Hand G as for steady problems are used onboth sides. Matrices 1P and 1] are also geometry-dependent square matrices (assuming,for simplicity, that the number of terms in expansion (5.18) is equal to the number ofboundary nodes), and 4>, q and a are vectors of nodal values.

By applying expression (5.18) at all boundary nodes and inverting, one arrives at

a = F-1;P

which, substituted into Eq. (5.24) results in

C4>+ H4> = Gq

with

(5.25)

(5.26)

System (5.26) can be integrated in time using standard time-stepping procedures.It should be stressed that the coefficients of matrices H, G and C all depend ongeometry only, thus they can be computed once and stored.

Employing a general two-level time integration scheme for solution of Eq. (5.26),the following discrete form is obtained

(5.27)

where 0 is a parameter which positions the values of 4> and q between time levelsm and m + 1. The right side of Eq. (5.27) is known at all times. Upon introducingthe boundary conditions at time (m + 1)~t the left side of the equation can berearranged and the resulting system solved by using a standard direct procedure likeGauss elimination.

Previous works on dual reciprocity schemes have shown that although a yarietyof functions can in principle be used as a basis for the approximation of ¢, bestresults are normally obtained with simple expansions, the most popular of which isf = 1 + r, where r is the distance between pre-specified fixed points (poles) and theboundary nodes [14]. This choice is based on practical experience rather than formalmathematical analyses.

In the present work, it was decided to start with a simple form of particularsolution 'l/J and find the corresponding expression for f by direct substitution into(5.20). The resulting expressions are:


in which (Xk, Yk) and (x, y) are the coordinates of the k-th pole and a general point,respectively.

The above choice was motivated by a previous successful experience with axisymmetric diffusion problems in which a similar approach was used [18]. It is interestingto notice that the set of functions f produced depends not only on r but also on thediffusivity, velocity components and the reaction rate, thus it will behave differentlywhen diffusion or convection is the dominating process.

The proposed dual reciprocity boundary element formulation was applied to themoving bar problem previously analysed in a steady state, assuming the initial condition </J = O. The geometrical and physical parameters are the same as before, and thediscretization, again that of Fig. 5.1, with an extra internal pole at the centre of therectangular region. The discontinuity of the normal flux at corners was considered byallowing corner nodes to have 3 degrees of freedom, i. e. </J, [N/an before the cornerand a</J/an after the corner, and prescribing 2 of these values. It is noted that the useof double nodes is not permitted with the dual reciprocity scheme because it leads toa singular matrix F which cannot be inverted.

Results for the variation of </J along x, at several time levels, are presented in Figs5.7 to 5.9 for the velocity values V x = 0.2, 1.0 and 6.0m/s, respectively, compared toanalytical solutions [19]. The BEM results were obtained using () = 1 in (5.27) anda variable time step. The starting value of 6.t was 0.10s for V x = 0.2m/s, 0.025s forV x = 1.0m/sand O.Ols for V x = 6.0m/ s, but these values were increased during theanalysis. It can be seen from the graphs that the accuracy of the dual ~eciprocity

boundary element formulation is very good in all cases, with no oscillations or damping of the wave front.

5.4 HEM Formulation for Variable Velocity Fields

If the velocity components in Eq. (5.1) vary in space, the BEM formulation previouslypresented cannot be directly applied to the problem any longer. Thus, the variablevelocity components vx(x,y) and vy(x,y) have to be decomposed into average (constant) terms Vx and vy and perturbations Px = Px(x,y) and Py = Py(x,y), suchthat

V x (x, y)V y (x,y)

This permits rewriting Eq. (5.1) as

V x + Px (x,y)vy + Py (x,y) (5.28)


300.00

~

~

'-./200.00

(])~

~+-'o~

(])

0...E100.00(])I-

66666 t 0.5 5~ ~ ~ ~ ~ t 1.0 500000 t 2.0 5••••• t 5.0 s00000 t 40.0 5-- Analytical

600200 400

X (m)

o 00 +-,---rT"T....,.,---r"....,-,,=::;=f'-r-'t-rO:;::=;~h+-t+~~4

oeo

Figure 5.7: Results for Vx = a.2m/s.

300.00

~

~

'-./200.00

~ 10... ~

E 100.00 ~(]) lI-

200 400X (m)

••••• t 0.2 5

~·~··t 1.0500000 t 2.0 s••••• t 4.0 s00000 t 20.0 5

-- Analytical

600

Figure 5.8: Results for Vx = 1m/s.


300.00 -t-"I:-4=::t----*==~-------..

600

••••• t 0.1 5••••• t 0.3 500000 t 0.5 5

10 500000 t 3.0 5-- Analytical

200 400

X (m)

o 0 0 -+--r---.-r-r-..--r-r-T::,-,-"r-......~-+-O-..,:::=,...,...-jl--,--............,.,-4?,..o.,.......0.00

~

Y::'-...../200.00

Q)'--::J

-+--'

o'--Q)

Q

E 100.00Q)

I-

Figure 5.9: Results for Vx =6m/ s.

2 _ 8¢ _ 8¢ 8¢ 8¢D'l ¢ - V x 8x - v y 8y = Px 8x + Py 8y

Starting with the weighted residual statement

(5.29)

(5.30)

and integrating by parts twice the Laplacian and once the first-order derivatives onthe left-hand side, the following boundary integral equation is obtained after a limitanalysis:

c(~)¢(O-D £¢*~~dr+D £¢~~*dr+ £¢¢*vndr=- fn(Px~: +py~~)¢*dn(5.31)

where vn = v·n.A dual reciprocity approximation can now be introduced to obtain a boundary

integral equivalent to the domain integral in the above equation. Expanding thenon-homogeneous term on the right-hand side of the equation in the form,

8¢ 8¢Px 8x + Py 8y

the domain integral in Eq. (5.31) becomes

(5.32)


(5.33)

Applying a similar procedure as for transient problems, by considering particularsolutions 'l/Jk to the convection-diffusion equation with non-homogeneous terms fk, thedomain integral can be recast in the form

(5.34)

Substituting expansion (5.34) into Eq. (5.31), and applying integration by partsto the domain integral of the resulting equation, one finally arrives at a boundaryintegral equation of the form

C(O¢(O - Dt¢*~~dr +Dt ¢~~* dr +t¢¢*vndr =

t, ak [C(O'l/Jk(O - Dt ¢*:kdr +Dt 'l/Jk:*dr + t'l/Jk¢*Vndr] (5.35)

The above equation is discretized in a standard BEM way, and the resultingdiscrete equation applied to all boundary nodes using a collocation technique, resultingin the matrix equation

Htf> - Gq = (Hl/I - G1])O: (5.36)

The next step in the formulation is to find an expression for the unknown vector0:. Applying Eq. (5.32) to all M nodes, it is possible to write the resulting set ofequations in the following matricial form,

(5.37)

where P x and P y can be understood as two diagonal matrices with componentsPx(x;, y;) and Py(x;, y;) respectively, i.e.

Px(Xl' Yl) 0 00 Px( X2' Y2) 0

Px

0 0 Px(XM' YM)

andPy(xl,yd 0 0

0 Py(X2' Y2) 0Py =

0 0 Py(XM' YM)

(5.38)

(5.39)


while ?If and ~ are column vectors defined as

~(Xl' yd ~(Xb Yl)

~(X2' Y2) ~(X2'Y2)

04> ~(X3' Y3) 04> ~(X3' Y3)oX ox (X4' Y4) oy ~(X4'Y4)

~(XM,YM) %;(XM,YM)

Inverting expression (5.37), one arrives at

(5.40)

0' =

Substituting into Eq. (5.36),

(5.41 )

Defining a matrix S in the form

S = (H1jI - G17) F- 1

one can write Eq. (5.42) as

(5.42)

(5.43)

(5.44)

It can be noted that matrix S depends on geometry only, once the sequenceof functions fk has been defined. The coefficients of matrices P x and Pyare alsoknown. Thus, there remains to be found an expression relating the derivatives of 4>with nodal values of 4> to reduce Eq. (5.44) to a standard BEM form. Herein, thealgorithm suggested by Partridge and Brebbia [20] is adopted.

By expanding the value of IjJ at an internal point using a similar approximationas in expression (5.32), one obtains

(5.45)

where :Fk = :Fk( x, y) are known functions and 13k unknown coefficients.Differentiating the above equation with respect to x and y produces

01jJ f= o:Fk13k (5.46)-

ox k=l ox

01jJ f= o:Fk 13k (5.47)=oy k=l oy


Applying Eq. (5.45) at all M nodes a set of equations is produced that can be represented in matrix form by

</> = :Fj3 (5.48)

with corresponding matrix equations for expressions (5.46) and (5.47). Invertingexpression (5.48)

j3 = :F-1</> (5.49)

and substituting the expression for j3 into the matrix forms of Eqs (5.46) and (5.47)gives

a</> a:F :F-1</>- =ax axa</> a:F :F-1</>ay ay

(5.50)

(5.51)

and Eq. (5.44) takes the form

or

where

(H - P)</> = Gq

P = S (p a:F P a:F) :F-1

x ax + Y ay

(5.52)

(5.53)

(5.54)

The coefficients of the perturbation matrix P are all geometry-dependent only.Thus, boundary conditions can be applied to Eq. (5.53) and the resulting system ofalgebraic equations solved by a standard direct scheme such as Gauss elimination.

It is important to notice that, although the expansions (5.32) and (5.45) are inprinciple arbitrary, both lead to the inversion of a matrix (see Eqs 5.41 and 5.49).Several different types of approximation have been tested in [211 where it was concluded that using the same global interpolation functions fk in both cases leads toreasonable accuracy (at least for the problems studied) with the highest computerefficiency since only one matrix is then inverted as F and :F become the same matrix.

The present dual reciprocity boundary element formulation was applied to a twodimensional problem with uni-directional velocity field V x function of the y-coordinate according to the expression

The V y component is equal to zero and consequently the equation to be solvedreduces to

\J2</> _ A (y _ B)2 a</> = 0ax

A particular solution to the above equation is

(5.55)


q = 300 A ~ (B _ 1) eA! [A! (B - t) +xl

......

+~

~~N

~N

I ICl:l Cl:l'--"

V x = A(y - B)2 '--";::., ;::.,..... '" . -I",

""<: ""<:-I", -''''""<: ""<:<U <U

0 00 0M M

II II--0- --0-

Lx1

q = -300A~ BeA3X

Figure 5.10: Geometry, discretization and boundary conditions for two-dimensional problemwith uni-directional velocity vx(y).

- A~ [A~y (B - Ii) + xl¢ = ¢ e 2 (5.56)

Figure 5.10 presents the problem geometry, discretization and boundary conditions which were specified according to expression (5.56). The value of the constantB defines the symmetry of the velocity field with respect to the coordinate y. IfB = 0.5 the velocity and temperature profiles are both symmetric.

Figure 5.11 shows the BEM results for the temperature ¢ along the faces y = 0or y = 1m for the case B = 0.5, compared to the analytical solution. In this case,the quadratic velocity field has its minimum at the extremes y = 0 and y = 1m andmaximum at y = 0.5m. It can be seen in the figure that the results are satisfactoryfor small velocity values but begin to deteriorate as the velocity increases.

Next, the value of B was considered as B = 0 to make the velocity profilesignificantly non-symmetric. This gives velocity values of V x = 0 at y = 0 andvx =Am/saty=lm.

Figures 5.12 to 5.14 compare the BEM and analytical solutions for this case.The same pattern of results can again be observed, i.e. very good agreement for lowvelocities and an increase in the error as the velocity increases.

Based on the previous and other similar results, Wrobel and DeFigueiredo [22Jsuggested that the present dual reciprocity formulation can be used for problems with


1500.00

r---,

~

"--/1000.00

Q)'--::J

-+-'o'-Q)

Q.

E 50000Q)

I-

••••• BEM ~A = ~1 ~00000 BEM A =••••• BEM A =-- Analytical

~0,00 ~ iii iii iii Iii iii iii iii iii ii' iii iii iii i i III iii iii i i 1111 iIi iii I

000 0.20 OAO 0.60 080 1.00 1.20

X (m)

Figure 5.11: Variation of temperature </> along faces y =0 and y = 1m for B = 0.5.

1500.00

r---,

~

"--/1000.00

Q)'-::J

••••• BEM (y = 01)00000 BEM (y = )-- Analytical

o'-Q)

Q.

E 500.00Q)

I-

1.000800.60

(m)OAO

X

0.200.00 +r-rTrTTTTTTTTT-rnrTrrTrTTTTTTTTlTT1rTrrTrTTTTTTTTTTr1

000

Figure 5.12: Variation of temperature </> along faces y =0 and y = 1m for B = O,A = 1.


1500.00

r---.:::,:::'-../ 100000

Q)L

::J+-'oLQ)

0..E 500.00Q)

~

00000 BEM (y = 01)..... BEM (y = )-- Analytical

1000.800.60

(m)0.40

X0.20

0.00 +r-rrr-rrr-rrr-rrr-rrr-rrrrrrrrrTTTTTTTTTTTTTTTTTTTTT"TTT"1

000

Figure 5.13: Variation of temperature ¢ along faces y = 0 and y = 1m for B = 0, A = 2.

1500.00

r---.:::,:::'-../100000

Q)L

::J+-'oLQ)

0..E 500.00Q)

~

0000 0 BEM (y = 01)••••• BEM (y = )-- Analytical

1.000.800.60

(m)0.40

X

0.20o.00 +r~-rT"T,.,,~rrr-rrrTTTTTT,.,,~rrr-rrr~-rT"T~~~

000

Figure 5.14: Variation of temperature ¢ along faces y = 0 and y = 1m for B = O,A = 4.


low velocity values (i. e. diffusion-dominated problems). For convection-dominatedproblems, domain discretization may be necessary but rather than treating the domain integral in the form given in Eq. (5.31), integration by parts can be performedso that the final domain integral involves values of the temperature <p rather than itscartesian derivatives [22].

5.5 Conclusions

This chapter has presented a formulation of the boundary element method for generaltwo-dimensional convection-diffusion problems employing the fundamental solution ofthe steady-state equation with constant coefficients. Transient problems and steadyproblems with variable velocity fields have been dealt with using a dual reciprocityapproximation. Thus, the numerical solution in all cases can be performed withboundary discretization only. Results have shown that the dual reciprocity approximation of the perturbation velocity field is adequate for diffusion-dominated problemsbut for convection-dominated processes domain discretization may still be necessary.

Acknowledgements

The second author would like to acknowledge the financial support of CNPq, Brazil.

References

1. Tanehill, J.C., Hyperbolic and hyperbolic-parabolic systems, in Handbook ofNumerical Heat Transfer, Wiley, New York, 1988.

2. Roache, P.J., Computational Fluid Dynamics, Hermosa Publishers, Albuquerque,New Mexico, USA, 1972.

3. Hughes, T.J .R. (Ed.), Finite Element Methods for Convection Dominated Flows,ASME AMD-Vol.34, New York, USA, 1979.

4. Ikeuchi, M. and Onishi, K., Boundary element solutions to steady convectivediffusion equations, Appl. Math. Modelling, Vol. 7, pp. 115-118, 1983.

5. Okamoto, N., Boundary element method for chemical reaction system in convective diffusion, in Numerical Methods in Laminar and Turbulent Flow IV,Pineridge Press, Swansea, UK, 1985.

6. DeFigueiredo, D.B. and Wrobel, L.C., A boundary element analysis of convectiveheat diffusion problems, in Advanced Computational Methods in Heat Transfer,Vol. 1, Computational Mechanics Publications, Southampton, and SpringerVerlag, Berlin, 1990.

7. Ikeuchi, M. and Onishi, K., Boundary elements in transient convective diffusion problems, in Boundary Elements V, Computational Mechanics Publications,Southampton, and Springer-Verlag, Berlin, 1983.


8. Tanaka, Y., Honma, T. and Kaji, I., Transient solutions of a three-dimensional convective diffusion equation using mixed boundary elements, in AdvancedBoundary Element Methods, Springer-Verlag, Berlin, 1987.

9. Okubo, A. and Karweit, M.J., Diffusion from a continuous source in a uniformshear flow, Limnology and Oceanography, Vol. 14, pp. 514-520, 1969.

10. Brebbia, C.A. and Skerget, P., Diffusion-convection problems using boundaryelements, Advances in Water Resources, Vol. 7, pp. 50-57, 1984.

11. Tanaka, Y., Honma, T. and Kaji, I., Mixed boundary element solution for threedimensional convection-diffusion problem with a velocity field, Appl. Math.Modelling, Vol. 11, pp. 402-410, 1987.

12. Taigbenu, A. and Liggett, J., An integral solution for the advection-diffusionequation, Water Resources Research, Vol. 22, pp. 1237-1246, 1986.

13. Aral, M.M. and Tang, Y., A boundary-only procedure for transient transportproblems with or without first-order chemical reaction, Appl. Math. Modelling,Vol. 13, pp. 130-137,1989.

14. Partridge, P.W., Brebbia, C.A. and Wrobel, L.C., The Dual Reciprocity Boundary Element Method, Computational Mechanics Publications, Southampton, andElsevier, London, 1991.

15. DeFigueiredo, D.B. and Wrobel, L.C., A boundary element analysis of transient convection-diffusion problems, in Boundary Elements XII, Vol. 1, Computational Mechanics Publications, Southampton, and Springer-Verlag, Berlin,1990.

16. Brebbia, C.A., Telles, J.C.F. and Wrobel, L.C., Boundary Element Techniques,Springer-Verlag, Berlin and New York, 1984.

17. Telles, J.C.F., A self-adaptive coordinate transformation for efficient numericalevaluation of general boundary element integrals, Int. J. Num. Meth. Engng,Vol. 24, pp. 959-973, 1987.

18. Wrobel, L.C., Telles, J.e.F. and Brebbia, C.A., A dual reciprocity boundaryelement formulation for axisymmetric diffusion problems, in Boundary ElementsVIII, Computational Mechanics Publications, Southampton, and Springer-Verlag,Berlin, 1986.

19. Jensen, O.K. and Finlayson, B.A., Solution of the convection-diffusion equationusing a moving coordinate system, in Finite Elements in Water Resources 2,Pentech Press, London, 1978.

20. Partridge, P.W. and Brebbia, C.A., Computer implementation of the BEM dualreciprocity method for the solution of general field equations, Commun. Appl.Numer. Methods, Vol. 6, pp. 83-92, 1990.


21. DeFigueiredo, D.B., Boundary Element Analysis of Convection-Diffusion Problems, Ph.D. Thesis, Wessex Institute of Technology, Southampton, UK, 1990.

22. Wrobel, L.C. and DeFigueiredo, D.B., A dual reciprocity boundary elementformulation for convection-diffusion problems with variable velocity fields, Engineering Analysis, Vol. 8, pp. 312-319, 1991.

Chapter 6

Solving Coupled Problems InvolvingConduction, Convection and ThermalRadiationA.J. NowakInstitute of Thermal Technology, Silesian Technical University,44-101 Gliwice, Konarskiego 22, Poland

6.1 Introduction

Background

Heat transfer problems in industry are usually of a very complex nature, frequentlyinvolving different energy-exchange mechanisms. The most common configuration ofan industrial plant is the concave wall forming an enclosure (e.g. industrial furnace,combustion chamber etc.). Within the enclosure, heat is generated due to the combustion process, or alternatively due to electric current flow, or others. The purpose ofthis heat generation process is to melt metal ore and heating slabs for rolling systemsand many others.

The fluid filling the enclosure usually has the highest temperature in the system.Heat is transferred from the fluid to the wall by radiation, conduction and/or convection. Simultaneously the radiation resulting from the internal surface causes a stronginteraction between all points lying on that face (if they can 'see' each other). Thenheat is conducted through the walls and is eventually absorbed by the cooling systemor is dispersed into the environment. This last stage is mostly due to convection butdepending on the temperature level, radiation can also play an important role.

The process described above is frequently associated with chemical reactions whichcan be sources of very strong additional nonlinearities in the problem.

For conduction and convection the transfer of energy between two points is proportional to the temperature difference of the points to approximately the first power.The transfer of energy by thermal radiation, however, is proportional to differences ofthe individual absolute temperature of the points each raised to the power in a rangeof about 4 [1]'[2]'[3].

From this basic conceptual difference between radiation, conduction and convection, it is evident that the importance of radiation intensifies at high absolutetemperature levels. However, even when the temperature level is not very high but


temperature differences are considerable, radiative heat fluxes are still important andradiation should be taken into account during the analysis. It is worth stressingthat radiation problems cannot be solved neglecting other modes of heat transfer,i.e. conduction and convection. Although the combination of the three modes ofheat transfer leads to complex thermal problems, their solution is of importance inengineering practice.

It is important to point out the following features inherent in thermal radiationboundary value problems:

i) the energy transport phenomenon is governed by an integro-differential equation.This equation for some cases can be simplified and becomes only an integral equation. However, radiation is always associated with convection and/or conductionthus different types of governing equations have to be tackled simultaneously.

ii) radiative properties depend on the temperature and/or radiation wavelength.This causes very strong nonlinearities in the set of equations describing heattransfer problems and special care should be taken during the iteration process.

iii) if fluid filling the enclosure is a participating medium containing triatomic gases(e.g. water vapor and carbon dioxide in combustion gases) radiation is in principle of band structure. Thus the gray body model can produce significant errorsand monochromatic analysis needs to be applied.

General Concepts

Methods of solving coupled problems of heat transfer described in textbooks on radiation (e.g. [1],[2]) are sometimes based on rather unacceptable simplifications. Usuallyconduction in solids is treated as a 1 - D problem or a rough approximation is introduced in the radiative portion of analysis (e.g. neglecting radiative interaction betweenpoints lying on the surface of enclosure or assuming non-participating medium).

More advanced techniques such as the Hotel's zoning method [3],[4], finite elements[5-9], finite differences [10],[11] and the Monte Carlo method [4] have also been applied.A survey of possible approaches can be found in reference [12]. Frequently authorshave used combinations of different techniques e.g. [11],[13],[14].

So far BEM has not yet been widely used for solving thermal radiation boundary value problems. Some early works, e.g. [15J refer to the processes when heat isexchanged by radiation between surface and the gray body having uniform knowntemperature. However, strong radiative interaction between elements having differenttemperatures on the surface itself have been neglected. Hence, thermal radiation hasbeen modelled as a nonlinear convection process, with a heat transfer coefficient depending on temperature to the power of three. Simultaneously convection (linear aswell as nonlinear) can take place on the surface under consideration. Thus the mostgeneral form of the boundary condition on the surface can be written as

q = he (T - TJ ) +8.h (t) (T - TJ) +C1- r (T4 - Tr4) =

=he (T - TJ) +8.h(t) (T - TJ) +hr (T - Tr)


where:

q heat flux density,

T temperature of the surface,

Tf temperature of the fluid exchanging heat with surface by convection,

Tr temperature of the gray body exchanging heat with surface by radiation,

flh temperature dependent heat transfer coefficient,

hr radiative heat transfer coefficient,

he heat transfer coefficient - constant value,

C1- r radiative exchange factor.

Application of the BEM for solving this kind of problem leads to a set of equationsof the type [15],[16]

HT=Gq+V

which after reordering can be solved iteratively.This approach, although introducing significant simplifications for the thermal

radiation process, is frequently accurate enough to analyze some configurations, suchas radiation from the external surface into the environment.

Since the approach mentioned above has now become a standard BEM formulationfor thermal problems with nonlinear boundary conditions, it will not be discussed indetail here. Instead, we are going to concentrate on more complex thermal radiationproblems.

The idea of using BEM for solving radiation heat transfer problems proposed in1983 by Bialecki, Nahlik and Nowak [17],[18] arises in quite a natural way from thenature of the governing equation. No transformation of the boundary value probleminto an integral equation is required because the equation already exists in a suitableform. The general concept is to treat radiation as a nonlinear boundary condition forthe equation of conduction. A special solution strategy based on the pre-eliminationof the 'linear unknowns' makes this approach very efficient.

The aim of this chapter is to present the state-of-art in BEM to cope with coupledheat transfer problems with emphasis on efficient algorithms for solving the resultingset of equations. The analysis starts with a very simple case of enclosure filled by anon-participating medium. Then the complexity of the mathematical description ofthe physical model, referring to Bialecki's works [24]'[26]' is increased in order to beable to represent more realistic problems.

It should be pointed out that the present formulation still needs to be analyzedfurther and that more research is required in this important topic.

6.2 Coupled Thermal Problems with Non-ParticipatingMedium

Assumptions

The steady state temperature field within a solid forming an enclosure is being considered (Fig. 6.1). The solid can be a multilayered medium but, in this chapter, only


. ., ...

.. ... '" ..~.. .. I

... -

'" ,I...... " . ~., '" .... .. _.. ' .-: ...... ...

solid : \.waLL..

,.' .

.... , " .. - .. ~.- <II

- , - .-,. \

_ .... ..l' __ ' ..

,.. -- .. .,"... _,.".,". -

... - .. f - ... _.... ".." _"' / :

' ...--- ..:.,' ,... .." '-'-...., " • ..,'" -! -- .. ..'- ,. ..

.. ." - ..

, .. \

,.

Figure 6.1: Solid wall forming an enclosure.

linear heat conduction, i.e. involving constant heat conductivity for each subregion,is analyzed.

A transparent gas having temperature Tg flows through the enclosure. Heat isexchanged between the gas and the walls by convection, and simultaneously eachpoint on the inner surface of the enclosure is irradiated by all others. All radiationproperties are assumed to be temperature independent and the walls are treated asgray emitters-absorbers of radiant energy. As a consequence absorptivity is equal toemissivity and only the latter value needs to be considered in the analysis.

On the external surface of the enclosure a linear boundary condition of the thirdkind is prescribed, although it is still possible to consider this surface as a radiatingone. This assumption was made to account for the lower level of temperature on thissurface.

If the solid forms an open cavity a fictitious surface should be introduced to makethe internal surface a closed one. This surface is assumed to be a blackbody havingthe temperature of the environment. Thus all rays reaching this surface are not reflected but absorbed into the environment.

Radiation Equation

Consider [17],[18] two infinitesimal surfaces dS(x) and dS(y), lying on the internalsurface of the enclosure (Fig. 6.2), where x and y stand for vectors coordinates. Foreach infinitesimal area we define a radiosity 'b' [1],[2],[3] which is a sum of self-emissionof the surface and the radiative energy reflected by this surface (see Fig. 6.3)


d.S(~ )

Figure 6.2: Configuration of two infinitesimal surfaces.

br

4tSeT

.... - - ...

(4 -e)h '

Figure 6.3: Division of the densityof incident radiant energy h intothe reflected and absorbed energy.

Figure 6.4: Energy balance forany 'under-skin' surface.


b(x) = t(x) (J T4(x) +[1- t(x)] h(x)

b(y) = t(y) (J T4(y) +[1- t(y)] h(y)

where:

t - emissivity of the surface,

(J 5.667 .1O-8 W/m 2K radiation constant

h - incident radiant energy flux density

(6.1 )

(6.2)

The density of radiant energy dh(y) which leaves surface dS(x) and reaches surfacedS(y) can be expressed as

dh(y) = b(x) K(x,y) dS(x) (6.3)

with the function K (x, y) depending solely on the geometrical configuration. Thisfunction is defined as

where:

!cos ¢J(x) cos ¢J(y) ( )

I 12 ax,y

7r x - YK(x,y) =

cos ¢J(x) cos ¢J(y) ( )

I Ia x, y

2 x - y

in 3- D

in 2- D

(6.4)

¢J(x)

I x - y Ia(x, y)

- angle between normal to the surface and the line connecting points x

and y

- distance between points x and y

is a two - valued function

a(x ) = {I when x. can be seen from y (6.5), y 0 otherwIse

The total radiant energy density which arrives at area dS(y) from any direction canbe obtained by integration of Eq. (6.3) over the whole internal surface Sj, i.e.

h(y) = l b(x) K(x,y) dS(x)ls.

Introducing (6.6) into Eq. (6.2) one ends up with the expression

(6.6)


b(y) = ((y) aT4(y)+

+ [1 - ((y)J h. b(x) K(x,y) dS(x) (6.7)

The next step is to eliminate the radiosity from Eq. (6.7). This can easily be done byincorporating an energy balance for any 'under-skin' surface represented by a dashedline as shown in Fig. 6.4. This gives,

(6.8)

(6.11 )

where: qr is the radiative energy flux density supplied to the surface by externalmeans.

Taking into account the definition of radiosity (Eq. (6.1)) one arrives at

1 - ((x)b(x)=aT4(x)- ((x) qr(x) (6.9)

1 - ((y)b(y) = a T4 (y) - ((y) qr (y) (6.10)

Introducing Eq. (6.9) and (6.10) into Eq. (6.7) one obtains after simple algebra manipulation the following integral equation:

a T4 (y) - h. a T4 (x) J{(x,y) dS(x) =

1 i 1 - ((x)= -() qr (y) - () qr(x) K(x,y) dS(x)( y s. (X

This equation is the relationship between radiative heat flux density and the fourthpower of temperature and is treated as a nonlinear boundary condition for the conduction equation.

Equation of Conduction Within the Wall

Under the assumptions postulated in the section on Assumptions, the steady-stateheat conduction within the walls forming an enclosure is described by the followingLaplace equation

(6.12)

On the internal surface of the enclosure, radiation and convection are taking placesimultaneously and this is expressed by the following boundary condition

aT- k an = hi (T - Tg ) + qr (6.13)

On the external surface of the enclosure a linear boundary condition of the third kindis prescribed

(6.14)


where Tf is the temperature of the fluid exchanging heat by convection with theexternal surface.

Obviously the boundary condition on the external surface can also be nonlinearand/or radiation can be taken into account. However, the linear condition was chosenon purpose to reduce the dimension of the set of nonlinear equation.

Using standard BEM formulation [19],[20] the solution of Eq. (6.12) can be expressed in terms of an integral equation, i.e.

where:

c(y) T(y) = is [T(x) q·(x,y) - q(x) u·(x,y)] dS(x) (6.15)

c(y) position dependent constant;(c = 0.5 for smooth boundary at point y)S total surface of the body(S = Sj +Se)

u· fundamental solution

q. -k8u· /8n heat flux analog

q -k8u/8n heat flux

The fundamental solution has the form [19],[20]

Discretization

I1

41r I x - y I

u· = _1 In I x _ y I21rk

in3 -D

in 2-D

(6.16)

In this section Eqs (6.11) and (6.15) are discretized. In order to do this the surfaceof the solid forming the enclosure is divided into a number of boundary elementsSj; j = 1, ... , N, N +1, ... , M. For the sake of simplicity boundary elements lying onthe radiating surface are numbered from 1 to N, whereas figures from N + 1 to Mare assigned to elements on the external surface.

Over each boundary element the temperature T, the conductive heat flux density qand the radiative heat flux density qr are described using locally defined interpolationfunctions [19],[20]. Although any order of approximation can be applied we start fromthe step function approach (i. e. constant elements).

Under such assumptions the equation of radiation (6.11) results in the followingset of equations

Al Tt = B 1 qr

with the entries of matrices being defined as

(6.17)

(6.18)

k,j = 1,2, ... ,N


N

-

NII

I~ I

C1 1

'- ... ---+--II

Ii

g1£D= + ~1-_:1 __

~

Figure 6.5: Structures of the sets of equation describing the coupled problems under consideration.

(6.19)

where:

bkj Kronecker's symbol

Ek E(Yk)

Providing the classical fundamental solution u*(x, y) is replaced by the functionK(x,y) which can be interpreted as a 'fundamental solution' for radiation problems,the matrices Al and B I can be created using a standard BEM computer code in theway the influence matrices can be determined. However it should be noticed that thesingularity in Eq. (6.18) and (6.19) is of l/r type for 2-D problems and 1/r2 type for3 - D problems. Hence, dealing with quadratic elements or splines the appropriateintegration quadratures should be applied (for constant and linear elements singularintegrals are equal to 0).


Matrices T 4 and qr (see Fig. 6.5) contain respectively the temperatures of radiating elements to the power of four and the radiative heat flux densities.

Discretization of Eq. (6.15) leads to the following set of Eqs [19],[20]:

HT=Gq (6.20)

in which vectors T and q consist of two parts. The first parts (from 1 to N) areassociated with unknowns on the internal surface whereas the second ones (from N +1to M) correspond to unknowns on the external surface. The elements of matrices Hand G are given as

(6.21a)

(6.21b)

k,j = 1,2, ... ,N,N + 1, ... ,M

Substitution of the boundary conditions (6.13) and (6.14) into Eq. (6.20) yieldsthe following system after reordering,

(6.22)

Vector F 1 results from multiplication of matrix G by a vector containing the nonhomogeneous terms of boundary conditions.

The forms of the Eqs (6.17) and (6.22) are schematically presented in Fig. 6.5.It should be noticed that discretization of Eq. (6.11) can also be done using the

Galerkin approach [26]. To accomplish this Eq. (6.11) is multiplied by a weightingfunction and integrated over the internal boundary. If the weighting function is simplyequal to unity within element and zero outside, discretization leads to the followingcoefficients for the matrices A] and B]

(6.23)

(6.24)

k,j=I,2, ... ,N

1 1 - fj i j' ) ()bkj = Okj - Sk - -- I\(x,y dS x dS(y)fk fj 5k 5J

In the above formulae it was assumed that the emissivity f is constant within eachelement and is represented by values of fk and fj, derived for the appropriate temperatures of the boundary elements.

The double integral in Eq. (6.23) and (6.24) can be readily recognized as theview (angle) factor. This function is extensively tabulated in the textbooks of thethermal radiation [1],[2],[3] and the analytical formulae for the integral can be easilyincorporated within the computer code.

Whatever formulation is taken into account, the intervisibility of elements hasto be checked. This stage, although very simple from the conceptual point of view,is in fact very cumbersome and time consuming in practice. Some algorithms fromcomputer graphics like those applying 'hidden lines' and clipping technique [21] have


been found to be very useful [22].

Solution Strategy

The two sets of Eqs (6.17) and (6.22) form a coupled nonlinear system. The unknownsin this system are temperatures on the internal and external surfaces of the enclosure and radiative fluxes on the internal surface. The number of equations coincideswith the number of unknowns and the most straightforward method of solution is toassemble one set of equations and apply any nonlinear equation solver, e.g. NewtonRaphson or an incremental loading algorithm. Unfortunately such approaches are notnumerically efficient.

Taking into account the structure of the system of equations under consideration a more effective method has been developed [18]. This method is based onthe preelimination technique and makes the final nonlinear system of equations considerably smaller than the original system. The preelimination is carried out usingGauss-Jordan algorithm [23].

The first step of the procedure is the elimination of the unknowns associated withthe external surface of the enclosure from the equations of conduction (6.22). As aconsequence this set of equations is split into two subsets; this process is schematicallypresented in Fig. 6.6.

(6.25)

C12 T i +T e = D 12 qr +F12 (6.26)

The first subset (6.25) contains only unknowns corresponding to the internal surface.Due to the size of the matrices in Eq. (6.26) the coefficients are written into backstorage for later calculations, thus allowing the standard memory to be available forother calculations. Once T i and qr are determined the remaining temperatures T e

can be obtained from the subset (6.26) by multiplication of matrices by appropriatevectors.

The second step of the algorithm is the elimination of the radiative heat fluxdensities qr from Eq. (6.25) and from Eq. (6.17). The latter equation is linear withrespect to qr' Again the application of the Gauss-Jordan algorithm results in the twofollowing equations (see Fig. 6.7)

(6.27)

(6.28)

Subtraction of Eq. (6.27) from (6.28) leads to the final set of nonlinear equationshaving the form

A2 Tt - C 2 T; + F 2 = 0 (6.29)

It should be noticed that the number of equations in (6.29) has been significantlyreduced in comparison with the original sets (6.17) and (6.22). Thus, the iteration


MN

I

II~ I

Ci l-~---I--II

HI

I~ C"1

I0I... -- -- ~f.

C12, I~I 0 -'1

\

II

8TDi

Li- F!=

""" ----

Te

~

8t· ])11 ~..+=

~-----

"'e Dt2 ~Z

¥,

N

~c~0= D~B+B

I~B+~=~B t ~Figure 6.6: Steps leading to the splitting up of the conduction equations.

f 157. Heat Trans ernt Methods InBoundary Eleme

I at I

~G"

I ~.. I

II

"

I u.t\I I

II

o

II

I u.N I+


loop is carried out much more effectively. Furthermore it is worth pointing out thatthe temperatures on the internal surfaces are the only unknowns in Eq. (6.29). Theseunknowns are limited by the highest and lowest temperatures Tg and Tj, respectively.Thus, the appropriate temperature distribution at the beginning of the iteration loopcan be easily guessed. Prediction of heat flux densities would obviously be much moredifficult.

Eq. (6.29) can be solved using any nonlinear equation solver but it has beenproved that the Newton-Raphson algorithm is very effective. Using this technique therequired number of iterations was 4 to 5 to achieve the accuracy of 10-5

- the residualvector of Eq. (6.29).

Temperatures T j on the internal surface are determined once the iteration loopis completed, thus allowing the radiative heat flux densities to be obtained from Eq.(6.27) or (6.28). One can then go back to Eq. (6.26) and calculate the temperaturesT e on the external surface of the enclosure. From the boundary condition (6.14) heatflux densities q on the external surface can also be obtained.

If needed, the temperatures and heat flux densities at any internal point of thewall forming the enclosure can be determined from Eq. (6.15).

6.3 Coupled Thermal Problems with Participating Medium

In section 6.2 a method of dealing with coupled thermal problems with non-participating medium has been discussed. Although that approach is very important from theconceptual point of view the assumption referring to the behaviour of the gas fillingthe enclosure is not realistic. In this section the gas in the enclosure is treated as aparticipating medium which is assumed to emit and absorb only radiant energy (i.e.scattering is neglected).

In the first part of this section the gas is treated as a gray medium with a constantabsorption coefficient. In the second part the analysis is extended to real medium(monochromatic analysis). However, we will still assume that the gas has constantknown temperature Tg . If the temperature of the gas is not uniform then conductionwithin the gas has to be considered during analysis.

The equation of radiation (6.21) has to be modified in order to take into accountthe radiative properties of the participating medium. The analysis of the conductionthrough the wall of the enclosure is identical to that discussed in the previous sectionand will not be mentioned further.

Radiation Equation - Gray Gas Model

Inside the enclosure filled by the nonparticipating medium the radiant energy leavingone element and reaching another depends on the radiative properties of elements andon their configuration. When the gas is a participating medium it affects the amountof radiant energy exchanged between elements.

In order to describe the transmittance of radiant energy from one element toanother through the participating medium one needs to define the intensity as anincident radiant energy flux density per unit elemental solid angle centered aroundthe particular direction 1>, i.e.


.:;."' .

Figure 6.8: Configuration of elements in order to define intensity.

i(y) = dh(y)cos 4>(y) dw

(6.30)

(6.31)

(6.32)

where: i( ) - intensity of radiationh( ) - incident density of radiant energy flux4>( ) - angle between normal to the surface and the direction from y to x

(see Fig. 6.8)w - solid angle

1

cos 4>(x) dS(x) . 3 - D

I 12 IIIx-y

dw =cos 4>( x) dS (x). D

1ll2-21 x - y I

Inserting Eq. (6.31) into Eq. (6.30) one arrives at an expression which is very similarto Eq. (6.3), i.e.

1

'( )cos 4>(x) cos 4>(y)dS( ). -Dl Y 1 12 x III 3x-y

dh(y) =

i(y) cos ~(x) cosl4>(y) dS(x) in 2 - Dx-y

The Bouguer-Lambert's law [2] describes an attenuation of radiant energy asproportional to the magnitude of the local intensity. This causes an exponential relationship between incident intensity at any point ~ (lying on the considered direction


Figure 6.9: Attenuation of intensity along the line x - y.

x - y, see Fig. 6.9) and the intensity which is observed at the point y. On the otherhand the gas filling the enclosure is also a source of radiation. Keeping in mind thataccording to Bouguer-Lambert's law part of the radiant energy emitted by the gasis instantaneously absorbed by itself, the total intensity at point y is expressed as asum, z.e.

where:

i(y) = i(~) exp(- a I ~ - y 1)+

+ ibg [1 - exp (-a I ~ - y I)] (6.33)

a - absorption coefficient of the gas,

Zbg intensity of radiation of the blackbody having the temperature of the gas

In order to specify the incident intensity i(O let us move the point ~ along theline x - y to the point x. Then the intensity i(~) has to be replaced by the directionalradiant energy i(x) emitted by the surface dS(x). From the relationship betweendirectional and hemispherical quantities one finds that,

i(x) = { b~) in3 - D

b(x)in 2 - D

2

i b, = !CTT 4

-g in3 -D1r

CTT4-g in 2 - D

2

(6.34)

(6.35)


Substitution of Eq. (6.33-6.35) into (6.32) results in

dh(y) = {b(x) exp (-a Ix - y I) ++ aT: {[I - exp (-a! x - y 1m I«x,y) dS(x) (6.36)

where the function K(x, y) has exactly the same form as given by formula (6.4).The rest of the analysis is very much the same as presented in section 6.2. In

tegrating Eq. (6.36) over the internal boundary and introducing the energy balance(6.8) one ends up with the following integral equation of radiation [24],

a T 4 (y) - l a T 4 (x) K(x,y)exp(-a Ix - y I) dS(x)-ls,

-laT:I«x,y)[l-exp(-a!x-yl)1 dS(x) =ls,

= - r l-(E\X) qr (x) I«x,y)exp(-a Ix-v I) dS(x)+lSi E X

1+ E(y) qr (y) (6.37)

Eq. (6.37) is the extended form of Eq. (6.11). Assuming that the absorption coefficientis null (a = 0, transparent gas) one immediately generates Eq. (6.11). Hence, equation(6.37) can be discretized in a similar manner as before leading to the following system,

Al Tt = B1 qr +R 1 (6.38)

where the vector R 1 comes from the integral representing the energy emitted byradiating gas. The elements of the matrices are expressed by

akj = Dkj a - a r J«(x,Yk)exp(-a I x - Yk I) dS(x) (6.39)ls)

J1 - E(X) ,

- () !\(x,Yk)exp(-a 1x - Yk I) dS(x)sJ E X

(6.40)

N

.L:J I«X,Yk) [1- exp(-a I x - Yk 1)1 dS(x) (6.41)j=l SJ

Ek=E(Yk) k,j=1,2, ... ,N

The solution strategy is as presented in the section on Solution Strategy. Thefinal set of nonlinear equations takes the same form as Eq. (6.29)

(6.42)


with the vector E2 defined by

(6.43)

After elimination of the radiative heat flux densities qr from Eq. (6.38) the vector R1

becomes equal to R2 •

Numerical Example

The analysis described in the previous section and section 6.2 was used to solve acoupled problem of heat transfer in a semi-infinite channel (2 - D problem) as depicted in Fig. (6.10a) [17],[18],[241. The gas flowing through the channel has temperature Tg ( = 1300K) whereas the temperature of the surrounding environment isTJ(= 290K). The heat transfer coefficient for the internal surface is hi (= 30W / m 2K)and for the external surface is he (= 20W/m2 K). The heat conductivity of the solidwas assumed to have a constant value k equal to 1.8W/ mK.

To minimize the number of integrals to be calculated over the boundary elements,a Green's function satisfying homogeneous boundary conditions of the second kindon axes x and y was applied instead of the usual fundamental solution [251. As aconsequence only the surfaces FG, GH and the internal surface have to be discretized(see Fig. 6.10b).

Two cases were examined, first the gas was treated as a transparent nonparticipating medium (a = 0) and then as a gray medium with constant absorption coefficient.A value 0.8 was assigned to the emissivity t of the walls forming the enclosure.

The final set of nonlinear equations was solved by the Newton-Raphson algorithm.Some representative results are presented in Fig. 6.11 and 6.12.

Fig. 6.11 is a plot of temperature distribution along the perimeter of the channel.Curve 1 corresponds to the case when radiation within the enclosure is neglected,so the heat is exchanged only by convection and conduction. Curve 2 is associatedwith coupled problems in which the gas is treated as a transparent medium. As thedifferences of temperature along the internal surfaces between these two models aresignificant one can conclude that all modes of the heat transfer need to be taken intoaccount during the analysis.

Curve 3 represents temperature fields on the boundary for the emitting-absorbinggas having a = 0.71/m. Again differences between models 2 and 3 show that mathematical description of a heat transfer problem should be formulated very carefullyemphasizing the importance of assuming the correct representation of the mediumfilling the enclosure.

What should also be noticed is the fact that when radiation is taken into accountthe temperature distribution along the boundary is much more uniform than when itis neglected (see curve 1 and 2 and/or 3). Obviously the level of surface temperatureis higher when the gas emits energy (see curve 2 and 3).

Radiative heat flux densities for certain values of the absorption coefficient areshown in Fig. 6.12. It can be seen that the radiative heat flux densities follow similarvariations of temperature along the boundary. The greater the absorption coefficientof the gas, the greater the radiative heat fluxes are on the boundary (negative value


().)

.~"'" . .. .,..: -: .......". -; ..... ....... -: -..• '. ~~'''': .I. "-=.,'---..~:=---•.-.-='::....-.•......:..+-t'l-

I•• , ,' ...., (), . . , ·, . ·, ... , • 't ~,

~ I c) ~, ". .

... '" ... '" ...... : ..... • 1' • • , '" tC).. ,'" • ... I' ~ • • .. ....... • ~ ..

. \"'., -" . ti~ .. : ~;"~.;:• .... "'... ~ .~ •• ' "'" .. " ••, , 11

0.3 0.9

b)he =2.0 W/m 2K

r G ~~

A C ~0 ~

" C~ E ]) ~

II

ka 1.8 W/mK -&::~

+- U

1=0

Figure 6.10: 2 - D channel filled by a gas at temperature Tg . a) dimensions of the channelb) mesh and boundary conditions.


00

00

00

'OOK

fJOO

- 1l, I

@'nV

@J -4

Ok trnJ

r\' It. ,. I I \~. 1-0-. 1 ~

0 \ J 5

Ir ~ l I~

rN~ r-...~ I

~) 1\ WrJ ,

I--0 4

1\ fu

0 d

110

-120

130

400

f G H

Figure 6.11: Temperature distribution along the perimeter of the channel. 1 - linear problem;radiation neglected 2 - transparent gas; a=O. 3 - radiating gas; a=O.7 11m.


tt-----T+--+---+---it----1

D C B A E

Figure 6.12: Distribution of radiative heat flux density along internal surface of the channel.1 - transparent gas; a=O. 2 - radiating gas; a=O.7 11m 3 - radiating gas; a=10. 11m.

means that heat is transferred from the gas to the solid).

Monochromatic Analysis

As has been shown in the previous section, the assumptions relating to the gas fillingthe enclosure strongly affect the results. The gray body model becomes then unacceptable in many practical cases, especially when the gas contains triatomic compoundsand its radiative properties depend on the radiation wave length and have a bandstructure (e.g. see Fig. 6.13). In this case, monochromatic analysis should be applied.

Here due to the necessity of the monochromatic analysis, the radiation equationhas to be modified. One can still assume a uniform temperature of the gas fillingthe enclosure but now an infinitesimal wavelength band d>' positioned around anywavelength>' needs to be considered. All radiative properties vary with >., so thatin order to get Eg. (6.37) one needs one more integration over the whole spectrumof wavelengths. Keeping in mind that the total intensity has to be replaced by theintegral of blackbody emissive power eb one obtains [26]


10

S~nd

designation.~.

~m

,~15

; 10. 4.I

: r Q,4I ,

4,8-

..- 4. 3

/~2. 7

Figure 6.13: Spectrum of absorption bands for CO2 gas at 830](,10 atm, and for pathlength gas of 38.8 em.

100

eb(.\, T) d.\-

-100 r eb(.\,T) J{(x,y) exp(-a,\ Ix - y I) dS(x) d.\o 151

-100 r eb(,\,Tg ) I«x,y)[I-exp(-a>. jx-y I)] dS(x)d.\=o lSI

100 i 1 - t,\(x)- () qr,>.(.\,X) J{(x,y) exp(-a>.1 x - y I) dS(x) d.\

o 5; (>. x

100 1+ 0 ((y) qr,,\(.\, y) d.\

where eb(.\, T) is given by Plank's law

(6.44)

1](6.45)

with constants C1

C2

= 3.74126. 1O-6 Wm 2

1.4387· 1O-2m !J{

The radiative properties (I' and a) are determined experimentally. Hence, thecommon assumption is to treat these functions as constants within certain bands ofthe whole spectrum of wavelength. Under such condition Eg. (6.45) can be easilyintegrated with respect to the wavelength providing that the spectrum is properlydivided into a finite number of bands


boAi = Ai - Ai-l i = 1,2, ... ,J

where limits are respectively equal to

After integration one arrives at the following integral equation

(6.46)

<Pi (y,T) - hi <pi(x,T) I«x,y) exp(-ai I x -y I) dS(x)

- l <Pi (x,Tg ) I«x,y) [1 - exp(-ai I x - y I)] dS(x) =is.

- r 1- t~x) qT,i(X) I«x,y) exp(-ai 1x - y I) dS(x)is. <Pi X

1+ -(-) qT,i (y) (6.47)

toi Y

where the blackbody emission with i-th band <Pi and radiative heat flux density transferred within i-th band qT,i are defined as

(6.48)

(6.49)

The function <Pi corresponds to the quantity known as blackbody fraction within thewavelength interval 0 - A [2]

fA 1

FO->'T = 0 aT4 eb(A, T) dA (6.50)

Taking into account the additive property of the integral operator, function <Pi can beexpressed as

<pi(T) = (FO->'iT - FO->'i_1 T) a~4 (6.51)

The blackbody fraction, according to [27],[28], is a function which is satisfactorilyapproximated by the following formulae

15 00 exp( -mv)FO->'T = "4 L 4 {[(m v +3) m v +

1r m=! m

6] m v +6} for v ~ 2 (6.52a)


VB )

13,305,600for v < 2 (6.52b)

C2where v = AT

The discretization of Eq. (6.47) can be performed as previously described (section onDiscretization) and leads to the matrix equation of type (6.38). However, temperatures to the fourth power and radiative heat flux densities have to be replaced by bandquantities <Pi and qr,i. Moreover instead of one, J sets of equations are now obtained,l.e.

i = 1,2, ... , J (6.53)

This system of equations should be coupled with the equation of conduction (6.22).Obviously, the vector qr in Eq. (6.22) is the sum of all vectors qr,i, i.e.

J

qr = L qr,ii=l

(6.54)

From the numerical point of view this simply means that preelimination of radiationEq. (6.53) must be carried out J times (for each band separately)

J

qr = L (A; <Pi - R;)i=l

(6.55)

Then the appropriate matrices can be collected to build up the final system of nonlinear equations

J J

L (A~ <pd - C2 T i + (F2 - L R~) = 0 (6.56)i=l i=l

This set of equation with very severe nonlinearity has to be solved iteratively for theunknown temperatures on the internal surfaces.

6.4 Concluding Remarks

An efficient approach to cope with coupled heat conduction, convection and thermalradiation in enclosures has been presented in this chapter. The main advantage of themethod is that it leads into a boundary-only formulation of the problem. Hence, usingstandard boundary element codes the effect of radiation can be easily calculated. Thetechnique is equivalent to introducing into the computer code a 'new fundamentalsolution'.

Due to the special numerical treatment of coupled conduction and the radiationsystem of equations the final set of nonlinear equations has been significantly reduced.As a result the present technique saves considerable computing time.


The spectral band analysis and shadow option discussed in this chapter permits tosolve a wide range of practical thermal problems, although the participating mediumwas assumed to have uniform temperature. Future research should incorporate theconduction for the fluid filling the enclosure. Nonlinear heat conduction in walls aswell as transient problems should also be analyzed in more detail.

Acknowledgment

The author's sincere gratitude and thanks go to Dr. C.A. Brebbia of the Computational Mechanics Institute. His valuable suggestions have stimulated the author towrite the above chapter.The financial support of The British Council is also gratefully acknowledged.Finally, thanks are due the many colleagues who have directly or indirectly contributedto the quality of this work.

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20. C.A. Brebbia and J. Dominguez, Boundary Elements - An Introductory Course,Compo Mech. Publications, Me Graw-Hill Book Co., 1988.

21. J. Encarnacao and E.G. Schlechtendahl, Computer Aided Design, SpringerVerlag, 1983.

22. P.H.L. Groenenboom, C.A. Brebbia and J.J. De Jong, New developments inengineering applications of boundary elements in the transient wave propagation,Engineering Analysis, vol. 3,1987, pp.201-207.

23. A. Bjorck and G. Dahlquist, Numerical Methods, Prentice-Hall, 1974.


24. R. Bialecki, Applying BEM to calculations of tern perature fields in bodies containing radiating enclosures, Proc. 7th Intern. BEM Conference (C.A. Brebbiaand G. Maier editors), Como, Italy, Springer-Verlag, Berlin and New York, 1985,vol. 1, pp 2-35 to 2-49.

25. R. Bialecki, A.J. Nowak, R. Nahlik, Applying Green's function for the semi-planewith boundary conditions of the third kind in BEM, Proc. 7th Intern. BEMConference (C.A. Brebbia and G. Maier editors), Como, Italy, Springer-Verlag,Berlin and New York, 1985, vol. 1, pp 2-99 to 2-105.

26. R. Bialecki, Radiative heat transfer in cavities. BEM formulation, Proc. ofthe 10th BEM Conference, Southampton, vol 2, Heat Transfer Fluid Flow andElectric Applications (C.A. Brebbia - Editor), Springer-Verlag, Berlin, 1988, pp.246 - 256.

27. M. Pivovonsky and M.R. Nagel, Tables of Blackbody Radiation Functions, TheMacmillan Company, NY, 1961.

28. J.A. Wiebelt, Engineering Radiation Heat Transfer, Holt, Rinehart and Winston,Inc., NY,1966.


Notation

a

b

c(y)

e

Fa-AT

h

b.h(T)

J

I«x, y)

M

n

N

q

q*

u*

x,y

I x - y Ia(x,y)

8kj

¢>(x)

absorption coefficient

radiosity

function of the internal angle the boundary S makes at point y

radiative interchange factor

constants in Planck's spectral energy distribution

emIssIve power

fraction of total blackbody intensity or emissive power lying inspectral range 0 - A

incident radiant energy flux density

constant heat transfer coefficient

temperature dependent heat transfer coefficient

intensity

number of bands in whole spectrum of wavelength

integrand in view (angle) factor

number of unknowns in the problem

outward normal direction

number of radiating surface elements

heat flux density

heat flux analog associated with fundamental solution

surface area

surface area of j-th boundary element

temperature of the fluid exchanging heat with surface by convection

temperature of gas filling the enclosure

temperature

fundamental solution

points or vector coordinates

distance between points x and y

function defining intervisibility of points

Kronecker's symbol

emissivity

azimuthal angle measured from normal of surface

w

faO/an


wavelength

band of wavelength

Stefan-Boltzmann constant

solid angle

point or vector coordinate

outward normal derivative

Subscriptsb blackbody

e

r

external surface

internal surface

radiative

spectrally dependent

MatricesAll matrices are marked by bold characters

Chapter 7

Advanced Thermoelastic Analysis

v. Sladek, J. SladekInstitute of Construction and A rchitecture, Slovak Academyof Sciences, 842 20 Bratislava, Czechoslovakia

7.1 Introduction

The boundary element method (BEM) is now a versatile and powerful tool of computational mechanics which has become a popular alternative to the well-establishedfinite element method (FEM). Recently, some authors indicated that the presence ofany domain integrals undermines most advantages of the BEM. We believe that thisis not true if the domain integral contains known quantities (such as body sources orinitial values). The actual sources of advantages of the BEM formulation are not determined by the question of if it is necessary to discretize the domain or not. The BEMformulation for the solution of a boundary value problem exhibits pure boundary character if the solution at any internal point can be expressed in terms of the boundaryintegrals of relevant quantities and the domain integrals of known body sources andinitial values, if any. In other words, the solution can be expressed at an internal pointwithout the need to know the solution at any other internal point. If the boundaryvalues of the relevant quantities were known exactly and the integrations were performed with absolute accuracy, the integral representation would present the exactsolution of the boundary value problem. That is why in the BEM solution, extremeemphasis is put on the accuracy of the approximations of boundary values and computation of boundary integrals. Since in the domain discretization techniques (suchas FEM, finite difference method, collocation methods) the unknowns at all nodalpoints are computed simultaneously, the computational error is accumulated. Thus,the pure boundary character of the BEM formulation for solution of boundary valueproblems implies advantages of the BEM in comparison with domain discretizationtechniques such as the possibility of higher accuracy and the reduction of dimensionof the problem (the unknowns, which are to be computed all at once, are localized onthe boundary alone). The reduction of dimension can result in the improvement ofeconomy of computation (simpler preparation of input data and processing of results,less computational effort). The most important restriction on the BEM formulation


is given by the fact that the fundamental solutions are not available in closed form forevery partial differential equation. Nevertheless, the BEM formulation has also beendeveloped successfully for solution of boundary value problems in nonhomogeneousmedia or inelastic problems (for which the fundamental solutions are not available)in connection with iterative schemes of solution. On the whole the BEM formulationdoes not exhibit pure boundary character in that approach, but it does within eachiterative step.

The physical foundation of the theory of elasticity is the thermodynamics of irreversible processes. The general theory of thermoelasticity includes such branches asthe theory of thermal stresses, quasi-static coupled thermoelasticity, quasi-static uncoupled thermoelasticity, stationary thermoelasticity, elastodynamics, and elastostatics. Moreover, the theory of heat conduction is also a part of thermoelasticity whichis a more general theory of continuum. A great contribution to the boundary integrallformulation of the solution of boundary value problems in thermoelasticity was givenby Nowacki and Ignaczak in the 1960's. The results of their works are collected inthe monograph [1]. They derived the integral representations of displacements andtemperature, the fundamental displacements and temperatures as well as the singular boundary integral equations (BIE). Kernel singularities were not discussed anda numerical implementation was not attempted. Nevertheless, their work is a goodtheoretical basis for the extension of the BEM to solution of the boundary value problems of thermoelasticity. Tanaka and Tanaka [24],[25] presented the boundary elementformulation for the time-domain coupled thermoelastic problem. Although they introduced the discretization formally, the kernel functions were not discussed and nonumerical results were included. More recently, Tanaka et ai. [26] chose to implementthe BEM formulation for 2 - d problems of quasi-static uncoupled thermoelasticity.Note that all the mentioned formulations for uncoupled thermoelasticity include thedomain integral of temperature gradients. Since the thermal unknowns are computedindependently of the mechanical ones in uncoupled thermoelasticity, the advantagesof the pure boundary formulation are wiped out only partially.

A great deal of work has been devoted to the question of removal of domain integrals from the BEM formulation. Rizzo and Shippy [27] transformed the volumeintegral of temperature gradients into surface integrals of the boundary temperatureand heat flux in 3 - d problems of stationary thermoelasticity. The pure boundaryformulation for any problem of uncoupled thermoelasticity was derived by Sladekand Sladek [6]. Masinda [28] transformed the volume integral in 3 - d quasi-staticuncoupled thermoelasticity into surface integrals and the free term of the gradientof temperature. Due to the last term the formulation is not quite pure boundaryformulation. The kernels in the surface integrals of his formulation are r- 2 and r- 3

singular in contrast to the singularities rO and r- 1 in [6],[2]. This is the cost whichshould be paid for the time independence of the kernels found in Ref. [28]. The pureboundary formulation for quasi-static problems of uncoupled thermoelasticity wasalso derived more recently by Sharp and Crouch [29],[30]. Using the thermoelasticpotential, Koizumi et ai. [31] derived the indirect BEM formulation for 3 - d problemsof stationary thermoelasticity which is free of volume integrals. The solution for displacements and stresses are composed of the particular solution (thermal part whichis expressed in terms of thermoelastic potential) and the solution of the homogeneous


differential equation. In order to satisfy the boundary conditions, one should use theBIE derived from both the integral representations of displacements and stresses. Theformer is nonsingular, while the latter includes the integral which should be consideredin the sense of Cauchy principal value (CPV). Koizumi et al. [31] aim to avoid thesingularities by using a fictitious boundary. Thermal stress analysis for 2 - d problemsof quasi-static uncoupled thermoelasticity was performed by Ochiai and Sekiya [32]using the thermoelastic potential and BEM. Henry and Banerjee [33] proposed theboundary only integral formulation for 2 - d and 3 - d problems of thermoelasticityin which the solution is decomposed into particular and complementary parts. Thecomplementary solutions are given by the integral representations of displacementsand stresses corresponding to the solution of the homogeneous differential equation.The particular solution is expressed in terms of the thermoelastic potential which iseither known a priori for simple temperature distributions or determined by the collocation method for complex temperature distributions. Internal nodes should be usedin addition to the boundary ones for a better representation of the particular solution.The number of additional points is dictated by the complexity of the temperature distribution. Since temperature should be computed for a pattern of interior nodes, theformulation does not exhibit the pure boundary character any more. Thus, the aimto remove the domain integral of known body heat sources (or initial temperaturesin unsteady problems) led to the loss of actual pure boundary character of the BEMformulation (e.g. Ref. [2-7]).

In order to utilize the advantages of the BEM offered by the pure boundary character as much as possible, one should aim to remove or decrease the singularities of theintegral kernels before the numerical integration. This question, however, is not discussed in literature so widely as the removal of domain integrals. Using the rigid bodymotion idea in elastostatics, one can arrive at the nonsingular BIE. This idea can beapplied only to steady problems in thermoelasticity. The nonsingular BIE, however,can be derived for any problem of linear thermoelasticity (Balas et al. [2],[3]). Moreover, the regularized integral representations of stresses and temperature gradientshave been derived for any class of two- and three-dimensional thermoelastic problems(Ref. [2-7]). Due to the regularization, the r- d singularities are decreased analyticallyto r 1- d singularities. The regularized integral representations can be used successfullyin the computation of the secondary fields (stresses and temperature gradients) andalso at internal points which lie near the boundary (the 'boundary layer effect' isremoved) (Sladek and Sladek [20],[21],[15]). Another application of the regularizedintegral representations of the secondary fields has been found in the derivation ofthe unique formulation of the BEM solution of crack problems in general (Sladek andSladek [3],[5],[2],[16]). Seeking to decrease r- 3 singularity in the integral representation of internal stresses for 3 - d problems of quasi-static uncoupled thermoelasticity,Chaudouet [34] transformed the thermal volume integral with a kernel varying as r- 3

to two surface integrals of temperature and heat flux (with kernels varying as r- 2

and r- 1) and the volume integral of the time derivative of temperature (with a kernelvarying as r- 1

). Despite the r- 1 singularity in the volume integral (this can be removed by using spherical coordinates on singular elements), the formulation does notexhibit pure boundary character. Moreover, the boundary integral of displacementscontains the kernel with r- 3 singularity (nonregularized integral representation). All


the integral kernels in that formulation are time independent. In the past, the nonstationary problems of thermoelasticity have been solved using the BEM formulation inthe Laplace transform domain even if the time-dependent fundamental solutions wereavailable (e.g. Ref. [16],[35],[36]). The transform domain formulation is sensitive tothe selection of values of the transform parameter, it requires a numerical inversion ofthe transform, and is strictly limited to linear problems. Therefore, time-domain formulations have become more popular recently. Tanaka et al. [37] have analyzed 3 - dproblems of quasi-static uncoupled thermoelasticity using the BEM formulation withthe thermal volume integral. Then, the kernels in the BIE for mechanical unknownsare independent of time. In the numerical solution of the heat conduction equationthey employed constant time interpolation during time steps. Constant time interpolation was also employed by Dargush and Banerjee [38] in the numerical solution ofquasi-static coupled thermoelastic plane problems using the pure boundary integralformulation. The BIE are written in a singular form (the CPY integrals are included)and it is not quite clear how the 'rigid body' technique was employed in these nonstationary problems. The nonsingular BIE for nonstationary thermoelasticity have beenpresented by the present authors [3],[2],[16],[18].

In addition to the BEM formulation in the Laplace transform domain and thetime-domain formulation for solution of nonstationary thermoelastic problems thedual reciprocity BEM [39] was also developed. In this approach the partial differentialequations of hyperbolic and/or parabolic type are converted to a set of ordinarydifferential equations which describe the time evolution of unknowns at boundarynodes.

In this chapter we start briefly with classification of thermoelastic problems. Then,we define the fundamental solutions which are presented explicitly for the Laplacetransforms in general coupled thermoelasticity because the inverse transformationcannot be performed analytically in general. We also present the time-dependentfundamental solutions when these are available. The temperature and displacementfields are expressed in terms of the boundary integrals of temperature, heat flux,displacements, and tractions. The domain integrals of the known densities of bodyforces and heat sources as well as the prescribed initial values are included, if theseare different from zero. The pure boundary formulation is given for both the Laplacetransforms and the time-dependent fields. The BIE are written in an advanced formwithout any singular integrals. The integral representations of the secondary fields(temperature gradients and stresses) are presented in both the regularized and nonregularized forms, which enables us to develop an efficient method for computation ofthe secondary fields also at internal points which lie very near the boundary. Althoughall the integrands in the present nonsingular BIE are bounded elsewhere, the integralkernels are singular. The removal of all these singularities is discussed in the sectionon numerical implementation. Advanced time-marching schemes are presented forsolution of the BIE for heat conduction and quasi-static problems of uncoupled thermoelasticity. Finally, we also present the BEM formulation for solution of stationaryproblems in media with temperature dependent Young's modulus and coefficient ofthermal expansion. The use of the advanced BEM formulations presented in thischapter is illustrated by numerical examples.


7.2 Governing Equations

Consider a perfectly elastic, homogeneous and isotropic continuum. The equilibriumof forces and the conservation of energy together with the constitutive laws yield thefollowing system of governing equations within the linear theory of thermoelasticity(e.g. Nowacki [11, Balas et al. [2])

1 '. Qo kk - -0 - Wk k = --'K ' Ko

(7.1 )

(7.2)

in which Ui(X, t) and 0(x, t) are the displacement and temperature fields, respectively.Strictly speaking 0 is the deviation of the current temperature T from the value Tocorresponding to the undeformed state (fij = 0). The symbols Xi(x, t) and Q(x, t)denote the densities of the body force vector and heat sources. The physical meaningof the coefficients employed in Eqs. (7.1) and (7.2) is given as follows.

Parameter X 1 f I/K3 - d problemsplane strain A 10 fo 1/Ko

plane stress 2 v 2Jta~ f!::£V ..! +~foa1l 1- v o l-v " -v

with

10 = (2Ji +3A)a,

Ji, A Lame constants (under isothermic conditions)

a coefficient of linear thermal expansion

Ao coefficient of heat conduction

p mass density

c, specific heat (under constant strains)

K o thermal diffusivity

v Poisson's ratio

Then, the tensor of elastic constants for a homogeneous and isotropic continuum isgiven by

Cijkl = X bijbk/ +Ji(bikbj/ +bi/bjk) (7.3)

The constitutive law giving the dependence of stresses on deformations and temperature is known as the Duhamel-Neumann relation


(Tjj = 2flfjj +(Xfkk - 18 )Ojj

which can be rewritten into a more compact form

(7.4)

(7.5)

Substituting the Cauchy infinitesimal strain tensor into (7.5), one can express tractionsat the boundary points as

(7.6)

where nj(11) are the components of the unit outward normal vector at a boundarypoint 11 E S.

The heat flux is proportional to the normal derivative of temperature

a8q(11,t) = -a(11,t) = nj(11)8 j(11,t)

n '(7.7)

The solution of a boundary value problem in linear thermoelasticity satisfies the governing equations and, furthermore, it is specified by boundary and initial conditions.There is just one quantity prescribed of each pair (Uj, tj) for i = 1, ... , d (d is thedimension of the problem) and one quantity of the pair (8, q) at any boundary pointby the boundary conditions. In the case of the heat conduction equation anotherphysically meaningful boundary condition exists, in which the heat flux is expressedin terms of the unknown boundary temperature 8 as

(7.8)

where 8 a is the ambient temperature and h is the ratio of the heat - transfer coefficientand the heat conduction coefficient. The boundary conditions are prescribed at anypoint on the boundary and at any time interval.

On the other hand, we prescribe the displacements, velocities and temperaturesat any point of the body, x E(V uS), at the initial time to (usually to = 0) by theinitial conditions

Uj(X,O+) = uf(x), ~~j(x,o+) = vf(x), 8(x,0+) = 8°(x) (7.9)

The time variable can be eliminated by using the Laplace transform technique, whenthe equations of motion become

(7.10)

(7.11)

where the superimposed bar denotes the Laplace transform, p is the transform parameter and (3; = piK ; the quantities B j and H are defined as


Bi(x,p) = Xi(X,p) +p[pui(x) +vi(x)]

H(x,p) = Q(x,p) + eO(x) + Wk,k(X) (7.12)

In the case of forced harmonic oscillations the amplitudes are governed by the equations

(7.13)

e~kk +iw(0* /,; +Wk,k) = -Q* /';0 (7.14)

which differ from (7.10) and (7.11) formally in parameters (p +-+ -iw, where w is thefrequency of oscillations).

Note that one can distinguish five physically different classes of problems withinthe thermoelasticity. The formulations corresponding to the special classes of problems can be obtained from that of general thermoelasticity by the physical specifications shown in the following scheme

ClassicalElastodynamics

Thermalstresses

Q = 0, e = -f';Uk,kX ~ Xs = X +,f';

Xi = O,fo = 0

General thermoelasticity

Quasi - static p ~ 0Thermoelasticity

Quasi - static Xi = 0, fO = 0UncoupledThermoelasticity p ~ 0

Stationary

Thermoelasticity

According to this scheme one can easily deduce the governing equations for each classof problems (Balas et ai. [2]).

7.3 Fundamental Solutions

According to the definition of fundamental solutions, they represent the physical response of an infinite solid to point sources. Two groups of fundamental solutions canbe distinguished with respect to their origin. The first of these is the fundamentaldisplacements and temperatures {Ui(x - Y, t - T), T(I x - y I, t - Tn generated bythe point sources {H (x, t) = tS(x - y )tS( t - T), Bj (x, t) = O}. The Laplace transformsof these fundamental solutions satisfy the equations


- - 2--p,Ui,kk + (X +p,)Uk,ki - pp Ui - ,T,i = 0

- 2- - 1T kk - 13 T - f.pUkk = --8(x - y) (7.15)to, /\'0

The point sources {H(x, t) = 0, B~j)(x, t) = 8ij8(x - y)8(t - Tn are the generators ofthe fundamental displacements and temperatures {Uij(x - y, t - T), 8 j(x - y, t - Tn,the Laplace transforms of which obey the equations

p,Uij,kk + (X + p,)Ukj,ki - pp2Uij - ,8j,i = -8ij8(x - y)

- 2- -8 j,kk - 130 8 j - f.pUkj,k = 0 (7.16)

The Fourier transform can be employed successfully in the derivation of the fundamental solutions (Balas et al. [2]). There is a reduction in the number of independentsolutions, since

(7.17)

In view of (7.6), we may write the fundamental tractions Ti associated with thedisplacements Ui and temperature T as

T i(1],y,p) = cijk/nj(1])8; Ud1] - y,p) - ,ni(1])T(I1] - Y I,p) (7.18)

where 8; = 8/81]/.

Similarly, we obtain the fundamental tractions Tik corresponding to the displacementsUik and temperature 8 k. Making use of Eqs. (7.6) and (7.17) we have

Other important kernels are defined as the normal derivatives

F(1J,y,p) = ni(1J)a;T(I1J - Y I,p)

Zk(1],y,P) = ni(1])8; Uk(1] - y,p) (7.20)

For more detailed derivation of the fundamental solutions we refer the reader to Ref.[2]. Herein, we present only the final results for the fundamental solutions as follows

- U6T(r,p) = d '2(d - 1)1l"Kor -2

- r,ini(X) -F(x,y,p) = - 2(d _ 1)1l"K

ord - 1 Ug

- 1 - -Zk(X,Y,p) = 2(d _ 1)1l"K

ord- 2 (U 58ik +U7r,i r ,k)ni(X)


+2pU7r,ir,jnj(x)}

- 1 - -Uij(r,p) = 2(d _ 1)1rprd-2 (U 1bij +U2r,i r,j)

OkUij(r,p) = 2(d _ 1~1rprd-l {(U2+U3 )r,kbij +U2(r,jb;k + r,ibjk)+

+[U4 - U3 - (d +2)U2]r,ir,jr,k}

Tik(X,y,p) = 2(d _ ~)1rrd-l {[ (;U 4 +2U2) - usr2] r,kni(X) +

+(2U2+U3)r,ink(X) + [(2U2+U3)bik+

+2 (U4 - U3 - (d +2)U2) r,ir,k] r,jnj(x)}

where d = 2 and 3 for two- and three-dimensional problems, respectively, and

(7.21 )

1/2 --ri = Xi - Vi, r = (riri) , r,i = r;/r, Us = (pUs

The coefficients Ua(r) for a = 1,2, ... ,9 are collected in Appendix A.Due to the complexity of the dependence of the fundamental solutions given by

(7.21) on the Laplace transform parameter p, it is impossible to invert the Laplacetransform analytically in general coupled thermoelasticity. The time-dependent fundamental solutions are available in closed form only for particular classes of thermoelastic problems. The explicit expressions for the fundamental temperature anddisplacements are presented in Appendix A. The other kernels can be obtained fromthose by differentiation.

For the sake of brevity we do not present the explicit expressions of the fundamental solutions for all the classes of thermoelastic problems. Note that these can befound elsewhere (Balas et al. [2]). Most of the fundamental solutions are singular suchas r ~ O. Since the singular behaviour of the fundamental solutions for each class ofthermoelastic problems is the same as for the case of stationary thermoelasticity, wepresent the stationary fundamental solutions. These are

U6 r .us -r 'n(x)T(r) = 2(d _ 1)1rKord-2' Ui(r) = 2(d -1')1rKord-3' F(x,y) = 2(d _ ;);Kord-1

Zk(X,y) = 2(d _ 1{1rKord- 2 (USbik - ; r,ir,k) ni(x)

Ti(x,y) = 2(d _ 1{1rKord-2 {[(2p + Xd)Us - x-;- - ,U6] ni(x) - pmr,ir,jnj(x)}


1Uij(r) = 4(d -1)1rJl(X +2Jl)rd-2 [(X +3Jl) U6hij + (X + Jl) r,ir,j]

18kUij(r) = 4(d _ 1)1rJl(X +2Jl)rd-1 {-(X +3Jl)r,khij + (X + Jl)(r,jhik + r,ihjk-

-dr,ir,jr,k) }

(7.22)

where the coefficients Us and U6 are defined for 3 - d problems as

Us = m/2, U6 = 1

and for 2 - d problems as

Us = -m(1 +2Inr)/4, U6 = -lnr .

Now the asymptotic formulae can be written as

T(r,p) = T(r) +0(1), Ti(x,y,p) = Ti(x,y) +0(1)

for both the 2 - d and 3 - d problems, while the asymptotic expressions for the otherkernels should be distinguished:(i) for 3 - d problems

F(x,y,p) = F(x,y) +0(1), Tik(X,y,p) = Tik(X,y) +0(1) (7.24)

(ii) for 2 - d problems

Ui(r,p) = Ui(r) +0(1),

F(x,y,p) = F(x,y) +O(rlnr), Tik(X,y,p) = Tik(X,y) +O(rlnr) (7.25)

Finally, we note that the fundamental solutions for the case of harmonic oscillationscan be obtained from the Laplace transforms by changing p to -iw.


7.4 Integral Representations of the Temperature and theDisplacement Fields. Boundary Integral Equations.

Consider two systems of causes I = {Xi(x,t),Q(x,t), boundary conditions} andl' = {XI(x, t),Q'(x, t), boundary conditions} in the linear theory of thermoelasticity. The responses to these causes are denoted as R = {Ui(X, t), 0(x, t)} andR' = {u;(x,t),0'(x,t)}, respectively.

From the Duhamel-Neumann relations

a;j = 21llij + (Xlkk - ,0')8ij

one can deduce the local identity

aij lij - aiiij = -,(0lkk - 0 'lkk )

Equations of motion can be written as

- +B 2-(1ij,j i = PP Ui

-, B' 2-'(1 ... + .= PP u·tJ,] I t

(7.26)

(7.27)

(7.28)

(7.29)

Integrating Eq. (7.27) over the region V and using the Gauss divergence theorem aswell as Eqs. (7.28), we obtain the integral identity

r(Bi~ - B:Ui)dV + r(ti~ - t:ui)ds + I r(0u',. k - 0'Uk,k)dV = 0Jv Js Jv '

Similarly, starting from the equations

e·· - Q20 - tPUkk = -HI'",11 1-'0 1 0

-, 2-' -,0,ii - (30 0 - tPU~,k = -H 1"'0

one can derive another integral identity

(7.30)

r -,- --, r(-a0' -,a0) r --/ -,J)H 0 - H0 )dV + "'0 Js

0 an - 0 an ds - "'otp J)0Uk,k - 0 uk,k)dV = 0

(7.31)Equations (7.29) and (7.31) are the integral representations of the reciprocity theoremin the coupled thermoelasticity. Eliminating the last term in these equations, oneobtains the integral identity


[r --, -,- r(-,ae -ae') ]=, J)H e - H e)dV +KoJs e an - e an dS (7.32)

If we assume the causes I' to be the point heat source in an infinite medium, i.e.I' = {X: = 0, Q' = b(X - y)}, then R = {Ui(x - y,p), 1'(1 x - Y I,p)}. Let thecauses 1 = {B i , H, boundary conditions} be unspecified in a finite body. Then, from(7.32) we have

Ll(y)e(y,p) = Koi [q(17,p)1'(I17 - y I,p) - e(17,p)F(17,y,P)] dST/

where

{I y E V

Ll(y)= 0: yf/ (VuS)

Y(y,p) = i [H(X,P)1'(! x - y I,p) - K~P Bi(x,p)Ui(x - y,P)] dVx

Similarly, taking into account the fact that the response to the point causes l' ={~(k) = bikb(X - y), H' = O} in an infinite solid is R = {Uik(X - y, p), ek(x - y, p) =KotPUk(x - y,p)}, we obtain from Eq. (7.32) the integral representation of displace-

'Yments

Ll(Y)Uk(Y,P) = is [Ii(17,P)Uik (17 - y,p) - Ui(17,p)1'ik (17,Y,P)] dST/ +

+Kois [e(17,p)Zd17,Y,P) - Q(17,P)Uk(17 - y,p)] dST/ +Vk(y,p) (7.34)

where

Vk(y,p) = i [Bi(x,p)Uik(X - y,p) - H(x,p)Uk(x - y,p)] dVx

The fields e(y,p) and Uk(Y,p) given by Eqs (7.33) and (7.34) are the response R to thecauses 1 = {Bi , H, boundary conditions} in the finite body. It is seen from the integralrepresentations (7.33) and (7.34) that the solution of boundary value problems at anyinternal point Y E V is expressed in the integral form in terms of the boundary values{ui(17,P),ti(17,p),e(17,p),Q(17,P)} and the body sources {H(x,p),Bi(x,p)}. In otherwords, in order to have the solution at an internal point, it is not necessary to know thesolution at any other internal point (the densities of body sources are prescribed). Onehalf of the boundary quantities (Ui, Ii, e, Q) at any boundary point are prescribed byboundary conditions and the others are unknown. Thus, the solution of the boundary


value problem in a body with the boundary S is reduced to the solution of a setof equations for computation of unknowns on the boundary S. These equations canbe obtained from the integral representations (7.33) and (7.34) by taking the limitapproach Y -+ ( E S and subsequent substitution of prescribed boundary quantitiesinto these integral relations.

Taking into account the definition of the kernels F(fJ,Y) and Tik(fJ,Y) as well asthe governing equations for the fundamental solutions, one can derive the auxiliaryidentities (see Sladek et al. [3])

Kol F( fJ, y) dST/ = E - 6.(y)

l Tik(fJ, y) dST/ = [E - 6.(y)]6ik

in whichE _ { 0, for internal boundary value problems

- 1, for external boundary value problems

In view of Eq. (7.35), we may write

(7.35)

(7.36)

(7.37)

6.(y)8(y,p) +Kol 8(fJ,p)F(fJ,Y,p) dS = E8(y,p) +Kol [8(fJ,p)F(fJ,Y,p)

-8(y,p)F(fJ, y)] dST/ = 8(y,p) {E +Kol [F(fJ, y,p) - F(fJ,y)] dST/} +

+Kol [8(fJ,p) - 8(y,p)] F(fJ,y,p) dST/ (7.38)

According to the asymptotic behaviour of the integral kernels, the difference (F - F)is bounded elsewhere and the boundary integral of this quantity is continuous acrossthe boundary S. Assuming the temperature 8(y,p) to be Holder continuous, the lastintegral on the r.h.s. of Eq. (7.38) is continuous across the boundary too. Furthermore,taking into account the fact that the kernels 1\, Ui, and T contain at most weaksingularities (r-1 in three dimensions and 1m in two dimensions), we can perform thelimit y -+ ( E S in Eq. (7.33) with the result

8((,p) {E +Kois [F(fJ,(,p) - F(fJ,O] dST/} +

+Kol [8(fJ,p) - 8((,p)] F(fJ,(,p)dST/-

-Ko rq(fJ,p)T(lfJ - ( I,p) dS" + KoEp 1[li(fJ,p)Ui(fJ - (,p)-is , s

-ui(fJ,p)Ti(fJ,(,p)] dS" = Y((,p) (7.39)

Analogously, making use of Eq. (7.36), we may write


+ Is [Ui(1J,p)Tik(1J,Y,p) - Ui(y,p)Tik (1J,Y)] dStj = ui(y,p){E8ik +

+ Is [Tid1J,Y,p) - Tik (1J,Y)] dStj} +

+1[Ui(1J,P) -Ui(y,p)]Tik(1J,Y,p) dStj (7.40)

Substitution of Eq. (7.40) into (7.34) yields the integral representation of displacements in which each integral is continuous across the boundary. Hence, in the limitY --+ ( E S, we obtain the boundary integral relation

Ui((,P) {E6ik + .Is [Tik(1J,(,p) - Tik (1J,O] dStj} +

+ .Is [Ui(1J,P) - Ui((,p)] T ik (1J, (,p) dStj - .Is Ii(1J,p)Uik (1J - (,p) dStj +

+K0.ls [q(1J,p)Uk(1J - (,p) - 8(1J,P)Zk(1J,(,P)] dStj = Vd('p) (7.41)

Inserting the prescribed boundary conditions into the integral relations (7.39) and(7.41), one obtains the boundary integral equations (BIE) for computation of unknowns on the boundary. Note that these BIE are written in a form which is free ofCauchy principal value integrals.

It is no problem to perform the Laplace inversion formally in the integral representations and BIB. Thus we obtain the time-domain formulation for:

integral representation of the temperature field

~(y)8(y, t) = l {Ko.Is [q (1J, r)T(I1J - Y I, t - r) - 8(1J, r)F(1J, Y, t - r)] dStj+

Kof1[. .]+- ui(1J,r)Ti(1J,y,t-r) - ti(1J,r)Ui(1J-y,t-r) dStj}dr+Y(y,t) (7.42)'Y S

where

Y(y,t) = l i [Q(x,t)T(1 x - y I,t - r) - K;f Xi(x,r)Ui(x - y, t - r)] dVxdr +

r{[ ] KofP [ fP+ lv 8°(x) + futk(X) T(I x - y I, t) - --:;- u~(x)8t2 +

+v~(x) ~] Ui(x - y, t)} dVx


integral representation of the displacement field

~(Y)Uk(Y, t) =l {is [ti(fl, r)Uik(fl- Y, t - r) - ui(fl, r)Tik(fl,Y, t - r)] dS')+

+Ko is [8(fl, r)Zk(fl, Y, t - r) - q(fl, r)Uk(fl- Y, t - r)] dS')} dr +Vk(y, t) (7.43)

where

Vk(y, t) = l fv [Xi(x, r)Uik(x - Y, t - r) - Q(x, r)Uk(x - Y, t - r)] dV')dr +

+ fv {p [ur(X)~ +Vf(X)] Uik(X - Y, t) - [8°(x) +wj)x)] Uk(x - Y, t)}

boundary integral equations

E8((, t) +Ko l {8((, r) is [F(fl, (, t - r) - F(fl, ()c5(t - r - 0)] dS') +

+ is [8(fl, t) - 8((, r)] F(fl, (, t - r)dS') - is q(fl, r)T(lfl- ( I, t - r)dS') +

+.:. r[ti(fl, r)Ui(fl- (, t - r) - ui(fl, r)1';(fl, (, t - r)] dS')}dr = Y((, t) (7.44)I Js

and

EUk((, t) +l {Ui( (, r) is [Tik(fl, (, t - r) - Tik(fl, ()c5( t - r - 0)] dS')+

+ is [ui(fl, r) - Ui((, r)] Tik (1], (, t - r)dS') - is ti(1], r)Uik (1] - (, t - r)dS') +

+Ko is [q(fl, r)Uk(fl- (, t - r) - 8(fl, r)Zk(fl, (, t - r)] dS')} dr = Vk((, tX7.45)

Note that the time-dependent fundamental solutions are available only for specialclasses of thermoelastic problems, but not in general. We do not present the BEMformulations (integral representations and BIB) for physically special classes of thermoelastic problems since such formulations can be obtained simply from those of thegeneral thermoelasticity by omitting relevant terms (Balas et al. [2]).

It is worth mentioning the BEM formulation for uncoupled thermoelasticity problems, in which the temperature gradients play the role of body forces (Sladek andSladek [4-7]). Note that such a formulation requires the computation of the temperature field in the interior of the body too. Although the computation of temperaturesat internal points is not mixed with the computation of unknowns on the boundary, the BEM formulation with temperature gradients partially deteriorates the pureboundary character of the present BEM formulation.


7.5 Integral Representations of the Temperature Gradientsand Stresses

It is known that designers are often interested in the computation of the secondaryfields (stresses and temperature gradients) in addition to the primary fields (displacements and temperature fields). Apparently, if the primary fields are known,the secondary fields can be computed by differentiation. Differentiating the integralrepresentations of primary fields, we obtain the integral representations of secondaryfields in terms of boundary values (8, q, Ui, t i ) and the densities of heat sources andbody forces. Then the kernels in the integral representations of secondary fields contain singularities of the type r- d

• Due to these strong singularities the numericalcomputation of secondary fields fails at internal points very close to the boundary.Fortunately, this 'boundary layer effect' has been removed by the development of theregularization procedure (Sladek and Sladek [4-7]).

The nonregularized integral representation of the gradient of temperature fieldresulting from (7.33) is given by

~(y)8,j(Y,p) = 1>:0 L[e(1J, p)F,j(1J, y,p) - q(1J,p)T,j(1J - y,p)] dST/ +

+I>:ofp f [ti(1J,P)Ui,j(1J - y,p) - ui(1J,p)I';,j{1J,y,p)] dST/ +Yj(y,p) (7.46), isin which

T,j(r,p) = ojT(r,p) = - 2(d _ l~'~I>:ord-t Us

- - 1 - -Ui,j(r,p) = ojUj(r,p) = 2(d _1)trl>:ord-2(U56ij +U7r,jr,j)

where

_ {m/p.i - A~) [Ai(1 +Atr)e->'1r - A~(l +A2r)e->'2r] ,d = 3Ug =

mr/Pi - AD [A1!<d Atr) - A~J(t(A2r)1, d = 2

In the statical limit p --t 0, when At, A2 --t 0, we obtain

(7.47)

(7.48)

(7.49)


It can be seen that the following asymptotic relation is valid

d=3_ { 0(1),Ti,j(x,y,p) - Ti,j(x,y) =

O(rlnr), d = 2

where ri = Xi - Yi and

(7.50)

(7.51 )

pmTi,j(x,y) = 2(d _ 1)1rll:ord-1 [r,jni(x) - r,inj(x)-

-(hij - dr,ir,j)r,knk(x)]

The r- d singularity is contained only in the kernel F,j , since

- 8 - -F)x,y,p) = - Byj F(x,y,p) = ni(x)8i8jT(1 X - Y I,p)

According to the regularization formula (B. 1) we may write

is 6(1],p)ni(1])8;8jT(I1] - y l,p)dS'1 = is Dji6(1],p)8;T(I1] - y l,p)dS'1+

+ is 6(1],p)nj(1])V'2T(I1] - y l,p)dS'1 (7.52)

From Eq. (7.15) we have

- 1 -V' 2T(r,p) = --5(r) + ll1(r,p)

11: 0

where

ll1(r,p) = f3;T(r,p) + tpUk,k(r,p) =

= 2(d _ 1;1rll:ord-2 [13;U6 + tp(dU5 + U7)]

Introducing the notation

Hj(1],y,p) =T,j(1] - y,p)Dji

we may rewrite Eq. (7.52) as

(7.53)

(7.54)

is 6(1],p)F,j(1],y,p)dS'1 = is [nj(1])Il1(I1] - y I,p) +Hj(1],y,p)] 6(1],p)dS'1

(7.55)Substituting Eq. (7.55) into (7.46) yields the regularized integral representation of thetemperature gradients

~(y)6,j(Y,p) = 11: 0 .Is {[nj(1])Il1(I1] - y I,p) +Hj (1], y, p)] 6(1],p) -

-q(1],p)T,j(1] - y,p)}dS7J + lI:otp [ [li(1],p)U i,j(1] - y,p)-I 1s

-ui(1],p)Ti)1],y,p)] dS'1 +Yj(y,p) (7.56)

(7.57)


Note that the integral kernels in (7.56) are only r 1- d singular at most.A similar procedure can also be repeated in the case of stresses. Inserting (7.34)

into the Duhamel-Neumann relations, we obtain the nonregularized integral representation of stresses

~(y) [a/p(y,p) + ')'h/p8(y,p)] = C/pjr {- is ti(11,p)a~Uij(11- y,p)dST)+

+11:0 is [q(11,P)~Uj(11- y,p) - 8(11,p)ni(11)~a:Uj(11- y,p)] dST)

-lI:otp is ui(11,p)ni(11)~Uj(11- y,p)dST) +

+Ciskt is Ui(11, p)n.(11)a:a~u kj (11 - y, p)dST) } +W/p(y, p)

where

with

+(2U2+U3 )(r,A/ +r,/hij ) +2 [U4 - U3 - (d +2)U2]r,ir,jr,d (7.59)

F/j(r,p) == C/jkmamUk(r,p) = 2(d _ 1;7rIl:ord-2 {hj/ [(xd + 2J.l)Us + XU7 ] +

(7.60)

The last integral in Eq. (7.57) contains a kernel which is r- d singular. According toEq. (7.16) and (7.17), we may write

, ,- 2--Cisktat as Ukj(11- y,p) = -hijh(11 - y) + pp Uij (11- y,p) + lI:otpff; Uj(11- y,p)

Hence and from (B.l), we have for y rlS

Ci.kt is Ui(11, p)ns(11 )f); a~ Ukj( 11 - y, p)dST) =

=Ciskt is iJrs Ui(11,p)a: Ukj (11- y,p)dST) +

+ is Ui(11,P)nr(11) [Pp2Uij (11- y,p) +11: 0 tpa: Uj(11- y,p)] dST) (7.61)

Finally, substituting (7.61) into (7.57) yields the regularized integral representationof stresses


~(y) [lflj(Y,P) + ,h/j8(y,p)] = is [ti(11,P) + ,ni(11)8(11,P)] D/ji (11- y,p)dSf/ +

+ is [Pp2 E1ji(11,Y,P) + Ko ipG1ji(11,Y,P) +T/ji (11,Y,P)] ui(11,y,p)dSf/ +

+Ko is [Q(11,p)F/ j(11- y,p) - 8(11,P)H,A11,P)] dSf/ +W,j(y,p) (7.62)

in which

- - 1 X[-E/ji(x,y,p) == c/jkT nr(x)Uik(X - y,p) = 2(d -1)7lTd- 2 {; U1ni(X)+

+U2r,ir,knk(X)] hj/ +U1 [8i/nj(x) +hijn/(x)] +

+U2r,i [r,lnj(x) + r,jn/(x)]} (7.63)

= 2(d _ ~)7rTd-l {hjlX (;U4 - U3) r,kDki + (XU4+2jLU2)(r,/Dji +r,jD/;) +

+jL(2U2+U3)r,k(8i/Djk +hijD/k) +

and

+2jL [U4 - U3 - (d + 2)U2]r,ir,k(r,/Djk + r,jD1k )}

- -, -H/j(x,y,p) == C/jkr ni(x)8r8iUk(X - y,p) + -ni(x)D/ji(x - y,p) =

Ko

= 1 {[(2jLU 7 _ XUg - X, U4 - 2,U2)8/-2(d - 1)7rK ord- 1 jL )

-2 ((d +2)(jLU7 -,U2) + JLUg + ,(U4 - U3)) r,jr,dr,ini(x) +

+ [2jLU7 -,(2U2 +U3)] lr,tnj(x) +r,jnl(x)]}

® {hj/ [(x(1- d) - 2jL) USni(X) + XU 7 (r,ir,knk(x) - ni(x))]

-2jLU7r,jr,/ni(x) + JLUs [8 ilnj(x) +8ijn/(x)] +

+jLU7r,i [r,lnj(x) + r,jn/(x)]}

(7.64)

(7.65)

(7.66)


It is seen that both the nonregularized and regularized integral representations ofsecondary fields preserve the pure boundary character of the present BEM formulationfor solution of thermoelastic problems. The difference is only in the strength of thesingularity contained. In the nonregularized scheme the kernels are r- d singular, whilein the regularized scheme only r1- d singular.

The regularized integral representations of the time-dependent secondary fieldscan be obtained by carrying out the Laplace inversion (7.56) and (7.62). Hence,

(7.67)

and

~(Y)[(7/j(Y, t) +,bj/8(y, t)] = l {.Is [ti(71, r) + ,ni(71)8(71, r)] 12)

12)D/ji (T/ - y, t - r)dST/ + .Is [pui(71, r)E/ji (71,y, t - r)+

+KoWi(71, r )G/ji (71, y, t - r) +T/ ji (71, y, t - r )Ui( 71, r)] dST/ +

+K0.ls [q(71, r)F/j(71- y, t - r) - 8(71, r)H/j(71,y, t - r)] dST/} dr +W/j(y, t)

(7.68)

where

1t i K EYj(y,t) = [_0 Xi(x,r)Ui,j(x-y,t-r)-os,

-Q(x, r)T,j(x - y, t - r)]dVxdr + fv (~EP [Ui(X) ~22 + vf(x) %t] 12)

12)Ui ,j(x - y, t) - [8°(x) +W%,k(X)] T,j(x - y, t)}dVx (7.69)

W/j{y, t) = l fv [Xi(x, r)D/ji(x - y, t - r) +Q(x, r)F/j(x - y, t - r)] dVxdr +

+p fv [Uf,k(X)F/jik(X - y, t) +vf(x)D/ji(x - y, t)] dVx +

(7.70)


The kernel F1jik is defined as

and the other time-dependent kernels are expressed in terms of the fundamental solutions T(r, t), Ui(r, t) and Uik(r, t) according to the definitions resulting from thosefor the Laplace transforms given by Eqs (7.59), (7.60) and (7.63-7.66) by performingthe Laplace inversion.

The regularized integral representations can also be successfully employed in thederivation of the boundary integrodifferential equations (by taking the internal pointto the boundary) which appear to be helpful in unique BEM formulation of solutionof crack problems. These questions are discussed elsewhere (Ref. [21,[3],[5],[7]). Notethat the boundary integrodifferential equations can be made free of Cauchy principalvalue integrals.

7.6 Stress Tensor and Temperature Gradient on Boundary

It is known in elastostatics, that the stress tensor components at any boundary point (can be expressed in terms of the tractions and tangential derivatives of displacementsat this point. Let us assume to know the temperature 0((, t), displacements Ui( (, t)and tractions ti((, t) at any boundary point (. If we denote (;ij = (Tij +,8ij 0 , then

(;ij(X,t) = CijkIUk,I(X,t) (7.71)

Making use of the analogy with elastostatical constitutive relations, the boundaryvalues of (;ij can be expressed immediately in terms of tk and the tangential derivativesof Uk. Hence,

(7.72)

where, in three dimensions

Aijk (() = [1 ~ )TiTj +PiPj) +ninj] nk + (Tinj +Tjni)Tk +

+(Pinj +Pjni)Pk

(7.73)


and in two dimensions

(7.74)

Note, that the unit vectors (T,p,n) in (7.73) and (T,n) in (7.74) are taken at point (.For plane stress problems v should be replaced by v/(l +v), with v being the Poissonratio.

Similarly, we may easily express the temperature gradient on the boundary interms of the heat flux and tangential derivatives of temperature. Thus, in threedimensions

ae aee,i((, t) = ni(()q((, t) +Ti(() aT ((, t) +Pi(() ap ((, t)

and in two dimensions

(7.75)

aee,i( (, t) = ni(()q( (, t) +Ti( () aT ((, t) (7.76)

It can be seen that Eqs (7.72), (7.75), and (7.76) have a local character, and donot require any integration. Once the secondary fields on the boundary and at theinternal points sufficiently close to the boundary are known one can compute thesefields at internal points very close to the boundary by using interpolation. In orderto find the points at which the results computed at internal points are still valid, onemight employ both the nonregularized and regularized integral representations withthe former being the comparative scheme. When the results received by both theseschemes start to deviate significantly as the internal point approaches the boundary,the results obtained at this point by the regularized scheme can be adopted as reasonable.

7.7 Numerical Solution

Basically the numerical schemes of solution do not depend on the physical natureof the problem considered and they can be developed according to the classificationof the partial differential equations into elliptic, hyperbolic and parabolic types. Itshould be noted that these schemes have been developed technically elsewhere (seee.g. Brebbia et ai. [9]). Therefore we shall not discuss them from the technical pointof view. We want only to document the nonsingular character of the BIE developedin the previous sections. However, we also recall the pure boundary character of thepresent formulation and the regularization of the integral representations of the secondary fields.


Laplace Transform Domain Formulation

As mentioned above, the time-dependent fundamental solutions are not available inclosed form in the case of general thermoelasticity. One of the approaches to solvingsuch problems is the use of the BEM formulation in the Laplace transform domain inconnection with the numerical inversion of the Laplace transform. Note that this approach can also be applied to any class of time-dependent thermoelastic problem andto problems of stationary thermoelasticity. Now, the governing equations are givenby a system of partial differential equations of elliptic type. The numerical schemefor the solution of these equations closely resembles that of the Navier equations inelastostatics or the Laplace equation in potential problems. The numerical implementation of the BIE as well as that of the integral representations of the primary andsecondary fields consists of:(i) The BEM discretisation and modelling of the boundary

MS = U Sq,

q = 1

n

11; Isq= L 11~q Na(Oa=!

(7.77)

where 11~q are the coordinates of the a-th nodal point on the element Sq and Na(oare the interpolation polynomials of the intrinsic coordinate ~ (~ should be replacedby two independent coordinates 6, ~2 in three dimensions).(ii) Polynomial interpolations of the boundary values of displacements, tractions, temperatures, and heat fluxes

n

9(11) Isq= L9(11aq

)Na(o

a=!

where 9 stands for any of the boundary quantities (u;,I;, e, q).(iii) Integration over the boundary elements(iv) Creation of the discretized BIE and integral representations.

(7.78)

(7.79)

The discretization of the boundary and the polynomial interpolations within theboundary elements are standard procedures. Note that we used largely quadraticapproximations of both the internal coordinates and boundary quantities within thecontinuous boundary elements. In two dimensions we used quadratic elements withtwo nodes at the ends points and one node in the middle of the element. In threedimensions we employed both 6-node triangular elements and 8-node quadrilateralelements. Having introduced the discretization and interpolation, each boundaryintegral in the BIE and integral representations is replaced by a sum. In general,

h9(11)J((11,()dST/ = ~9(11aq) jU Na(OJ((11 ISq,()Gq(Od~

where Gq(O is the Jacobian of the transformation from the global coordinates 11; tothe local one ~, and I, u are the integration bounds.

As mentioned above each integral in the BIE given by Eqs (7.39) and (7.41) existsin the ordinary sense. Nevertheless, the kernels F and T;k are r 1- d singular and the


kernels T, Ti , Uik and Zk are logarithmically singular in two dimensional problems,while in three dimensions they are r-1 singular. The logarithmic singularity canbe removed by using the transformation r = 8

2, when 1m dr = 4s/ns ds. In the

case of linear interpolation the logarithmic singularities can be treated directly byanalytical integration. The other singularities are cancelled out by introducing specialtransformations for intrinsic coordinates on singular elements. In two dimensions thistransformation is given by the shift of the origin into the singular node, while inthree dimensions the intrinsic coordinates on singular elements are transformed topolar coordinates with the origin in the singular node (see Balas et al. [2]). Thus, allthe integrands of the integrals over each element (including singular) in the BIE andintegral representations are bounded. Consequently, all the prescribed integrationscan be carried out sufficiently accurately by using the regular Gaussian quadrature.

In view of Eq. (7.79) the discretized BIE gives a system of linear algebraic equations for computation of unknowns at nodal points. The matrix of coefficients of thissystem is given by the integrals of the type shown on the right-hand side of Eq. (7.79).The prescribed nodal values of boundary quantities (Ui, Ii, 0, q) contribute to theright-hand side vector of the system of the discretized BIE.

The discretized integral representations are given by the linear combination of thenodal values of the boundary quantities (Ui, Ii, 0, q) with the expansion coefficientsbeing the integrals of the type shown on the right-hand side of Eq. (7.79). Note that inthe case of integral representations all the boundary elements are nonsingular and thecomputation of boundary integrals is free of the manipulations made on the singularelements in the case of the BIE. Now, the accuracy of the numerical computation of theboundary integrals depends on the distance from the internal point y, (at which thefield is evaluated) to the integration point 1](0. In order to eliminate the 'boundarylayer effect', one can interpolate between the results computed at internal points by theintegral representations and the results obtained on the boundary. In this approach itis important to determine the shortest distance of the internal points so that the resultscomputed by the integral representations could be considered to be reasonable. In thecase of secondary fields the 'shortest distance' can be determined by the simultaneoususe of both the regularized and nonregularized integral representations as mentionedat the end of Section 7.6. Since the order of the strongest singularity of the kernelscontained in the integral representations of primary fields is the same as that ofsecondary fields, one can utilize this consistency in the extension of the use of thedetermined 'shortest distance' also to primary fields.

For the numerical evaluation of inverse Laplace transforms there exist variousalgorithms. A comprehensive comparative study by Narayanan and Beskos [10] revealed that for elastodynamic problems, out of eight well known algorithms the bestis that of Durbin [11], which is based on the sine and cosine transforms. Note thatcomplex arithmetic is required since the values of the Laplace transform parameterare complex in the Durbin algorithm.

Now we present some illustrative numerical examples. In the first and secondexamples it is sufficient to solve the BIE for boundary value problems in two dimensional stationary thermoelasticity.


2

U,· 0tz=Oq=O

R,

Figure 7.1.

T=oT =10'C

EXAMPLE 1In this example we present the comparison of the numerically computed radial distributions of the temperature and stresses with analytical results in a tube subjectto a steady temperature gradient. By nature, the problem is one-dimensional. Inthe numerical solution we have considered a quarter of the cross-section of the tube,as shown in Fig. 7.1. The boundary contour is divided into 20 quadratic elementswith 40 nodal points. The radial distribution of the temperature and the angularcomponent of the stress tensor are given (Boley and Weiner [12]) as

(7.80)

(7.81 )

where

The computational results shown in Tables 7.1 and 7.2 correspond to the numericalvalues employed: R1 = 8, R2 = 10, E = 2 x lOsMPa, v = 0.25, and C\' = 1.67 X

lO-S(OC).The relatively large error in the computation of the stresses at r = 9 is due to the

small value of (J<p<p at this point in comparison with (J<p<p at the other points.

EXAMPLE 2Consider an infinitely long edge-cracked bar with rectangular cross-section. The baris subjected to a linear temperature gradient as shown in Fig. 7.2.The surface of the crack is assumed to be free from tractions and thermally insulated.Owing to the symmetry of the problem with respect to the crack plane, it is sufficientto analyse only half of the cross-section. The boundary conditions on the imaginary


Table 7.1.

r 8[OC] %errorexact BIE

8.5 2.7168 2.72 0.119 5.278 5.279 0.019.5 7.701 7.699 0.03

Table 7.2.

r O'cpcp[M Pal %errorexact BIE

8 23.917 23.90 0.078.5 10.805 10.82 0.149 -0.817 -0.868 8.549.5 -11.221 -11.06 1.4310 -20.616 -20.68 0.31

L

o~~ ~--=:!'_--I

-10'C

1__--=.b---<J

Figure 7.2.


040

Llb- 2035

030

025

F(

020 Lib -1

015

010

DOS

0 02 04

alb

Figure 7.3.

06 O.B

(7.82)

cut result from the symmetry of the problem. It is obvious that, at these points, thenormal component of displacements, tangential component of tractions and the heatflux vanish.

Having solved the boundary-value problem, the stress intensity factor can be computed by extrapolation using the well-known asymptotic formula for displacementsnear the crack tip (Sih and Liebowitz [13]). Hence,

]{[ = lim jl(21l")1/2 un (11<)(; -. 0 2(1 - v)~

where un ( 11<) is the normal component of the displacement computed at the crackpoint 11< lying at the distance f. from the crack tip.

Figure 7.3 shows the dependence of the dimensionless SIF on the crack length a,where

F[ = ]{/ 1 - v(1l"a )1/2 eoEn:

Computations have been carried out for two different values of the ratio Lib.

EXAMPLE 3In this example we shall consider a boundary value problem of stationary thermoelasticity for which in addition to the numerical solution of the BIE the stresses atinternal and boundary points are also computed numerically. In contrast to the otherexamples presented in this chapter we have used a linear approximation within theboundary elements. The numerical integrations have been carried out using 6 or 12Gaussian points with respect to the mutual distance of the field and source points.

Consider an infinite strip subject to thermal loading. In order to avoid bendingeffects, we considered the temperature to be constant in the body. In the numerical


21 8=400 t, =0q=O t2=0

8=400 8=400q=O b=Q.S q=Ou,=O L=2 u,=Ot2=0 h=O

•8=400 t, =0 1

q=O t2=0

Figure 7.4.

computation, we analyzed a finite part of the strip with the boundary conditionsshown in Fig. 7.4.

The boundary of the body was divided into 44 linear elements. The solution ofthe boundary value problem for thermal unknowns is trivial and was not performednumerically. The mechanical unknowns on the boundary were computed with an errorsmaller than 0.001 %. The exact values of the thermal stresses 0"11 are given by theanalytical solution

(7.83)

Figure 7.5 shows a comparison of the accuracy of the numerical computation of internal stresses by the regularized and nonregularized integral representations as y2lb -+ O.It is seen that the nonregularized scheme can actually play the role of a comparativecomputational method. The 'shortest distance' (introduced above) takes approximately 0.3 of the length of the boundary element.

The simplicity of the proposed method of computation of stresses (secondaryfields) results from the fact that the computational effort spent in the comparativecomputation is minimal, since the nonregularized integral representation differs fromthe regularized one only in one boundary integral.

Other numerical examples illustrating the solution of boundary value problems ofstationary thermoelasticity have been analyzed using the present BEM formulationelsewhere (Sladek and Sladek [14],[15]).

EXAMPLE 4In this example we present the numerical solution of a three-dimensional boundaryvalue problem of quasi-static uncoupled thermoelasticity using the BEM formulationcombined with the Laplace transform (Sladek and Sladek [16]).

Consider a semi-elliptical surface crack in an infinitely long layer of thickness L.The layer is initially kept at a uniform temperature To, and at time t = 0 the cracked


1800

1600

1400

- analytical- regularized BEM--- nonregularized BEM

600

BOO

'0 1200a..~

- 1000 .._ _._ _._ == ----:,,=.....-.-------i~ //

(,,/

IIIIII

400 10L---O~0-3-....l.....-0~06-:---~OO-:-:9:---0:-'-.12----=-'0:15

~/b

Figure 7.5.

CzltDL 20 J

Figure 7.6.


surface is suddenly cooled to the temperature Ta , while the rest of the surface is thermally insulated. According to Saint-Venant 's principle, the infinitely long layer canbe replaced by the specimen shown in Fig. 7.6 with the lateral surfaces at a distance5L from the crack. The lateral surfaces are assumed to be traction-free and thermallyinsulated. Due to the symmetry of the problem only a quarter of the specimen isanalysed, with the following boundary and initial conditions:

0(x, t = 0) = T(x, t = 0) - To = 0, x EV

0(11, t) = Ta - To = -0 0 , t > 0, 11 E ABFE

Q(11,t) =0, t>O, 11ES\ABFE

t(11, t) = 0, t > 0, 11 E (Scr U ABFE U BCGF U CDHG U EFGH)

Having solved the BIE for the Laplace transforms, the inversion was performed by theDurbin algorithm. The computed displacements on Scr can be used in the evaluationof the stress intensity factor at a point ~ on the crack contour by

}' (t t) 1· 1l(21r)1/2 U3(11" t)\f \" = 1m

'--+02(1-v) Jtwhere 11, E Scr lies on the perpendicular to the crack contour at ~, with t =111, - ~ I.The dimensionless stress intensity factor Ff and time t* are defined as

F ' 1 - v E( k) * tet1= l\f 00EO' (1rb)l/2' t = U

where E( k) is the complete elliptic integral of the second kind with k = [1 - (bl a)2]1/2.The computed results have been used in the preparation of two plots, which show

the angular dependence of the SIF at various times (Fig. 7.7) and the time-dependenceof Ff at two different points on the crack contour (Fig. 7.8). Recall that the angle t.p

is usually defined as shown Fig. 7.6.Note that in most materials the mechanical equilibrium is established faster than

the thermal equilibrium. Thus, in the case of thermal loading with the Heavisidefunction time dependence, the time evolution is well described within the quasi-staticapproximation except at very early times. That is why we computed the last exampleunder conditions of the quasi-static approximation, though it could also be computedwith the same effort within the uncoupled theory of thermoelasticity. The computational effort would increase, if we used the time-domain formulation.

028


b/o:02b/L=02S

o zo 40

Figure 7.7.

60

b/L=Q.Z5- \I =0---- ~ • Tl/2

o 01 OZt·

Q.J 0.4 05

Figure 7.8.


Time-marching solution of BIE for heat conductionConsider the heat conduction equation in the uncoupled theory of thermoelasticity

2 1 . Q\7 8 - -8 =--

K K(7.84)

(7.86)

Making use of the time-domain formulation for the BEM solution of thermoelasticproblems developed in Section 7.4, we may write the integral representation of thetemperature field as

~(y)8(y, t) = KII [q(11, T)T(I T/ - y I, t - T) - 8(11, T)F(11,Y, t - T)] dS'7 dT +

+18°(x)T(1 x - y I, t)dVx (7.85)

and the BIE as

E8((,t) +K l {8(,T) l[F(11,(,t - T) - F(11, ()8(t - T - O)]dS'7+

+ l [8(11, T) - 8(, T)] F(11, (, t - T)dS'7 -l q(11, T)T(I 11 - ( I, t - T)dS'7} dT =

= 18°(x)T(1 x - ( I, t)dVx

since t = to = °and K = K o in the uncoupled theory. For the sake of brevity weassume Q(x, t) = o.

In what follows we shall illustrate the time-marching scheme for solution of twodimensional problems when

ll(t) (r 2)T(r,t) = -exp --

41rKt 4Kt

H(t)rr,i ni(11) (r2 )F(11,Y, t) = - 81r(KtF exp - 4Kt ,ri = 11i - Yi

(7.87)

(7.88)

Let us be interested in the solution in the time interval < 0, tF > and divide thisinterval into subintervals < t j-ll t j > for f = 1,2, ... , F , i.e., into F time steps withthe length b.t j = t j - t j-I' We assume that to = 0 and the boundary values of thetemperature and heat flux vary linearly within each time step, i.e.,

g(11,T) = g(11,tj-dw I(T) +g(11,tj)W2(T), T E< tj-btj > (7.89)

where 9 stands for 8 and q, and the interpolation polynomials are defined as

(7.90)

There are usually two different time-marching schemes (Brebbia et al. [9]). Scheme1 treats each time step as a new problem and so, at the end of each step, computesvalues of the temperature at a sufficient number of internal points in order to use


them as pseudo-initial values for the next step. In Scheme 2, the time integrationprocess always restarts from time t = O.

Introducing the notations

TOII(r,tF) =I t' WOI(r)T(r,tF - r)dr

t/_1

FOII(fI,y,tF) =I t' WOI(fI,y,tF - r)dr

t/_ 1

for 0: = 1,2 and f = 1,2, ... , F , we may rewrite the BIE (7.86) as

8((, tF) {E +,. is [F2F (77' (, DotF) - F(fI, oj dST}} +

+,. is [8( fI, tF) - 8((, tF)] F 2F(fI, (, DotF )dST} -

-,. is q(fI, tF)T2F (1 fI - ( I, DotF)dST} =

(7.91 )

F-1+,. L 1[q( fI, t1)T21(I fI - ( I, tF) - 8(fI, t1)F2

1(fI, (, tF )]dST}+1=1 s

(7.92)

using Scheme 2, while in the case of Scheme 1 the BIE becomes

8((, tF) {E +,. is [F2F(fI, (, DotF) - F(fI, oj dST}} +

+,. is [8( fI, tF) - 8((, iF)] F 2F(fI, (, DotF )dST} - ,. is q( fI, iF )T2F (I fI - ( I, DotF )dST}

=,. is [q(fI,tF_dT1F (1 fI - (I,DotF) - 8(fI,tF_dF1F(fI,(,DotF)] dST} +

(7.93)

where the pseudo-initial temperatures at internal points, x EV, can be computed by

8 (x, tF) = ,. is [q( fI, tF_dT 1F(I fI - x I, DotF) +q( fI, tF)T2F (I fI - x I, Do tF)

-8(fI,tF_dF1F(fI, X, DoiF) - 8(fI,iF)F2F(fI,X, DoiF)) dST} +

(7.94)


If a linear time interpolation is employed, one can perform the time integrations in(7.91) in closed form. The time integrals of the fundamental solutions are expressed interms of exponential functions and exponential-integral function Ei ( x)(see Gradshteynand Ryzhik [17]). The explicit expressions of the time integrated kernels are givenin Appendix C together with the asymptotic behaviour as r = IT/ - (I --+ O. Thelogarithmic singularity contained in T 2F can be removed by the substitution r = 82•

Although the kernel F 2F contains l/r singularity, the kernel F2F - F is bounded andthe integrand

is also bounded due to the assumption of the Holder continuity of the temperatureon the boundary.

Now one could apply the standard discretisation of the boundary and interpolation of the boundary values of temperature and heat flux within each boundaryelement. Since all the integrands in the BIE are bounded everywhere, the prescribedintegrations can be performed by regular Gaussian quadrature. It is worth mentioning that the present BIE are free of any singularities independently of the order ofthe polynomial approximation, in contrast to the more commonly used BIE with freetemperature term (e.g. Brebbia et al. [9]). In general, this BIE contains the Cauchyprincipal value integral, which disappears only if constant or linear approximation isemployed.

Note that in the case of Scheme 1 it is necessary to also discretize the domain andto compute the temperatures at internal nodes too. If the length of each time step isassumed to be constant, it is sufficient to compute all the integrals over the boundaryelements only once at the beginning of the time-marching procedure. In the caseof Scheme 2 the temperature is not required to be computed at internal points, butthe number of boundary integrals is always proportional to the number of time steps F.

EXAMPLE 5Consider an infinite layer with the initial temperature 0°(x) = 0 at time to = 0, whichis subjected to a sudden heating to the temperature T on both the surfaces, i.e.,

0(T/,t) = TH(t), T = 400°C

In the numerical computation we considered the region b x b with b = 0.05m andK, = 8.19 x 10- 5m 2

8-1

. The heat flux is zero on the lateral surfaces and the Heavisidefunction is modelled as H(t) shown in Fig. 7.9. Such a modelling is necessary becausewe need to know temperature and heat flux simultaneously at t = O. The length of thefirst time step is taken ~t = 0.05 and the next 20 steps have the length ~t = 0.25s.The boundary contour is divided into 16 boundary elements and the interior into 46triangular elements with quadratic variation. Scheme 1 was employed.

The distribution of the temperature field at discrete time moments is shown inFig. 7.10, while Fig. 7.11 shows the evolution of the temperatures at four internalpoints with different distances from the boundary surface. The discrepancies betweenthe numerical and analytical results can be explained by modelling the Heaviside step.

8=TR(tl

b


q=O

2

1 8- TR(tl

bq=O R

1

0.05s t

Figure 7.9.

400 - arolytical• BEM

360

320

280

240

Sl·C]200

160

120

80

40

00 01 Q2 OJ 04 0.5

Y2/b

Figure 7.10.


400,-----------------,

54

- analyticalo BEM

321o~~---'----~---~-------'-------'

o

40

80

360

120

160

280

320

t[s]

Figure 7.11.

The small values of the temperature are affected more remarkably. Hence, the influence of the modelling is more expressive at early times (it appears everywhere) andfor deeper points the influence is observable also at later times.

Time-Marching Solution of Quasi-Static Problems in Uncoupled Thermoelasticity

In the preceding section we have performed a part of this analysis, since the temperature field in quasi-static uncoupled thermoelasticity is governed by the heat conductionequation (7.84). What remains is the analysis for mechanical fields. Note that thefundamental displacements Uik and tractions Tik are now the same as in elastostaticsand are given by Eq. (7.22). Thus, inverting the Laplace transform in Eq. (7.41), weobtain the BIE (Sladek and Sladek [19])


where the body sources and the initial temperature eo are assumed to be absent, and'" = "'0 in the uncoupled theory. The time convolution integrals of the temperatureand heat flux with the corresponding kernels are converted to the summations overthe time steps, assuming a linear time interpolation of the temperature and heat fluxwithin each time step as it was done in the section on Time Marching Solution of BIEfor Heat Conduction.

Similarly, from Eq. (7.62), we obtain the regularized integral representation ofstresses

(7.96)

The time integrated kernels U;J, Z~J, F[cjI , and Hf/ are defined symbolically as

(7.97)

where 111",(7) for a = 1,2 are given by Eq. (7.90).We have not yet presented the explicit expressions for the integral kernels. In two

dimensions these are

mr k ( G) r2

Uk(r,t) = 211"~ 1- e- , G = 4",t

mn(x) [ r 'r k ]Zk(X,y,t) = 2~ l/r 2(fJik - 2r,ir ,k)(l- e-G

) + ;",t e-G

1D/ji(r) = 411"(1 _ v)r [(1 - 2v)(fJjir,j + fJijr,/ - fJjjr,i) + 2r,ir ,jr,d

1Tjji(r)= ( ) {4vfJJjrkfki3+(1-2v)[rjfJ'i3+rJ'f/i3-

411" 1 - vr' "


Ilm { ( v ) rr,ini(X)Hlj(x,y, t) = 7 - 1 _ 2v hjl +r,jr,l (2I1:t)2 +

+l/r [r,jnl(x) +r,lnj(x) + (hjl - 4r,jr,l)r,ini(X)] @

(7.98)

The time integrations in the definitions of kernels Uri, Z;I, F//, and Ht/ can becarried out analytically and the explicit expressions of these kernels are given inAppendix D.

Although the kernel Tik is l/r singular, the integrand which contains this kernelin Eq. (7.95) is finite due to the assumption of the Holder continuity of boundarydisplacements. Taking into account the asymptotic behaviour given by Eqs (D.4) and(D.5), one can see that the only singularity in the BIE given by (7.95) is the logarithmicsingularity contained in Uik and Z~F. This singularity is integrable and can be removedby a substitution. Thus, all the integrands in the BIE are bounded anywhere and theintegrations over the boundary elements can be carried out numerically by regularGaussian quadrature.

In the regularized integral representation of stresses the strongest singularity isl/r singular and it is contained in the kernels Tlji ,Dlji , and HI~F. The comparativenonregularized integral representation differs from the regularized one only in oneterm and it can be obtained from (7.96) by the simple change

auTlji (1] - y) ar' (1], tF) ~ -Ui(1], tF )5Iji (1], y)

where the kernel 51ji is l/r2 singular and the explicit expression of this kernel is knownfrom elastostatics (see e.g. Ref. [2],[9],[20],[21]).

EXAMPLE 6Consider an infinite layer with the initial temperature eO(x) = 0 and subject tosudden warming e(1], t) = TH(t) on both its sides. Let the second axis of the coordinate system be orthogonal to the surface planes of the layer. Then, the temperaturedistribution is

e(X2' t) = TH(t) {I +2 f= _1 [(-It - 1] sin (n1r X2)@n=ln~ b

(7.99)

where b is the thickness of the layer.The heat fluxes on the layer surfaces X2 = 0 and X2 = bare

aeq(b,t)=-a(b,t),

X2ae

q(O, t) = --a (0, t)X2


T(t)

400

internalpoints

b

1 0.02 (s] t

Figure 7.12.

Hence and from (7.99), we have

q(b, t) = q(O, t) = ~E[1 - (-ltl exp [_ (~1r)2Kt]

The thermal stresses induced in this layer are

(7.100)

Eex0"11(X2, t) = -1 _ /I 0(X2' t) (7.101)

In the numerical computation we considered the finite body as shown in Fig. 7.12with the temperature and heat flux being taken from the analytical solution. TheHeaviside step function was modelled as shown in Fig. 7.12 and we employed thematerial parameters:

E =5.668 X1010Nm- 2 , /I =0.37, K =8.19 x 1O-5m28-1, ex = 2.786 x 1O-5(OCt1,b = 0.05m, L = 0.2m.

The boundary contour was divided into 44 linear boundary elements and the lengthof the first time step was 0.028, while that of the next step was 0.2s. Figure 7.13shows the results of numerically computed stresses in nine internal points and at fivedifferent times. The solid lines correspond to the analytical solutions. Three differentaspects can be drawn from the analysis of the accuracy of the numerical computations.

Figure 7.14 shows the percentage errors of the numerically computed stressesat internal points, while Fig. 7.15 shows that of the numerical solution of the BIE,because t1( 112. t) are the tractions computed in the boundary points (111 = 0,112) or(111 = L,112)' Firstly, it is seen that the accuracy of numerically computed stressesfails abruptly as the internal point approaches very near to the boundary. Such adecrease in accuracy can be seen at any time (t > 0) because it is due to the singularbehaviour of the integral kernels at subsurface points. This aspect is also known instationary theories (Sladek and Sladek [20],[21],[15]).


o

61.1 )C

o

Figure 7.13.

0.4 05

20

16....

'" 12..-oIot: 8(1J

~

4 -

o

• - (o ,+)• - ( .. ,0 ,+1• - (0,.,0,+1

Figure 7.14.


10.-(6,+,0)

8~-_ 6oL-ot 4OJ

~o

I

o 0·1

Figure 7.15.

0.5

Secondly, in both Figs 7.14 and 7.15, one can find that the error of the numericalcomputation in any internal and/or boundary point at short times is higher than atlater times. Since this aspect can be seen at both the internal and boundary points, itcan be explained by the fact that the Heaviside instantaneous jump in the prescribedboundary temperature has had to be modelled by a finite rate jump. Obviously, thisnon-stationary aspect plays a role only at short times.

Finally, the last aspect, which can also be seen in both Figs 7.14 and 7.15, is the'strange' increase of the relative error of the numerical computations at the midpointof the layer at the time t = 0.82s. This increase can be explained by the very smallvalue of the computed stresses at the midpoint at that time.

The following notations have been used in Figs 7.13-7.15 for the numerical computations corresponding to different times: x at t = 0.828, 0 at t = 1.828, ~ att =3.828, + at t =5.828 and 0 at t = 7.82s.

7.8 Stationary Problems in Media with TemperatureDependent Young's Modulus and Coefficient ofThermal Expansion

If the coefficient of thermal expansion is dependent on temperature, the DuhamelNeumann relations in an isotropic continuum are given as

(7.102)


where

. fef(8) =1

0a(T)dT

Let the Poisson ratio be constant and the shear modulus J-l depend on temperaturethrough the Young modulus J-l(8) = E(8)j2(1 +v). Then, Eq. (7.102) can be rewritten as

J-l ° l+v·(Jij = -Cijk/Uk,/ - 2J-l--2-bij f

J-lo 1 - v

where J-lo is an arbitrary constant and

(7.103)

cfjk/ = J-lo C:v2v

bijbk/ + bikbj/ + bi/bjk ) (7.104)

For the sake of brevity we assume the body sources to be absent (Xi = 0, Q = 0).Then, the governing equations take the form

(7.105)

° J-l,j [ 1 + v • 0 ] 1 + v •Cijk/Uk,/j = -;; 2J-l0 1 _ 2v bijf - CijklUk ,/ +2J-l° 1 _ 2v f ,i (7.106)

Let Ukm be the elastostatical fundamental displacements in the medium with materialconstants J-lo and v, i.e.,

Cijk/8j8/Ukm(x - y) = -bimb(x - y)

From Eqs (7.106) and (7.107), one can easily derive the integral identity

In view of the Gauss divergence theorem, we have

(7.107)

(7.108)

2J-lo / ~;v [1 ni(11)!(8)Uik (11- y)dST) - i !.i(8)Uidx - Y)dVx ] =

1 + v r .= 2J-lo 1 _ 2v Jv f(8)8iUik (X - y)dVx (7.109)

Further, we can use the relation

(7.110)


known from stationary thermoelasticity (Sladek and Sladek [6]), where a o is an arbitrary constant representing the coefficient of thermal expansion of a reference mediumwith the material constants ""0' v, a o, 1\.. Now, in view of Eq. (7.110) and the divergencetheorem, we may write

(7.111)

where the kernel Zk is the normal derivative of Uk.Apparently,

(7.112)

where

a'(0) = da(0)/d0

Although the temperature is high enough to consider the temperature dependence ofsome material parameters, the heat fluxes are assumed to be small so that a linearapproximation is valid, i.e., 0,i0,i ~ O. Hence and from Eqs (7.105) and (7.112), wehave

(7.113)

(7.114)

Finally, in view of Eqs (7.109), (7.112) and (7.113), we may rewrite Eq. (7.108) as

r ~ K. •~(Y)Uk(Y) = 1s {J-l(0) ti(TJ )Uik (TJ - y) - Ui(TJ)Tik (TJ, y) + a

o[f(0)Zk( TJ, y) -

-a(0)q(TJ)Uk(TJ - y)]}dS" + fv J-l~~)J-l'(0) [1 :V2v 0,i(X)Uj,j(x)+

1 + v .]+0,Ax) (Ui,j(X) + Uj,i(X)) - 21

_ 2v 0,i(X)f(0) Uik(X - y)dVx

If the shear modulus were temperature independent, Eq. (7.114) would be the integralrepresentation of displacements.

Taking the limit y --+ ( E S in Eq. (7.114), we obtain (for an internal problem)

Is {[Ui(TJ) - Ui( OJ Tik (TJ, 0 - J-l~~) ti(TJ )Uik (TJ - O} dS" =

= .!!:- r [J(0)Zk(TJ, O- a(0)q(TJ)Uk(TJ - 0] dS" + r J-l(~)J-l'(0) ®ao 1s 1v J-l \":}


[2v

t<:lI --8 ·(x)u· ·(x) +8 (x) (u· ·(x) +u ·(x))-I(y 1 _ 2v ,I J,J ,J I,J J,I

(7.115)

(7.116)

Since this integral identity includes the unknown gradients of displacements at internal points, it cannot play the role of the BlE for computation of unknowns on theboundary. In order to resolve the boundary value problem in the nonhomogeneousmedium, it is necessary to solve the system of Eqs (7.114) and (7.115) iteratively. Inthe current iterative step the gradients of displacements in the domain integrals areevaluated from the displacements computed at the preceding iterative step. In thefirst step the nonhomogeneous terms are ignored.

Having known the displacements at both the internal and boundary points, thestresses at internal points can be evaluated by the regularized integral representation

+~ r [a (8(11)) q(11)Fij(11- y) - j (8(11)) H/j (11,Y)] dST/ +ao is

+1, p, (~(x))p,' (8(x)) [1 :v2v 8,i(X)Uk,k(X) +8,k(X) (Ui,k(X) +Uk,i(X))-

1 + v . ]-2--8 ;(x)f (8(x)) D/J·;(x - y)dVx }1 - 2v '

Note that this integral representation is valid in two dimensions, while in three dimensions it is necessary to make the replacement

where 'ilji is the elastostatical variant of the kernel-operator defined by Eq. (7.64).The kernel unknowns on the boundary can be computed from the BlE

". is [8(11) - 8(()] F(11,()dST/ -". is q(11)T(I11- ( I)dST/ = 0

and the temperature at internal points is given by the integral representation

(7.117)

8(y) = ". is [q( 11 )T(I11 - Y I) - 8(11)F(11, y)] dST/ (7.118)

The integral kernels T(r), F(x, y), Ui(r), Zk(X, y), Uik(r), and Tik(X, y) are given byEq. (7.22), further the kernels D1ji(r) and Tlji(r) by Eq. (7.89), and the kernels Fij(r)and H/j(x,y) are defined in two dimensions as


Po 1 +v ao [ 8/j ( ) 1F/j(r) =-- -- - -- 1 +2v+21nr +2r,jr,/471" 1 - v '" 1 - 2v

Po 1+v aoH/j(x,y) = - 7I"r 1 _ 2v --;: (8/j - 2r,jr,d r,ini(x) (7.119)

Note that in the kernels employed in this section the material constants P, a, "'0 shouldtake the values Po, ao, "'.

The components of the stress tensor on the boundary can be computed by

OUk 1 + v •+ Bijk(()a:;(() - 8ij21

_ 2v P (8(()) f (8(()) (7.120)

where the coefficients Aijk and Bijk are given by Eq. (7.74) with P being replaced byp(8(()).

It can be seen that the only singularity in Eqs (7.115) and (7.117) is the logarithmicsingularity in the kernels T, Uik , and Zk. This singularity however, can be removed bya substitution preceding the numerical integration or by analytical integration in thecase of linear or constant approximation on the boundary elements. Equation (7.114)is employed for computation of displacements at internal nodes. Then, the logarithmicsingularity of the kernel Uik in the domain integral can be removed by transformingthe intrinsic coordinates on singular elements to polar coordinates. In this way onecan remove also the l/r singularity of the kernel D/ji in the domain integral of Eq.(7.116). Owing to this singularity we may compute the internal stresses by (7.116)only at internal nodes. Since the internal nodal points lie at a sufficient distance fromthe boundary, we do not need to use the nonregularized integral representation asa comparative method. For more details of the numerical computation we refer thereader to Ref. [22].

Note that Ghosh and Mukherjee [23] have to compute internal stresses differentiating displacements elementwise, because the integral representation of internalstresses developed by them contains the gradients of the fundamental displacementsUik in the nonhomogeneous domain integral, which is l/r2 singular and so such anintegral representation is impracticable.

The dependence of material parameters on temperature can usually be modelledpolynomially, for instance as

p(8) = Po - Bt 8 - B28 2

a(O) = ao +Gt 8 - G28 2 (7.121)

where 8 = T - To and the material characteristics Po, Bt , B2, a o, Gt , and G2 are relatedto the reference temperature To at which equilibrium occurs. This model satisfies thetemperature dependences employed in the present numerical examples.


q=O q..O~·O t----+-----<f--~---t----___+----_i u1=0

a t2=0 ttO

L...--....-.- ....-.- ....-.- ~-_~

Figure 7.16.

EXAMPLE 7In this example we assume a linear temperature dependence of the shear modulus(B2 = 0) and the coefficient of thermal expansion to be constant (CI = C2 = 0):

J1.(0) = J1.o - B10, 0 = T - To

J1.o = 7.95683 X 104M Pa, BI = 15.23733 M Parc

a(0) = ao = 0.125 x 1O-4(OCt l

v =0.3, To = 25°C

The boundary value problem (Fig. 7.16) is defined for a parallelogram subject to permanent temperature gradient (T(O) = TI , T(a) = T2) with the horizontal edges beingtraction-free (t l = t 2 = 0) and we assume UI = t2 = 0, q = 0 on the vertical edges.This example is equivalent to computation of displacements and stresses in a layersubject to steady temperature gradient. Although the shear modulus is dependent ontemperature, the analytical solution is available for this problem and it is given by

(7.122)

with

The displacements are determined with respect to the lower edge (U2(0) = 0). In thenumerical solution we employed the values TI = 30°C, T2 = 125°C, a = 1, and thediscretization mesh shown in Fig. 7.16. The results obtained after two iteration stepsare presented in Table 7.3.


Table 7.3.

X2 U2(X2) X 104 -l1n(X2) 8(X2)exact BIE % error exact BIE %error

O. O. O. O. 18.4533 18.73 1.49 5.0.25 0.9793 0.9747 0.46 105.623 106.3 0.64 28.750.50 3.337 3.332 0.15 191.995 193.1 0.57 52.500.75 7.073 7.066 0.01 277.569 281.7 1.48 76.251.00 12.187 12.18 0.05 362.345 367.9 1.53 100.

Figure 7.17.

EXAMPLE 8This example is devoted to the computation of thermal stresses in a thick-walled tubesubject to a permanent temperature gradient (Fig. 7.17).

In this example we considered four different material media:

(2) C1 = C2 = B2 = 0

(3) C1 = C2 = 0

(4) {CI, C2 , BI, B2 } i: 0

If the material constants are different from zero, they are given as


0.

-1ao. '--~~=-----------,-.:-~---:-==---"---:'1. 1.125 125 1.375 1.5

r/R,Figure 7.18.

Bo= (4.9861498 +0.0050667 To)To x 2.6288 x 106Nm-2

B1 = (4.9861498 +0.0101334 To) x 2.6288 X 106Nm- 2(OCt l

Co = (1.9491111 X 10-8 - 1.6736868 X 10-11 To)To(°C)-1

CI = (1.9491111 X 10-8 - 3.3473736 X 10-11 ToWCt 2

C2 = 1.6736868 x 1O-11 (OCt3

where the material parameters III and QI are the shear modulus and the coefficientof thermal expansion of aluminium at temperature T = oDe.

In the numerical computations we employed the values To = 25°C, TI = 30°C, T2 =200°C, and the following feature can be seen. If the shear modulus is temperaturedependent and the coefficient of thermal expansion is constant, then the profile ofstresses across the tube appears to be same as the shifted profile in the homogeneoustube.

Moreover, we computed the stress intensity factors for two axial cracks in thethick-walled tube considered above. Since the cracks are assumed to lie in the sameplane passing through the axis and both cut the internal surface, it is sufficient todiscretise only a quarter of the cross-section. The stress intensity factors are evaluated


crock (length ojq=O,t,= trO

Figure 7.19.

21 f.cL

1

Table 7.4.

case a =O.5b a =0.3bF[ F[

(1) C1 =C2 = B1 = B2 =0 0.585 0.5218(2) C1 = C2 = B2 =0 1.1276 0.8683(3) C1 = C2 =0 1.2577 0.9459(4) {C1,C2 ,B1 ,B2 } =f 0 1.598 1.2740

from the computed displacements on cracks by Eq. (7.82) and the dimensionless SIFis defined as

where a is the crack length and E is the Young modulus.We considered two different values for the crack length (a = 0.5b and a = 0.3b)

and four different material media. The computational results are collected in Table7.4.

In conclusion, we may say that the effect of thermally induced nonhomogeneityis expressive and should not be neglected in aluminium components of constructionssubject to thermal loadings.

Appendix A

In this appendix we present some formulae concerning the explicit expressions ofthe fundamental solutions. The coefficients Ua (r) for a = 1,2, ... , 9 occuring in thedefinition of fundamental solutions in general thermoelasticity are given as:


(i) for 3 - d problems

UI = [1 + (C2/pr)2 (1 +pr/C2)] e-pr/

c2 +r-2 [AI(1 +AIr)e- Atr

-A2(1 + A2r )e-A2r]

U2 = - [1 +3 (C2/pr)2 (1 +pr/c2)] e-pr/

C2- Al [Ai +3(1 +AIr)/r2] e-Atr +

+A2 [A~ + 3(1 + A2r )/r2]e- A2r

U3 = - (1 +pr/C2) e-pr/

c2

U4 = AIAi(1 +AIr)e- Atr - A2A~(1 +A2r)e-A2r

Us = (A~ -~i)r2 [(1 +AIr)e-Atr

- (1 + A2r )e-A2r

]

(A.l)(ii) for 2 - d problems

- 2 2U2= -J(2(pr/C2) - AIAIJ(2(AIr) +A2A2J(2(A2r)

U3 = -(pr/C2)J(I(pr/C2)


(A.2)where Kn(z) is the modified Bessel function of the second kind and the n-th order.Furthermore, we have used the following notations

Ai.2 = ~ {a2+ (picd2± [(,82 +p2Icn 2- 4,80 (piCl)2r/2}

,82 = (1 + mEK,),8;, ,8; = plK,

m - -'- Cl = (x +p21l ) 1/2, C2 = (Ill p//2- x+ 21l'

(A.3)Now we present the time-dependent fundamental temperatures and displacements forparticular classes of thermoelastic problems.

Thermal stresses(i) 3 - d problems

18' {2erfc(ro) - eb2t [(1- rcd K,)erC]I/t<erfc(ro+bVt)+

+(1 + rcd K,)e-rC]/t<erfc(ro - bVt)]}

Uik(r, t) = _1_{ [t5(C2 t - r) + (c2 tl r2 )(H(C2 t - r) - H(Clt - r))] t5ik 41rPC2r

-r.ir.k[t5(C2t - r) - (cdcdt5(Clt - r) + (3c2tlr2) (H(C2t - r)-

-H(Clt - r))]}

(AA)where b = cd.JK"H(t) and t5(t) are the Heaviside unit step function and the Diract5-function, respectively, and the error functions are defined as


erf(x) = 2/Vil x

e-t2 dt, erfc(x) = 1- erf(x)

(ii) 2 - d problems

H(t) 2T(r,t) = __e-ro

41rK ot

Uik(X,t) = 21l"~C2 {8ik [(1/~2 +~2/r2) H(C2t - r) - (C2~tlc1r2)H(C1t - r)] +

+r,ir,k [(1/~1 +2~tlr2) C2/C1H(CIt - r) - (1/~2 +2~2/r2) H( C2t - r)]}

(A.S)where

Unfortunately, the explicit expression for the fundamental displacements Uk(r, t) in2 - d problems is not available.

Quasi-static thermoelasticity(i) 3 - d problems

_ H(t)e _(2 1/2T(r, t) - (41l"Ket)3/2e , (= r(4Kett

Ui(r,t) = -H(t) me r,i [(1l"Kett1/2e-(2 -l/r erf(O]41l"r

Uij(r, t) =~ {x 7(2 + e~JL 8(t) +H(t)B/r2 [erf(() - 2(/.fi e-e]} -41l"JLr 2 X + 2JL

-3/r2 erf(0] }(A.6)

where

(ii) 2 - d problems

T(r, t) = H(t) e-(241l"Kot

meK ( (2)Ui(r, t) = H(t)-2-r,i 1 - e-1l"Kor


Uik(r, t) = __I_bij [x ~ (2 +ejp b(t) In(r) _ B/r2H(t) (1 _e-(2)] +21rp 2 X +2p

+r,ir,j {X + (2 - e)p t5(t) _ BH(t) [2/r2 _(_1_ +2/r2) e-<2]}21r p 2(X +2p) 2Ket

(A.7)Quasi-static uncoupled thermoelasticity(i) 3 - d problems

(A.8)

(ii) 2 - d problems

(A.9)The fundamental displacements Uik(r, t) are equal to the elastostatical fundamentaldisplacements (Kelvin solution) Uik(r) in both two- and three-dimensions.

Appendix B

In this appendix we shall deal with the regularization procedure developed in thederivation of regularized integral representations of the secondary fields. Considera domain V bounded by the boundary 5, though the regularization formula can bederived also for an open boundary (see e.g. Ref. [2],[8]). Making use the Gauss divergence theorem repeatedly, one can derive the following integral identity

in which Ui(1'/) are integrable functions, ni are the components of the outward normalvector, and Ukj ( 1'/, y) are the two-point integral kernels. The differential operator Dij

is defined as

(B.2)

It can be shown (Balas et al. [2]), that this operator can be written in terms oftangential derivatives as:


(i) for 3 - d problems

(B.3)

where

Eij = ni(11 )Tj{11) - nj(11 )Ti(11) = fijkPk( 11)

Fij = ni(11)pj(11) - nj{11)Pi(11) = -fijkTk(11)

(B.4)in which (n, T, p) is the triad of ortonormal vectors, with T being a unit tangent vectorand

(ii) for 2 - d problems

(B.5)

where(B.6)

with T being the unit tangent vector related with n by

Appendix C

In this appendix we present the time integrals of the fundamental solutions occuringin the section on the Time-Marching Solution of BIE for Heat Conduction. Makinguse of the notations

r 2 ~ ~O:j = iF - tj, f3j = O:j + f:).t j , A = --, B - -- E =-- (C.1)

41wj - 4K,f3j' 4K,f:).tFwe may write (Sladek and Sladek [18])(i) for f < F:

T1f(r, tF) = If:). {f3je- B - O:je-A +(r2 /4K, + O:j) [Ei( -B) - Ei( -A)J}411" K, t j


(ii) for f = F:

(C.3)where Ei(x) is the exponential-integral function (see Ref. [17]).It can be seen (Sladek and Sladek [18]), that in the limit r =1 1] - Y 1-+ 0 theseintegrated kernels behave as(i) for f < F: Ta!(r,tF) = O(rO), Fa!(1],y,tF) = O(r) (C.4)

(ii) for f = F:

TIF(r,b.tF) = O(rO),F1F(1],y,b.tF) = O(rlnr)

(C.5)

wherer,ini(1])

F(1],y)=- 2 ' ri=1]i-Yi1u"r

is the fundamental heat flux in the stationary theory.

Appendix D

In this appendix we present the time integrated kernels U~!, zf!, Flj!, and Ht/ occuring in the section on Time-Marching Solution of Quasi-Static Problems in UncoupledThermoelasticity. Making use of the notations given by (C.1) we may write (Sladekand Sladek [19])


Hlf(r,i F) = flm {[blj (l/r I{(r,iF) - -V-I{(r,iF))-J 7rbaij 1 - 2v

-r,jr,1 (I{(r,i F)+4/r I{(r, iF))]r,ini(X) +

+[r,jnl(x) +r,lnj(x)]1/r !t(r,iF)}

H1Y(r,iF) = :;;;j {[blj (l/r I.{(r,iF) + 1 ~2VI'(r,iF))

-r,jr,1 (II(r,iF) - 4/r I.{(r,iF))]r,ini(x) +

+ [r,jnt(x) +r,lnj(x)] l/r I.{ (r, iF)}

(D.1)where I!(r,iF) for a = 1,2, ... ,8 are given as(i) for f < F:

+r2 /2K(aj + r2 /8K) [Ei( -B) - Ei( -A)J}

It = ;r {,8j(l - e- B) +aj(l - e- A

) - 2aj,8j(1 - e- A) +r2 /4K(aje- A

- ,8je-B )+

+r2/2K(,8f +r 2 /8K) [Ei( - A) - Ei( - B)J}

II =L{,8je- B- aje- A +r 2 /4K [Ei( -B) - Ei( -A)]} - 1/r2 [,8j(l - e- B )+

+aj(l - e- A) - 2aj,8j(1 - e- B

)]

I{ =L{aje- A- ,8je-B + r2 /4K [Ei( -A) - Ei( -B)J} - 1/r2 [,8j(l - e- B )+

+aj(l - e- A) - 2aj,8j(1- e- A

)]

I{ = r/4K2 [Ei( -A) - Ei( -B)] +aj/ Kr (e- A- e- B

)

II = r/4K2 [Ei( -A) - Ei( - B)] + ,8j / Kr (e- A- e- B

)

I.{ = I{, It = It

(D.2)


(ii) for f = F:

Ii = 21r [(~tF)2(1 - e-E) +(r2/4K)~tFe-E + (r2/4K)2Ei(-E)]

If = ;r [(~tF)2(1 - e-E) - (r2/4K)~tFe-E - r2/2K(~tF +r2 /8K)Ei( -E)]

If = L[~tFe-E +r2/4K Ei( -E)] - (~tF/r)2(1 - e-E)

If = - L[~tFe-E +r2/4K Ei( -E)] - (~tF/r)2(1 - e-E)

I{ = -r/4K2 Ei(-E)

I F IF A / IF _ IF _ 1 - v ~tF7 = 4 +utF 2K, 8 - 6 v(1 - 2v) Kr

(D.3)It can be seen (Sladek and Sladek [19]), that in the limit r --t 0 the integral kernelsbehave as(i) for f < F:

u:J(r, tF) = O(r), z:J (r, tF) = O(rO)

Hf/(r,t F) = O(r), ~(Y(r,tF) =)(rO)

(D.4)(ii) for f = F:

UV(r,tF) = O(r), UfF(r,tF) = O(rlnr)

ZV(r,tF) = O(r°), Zt(r,tF) = O(lnr)

F/}F(r,tF) = O(rO), F/;F(r,tF) = O(lnr)

H/~F(r,tF) = O(rlnr), HI~F(r,tF) = O(r- l)

(D.5)


References

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3. Sladek,V., Sladek,J. and Balas,J. Boundary Integral Formulation of Crack Problems, ZAMM, Vol.66, pp. 83-94, 1985.

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8. Sladek,V. and Sladek,J. Three Dimensional Crack Analysis for Anisotropic Body,Appl. Math. Modelling, Vol.6, pp. 374-380, 1982.

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11. Durbin,F. Numerical Inversion of Laplace Transforms: An Efficient Improvementto Dubner and Abate's Method, Comput. J., VoLl7, pp. 371-376, 1974.

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13. Sih,G.C. and Liebowitz,H. Mathematical Theories of Brittle Fracture. Chapter2, Fracture, An Advanced Treatise, (Ed. Liebowitz,H.), Vol. 2, pp. 67-190,Academic Press, New York, 1968.

14. Sladek,V. and Sladek,J. Computation of the Stress Intensity Factor in 2 - dStationary Thermoelasticity, Acta Technica CSAV, Vol.32, pp. 217-229, 1987.

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17. Gradshteyn,I.S. and Ryzhik,I.M. Table of Integrals, Series and Products, Academic Press, New York, 1965.

18. Sladek,V. and Sladek,J. Nonsingular BIE for Transient Heat Conduction, Engn.Analysis (to appear)

19. Sladek,V. and Sladek,J. Computation of Thermal Stresses in Quasi - Static NonStationary Thermoelasticity Using Boundary Elements, Int. J. Num. Meth.Engn., Vol.28, pp. 1131-1144, 1989.

20. Sladek,V. and Sladek,J. Improved Computation of Stresses Using the BoundaryElement Method, Appl. Math. Modelling, VoUO, pp. 249-255, 1986.

21. Sladek,J. and Sladek,V. Computation of Stresses by BEM in 2 - d Elastostatics,Acta Technica CSAV, Vol.31, pp. 523-531, 1986.

22. Markechova,I., Sladek,J. and Sladek,V. Computation of Stresses in Nonhomogeneous Bodies Using BEM, Acta Technica CSAV, Vo1.34, pp. 487-511, 1989.

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25. Tanaka,M. and Tanaka,K. A Boundary Element Formulation for Coupled Thermoelastic Problems II. Boundary Element Discretisation, Rev. Roum. Sci.Techn.-Mec. Appl., Vol.26, pp. 257-263, 1981.

26. Tanaka,M., Togoh,H. and Kikuta,M. Boundary Element Method Applied to 2D Thermoelastic Problems in Steady and Non-Steady States, Engn. Analysis,YoU, pp. 13-19, 1984.

27. Rizzo,F.J. and Shippy,D.J. An Advanced Boundary Integral Equation Methodfor Three-Dimensional Thermoelasticity, Int. J. Num. Meth. Engn., Vol. 11 , pp.1753-1768, 1977.

28. Masinda,J. Application of the Boundary Element Method to 3 - D Problems ofNon-Stationary Thermoelasticity, Engn. Analysis, YoU, pp. 66-69, 1984.

29. Sharp,S. and Crouch,S.L. Boundary Integral Methods for Thermoelasticity Problems, J. Appl. Mech., Vol.53, pp.298-302, 1986. 30. Sharp,S. and Crouch,S.L.Heat Conduction, Thermoelasticity and Consolidation. Chapter 10, Boundary Element Methods in Mechanics, (Ed. Beskos,D.E.), Vol.3, Elsevier SciencePubl., Amsterdam, 1987.


30. Koizumi,T. et al. Boundary Integral Equation Analysis for Steady ThermoelasticProblems Using Thermoelastic Potential, Jour. Thermal Stresses, Vol.lI, pp.341-352, 1988.

31. Ochiai,Y. and Sekiya,T. Unsteady Thermal Stress Analysis in Compound Region by Means of Thermoelastic Displacement Potential and Boundary ElementMethod, Jour. Thermal Stresses, Vo1.l2, pp. 57-65, 1989.

32. Henry,D.P. and Banerjee,P.K. A New Boundary Element Formulation for Twoand Three-Dimensional Thermoelasticity Using Particular Integrals, Int. J.Num. Meth. Engn., Vo1.26, pp. 2061-2077, 1988.

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35. Rizzo,F.J. and Shippy,D.J. A Method of Solution for Certain Problems of Transient Heat Conduction, AIAA, Vo1.8, pp. 2004-2009, 1970.

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Chapter 8

Integral Equation Analyses of NaturalConvection Problems in Fluid FlowN. Tosaka (*), N. Fukushima (**)(*) Department of Mathematical Engineering, College of IndustrialTechnology, Nihon University, Narashino, Chiba, 275 Japan(**) 4th Development Department, Basic Software DevelopmentDivision, NEC Corporation, Fuchu, Tokyo, 183 Japan

Abstract

The applicability of an integral equation method to the numerical modelling of thermal phenomena due to buoyancy-driven flow under heating of incompressible viscousfluids is presented. The governing differential equations in terms of primitive variables by using velocity-pressure-temperature are transformed into the correspondingnonlinear system of integral equations. The related fundamental solution tensors fortwo-dimensional case are constructed explicitly. The solution procedure to solve thesteady problem is presented from the standpoint of the boundary-domain elementapproach. In unsteady problems, the solution procedure by using a time marchingscheme in conjunction with the above space discretization is proposed. In order toshow the accuracy and efficiency of the integral equation method, several numericalexamples of steady and unsteady two-dimensional natural convection problems in asquare and some nonrectangular enclosures are illustrated.

8.1 Introduction

In many fields of natural science and engineering we encounter convective heat transfer phenomena. These include many practical problems such as thermal insulation ofbuildings, cooling of electronic equipment and general circulation of planetary atmospheres. The convective and conductive heat transfer problem due to the buoyancydriven flow under heating of incompressible viscous fluids is called the natural convection problem.

The mathematical model of this problem is given as a boundary-value problem oran initial-boundary-value problem of a highly complicated nonlinear coupled systemof the Navier-Stokes equations and the equation of energy. Because of difficultiesin obtaining analytical solutions of the problems, several numerical solution proce-


dures based on the finite difference method and the finite element method have beendeveloped with the advance of electronic computers by many researchers.

On the other hand, the boundary element method has been successfully appliedto various kinds of problems. However, effective applications of the integral equationmethod, which contains the boundary element method, to viscous fluid flow problemshave been lagging behind the other two methods at the present stage. In particular,limited attempts have been made to solve natural convection problems by using theintegral equation method [1-9].

lt is well-known that several kinds of formulation exist for solving the naturalconvection problem. Among boundary element analyses of the problem, a streamfunction-vorticity-temperature formulation was used by Onishi et al. [9], a velocitytemperature formulation based on the penalty function method was used by Kurokiet al. [11] and Kitagawa et ai. [10], a velocity-pressure-temperature formulation wasused by Tosaka and Fukushima [7-9], and a velocity-vorticity-temperature formulationwas used by Skerget et al. [12].

In paper [5] three integral equation formulations for the two-dimensional problem, which are the primitive variables one (i.e., velocity-pressure-temperature), thevorticity-stream function-temperature one and the stream function-temperature one,were derived systematically. And recent enhancements in the application of theboundary integral equation method to numerical solutions of two-dimensional natural convection problems were presented in [13].

The aim of this chapter is to collect recent results obtained in our study onfurther development of the integral equation method based on the boundary-domainelement discretization to numerical simulation of two-dimensional thermal fluid flowproblems. Among the formulations discussed in this study is the primitive variablesformulation which is applicable not only to two-dimensional problems but also threedimensional ones and adaptable to any types of prescribed boundary conditions. Thenonlinear system of integral equations for steady and unsteady natural convectionproblems are derived systematically from the integral identity of the governing systemof differential equations. The related fundamental solution tensors are constructedfor the corresponding differential operator of a linearized equation set. Approximatesolution procedures of the nonlinear system of integral equations derived herein arealso described for each problem. Finally, several numerical examples of convectionproblems in a square and some rectangular enclosures are presented to demonstratethe efficiency of the proposed method.

Throughout this chapter, the summation convention on repeated indices is employed. A comma (,) following a variable and a dot (") over a function are used todenote partial differentiations with respect to space and time variables, respectively.

8.2 Natural Convection Problems

Let us consider a two-dimensional domain n filled with an incompressible viscousheat-conducting Newtonian fluid. The domain n has a piecewise smooth boundary rwith outward unit normal vector ni. And let T denote the open time interval (0, T)in question.

We assume that the flow conditions are such that the well-known Boussinesq ap-


proximation holds, and the viscous dissipation is negligible. Under these conditions,the governing equations can be expressed with non-dimensional variables as follows:

Equations of motion

Re(u· +u·u··) = T'" + Gr 06'2 m 0 x T1 ) I,) I),) Re 1

Equation of continuity

(8.1)

Uj,j =0

Constitutive equations

T" - -Rep6· +u· .+u··I) - I) I,) ),1

Equation of energy

m OxT

m OxT

(8.2)

(8.3)

. 1Pr(O +u·O .) = -0·· m 0 x T (8.4)) ,) Re ,))

in which Uj is the velocity vector component, Tjj is the Cauchy stress tensor, p is thepressure and 0 is the temperature. The physical numbers used in the above set arethe Reynolds number, Re the Grashoff number, Gr the Prandtl number, Pr and theRayleigh number, Ra == PrGr.

In addition to the above equations, the following initial and boundary conditionsare prescribed:

Initial conditions (at time t = 0)

U;(x,O) == u?

O(X,O) = (/J

Boundary conditions (on the boundary r)

U;(X,t) = u;

OJ(X,t) = /}

mO

in 0

on rr x T

on f o x T

(8.5)

(8.6)

(8.7)

(8.8)

(8.9)

qj(X, t) == O,jnj = q on r q x T (8.10)

where Tj denotes the components of surface traction and q is the heat flux.In this place, let us rewrite the above Eqs (8.1-8.4) in terms of the so-called

primitive variables Ui,P and 8 as follows:

L/JUJ = B[ (I, J = 1,2,3,4) (8.11)


in which [Lu] denotes the matrix ofthe linear differential operators appearing in Eqs(8.1-8.4), {UJ} is the unknown vector and {Bd is the forcing vector of nonlinearconvected terms. They are given as follows:

(-ReDt + ~)8ij +DiDjGr

-ReDi-8"2Re •

[Lu ]= 1 (8.12)0 -PrDt+-~ 0Re

D 0 0J

(8.13)

(8.14)

in which D i == a/aXi, Dt == a/at and ~ == DiDi denotes the two-dimensional Laplacian. It is evident that the matrix [Lu ] in the case of steady problems reduces to

~8""+D"DGr-8"2 -ReDi'J • J Re •

[LuJ = 0 ~~ 0 (8.15)Re

D 0 0J

After all, we can formulate in terms of the primitive variables the initial-boundaryvalue problem of Eq. (8.11) with (8.12-8.14) in unsteady case, and the boundary-valueproblem of Eq. (8.11) with (8.13-8.15) in steady case.

8.3 Steady Analysis

Integral Equation

Let us transform the governing Eq. (8.11) with (8.15) into the related integral equationfollowing the methodology proposed in Tosaka [8] and Tosaka and Fukushima [3J. Westart with the following integral identity of Eq. (8.11) over the domain for the tensorVii< which will be specified later:


k(LIJUJ - BI)VIKdO =0 (I,J,K =1,2,3,4)

Integrating by parts we arrive at

UK(y) =£{uj(x)E:K(x,y) - Tj(X)V;K(X,y)}df(x)

f 1 8V;i< • ( )}+ir Re {O(x) 8n(x) (x, y) - q(X)V3K x, Y df(x)

+ in BI(x)VIK(x,y)dO(x) (y E 0°)

(8.16)

(8.17)

where the traction vector and the pseudo traction tensor I::K are defined by, respectively

T' - (-Reph· +u· . +u· ·)n·• - 'J ',J J,' J (8.18)

(8.19)

We consider the expression of the above Eq. (8.17) as the interior point y tendstowards a point on the boundary. From the discontinuity of double layer kernel I::Kacross the boundary, we have the following so-called boundary-type integral equation:

CKI(y)UI(Y) =t{uj(x)E:K(x, y) - Tj(x)V;K(x, y)}df(x)

f 1 8V3K (X, y) • }+ir Re {O(x) 8n(x) - q(x)V3K(X,y) df(x)

+ in BI(x)Vii«x,y)dO(x) (y E f) (8.20)

where CKI denotes the shape coefficient tensor which depends on the geometricalproperty of the boundary point y. Consequently, we succeed in a formulation of theproblem with use of the nonlinear system of integral Eqs (8.17) and (8.20).

Fundamental Solution

Let us discuss how to construct the fundamental solution tensor VIi< which is indispensable for the derivation of integral Eq. (8.17), and its computer implementationwhich will be treated later.

The tensor must be determined as a solution of

(8.21)


where h(x - y) is the Dirac delta function with a pole at x = y and £IJ, which is theadjoint operator of LIJ, is given by

6- +D~ D2D} 0 -D}

D}D2 6-+ D~ 0 -D2

[.c IJ ] = Gr _1 6-(8.22)

0 - 0Re Re

ReD} ReD2 0 0

Let us seek a solution of Eq. (8.21) in the following potential form:

VjK = M JK ¢>" (8.23)

in which [MJK] denotes the transposed cofactor matrix of [£KJ] given by

D26- -D}D26- 0 _1 D}6-22 Re

-D}D26- D26- 0 _1 D26-21 Re

[MJKI= (8.24)

D}D2Gr -D;Gr 6-2ReGr

--D26-Re

-D I 6-2 -D26-2 0 2.6-3

Re

and the scalar function ,p" is the fundamental solution for the differential operator.c == det[.cKJ ] which satisfies

.c¢>" = c5(x - y) (8.25)

After all, if the solution ¢>" for the operator .c == 6-3 in this problem can be sought,then we can determine the fundamental solution tensor VJK from the expression (8.23).The explicit form of the solution for the tri-harmonic differential operator 6-3 is givenby

,p"(x, y) = 12~7/4In r (8.26)

where r =11 x - y II denotes the distance between x and y.Therefore, the fundamental solution tensor VI} can be determined explicitly from

the expression (8.23) with (8.24) in conjunction with (8.26) as follows:


~j = 4~ {(In r +2)hji - r,jr,i}

1~; = --r

211"rRe "

Gr 2-r (41nr +3)r lr 26411" ' ,

GrV32 = -12811" r2 {(121n r + 7)r~1 + (41n r + 1)r~J

Re-(21nr +3)411"

Gr--R r(ln r + 1)r,2

411" e

~3 ~; = V4*4 = 0

Solution Procedure

(8.27)

In this place we shall develop the boundary-domain element discretization of oursystem (8.17) and (8.20), and the related solution procedure to solve the resultingnonlinear system. For that purpose let nand r be subdivided into a finite number oftriangular cells and linear boundary elements, respectively. The unknown functionsinvolved in our integral equations are discretized at each element or cell in the followingform:

On the boundary element,

Uj(X) = 4>N(X)Uj(XN)

Tj(X) = 4>N(X)Tj(XN)(8.28)

O(X) = 4>N(X)O(XN)

q(x) 4>N(X)q(XN)

In the interior cell,Uj(X) = t/W(X)Uj(XM)}

O(x) = tPM(X)O(XM)(8.29)

where 4>N(X) and tPM(X) are linear interpolation functions on the boundary elementand the interior cell, respectively, and Uj(XN), T;(XN)' O(XN), q(XN) and Uj(XM), O(XM)denote nodal values of the corresponding functions at the boundary element node and


the interior cell node, respectively. Indices Nand M refer to the number of nodeswithin each boundary element and within each interior cell, respectively.

Substituting these discretized forms (8.28) and (8.29) into Eqs (8.17) and (8.20)for I, K = 1,2,3, we obtain the following discretized expressions:

TeO = -GeT +Heu - Seq +Ne(u)u +Me(u)O

(8.30)

(8.31 )

(8.32)

(8.33)

where matrices H, G and T, S with indices u and 0 are the so-called influence matrices,N denotes the nonlinear mappings, u, T, 0 and q denote the nodal velocity vector, thenodal traction vector, the nodal temperature vector and the nodal heat flux vector,respectively and the quantities with the upper-bar (-) represent the correspondingones defined at interior points.

Setting up the above equations at all nodal points and applying the boundaryconditions we can arrive at the following final nonlinear system:

A(u)X=B (8.34)

(8.35)

where A(u) is the system matrix whose components depend on the unknown vectoru, X is the vector of nodal unknown values on the boundary and interior domain andB is the known vector.

It is well-known that the method of solution for the nonlinear system given byEq. (8.34) greatly depends on the problem under consideration. In this study we employ the well-known Newton-Raphson iterative method in order to achieve accuratenumerical solutions of problems at high Rayleigh numbers.

8.4 Unsteady Analysis

Integral Equation

Applying the methodology developed in the preceding section on steady analysis inthe same way, we can derive a necessary integral equation of an unsteady problem(see Tosaka and Fukushima [6J,[8]). Let us start with the following integral form ofEqs (8.11) and (8.12) over the space-time domain for the time-dependent tensor VjK:

!r k(L/JUJ - BI)ViKdOdt = 0

Integrating by parts we can get

(8.36)


UK(y,s) = Ir£Uj(x,t)E:K(x,t jy ,s)df(x)dt

- irk Tj(X, t)V;i«x, tj y, s)dr(x)dt

+ irk ~eO(x, t) OV3K~:(~j(, s) dr(x)dt

- irk ~e q(x, t)\!;i«x, tj y, s)dr(x)dt

+ in[ReUi(x,t)V;i«x,t;y,s)]~dO(x)

+ ~[PrOi(x, t)V3i«x, t; y, s)]~dO(x)

+ ir~BI(X)VtK(X,tjy,s)dO(x)dt

in which we introduce the same traction vector Tj and pseudo-traction tensor L:K thatwe defined with (8.18) and (8.19). However, the tensor VjK in this case is defined bythe time-dependent fundamental solution tensor which satisfies

(8.37)

(8.38)

Taking a limiting process as the source point y approaches a boundary point and taking into consideration the discontinuity of the derivative of VjK across the boundary,we can arrive at the following system of so-called boundary-type integral equations:

CKI(y)UI(Y,S) = lrkUi(x,t)EiK(X,tjy,s)dr(x)dt

- lrkTj(X, t)V;i«x, tj y, s)dr(x)dt

+ irk ~eO(x, t) OV3*K~:(~~y, s) dr(x)dt

- irk ~e q(x, t)V3i«x, t; y, s)dr(x)dt

+ ~[Reuj(x, t)V;i«x, tj y, s)l~dO(x)

+ ~[PrOj(x, t)\!;i«x, tj y, s)l~dO(x)

+ ir in B1(x)V1i«x, t; y, s)dO(x)dt


Time-Dependent Fundamental Solution

We construct the fundamental solution tensor ViK in this time-dependent problem.In this case the tensor is sought as a solution of the singular differential equation,

£J/V/K =6JK6(x - y)6(t - s)

where

ReDt +~ +Dr D2D} 0 -D}

D}D2 ReDt +~ +D~ 0 -D2

[£u] = Gr 10 - PrDt + Re~ 0

Re

ReD} ReD2 0 0

(8.39)

(8.40)

The solution Vii< of (8.39) is expressed in the same potential form as (8.23). However,the transposed cofactor matrix of (8.40) is given by

o

o(8.41 )

D}Re £}£2

D2Re £}£2

GrRe~L2 --D2L2Re

-GrD2}

[Mul =

o

£} _ (RePrDt +~)

(8.42)

Re£3 - (TDt +~)

and a scalar function ¢J* must be determined as the time-dependent fundamentalsolution which satisfies

£¢J* = 6(x - y)6(t - s) (8.43)

where£IJ == det[£] = ~(ReDt + ~)(RePrDt +~) (8.44)

The solution of Eq. (8.43) and (8.44) can easily be given, for the case that Pr isnot equal to one, as the following integral expression:


</>*(X,t;y,s)

_ -H(s - t) IS _1_E- -PrRe r2 _ E- -Re r 2 dt- 41rRe(Pr -1) t {PrRe '[4(s - t)] '[4(s - t) ]} (8.45)

(8.46)

where H denotes the Heaviside step function and Ei is the exponential integral function defined by

Ed-a] = -100

e-X

dxa X

Time Stepping Solution Procedure

In this section we consider the space-time discretization to solve our system of integralEqs (8.36) and (8.38). In addition to the boundary-domain element discretizationdeveloped in the steady analysis, we must introduce the discretization in time interval,T. To this end, let us subdivide the time interval into steps tk+I = tk + l:1t( k =0,1,2,···) with time increment l:1t. Then, the unknowns involved in our system(8.36) and (8.38) are discretized as follows:

On the boundary element

In the interior cell,

Ui(X, t)

Ti(X, t)

O(x, t)

q(x, t)

MQ(t)¢N(X)Ui(XN' tQ)

MQ(t)¢N(X)Ti(XN, tQ)

MQ(t)¢N(X)O(XN, tQ)

MQ(t)¢N(X)q(XN, tQ)

(8.47)

Ui(X, t) = MQ(t)'l/JM(X)Ui(XM' tQ) }

O(x, t) = MQ(t)'l/JM(X)O(XM, tQ)(8.48)

where MQ(t) is the interpolation function in time step and the index Q refers to thenumber of nodes within time element. In the case of constant time element, we assumethat all unknown values could have been calculated at the time level tk.

In this place we adopt the time stepping approach in our time-dependent analysisin order to simulate unsteady behaviour with accuracy. To this end, replacing thetime interval T with the (k + 1)-th time interval [tk, tk+d in (8.36) and (8.38) andsubstituting the discretized form (8.47) and (8.48) into the resulting integral equations,we can obtain the following discretized expressions at the (k + 1)-th time interval:

Uk+l = HuU k+1- GuTk+1 +TuOk+I - Sul+I

(8.49)


(8.50)

(8.51 )

(8.52)

Setting up the above discretized equations at all nodal points and taking intoconsideration the prescribed boundary conditions, we arrive at the following nonlinearsystem for the k-th time interval:

(8.53)

where B is the column vector of the prescribed boundary data and components contributed from the k-th time step, which can be regarded as a known quantity in thetime stepping scheme.

Since the final system (8.53) is nonlinear at each time step, we adopt the NewtonRaphson iterative procedure in conjunction with the time stepping scheme in whichthe value of velocities and temperature at the end of each time interval are used asinitial values for the next time step in order to effectively solve the system (8.53).

8.5 Numerical Examples

In order to show the adaptability and effectiveness of the method, typical sample problems of two-dimensional thermally driven cavity flows are discussed. All the numericalresults were obtained by using linear boundary elements, linear triangular cells, andconstant time elements. As the numerical integration formula used in numerical performances, we adopt a 8-point Gauss quadrature for linear boundary elements anda 7-point quadrature suggested by Hammer et aI. for triangular cells. Further, wefix the Prandtl number and the Grashoff number as Pr = 0.7 and Gr = 1.43 x 103 ,

respectively, in the following numerical computations for both steady and unsteadyanalyses. Three Rayleigh numbers Ra = 103 ,104 and 105 are used in the presentstudy.

Natural Convection in a Square Enclosure

As the first example, we consider the buoyancy driven natural convection in an enclosed square cavity subjected to different side heating. Fig. 8.1 shows the geometryand boundary conditions of the problem. The element mesh used in steady and unsteady analyses is shown in Fig. 8.2.


I. 0

q'" 0

oB = I

on ,II th. bound,rl'l

q '" 0

B = 0

Figure 8.1: Geometry and boundary conditions.

Figure 8.2: Element mesh.

First of all, let us show numerical results of a steady problem. Fig. 8.3 showsisotherms and the velocity vector fields for three different Rayleigh numbers rangingfrom 103 to 105 (for fixed Prandtl number, Pr = 0.7). The relation of the averageNusselt number-Rayleigh number is shown in Table 8.1 through the comparison withother results (G. de V. Davies f3 Jones [7) and Phil/ips [1}) obtained by differentmethods. The results are in good agreement with other values.

Next, let us present time-dependent results for Ra = 103 ,104 and 105. All re

sults are obtained by computation with fixed time increment tit = 0.01. Calculated time-dependent results are shown in Figs 8.4, 8.5 and 8.6. Evolutions of temperature and horizontal velocity component U2 at X2 = 0.5 for unsteady flow atRa = 103 ,104 and 105 from the first time step to the final time step which is considered to be a steady state are illustrated in Figs 8.7, 8.8 and 8.9 respectively.


Table 8.1: Average Nusselt number-Rayleigh number.

Average Nusselt NumberRa Present Copper[7] Stevens[7] Phillips[l]103 1.11 1.12 1.115 1.118104 2.24 2.26 2.241 2.250105 4.60 4.55 4.497 4.573

isotherms

.• \ \ " - - ..- / / I "

.. \ " ....... - - -

----, \"

-\

~ - - - ... - .

- - - - , \'- - - - \ I'

- , I III I'

I I I'/ /' I'/' / "

- - - - - - / ,.,\ -

., / - - - - -,I / /'

" / /

,I I ,,I III I , _,I \ _ _ _ _ , _,I \ _

- - - - - - - .... ,

~ - - - - - - ,"

f / ...... ---- " \ I'

I / / - - - "I /\ I I'

I ~ - , ,\ I I'I I I ,

I I I I I'\ , , - ~ I / I I'

I \ " - - - / / I,.

\ " ---- ...... / f .',. , - - - - - - ~, ..

.. - - - - - - ,

..- / ,

, \ \ I I'

I 1 I I"/ / I I ,.

/ / I I

\ , -

,. I /

,. , I

,I I I

" I \

velocity vectors

(a) Ra = 103 (b) Ra = 104 (c) Ra = 105

Figure 8.3: Isotherms and velocity vectors of the steady problem.


I

I

T= O. 01

T= 0.07

T= 0.03

T= O. 10

(a) isotherms

T= 0.05

T= 0.20

I _

I I I ,

" I

,I I,I \, \ ,, ,

. \

- - ,

I / ~ - "- , ,I I .- - .... \ \ \ ,

, , I I , \ \ I ,I I I I , I I I ,I I \ , - , I I I ,, \ \ , - .- / I I

\ \ -T= O. 01 T= O. 03 T= 0.05

,\,'-- .... //1

, \ " -

I I " -- - "- \ I, I I I - - , \ I ,, , I I I , I , ,, I \ \ , - , I I

,., I \ \ - - -- / I I

\ \ - , ,

- - - -

I / --, I I " "- \ \ ,I I , \ \ \ I

,, I I

I I I I

I I \ \ - I I I I, \ \ "- - ~ .- / I ,\ "- -

I I __

I I " ~

I , I I ,

,I I 1

, I \ \ ,

- .... \

- \ \

I I

- I I

\ I

\ I'

/ I

I I

, I __

- - ...... ,

T= O. 07 T= O. 10

(b) velocity vectors

T= 0.20

Figure 8.4: Time-dependent solutions (Ra = 103).


T= O. 01

T= 0.07

.,I I

I •I ,

I I ,I\

I \ _

T= O. 01

-------,.

\ .... _----- ,

T= 0.07

T= 0.03

T= O. 10(a) isotherms

1/ / ....

,'-------

T= 0.03

,,------'1

, ... _----- .....

T= O. 10


T= 0.05

T= 0.20

• 1/ ...

\,-----

T= 0.05

------- ....

,,------ ....

T= 0.20

Figure 8.5: Time-dependent solutions (Ra = 104 ).


T= O. 01

T= 0.07

----_ ...

,r=-:-:: ~ ::::... , , I , , , , I

, " .... .. ... , I I , •

\ , , I \

1

I \ 1

T= 0.01

1/ .... --"

ii' :·---~~:iiI , ..... I II11'._""1 /1\----_·,/1\\------ .... "\_-------

T= 0.07

T= 0.03

T= O. 10(a) i otherm

- ---_ ..1 (/ ,

I /_ ......... __ .. ,\

I • , , " .... , \ I, .. I , I , \ , I- - - . . . , I'

\·----'1/1\,-----"'"1\,. - ... /'

\- , - - - - - ,

T= 0.03

,/,,------ ..... ,

;II:~::~:~~:I I • • , I II , ..... , II11 •..... ', 111 ""1

1\,------ .... ",--------"

T= O. 10


T= 0.05

T= 0.20

, I .... ,

'Il-:~:::i:, I II . _ , I I

'\------1/1\\,-----,."\ ..... - - - - - - -

T= 0.05

·,--------1,1 .... -\\

I{I····--II

:I , . . . . ~ ~ )\ II

1 \ • . . . • . 1

11' ."/1\\------ .... '1

, ..... _-----_/1

T= 0.20



\·00O·UU0·60U· 40O·7.U

oo -+----r-91.00

00

..:.C> T= 0.0106 T= 0.030

0 + T= 0.050CD

0 x T= O. 070<) T= O. 100'I' T= O. 200

0 • steady'"...J' solution

<0

:a:::a:w:r o......

0

0

'"0

(a) variation of temperature

o

>....... 0o .---~.".....---\--\o 0

..4Q/

>

C) T= 0.0106 T= O. 030-I- T= O. 050XT= 0.070oT= 0.100'I' T= 0.2UO• steady

solution

•o I---r----, ,-\f. 00 O· 711 11.411 (1·(;0 O. (III 1.110

(b) variation of U2

Figure 8.7: Evolutions of temperature and Uz at Xz =0.5 (Ra = 103).


oo "1.-----------------,

C) T= O. 0 I 0.do T= O. 030... T= O. 050x T= 0.070¢T= 0.100'" T= 0.200• steady

solution

----.--~

C A ...~A. I

o +----r- -.....,...-1",:=." , , • ..~t"

9J.oo 0.2U 0.40 0.60 O. no 1.00

Xl

o

'"o

oCD

o

o

'"~oJ:lCW:1: 01- ..

o


0

(!) T= 0.010A T= 0.030+ T= O.OSOx T= 0.0'10¢ T= O. 100'" T= O. 200• steady

solution

,-0.00 I. UUU.ooOdUU.2U

o

~I--.--.,-I-I-.-,-~-.

d.ooX,

(b) variation of U2

Figure 8.8: Evolutions of temperature and Uz at Xz = 0.5 (Ra = 104).


8 ~ ---,

1.000·60

<!> T= 0.010A T= O. 030+ T= 0.050X T= 0.070~T= 0·100'1' T= O. 200• steady

solution

0-40O. 20-"--,--'--,.--r----,--,

oN

c:i

oo

91.00

o...c:i

oID

...J'<0J:a:l&J:1: 0>--.

c:i


C) T= O. 0 I 0AT= 0.030+ T= 0.050XT= 0.070~ .T = O. 100'1' T= 0.200• steady

solution

o

o r-----------------,lI'lrol

>.4J.~

UOo .rolOQ)

>

o

0.2U tI.~1J U·GO

o~ 1---.---. --,----,-,b.oo

-,.--.,---, -,--g.lIt1 I. UU

(b) variation of U2

Figure 8.9: Evolutions of temperature and U2 at X2 = 0.5 (Ra = lOS).


o

q = 0

0 = I e = 0

u = u = 01 2

on a 11 the boundaries

q =()


Natural Convection in a Nonrectangular Enclosure

As the second example, let us show numerical results for some nonrectangular enclosures. The geometry and boundary conditions are shown in Fig. 8.10 and Reddy'spaper [2]. The element mesh adopted in our computation is depicted in Fig. 8.1l.

First of all, we would like to show numerical results of steady problem. Plotsof the isotherms and the velocity vector fields for the natural convection in a nonrectangular enclosure are given in Fig. 8.12. Isotherms and velocity vectors based onunsteady solutions for Ra = 103 ,104 and 105 are presented in Figs 8.13,8.14 and 8.15,respectively.

Natural Convection in a Circular Annulus

Let us analyze natural convection problems in an annulus between two concentriccircular cylinders as shown in Fig. 8.16. The ratio of the outer to the inner radii istaken to be ro/r; = 2.6 following Reddy's paper [2]. Taking into consideration thesymmetry with respect to the vertical centreline of the problem, we may consider onlyhalf of the annulus in our analysis. The geometry and boundary conditions are shown



in Fig. 8.16 and the element mesh of the problem is shown in Fig. 8.17.Plots of steady isotherms and velocity vectors for each Rayleigh number are shown

in Fig. 8.18 (a), (b) and (c). Time-dependent solutions for each Rayleigh number arepresented in Figs 8.19, 8.20 and 8.21.

8.6 Conclusions

An integral equation analysis is presented for the numerical solution of the coupledsystem of the Navier-Stokes equations and the energy equation governing laminar motion of an incompressible viscous fluid. Integral equation formulations are given forboth the steady and the unsteady problems in terms of the primitive variables, whichare the velocity vector, the temperature and the pressure. The related fundamentalsolutions for each problem are constructed explicitly. Numerical results are presentedfor the two-dimensional square cavity and non-rectangular enclosure problems and aconcentric cylindrical annulus problem at Ra = 103 ,104 and 105

• It is noted that theglobal behaviour of solutions obtained by our method is very accurate despite the useof comparatively coarse meshes.


velocity vectorsisotherms (a)

I /I

\ \\

, "

- .......... ,

\\ \

/I I

// /

: _L_-------'...;.. :::

L-':"'-""::'- --

\ I

" \ I

II

\\ \

II /

II

"" -I \

I /

II I

isotherms (b) velocity vectors, / - -

~ -I , I

I I\ I

\ ~ ::::. ~ - - l/ Y \ II / - - - - - ---/

\/ I

I " - - - - - -I I \ I

, .. - - - ,I I

I \ - - - - , -- I I II \ - - "- - - / / I

\ - - - - - - --- I ,I - - .. - - - - ,

isotherms (c) velocity vectors

Figure 8.12: Numerical results for a nonrectangular enclosure (a) Ra = 103; (b) Ra = 104

;

(c) Ra = 105.


T= 0.01

T= O. 07

T= O. 01

:..

, ,-, , ", \ "

. \ ... - - - ...

T= 0.07

T= 0.03

T= O. 10

(a) isotherms

:..-----:-.-.-.:

, , -, , ..I , "

T= D.03

, , "I , ,.

, ... -. - - -

T= n. 10


T= 0.05

T= 0.20

,~.-,--;--:--'.. .

, , -I , ••

I • _

T= 0.05

, , II

I I \\

, -. - - -

T= U.2D



T= o. 01 T= 0.03 T= 0.05

T= O. 07 T= O. 10

(a) isotherms

T= O. 20

I I I IeI I

, I

I I, I

I \

I ,~ ~ ..

/ ---- -I I / - - .... ....I I I " \ \

I \ \ - / I I 1/ I, \ .... - -. .... -- - - -

/ ------ ...I 1/_. ........ , \1 I I II ... \ \\ \ I

'\1 __ "1111

,\ ...... ---_///1

· .... -- - - -

T= 0.01 T= 0.03 T= 0.05

~., .. 17· . , . .. . . .

· . . . .

: I :. ~ .. _ , ..!- I I' _1// ...... "

I I I " _ .. .. \\ I,11 __ ""1

... _----- ....'-----/ I

, -------, I

J / -' , \ I

I , I II ". .... \\, I I

,11"' __ '"//, \

,--------

' _//-----.".

I I I I' , ..... \\ \ I I

,11 '11//

,\ ...... -----/ I- I I

1\ ...... _

T= 0.07 T= O. 10 T= 0.20




T= 0.01

T= 0.07

T= 0.03

T= O. 10(a) i otherm

T= 0.05

T= 0.20

I I\ I \ I

I\~

.: I ,~ . ~ ~ \ I I I ~ .,:,f \ II ... - I / --......-- --- \ I I I -..-...-- - --", \ II . "

, . . I / , - - - - - .. I/ "- -- - - ~ \I - "

I I . .. .. II . .. - . II \ I \

,\, - ~ - - ~ - - I I , ~ - ~

, · - / II , ,,- - \ \ -- -- - -/ I 0 I \ -- - - --/ I •1_

" - \- .- - - -- -., , '- .- - - ---~ I.. - . ·T= O. 01 T= O. 03 T= o. as

: : :., .. , , , I I .. ,I , I I

\ I. / , I

I , ,. ..vf \ I I I .. !JJf I I I , ,. !JJf \ I. I-~-- --" \ I

. I - -- - - -~,. \ I0 I -~-- --,;" \ II '- - - - -- \

I "- - - - - - II I "- - - - - - \I . .. - ,

II . .. . - . I

I . .. . - ,I\

~ - - - . - I II . - - . . - I I

I ~ - - . · - I II \ -- - -- I \ - - - I \ --/ I I -- -/ I • -- - --/ I I

,- -- - - - -- /

T= O. 07, - - .. -

T= O. 10(b) velocity vector

t ..... /

T= O· 20



u =01

u =02

o =0




(a) (b) (c)isotherms

1//.1\ "\,\ ,\ -"'" \ ,... ;\\, ,\\

\ \ \ ' \ '

\\\\\.0\11 'II I \ , . , I I I'II ,

1/ '-'1/ '\

, I;:'; /, ., I' •I I •

"

. ,, ," ...I, '

1\ " '-'.II, \ "1\ ,. \ I

" - " ,,\ ,". \ \ ' . ,,, \ \\\ \ \ \ . , II I'

1111"111 1 '

" I ,1/ • , II, / I "

, '1/. ,. ,:,~" I, ..,It'...

(a) (b) (c)velocity vector

Figure8.18: Steady isotherms and velocity vectors (a) Ra = 103; (b) Ra = 104; (c) Ra = 105,


T= o. 01

T= 0.10

T= 0.03

T= O. 15

T= O. 05

T= O. 20

T= O. 07

T= O. 30

(a) i otherm

T= O. 01 T= 0.03 T= 0.05 T= O. 07

T= O. 10 T= O.IS T= 0.20

(b) v !ocity vector

T= O. 30



T= o. 01

T= O. 10

T= 0.03

T= O. 15

T= 0.05

T= O. 20

T= 0.07

T= O. 30

(a) i otherm

T= o. 01

T= 0.10

T= 0.03

T= O. 15

T= 0.05

T= 0.20

T= 0.07

T= 0.30

(b) velocity v ctor



T= 0.01

T:: O. 10

T= 0.03

T= O. 15

T= 0.05

T= 0.20

T= 0.07

T= O. 30

(a) isotherm

T= 0.01

T= O. 10

T= 0.03

T= O. 15

T= 0.05

T= O. 20

T= 0.07

T= O. 30

(b) velocity vector



Acknowledgements

We wish to acknowledge the help of Mr. H. Sato in proof-reading the manuscript andfor typing it.

References

1. Phillips, T.N. Natural Convection in an Enclosed Cavity, J. of ComputationalPhysics, 54, pp. 365-381, 1984.

2. Reddy, J.N. Penalty-Finite Element Methods in Conduction and ConvectionHeat Transfer, Chapter 6, Numerical Methods in Heat Transfer, Vol II. (Eds.R.W. Lewis, K. Morgan & B.A. Schrefler), pp. 145-178, John Wiley & Sons,1983.

3. Tosaka, N. & N. Fukushima, Integral equation analysis of laminar natural convection problems, in: Boundary Element VIII, Vol. II, (Eds. C.A. Brebbia & M.Tanaka), pp. 803-812, Springer-Verlag, 1986.

4. Tosaka, N. & N. Fukushima, Integral equation analysis of laminar natural convection problems, in: Theory and Applications of Boundary Element Methods,(Eds. M. Tanaka & Q.H. Du), pp. 123-132, Pergamon Press, 1987.

5. Tosaka, N. & K. Onishi, Integral equation method for thermal fluid flow problems, in: Computational Mechanics '86, Vol. 2 (Eds. G. Yagawa & S.N. Atluri),pp. XI-I03-XI-108, Springer-Verlag, 1986.

6. Tosaka, N. & N. Fukushima, Numerical simulations of laminar natural convectionproblems by the integral equation method, in: Numerical Methods in ThermalProblems, Vol. V Part 1 (Eds. R.W. Lewis, K. Morgan & W.G. Habashi), pp.501-511, Pineridge Press, 1987.

7. G. de Vahl Davis & J.P. Jones, Natural Convection in a Square Cavity: AComparison Exercise, Int. J. Numer. Meth. Fluids, Vol. pp. 227-248, 1983.

8. Tosaka, N., Integral Equation Formulations with the Primitive Variables forIncompressible Viscous Fluid Flow Problems, Computational Mechanics, Vol. 4,pp. 89-103, 1989.

9. Onishi, K., T. Kuroki & M. Tanaka, An Application of a Boundary ElementMethod to Natural Convection, Appl. Math. ModeL, Vol. 8, pp. 383-390, 1984.

10. Kitagawa, K., C.A. Brebbia, L.C. Wrobel & M. Tanaka, Viscous flow analysisincluding thermal convection, in: Boundary Elements IX, Vol. 3 (Eds. C.A.Brebbia, W.L. Wendland & G. Kuhn), pp. 459-476, Springer-Verlag, 1987.

11. Kuroki, T., K. Onishi & N. Tosaka, Thermal fluid flow with velocity evaluationusing boundary element and penalty function method, in: Boundary ElementsVII, Vol. 2, (Eds. C.A. Brebbia & G. Maier), pp. 2-107-2-114, Springer-Verlag,1985.


2. Skerget, P., A. Alujevic, G. Kuhn & C.A. Brebbia, Natural convection flowproblems by BEM, in: Boundary Elements IX, Vol. 3 (Eds. C.A. Brebbia, W.L.Wendland & G. Kuhn), pp. 401-417, Springer-Verlag, 1987.

3. Onishi, K., T. Kuroki & N. Tosaka, Further Development of BEM in ThermalFluid Dynamics, Chapter9, Boundary Element Methods in Nonlinear Fluid Dynamics, Developments in Boundary Element Methods 6, (Eds. P.K. Banerjee &L. Morino), pp. 319-345, Elsevier Applied Science, 1990.

Chapter 9

Improperly Posed ProblemsTransfer

in Heat

D.B. InghamDepartment of Applied Mathematical Studies, The University of Leeds,Leeds, LS2 9JT England

Abstract

In this chapter the numerical solution of two inverse Laplace type problems which naturally occur in heat transfer and are improperly posed are investigated. Three differentmathematical models, namely direct, least squares and minimal energy methods, arepresented for the two problems. The Boundary Element Method is employed and itis found that the minimal energy method always gives a good, stable approximationto the solution, whereas the direct and least squares methods do not.

9.1 Introduction

One may regard a problem as being well posed if a unique solution exists whichdepends continuously on the data, otherwise it is improperly posed. In order to givea more precise definition one must indicate in what space the solution is to lie, aswell as a measure of the continuous dependence. In solving the Laplace equation,which describes a steady heat conductive problem for a physical variable, say T, thenif either T or oT/ on is specified at all points on the boundary of a region (T must bespecified at least at one point on the boundary), then T can be uniquely determinedat all interior points of the region. This class of problems can be solved using eitherFinite Difference, Finite Element or Boundary Element Methods. Lavrentiev [12]discussed bounded solutions of the Laplace equation in a special two-dimensionaldomain such that the Cauchy data is continuous. Further, Payne [14],[15] obtainedsolutions of more general second-order elliptic equations. Whilst Han [8] studied anenergy bounded solution of second-order elliptic equations and he proved this solutionis dependent on the Cauchy data being continuous. Falk and Monk [5] investigatederror estimates for a regularisation method for approximating the Cauchy problem forPoisson's equation on a rectangle. However, in numerous experimental situations it isnot always found to be possible to specify a boundary condition at all points on theboundary of the region. For example, in heat transfer problems many experimental

(9.1 )


impediments may arise in measuring or producing given boundary conditions. Thephysical situation at the boundary may be unsuitable for attaching a sensor or theaccuracy of a boundary measurement may be seriously impaired by the presence ofthe sensor. Frequently it is possible to determine, or specify, either the function TorfJTIon (i.e. the temperature or the heat flux) on part of the boundary of the regionand to be unable to give any information on the remaining part of the boundary.Clearly this is insufficient information in order to determine the function T everywherewithin the region of space. Experimentally however, in heat transfer applications,extra sensors may be inserted into the interior region of interest and the temperaturemeasured at these locations, in order to provide more information. The question thenarises as to whether, given Tor oTlon on part of the boundary and T at a numberof interior points of the domain, it is possible to determine uniquely the temperaturedistribution within the region of interest. One of the aims of this chapter is to usethe Boundary Element Method (BEM) to obtain the numerical solution to this classof improperly posed problems and we will refer to this as problem I.

In order to illustrate the numerical procedures for solving problem I, all the calculations have been performed in a square region, with either T or aTIon given on3 sides of the square. Further interior information has been given on a straight line.Extension of the work to more irregular shaped boundaries and to the interior information being given at random positions is straightforward and some solutions havebeen found in these situations. Therefore we let n c R2 be a square, such that eachside is of unit length, on = flU f 2, where f 1 is one side of the square and f 2 denotesthe other three sides. For the purpose of illustrating the solution procedures we letf o = {0.25 ~ x ~ 0.75, y = Yo} C n, where Yo is a preassigned value such that0< Yo < 1, see Fig. 9.1, and on n we consider a steady heat conduction problem

1::~~: ¢(x,y)(o' 8T/8n(x,y) ~,p) ;:,:; E r"

T(x,y) = g(x,y) (x,y) E fo.

Another problem, II, that we will consider in this chapter is the steady statesolution of the nonlinear heat conduction equation,

\7 . (J(T) '\'T) = 0 in n (9.2)

where f(T) is the thermal conductivity of the body which is temperature dependent.We will assume that at every point on the surface of the body either the temperatureor the heat flux is prescribed, although it is easy to extend the analysis to includelinear combinations of these quantities or even nonlinear boundary conditions, e.g.radiative conditions. If f(T) is known then the techniques as described in Inghamand Kelmanson [10] may be applied. However, frequently in practice the detailedvariation of the thermal conductivity with temperature is unknown but extra information in the interior of the body is known, e.g. the temperature may be measuredat a number of points within the body. This phenomena also falls into the generalclass of problems known as improperly posed heat conduction problems since more


r

J.

Figure 9.1: The square solution domain.

conditions are specified on a problem than that which is normally required but thereis an unknown function within the governing equation.

In this chapter we will show how the BEM may be modified in order to solveEquation (9.2) subject to the conditions

T(x,y) = </>(x,y) on an

}aT(9.3)

an (x, y) = 1/;(x, y) on an

T(x,y) =g(x,y) on r (9.4)

where </>,1/; and 9 are given functions and r is a set of interior points to the boundaryan.

A transformation of Equation (9.2) is employed such that all the nonlinear aspectsof the problem are transferred to the boundary of the solution domain. The functionf(T) is then represented by a piecewise quadratic function f(T) and a modified BEMdeveloped. In this chapter all the calculations have been performed in a square regionwith T given on the boundary an, but the extension to boundaries of arbitrary givenshape is trivial.

Both problems I and II are typical examples of the kinds of inverse ill-posed problems that occur in heat transfer. The systematic study of such problems for partialdifferential equations is of rather recent origin although consideration was already being given to such questions in the middle and latter half of the nineteenth century. Agood historical review of this work may be found in the book by Payne [16]. However,a more up to date review may be found in the excellent book by Beck, Blackwell and


St. Clair [1] but, unfortunately, most of the previous work in this field has mainlyconcentrated upon the unsteady heat conduction equation.

9.2 Formulation

There is an extensive range of published literature giving detailed descriptions of thevarious BEM formulations for obtaining solutions to plane potential boundary-valueproblems and for an up to-date state of the art in the method, see the previous chaptersin this book and the annual publication of the international conference organised byBrebbia (see Brebbia [3] and Tanaka et al. [17]). The fundamental basis of the BEMis Green's Integral Formula, see for example Jaswon and Symm [11], which, for anysufficiently smooth function, say the temperature T, which satisfies Laplace's equationin a plane domain n having a piecewise smooth boundary an, may be expressed as

r {T(q) in' Ip - q I -T' (q) In Ip - q I }dq = 1] (p) T(p) (9.5)Jon

where,

i) pEn + an, q E an

ii) dq denotes the differential increment of an at q

iii) the prime (') denotes the derivative in the direction of the outward normal toan at q

iv) if pEn then 1] = 211", and if p E an then 1] is the angle included between thetangents to an on either side of p.

If either T, T' or a linear combination of T and T' is prescribed at each point of anthen the solution of the integral equation,

J{T(q) In' Iq - q I -T' (q) In Iq - q I }dq -1] (q)T(q) = 0, q, q E an (9.6)

determines T and T' at each point of an. The temperature T at any point p E (n+an)can then be computed employing Green's Integral Formula, Eq. (9.5).

Thus, the application of Green's Boundary Formula, Eq. (9.6), enables a wellposed two-dimensional Laplacian boundary-value problem to be reformulated as anintegral equation in which the unknowns are the boundary-values of the temperatureT and its normal derivative T' complementary to those prescribed by the boundaryconditions. In practice the integral Eq. (9.6) can rarely be solved analytically. Consequently, various numerical techniques have been proposed in order to enable thedetermination of a solution.

In the classical BEM (CBEM), Fairweather et al. [4], the boundary an is firstsubdivided into N smooth intervals anj , j = 1, .... ,4N, on which T and T' areapproximated by piecewise-constant functions Tj and T;. Then, the correspondingdiscretized form of Green's Integral Formula,


4N

L {Tj i In'j=l &OJ

Ip - q Idq - T; r in Ip - q Idq} = 7J (p)T(p),Jaoj

pEn + an, q Ean (9.7)

is collocated at the midpoint qj of each interval.This generates a system of linear algebraic equations in the unknown Tj and T;.

Solution of this system of linear algebraic equations determines the values of both Tj

and T; on each interval. The temperature at any interior point may then be computedby a relatively simple quadrature, Eq. (9.7), if required.

The linear BEM (LBEM), Harrington et al. [9J, affords a slightly more sophisticated approximation of Green's Integral Formula than the classical BEM. On eachinterval anj , j = 1, .... ,4N, T and T' are approximated by piecewise-linear functions

T = (1 - 0 T (qj) + ~T(qj+l)

T' = (1 - 0 T' (qj) +~T'(qj+d

where qj and qj+l are the endpoints of an j , and ~ is a linear function which increasesfrom zero at qj to unity at qj+l' With these approximations Green's Integral Formulabecomes,

4N

L{Tj r .(1-0 In' Ip-qldq+Tj+l r . ~ In'lp-q!dq}j=l JaoJ JaoJ

4N

- L {T; 1 (1 - 0 In Ip - q I dq +T;+ll . ~ In Ip - q Idq}j=l aOJ &OJ

= 7J(p) T(p), pEn +an, q Ean (9.8)

where Tj and T; denotes T(qj) and T' (qj), respectively. A system of linear algebraicequations in the unknown Tj and T; is then generated by enforcing Eq. (9.8) at eachof the points qj.

More acurate approximation of the solution to the boundary integral equation canbe obtained using the quadratic BEM (QBEM). In this approach, on each intervalanj,j = 1, ... .,4N,T and T' are approximated by piecewise-quadratic functions,

T = M1 (~) T (q2j-d +M2 (~) T (q2j) +M3 (0 T (q2j+d

T' = M1 (0 T' (q2j-d +M2 (0 T' (q2j) +M3 (0 T' (q2j+l)

where q2j-l and q2j+l are the endpoints of an j,q2j is the midpoint of anj,~ is a linearfunction which increases from zero at q2j-l to unity at q;j+l and

M 1(0 = 1 - 3~ +2e

M 2 (O = 4~ - 4eM3 (O = -~-2e

On the basis of these approximations Green's Integral Formula becomes,


4N

L {T2j- 1 l M1 (~) In' Ip - q Idq +T2j l M2 (O In' Ip - q I dq~l 00) ~

+T2j+1 100. M3 (0 In' Ip - q Idq})

4N-f; {T~j_l fan) M 1 (0 In Ip - q Idq + T~j fanj M2 {O In Ip - q I dq

+ T~j+I [ M3 (0 In I p - q I dq} = 7J{p) T (p), pEn +an,q EOn. (9.9)JonjA system of linear algebraic equations in the unknown Tj and T; is then generated byapplying formula (9.9) to each of the points qj,j = 1, ... .8N. Thus for a 4N intervaldiscretization, the QBEM requires the solution of 8N equations in 8N unknowns,whereas the CBEM and LBEM methods only require the solution of 4N equations in4N unknowns.

If the interval anj is a straight line segment, then the integrals occuring in theformulae (9.7-9.9) can be evaluated exactly and the details can be found in Manzoor[13]. The evaluation of the integrals occurring in LBEM and QBEM formulationsrequires only a fraction of the computational time taken by an accurate numericaltechnique, and since, for a 4N interval discretization, each of the integrals has tobe evaluated 4N times for every point to which Green's Integral Formula is applied,it is apparent that the use of these analytical expressions will facilitate appreciablereductions in the computational times required by the LBEM and QBEM.

9.3 Non-Linear Formulation

In this section we will consider the non-linear elliptic Eq. (9.2) and the first step insolving this equation using a BEM formulation is the introduction of the transformedvariable A which satisfies

\7 A = j(T)\7T (9.10)

Equation (9.10) is a form of the Kirchhoff transformation and may be justified bynoting that the curl of the right hand side is identically zero for any functions j andT. Then, from Eqs (9.2) and (9.10), A satisfies Laplace's equation

(9.11)

The application of the BEM method to the solution of Eq. (9.11) is now as describedin the previous section.

If either T or T' is prescribed at each point q Ean then the solution of theboundary integral equation obtained by letting p = q E an in Eq. (9.5) determinesthe boundary distribution of both T and T'. Equation (9.5) with A replacing T, maynow be used to generate the solution A(p) at any point pEn +an.

Defining h(T) by


h(T) = JT f(f3) df3 (9.12)

and employing Eq. (9.10), we may write the Kirchhoff transformation in the form

A = h(T), A' =T' f(T) (9.13)

relating the original and transformed BEM variables. Combining Eqs (9.5) and (9.13)then gives

Joo {h[T(q)] In' Ip - q I - T' (q) f[T(q)] In Ip - q I}dq

-17(P)h[T(p)] =0 ; pEO+aO,qEaO (9.14)

as the non-linear integral equation on ao. Iterative solution of this equation (plusprescribed boundary conditions) constitutes the BEM solution to this problem.

Although the formulation is applicable to problems containing a general boundedfunction f(T), we shall restrict our study to a physical problem in heat transfer. Inthis case the function I is the thermal conductivity of the medium and it is usuallyobtained on the basis of experimental results which provide some form of empiricalrelationship with T. The present formulation permits any bounded variation of Iwith T but we shall illustrate the method with a particular example in which I is asimple function of T.

9.4 Existence of Solution of Problem I

Instead of considering the problem I we investigate the solution of the following relatedproblem:

r'T~Oinn,

T=</> on f 2 , (9.15)

lu-gl::;c: on f o,

where f > 0 is a preassigned small quantity. Clearly the solution of problem (9.15)may be considered as an approximation to the solution of the problem I. In this section we will prove there exists a solution to problem (9.15).

Lemma: Let 0 1 and n 2 be two rectangular domains such that 0 1 n O2= 0, and01n02 = f is a straight line (as show in Fig. 9.2). Let 11 (x, y) and !2(x, y) be harmonicfunctions in 0 1 and O2 which are continuously differentiable in 0 1 U f and O2 U f,respectively. If for all (x,y) E f,/dx,y) = 12(X,y) and al1(x,y)/ay = aI2(x,y)/aythen

{

!1(X,y) (x,y) E n1

F(x,y) = 12(X,y) (x,y) E O2

!1(X,y) = h(x,y) (x,y) E f(9.16)


A

r

Figure 9.2: The existence domain.

is an harmonic function in 0 1 U O2 U f*, where f* is f without its end points.

Proof: For any simple closed curve C which is divided into C1 C 0 1 and C2 C O2

by f. We let, denote the part of f which is common to C1 and C2 and the directionsof the curves C1 U , and C2 U ,are as shown by Fig. 9.2.

Let (xo, Yo) be an interior point of C1 U , then using Green's Formula we obtain

27r Il(xo,Yo) = r Il(x,y) 8G(xo,Yo;x,y)/8n ds-}C1U'"!

r 8II (x,y)/8n G(xo,yojX,y) ds}C2 U'"!

r 8h(x,y)/8n G(Xo,Yoix,y) ds}C2U'"!

(9.17)

(9.18)

where G(xo, Yo; x, y) = In J(x - xoF + (y - YoF is Green's function. Adding (9.17)and (9.18), we have, since the integrations along, cancel, that

27r II (xo, Yo) =LF(x, y) 8G(xo, Yo; x, y)/ 8n ds

L8F(x,y)/8n G(xo, Yo; x,y) ds (9.19)

Similarly, if (xo, Yo) is an interior point of C2 U " using Green's Formula, we have


211'h(xo,Yo) = !cF(x,y) oC(xo,Yo;x,y)/on ds

!c of(x,y)/on C(xo, Yo; x,y) ds (9.20)

(9.21 )

Now for any point (xo, Yo) which is the interior to C, and since F( x, y) is continuouson C, then the function

9(Xo, Yo) = 2111' !c F(x, y) oC(xo, Yo; x, y)/on ds

2- [ of(x,y)/on C(xo,Yo;x,y) ds211' Jc

is harmonic inside C. Hence

{

h(xo,yo) (xo,YO)E 0 1

9(Xo, Yo) = h(xo, Yo) (xo, Yo) E O2

f1(XO,YO) = h(xo, Yo) (xo,Yo) E /

and 9(X,y) == F(x,y) in C. Because C is an arbitrary curve we have 9(X,y) == F(x,y)in 0 1 U O2 U f·, which completes the proof.

We now proceed to prove the existence of the solution of the problem (9.15). Firstof all, we extend the line f o smoothly such that it meets f 2 at A and B, and this thendivides domain 0 into 0 1 and O2. Let us denote the extension of f o by 1'0' SinceCOO(ro) is the density in C2(rO) we can take a function 91(X,y) E COO(ro) whichsatisfies 191 Iro -9 I::; (/2. On the domain 01 we now consider the boundary problem

{

\J2T = 0T Ir2 = <P

T Iro= 91

It is well known that problem (9.21) has a unique solution Ut, see Gilbarg et ai. [6],and we know oTt!an on a01• Then on the domain 1\ we consider another problem,namely,

{

\J2T = 0T Ir2 = 1>Tiro = 91 (9.22)oT/on Iro = oTt!on Iro

There exists a solution of problem (9.21), see Hadamard [7], and we denote it by T2•

We now let(x,y) E 0 1

(x, y) E O2

(x,y) E 1'0

(9.23)

and in view of the lemma, we know that T satisfies Laplace's equation in 0 andT Ir2= <P, 1Tiro -9 I::; t. Thus we have proved that T is a solution of the problem(9.15).


9.5 Mathematical Models for the Solution of Problem I

We suppose that an is divided into 4N segments, where N segments belong to r1

and 3N to r 2 . Thus the CBEM gives 4N equations and 5N unknown variables. Inorder to solve this system of equations we must add N extra equations. These areobtained by using the known values of T(x), on ro, in Eq. (9.7). The boundary rois subdivided into k segments and the function T(x) is assumed to be constant overeach segment and takes its midpoint value. Then, by Eq. (9.7), we have

4N 4N

L Elij Tj - L Glij T; = 27l" T(Xi) i = 1, ... 4Nj=1 j=1

and

where

4N 4N

"" E· T - "" G·· T' - 0L...J I) ) L...J I) )-

j=1 j=1

i=I,2, ... 4N

(9.24)

Gij fa in IXi - Y Idyallj

Eij 1 in' IXi - y Idy - T/(Xi)Oijall)(9.25)

Glij = kll) in IXi - Y Idy

Eljj = kll! in' I Xi - y Idy

where Xj E ro and Xi E ao and are the mid points of the ith segments.

Direct Method

In this case we take k = N and therefore Eq. (9.24) results in a set of 5N linearequations in 5N unknown variables. We may now directly solve this system of equations to obtain T; (j = 1,2, ... ,4N) and the unknown values of Tj . By inserting thesevalues into Eq. (9.7) we are able to obtain the function T anywhere in O.

Least Squares Method

As observed, problem I is an improperly posed problem and frequently the data g(x)on r° is obtained by measurement and hence there will be some error between themeasured values of the function g( x) and the solution T(x) on ro. In order to overcomethis difficulty we use a least squares approach.

We let H 1(0) denote the usual Sobolev space in domain 0, and

* 1H (0) = {v E H 1(0); V = <P

* 1/2 * 1/2H (aO) = {The trace of von ao; v EH (On,


* 1/2 * 1/2H (f1) = {The restriction of v on f 1 ; v EH (an)},

* 1/2for any t/J EH (f1), we let

- {</>t/J= t/J

_ * 1/2then t/J EH (an). We now consider the following boundary value problem

in n,on an. (9.26)

(9.27)

* 1We know that problem (9.26) has a unique weak solution such that T(x) EH (n),see Gilbarg and Trudinger [6], and in domain n the function T(x) is harmonic. Hencewe may define an operator

{

* 1/2A :H (f1 ) ----+ H 1(n),

* 1/2Av = T(x) Vv EH (f1 ),

* 1/2where T(x) is the weak solution of the problem. If there is a t/J EH (fd such thatAt/J = 9 or II At/J Ira -9 IIHI (ra) is sufficiently small, then use this t/J in order toobtain an approximate solution of the problem (9.1) by the BEM. Now we use theleast squares method to determine the function t/J, i.e. we find

(9.28)

As described earlier the boundary and the expression (9.28) are discretised. IfG = (Gij),E = (Eij) are 4N x 4N matrices, G1 = (G1ij ),E1 = (E1ij ) are k x 4Nmatrices, k ~ N, and ¢" 71> are 4N vectors, then we have

¢' = C- 1 E 71>,

21l' At/J Ira= E1 71> - G1 71>'.

Hence21l' At/J Ira= E1 71> - G1 G-1E 71>,

= (E1 - G1 G-1E) 71>

1and writing W = 21l' (E1 - G1 G-1 E), which is a k x 4N matrix, then

(9.29)

(9.30)

(9.31)


where A'ljJ Irois a k vector and WI and W2 are k X Nand k x3N matrices, respectively.After discretising, problem (9.28) is equivalent to

(9.32)

(9.33)

which is a linear system of N equations with N variables.Because we have not added any restrictions on the solution, the calculated results

show very little improvement over those obtained by using the direct method whichare very inaccurate.

Minimization of Energy

Instead of considering the problem (9.1) we investigate the solution of the followingrelated problem:

{

V2T = 0T=4>IT Iro -g 1:S E,

where E > 0 is a preassigned small quantity. Clearly the solution of problem (9.33)may be considered as an approximation to the solution of the problem (9.1).

We write

a(T,v) = J10 VT·Vvdxdy,

J(T) = ~ a(T, T) = ~ Jr I VT 12 dxdy,

2 2 Jn

- ~ { T ot ds- 2 Jon an '

where n is the outward normal to on. We know that J(T) describes the 'kineticenergy' of the steady field and hence J(T) may be called the energy functional.

• 1/2Now considering a subset of H (rd

• 1/2

I< = {v; v EH (fI), I Av Iro -g 1:S E},

where E > 0 is a small preassigned constant and from the definition of the operatorA, we obtain the subset of HI(n)

S=AK. (9.34)

Clearly S is a closed convex set in HI(n). If the solution of problem (9.33) existsthen S is not empty. So problem (9.33) is equivalent to the variational problem:

J(T) = in! J(v).v E S (9.35)


Han [8] has proved that there is a unique solution of (9.35) and that the solution issmooth in n if the functions <p and 9 are sufficiently smooth. The variational problem(9.35) is now equivalent to

J(A1f;) = in! J(A1jJ),1jJEK

which on discretisation becomes

(9.36)

(9.37)

= r ¢t G-1E ¢J ds,Jan

-T - - - T - - - .where rP = (rPI'rP2' .... 'rP4N) and we assume that 1jJ = (rPI, ... ,rPN) IS an unknownvariable. The constraint condition 1jJ E J( may be written

z.e.I (EI - GI G-1E)rP - 9 I::; f. (9.38)

(9.40)

If we denote the first k columns of the matrix (EI - GI G-1E) by Wand the remaining 4N - k columns by WI, then expression (9.38) becomes

and if 9 = 9 - WI¢> then we have

IW1jJ - 9 I::; f.. (9.39)

It is clear that min J(T) = min r ¢JTG-1 E ¢J ds is equivalent to min J(T) =Jan

-T - - - -min rP G-1E rP and the problem reduces to finding 1jJ = (rPI' ... , rPN) which satisfies

{

j(A~) = min rP-TG- 1 E¢J,

IW1jJ - 9 I::; f..

Solving the constrained minimal problem (9.40), using the Nag routine E04UCF, weobtain the function T(x) on rIo Then we may obtain an approximate solution of(9.1) by the use of the standard BEM. Some test problems are presented and theseindicate that accurate numerical solutions of problem (9.1) may be obtained usingthis technique.


9.6 Mathematical Model for the Solution of Problem II

Normally in the solution of Eq. (9.2) the thermal conductivity is specified and oneboundary condition is enforced at every point on the surface of the body. However,frequently in practice, the detailed variation of the thermal conductivity with temperature is unknown but extra information in the interior of the body is known, e.g.the temperature may be measured at a number of points within the body. This phenomena falls into the general class of problems known as an inverse heat conductionproblem since extra conditions are specified on a problem than that which is normallyrequired but there is an unknown function within the governing equation.

In order to obtain the numerical solution of this problem we first use a k-piecewisequadratic function ](T) to represent the unknown function f(T). In view of the maximal principle there are points PI and pz E an such that

T(PI) = mm T(p) = mpEn

T(pz) = max T(p) = MpEn

Hence, the defining region of f(T) is a closed interval [m, M], and this is also true forthe function 7(T). Subdividing [m, M] into k equal intervals with the mesh points atm = to, t l , ... , tk-l, tk = M, then ](T) may be written in the form

where ai, bi and Ci, i = 1, ... k, are constants to be determined. In the transformation(9.12) we let ](T) represent f(t) and then we obtain a piecewise third-order polynomial, say h(t), which may be written in the form

_ { ~aIT3+!bl~2+CIT+dlh(T) = :

~ akT3+! bkTz +Ck T +dk

where dj , i = 1, ... k, are to be determined integral constants of the transformation(9.12)

Inserting the functions] and h into the linear system (9.14), we obtain

i=1, ... ,4N (9.41)

which includes 4N equations and 4N + 4k unknown variables. In order to solveproblem (9.41) we need 4k extra equations. The continuity of ](T), ](T) and h(T)at the mesh points, except the two end points T = m and T = M, may offer 3k - 3equations, and clearly d1 = 0 is another equation. Since (9.41) is a homogeneoussystem of equations we need some extra condition to solve it, hence ](T) has to be


fixed at one point. However, if only one point is fixed the accuracy of the solution isnot good, so we suppose that ](T) is fixed at two mesh points, for example at thetwo end points m and M. If the extra information on the problem is supplied on theline r which contains k points, then this will result in the remaining k equations.Combining all the conditions above, and (9.41), we obtain a new solvable system ofequations

E jj h(Tj ) - G jj Tj](Tj ) = 0

Eljj h(Tj ) - Gljj Tj](Tj ) =h(Tj )

aj t; +bj tj +Ci = ai+! t; +bj+! tj +c;+!

2aj t j +bi = 2aj+t t j +bj+1

~ aj t~ + ~ bj t; +Ci ti +di =

~ aj+!t~ + ~ bj+! t; +Cj+l tj +di+!

i = 1, ... ,4N

i=l, ... ,k

i=1, ... ,k-1 (9.42)

(9.43)

alm2 +btm +Cl = s, akM2 +bkM +Ck = 5

where s = f(m) and 5 = f(M). Unfortunately, it is not possible to obtain an accuratesolution by solving this problem directly and hence we introduce the minimal energymethod.

Instead of considering the original problem we investigate the solution of thefollowing problem

{

'\7.(J(T)"VT) = 0 in nT = 0 on anIT Ir -g I:S f on r

where f > 0 is a pressigned small quantity. Clearly the solution of problem (9.43)may be considered as an approximate solution of the problem with f = O. Insertingtransformation (9.12) into (9.43) we obtain

Again we write

{

"V2h(T) = 0 in nh(T) = h(</J) on anh(g - f) :S h(T) :S h(g +f) on r

a(u, v) = j ~ "Vu· \7v dxdy

J(u)=!a(u,u)=!j r l\7ul 2 dxdy2 2 In

= ! r u au ds2 Jan an

(9.44)(9.45)(9.46)


where J(u) is called the energy functional of the Laplace equation

inn

{

V'2u = 0

u(x,y) = <p(x,y) on 00(9.47)

It is known that the Laplace Eq. (9.47) is equivalent to the following minimal problem

J(u) = in! J(v)v E H*(O)

where H*(O) = {Uf H1(0); U 180 = <p}, and H1(0) is the normal Sobolev space.Similarly, the problem (9.44-9.46) is equivalent to the minimal problem

J(h(T)) = _ ini J(h(T))h(T) E Y*(O)

where hE C2 [m, M], and T E H 1(0), which on discretisation becomes

4N r aJ(h(T)) = i~1 Jan h(<Pi) an h(<pj) ds

4N

=L: 180 h(<Pj) a;/ Ekj h(<Pj) dsI,J=1

Inserting h(T) into (9.48) we obtain

(9.48)

(9.49)4N

J(h(T)) = L: fa h(<Pi) Gii} Ekj h(<pJ dsi,j=1 an

m 4N

It is clear that inf ij;l fan h(<Pi) Gii/ Ekj h(<Pj) ds is equivalent to inf ij;l h(<Pi) a;,}

Ekjh(<Pj), and we set the all the continuity conditions on ](T) and the inequality(9.46) as the constraining conditions and then the problem reduces to finding h whichsatisfies

!J(h(T)) = inf i~l h(<Pi) Gi/ Ekj h(<pj)

4N (9.50)h(tPi - f) :s L:Wij h(<pj) :S h(tPi + f) i = 1, ... , k

j=l

where Wij = Elij - Gli{ G~ Emj , tPi = T(Pi) and Pi E r. Solving the constrainedminimal problem (9.50), using the Nag routine E04UCF, we obtain the functionsh(T) and ](T). Then we may obtain an approximate solution of the problem by theuse of the BEM.


9.7 Solutions of Some Test Examples for Problem I

We described three methods of solving the problem (9.1) and we now give two examples to illustrate the accuracy of these methods. In one of the problems we comparethe results with a known analytical solution.

First we consider a problem for which there is a very simple solution, namely,T(x,y) = x2 - y2 and impose the following boundary conditions:

{

I - y2 X = 1, 0::; y ::; 1,¢J( x, y) = x2 - 1 y = 1, 0::; x ::; 1,

_y2 X = 0, 0::; y ::; 1,

g(x) = x2 - 0.25, x E f o,

where f 0 = {0.25 ::; x ::; 0.75, y = Yo} and Yo = 0.1,0.3,0.5, 0.7 and 0.9. Solutions areobtained with N = 10,20 and 40, where N denotes the number of segments on eachside of the square.

Figures 9.3-9.5 show the lines of constant T for Yo = 0.5 by using the direct,the least squares and the minimal energy methods, respectively. The first two setsof figures show that the solution, using the direct and the least squares methods,are not converging to the analytical solution as the value of N increases. In fact asthe value of N increases the solution near y = 0 deteriorates. In the case of thedirect method the solution remains reasonable away from y = 0 as N increases but ony = 0 the solution oscillates violently with errors of the order of ±106 being typicalwhen N = 40. For the least squares method the solution with N = 10 is reasonableexcept close to y = O. However, when N = 20 and 40 the predicted solution showsno similarity with the analytical solution with predicted values of u being 0(104

) ormore over a large region of the solution space. On the other hand, we observe fromFig. 9.5 that the minimal energy method gives a good solution everywhere and thesolution becomes more accurate as the value of N increases. The variation of theenergy functional J (u) as a function of N for the three numerical methods and theanalytical solution are given in Table 9.1 for Yo = 0.5 and the observations regardingthe accuracy of the methods made above are again seen to be borne out. Further,Table 9.2 illustrates the energy of the system when N = 40 for all three methods.These results indicate that when using the minimal energy method, the location ofthe interior boundary conditions is not very important and the function J(T) may bepredicted to within about 0.05%. However, the energy function is very erratic in itsbehavior when using the other two methods and is substantially different from thevalue as predicted by the analytical solution.

Finally, we consider a problem for which there is no simple analytical solution,namely

~(x,y) ~ {y - 1, x = 1,sin(27l"x), y = 11 - y, x = 0

O::;y::;l,0::; x::; 1,O::;y::;l,


7 6 5

a.

N = 10

2

3

4

]( 1)=0.0l{ 2)=0. 21(3)=0.51(4)=0.81 (5 )c.=-O. 21( 6 )=-0.51(7)=-0.8

7 6 5

b.

N = 20

l( 1)=0.0l( 2 )=0. 21 (3 )=0.51( 4 )=0. 81(5)=-0.21(6 )=-0. 5l( 7 )=-0. 8

7 6 5

c.

N = 40

1( 1)=0.01(2)=0.21 (3 )=0. 51( 4 )=0. 81(5)=-0.21(6)=-0.51( 7 )=-0. 8

Figure 9.3: The lines of constant T using the direct method.


7 6 5

a.

N = 10

N = 20

N = 40

1(1)=0.0

1(2)=0.2

1(3) =0.5

1(4)=0.8

1(5)=-0.2

1(6)=-0.5

1(7)=-0.8

1 (l) =0.0

1(2)=10

1(3)=100

1(4)=1000

1(5)=5000

1(6)=10000

1(7)=-100

1(1)=0.0

1 (2) =500

1(3)=10000

1(4)=100000

1(5)=500000

1(6)=-500

1(7)=-100000.

Figure 9.4: The lines of constant T using the least squares method.


7 6 5

a.

N = 10

1( 1 )=0.01(2)=0.21 (3)=0. 51(4)=0.81(5)=-0.21( 6)=-0. 51(7)=-0.8

7 6 5

b.

N = 20

2

3

4

1( 1)=0.01( 2)=0.21(3)=0.51(4)=0.81(5)=-0.21(6)=-0.51( 7 )=-0.8

6 5

c.

N = 40

1( 1 )=0.01(2)=0.21 (3)=0. 51( 4 )=0. 81(5)=-0.21(6)=-0.51(7)=-0.8

Figure 9.5: The lines of contant T using the minimal energy method.


Table 9.1: The variation of J(T) as a function of N for y = 0.5.

N Direct Method Least Square Minimal Energy AnalyticalMethod Method Value

10 3420 347.5 2.6623 2.666720 6.35 x 107 2.84 X 107 2.6659 2.666740 8.07 x 1010 2.19 X 101

;:1 2.6665 2.6667

Table 9.2: The variation of J(T) as a function of Yo.

Yo Direct Method Least Square Minimal EnergyMethod Method

0.1 5.55 x 103 3.32 X 1014 2.66810.3 170.78 1.55 x 101~ 2.66810.5 8.07 x lOW 1.19 X 1014 2.66650.7 1.88 x 1011 1.82 X lOI.: 2.66810.9 1.21 x 108 1.48 X 1010 2.6681

Yo = 0.5,

g(x) = 0.5 - x, 0.25 ~ x ~ 0.75

and solve, using the three numerical methods with N = 10,20 and 40. Again thedirect and least squares methods give inaccurate solutions which decrease in accuracyas N increases. Figure 9.6 shows the lines of constant T using the minimal energymethod with N = 40 but the results obtained using N = 10 and 20 are graphicallyindistinguishable. It is clear that as N increases, the solution is converging and isstable when the minimal energy method is applied.

9.8 Solution of Some Test Examples for Problem II

In order to illustrate the method all the calculations have been performed in a squareregion n, and the set r contains n x n points which are evenly distributed in achecker-board formation in the solution domain n. Only one test example is givenin this chapter but numerous other examples have been investigated and the resultsgiven with the test example are typical of all other problems investigated.

Here we take a simple function as the test function, namely, f(T) = sin(T), T =arccos [~(X2 - y2)] and impose the following boundary conditions

{

arccos(x2/2) 0 ~ x ~ 1,</>(x )_ arccos(!-y2/2) O~y~l,

,y - arccos(x2/2 -!) 0 ~ x ~ 1,arccos( _y2 /2) 0 ~ y ~ 1,

y=Ox=lY = 1x=O


43:'1576

Figure 9.6: The lines of constant T where (a) shows the numerical solution with N = 20and (b) shows the analytical solution.

The purpose of choosing this test problem was to investigate how the technique dealswith functions f(T) not being monotonic functions. Solutions were obtained withN = 10,20 and 40, k = 10, and r containing 100 points. The function j(T) was fixedat two points, namely T = 7r /3 and T = 7r /2, and N = 10 and 20. In the case ofN = 40, the solution was obtained with j(T) being fixed at one point, T = 7r /2 andat two points, T = 7r/3 and 7r /2.

Figure 9.7 shows the variation of f(T) as a function of T and the numerical solution ](T) with N = 40 with both one and two points being fixed. It is clear thatin order to obtain an accurate solution at least two points must be fixed and thisobservation is typical of numerous other numerical results we have obtained. Table9.3 shows the values of the analytical solution and numerical solution with N = 10,20and 40, respectively, at mesh points of j(T). It is seen from this table that, as expected, as the value of N increases then the closer the numerical solution approachesthe analytical solution. Figure 9.8 shows the lines of constant T and Table 9.4 illustrates the temperature at some interior points, using the analytical solution and thenumerical solution with N = 40. A rough estimate shows that the relative errors areabout 0.1 % for f(T) and about 0.005% for all values of T. All the results show thatthe accuracy of the numerical solution is very good and it become more accurate asthe value of N increases.

9.9 Conclusions

In this chapter we have illustrated, by example, the use of the BEM in the solution ofthe Laplace equation for improperly posed problems. Three methods have been used,namely the direct, the least squares and the minimal energy. It has been found thatthe direct and least squares methods give reasonable results with a small number ofdiscretisations but the accuracy decreases as the number of discretisations increases.


1~3!.... 2 1

I )( 1!

T1.2 1.4 1.6 1.8 2.0

0.96

0.98

0.92

o.~

0.90

0.88

0.86~------------

f(T) 1.00

Figure 9.7: The solution f(T) where 1 is the analytical solution, 2 is the numerical solutionwith two points fixed and 3 denotes the solution with one point fixed.

(a) (b)

Figure 9.8: The lines of constant T where (a) shows the numerical solution with N=20 and(b) shows the analytical solution.


Table 9.3: The values of j(T) with N = 10, 20 and 40, respectively.

Mesh Numerical Solutions Analyticalpoints Solution(T) N= 40 N = 80 N = 160

1.40720 0.86571 0.86571 0.86571 0.866031.15192 0.91402 0.91253 0.91259 0.913351.25664 0.95339 0.94877 0.94993 0.951061.36136 0.98032 0.97645 0.97715 0.978151.46608 0.99580 0.99388 0.99405 0.994521.57080 0.99849 0.99849 0.99849 1.01.66552 0.99173 0.99306 0.99330 0.994521.78024 0.97375 0.97565 0.97641 0.978151.88495 0.94593 0.94776 0.94897 0.951061.98967 0.91110 0.91160 0.91214 0.913352.09439 0.86528 0.86528 0.86528 0.86603

Table 9.4: The values of T.

Numerical Analytical(x,y) Solution Solution

(0.25,0.25) 1.57079 1.57030(0.25,0.5) 1.66463 1.66468

(0.25,0.75) 1.82338 1.82348(0.5,0.25) 1.47693 1.47691(0.5,0.5) 1.57077 1.57030

(0.5,0.75) 1.72754 1.72769(0.75,0.25) 1.31811 1.31812(0.75,0.5) 1.41398 1.41390

(0.75,0.75) 1.57078 1.57030


However the minimal energy method always gives an accurate, convergent and stablesolution with an increasing accuracy as the number of discretisations increases. Thedirect and least squares methods do not give reasonable approximate solutions becausewe do not add any restrictions to the solution and Hadamard [7] affirmed that one hasto add some restrictions in order to obtain a solution to improperly posed problems.Numerous other examples have been tried and we always come to the same conclusionsas those obtained using the examples given here. Clearly even better results couldhave been obtained if all arithmetic had been performed using quadruple, rather thandouble, precision accuracy.

Finally it must be emphasised that although in this chapter we have found thatthe Boundary Element Method may be extended to deal with improperly posed problems the extension of Finite Difference and Finite Element Methods for dealing withsuch problems is not straightforward.

Acknowledgements

Some of the work presented in this chapter has been as a result of ongoing researchinto BEM at Leeds University. The author would like to acknowledge the contributions made by Professor Han and Yuan Yong.

References

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10. Ingham, D.B. and Kelmanson, M.A. Boundary Integral Equation Analysis ofSingular, Potential and Biharmonic Problems, Lecture Notes in Engineering,Springer-Verlag, Berlin, Heidelberg, New York and Tokyo, 1984.

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boundary element methods in heat transfer

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