solution of nonlinear reaction-diffusion equation by using...

Solution of Nonlinear Reaction-Diffusion Equation by Using DualReciprocity Boundary Element Method with Finite Difference or Least

Squares Method

G. Meral1,a M. Tezer-Sezgin1,b

1 Department of Mathematics, Middle East Technical University, 06531,Ankara, Turkey.

a [email protected], b [email protected]

Keywords: Dual Reciprocity Boundary Element Method, Finite Difference Method, Least SquaresMethod, Nonlinear Reaction-Diffusion Equation.

Abstract: In this study, the system of time dependent nonlinear reaction-diffusion equations issolved numerically by using the dual reciprocity boundary element method (DRBEM). As the timeintegration method both the finite difference method (FDM) with a relaxation parameter and the leastsquares method (LSM) are made use of.

The DRBEM is applied for spatial derivatives keeping the nonlinear term and the time derivativeas nonhomogenity. The resulting time dependent system of ordinary differential equations (ODE) issolved by using the FDM with a relaxation parameter as well as the LSM for obtaining accurate andcomputationally efficient results.

The computations are carried out for one nonlinear reaction-diffusion equation which has an exactsolution and for one system of reaction-diffusion equations. The solution obtained with both methodsagree well with the exact solution in the first example and with the other numerical results in thesecond example. The comparison of both time integration schemes shows that the FDM with a relax-ation parameter gives better accuracy when the optimal value of the parameter is used. This makesthe solution procedure time consuming and computationally expensive comparing to the LSM whichis a direct application.

1. Introduction

The nonlinear reaction diffusion equation as well as the system of nonlinear reaction-diffusionequations are very attractive in recent years, since they have practical applications in many fields ofscience and engineering.

A number of combined methods for the time dependent partial differential equations is appliedin the literature. For solving these problems classical methods discretize the spatial domain of theproblem with one of the known methods such as boundary element method(BEM), finite elementmethod(FEM), differential quadrature method(DQM) and finite difference method(FDM); then theresulting system of time dependent equations is solved by using the time integration schemes suchas FDM, RKM(Runge-Kutta Method),FEM, LSM etc. In the linear case, for the one-dimensionalconvection-diffusion equation and for the two-dimensional diffusion equation, the FDM is used inboth the space and the time directions in [1]and [2], respectively. Also, the heat and the heat -conduction equations are solved by the DRBEM and with one and two step LSM in [3] and [4],respectively. They have also examined the three and four step LS schemes for the heat equationin [5] and have found that one and two step methods are more efficient. Recently, the nonlinearreaction-diffusion equation is solved by the combined application of DRBEM and FDM in [6] andwith DQM and FDM as well as the one-dimensional Fisher’s equation in [7]. In the study of [8],the two-dimensional reaction-diffusion Brusellator system is solved by using DRBEM for space andFDM for time derivatives which includes a linerization procedure.

In the present paper, the nonlinear reaction-diffusion equation as well as the system of nonlinearreaction-diffusion equations is solved by using the DRBEM for the spatial derivatives.Then for thetime integration two different schemes are introduced, namely the FDM with a relaxation parameter

Advances in Boundary Element Techniques IX 317

and the LSM. Then the effect on the convergence behaviour of the proposed FDM and the com-parison with the LSM are presented. We also compare the computational efficiency (based on thecomputational time and accuracy) of both methods.

2. Definition of the Problem

We consider the following system of nonlinear reaction-diffusion equations

∂uj

∂t= ν∇

2uj + pj (x, y) ∈ Ω, t > 0 j = 1 or j = 2 (1)

with the initial and mixed-type boundary conditions

uj(x, y, 0) = gj(x, y) (2)

βj(x, y, t)uj + γj(x, y, t)qj = 0 (x, y) ∈ Γ, t > 0 (3)

where Γ is the boundary of the domain Ω and qj =∂uj

∂n, n being the outward normal on the boundary.

Here the nonlinearity pj depends on the unknowns, i.e. for a single equation the nonlinearity is p1 (u1)and for the system, the nonlinearities are p1 (u1, u2) and p2 (u1, u2) for the first and second equationsin Eq. 1.

3. DRBEM Formulation

Eq. 1 can be weighted by the fundamental solution u∗ = 1

2πln 1

rof Laplace operator in order to

obtain

ν(ci (uj)i+

∫Γ

(q∗uj −u∗

qj)dΓ) =

∫Ω

(∂uj

∂t− pj)u

∗

dΩ (4)

after the application of the Divergence Theorem [9]. Here i denotes the source(fixed) point, q∗ = ∂u∗

∂n

and ci =∫

Ω∆(xi, yi, x, y)dΩ.

The nonhomogenity, i.e. the time derivative and the nonlinear term, can be approximated usingradial basis functions fj(x, y) resulting with a linear system of equations to be solved

[F ] α (t) = b (5)

where N and L are the number of boundary and selected interior nodes, respectively, and [F ] containsfj s as coloumns . The radial basis functions fj are related to other distance functions uj(x, y) throughthe relation ∇

2uj = fj in order to obtain the boundary integral only form after the application of the

Divergence theorem [10].This leads us to the following matrix-vector formulation after the usage ofthe relationship (5) and substitution of the nonhomogenity vector b

ν ([H] uj − [G] qj) =([H]

[U

]− [G]

[Q

])[F ]−1

(∂u

∂t

− p (u)

)(6)

where H and G denote the whole matrices of boundary elements with kernels q∗ and u

∗, respectively.U and Q compromise the coordinate function column vectors uj and qj . The sizes of all the matricesin (6) are (N +L)×(N +L) and the vectors are of size (N +L)×1. Defining a new (N +L)×(N +L)matrix C as

[C] = −

([H]

[U

]− [G]

[Q

])[F ]−1

. (7)

318 Eds: R Abascal and M H Aliabadi

Eq. 6 can be rearranged as

[C]

∂uj

∂t

+ ν [H] uj− ν [G] qj = [C] pj . (8)

4. Time Integration

4.1 Finite Difference Method

The system of ordinary differential Eq.s 8 for the unknown uj can be written as

∂uj

∂t

= pj−

(βj

γj

[G

]+

[H

])uj (9)

after the application of the boundary condition (3) with the matrices[G

]= ν [C]−1 [G],[

H]

= ν [C]−1 [H]. In this subsection we will employ Euler scheme for the time derivative [10]

u

m+1j

=

u

mj

+∆t

(p

mj

−

(βj

γj

[G

]+

[H

]) u

mj

). (10)

Since the method is explicit, the stability problems can be encountered and ∆t must be taken carefully.A relaxation procedure is employed with a parameter 0 ≤ µ ≤ 1 for the unknown uj in the formuj = (1 − µ)um

j + µum+1j positioning the values of uj between the time levels m and m + 1. Then

the Eq. 10 takes the form

(I + ∆tµ

(d

m+1j

[G

]+

[H

])) u

m+1j

=

(I − ∆t (1 − µ)

(d

mj

[G

]+

[H

])) u

mj

+ ∆t

p

mj

(11)

where dmj =

[βm

j

γm

j

]evaluated at the time level t

m.

4.2 Least Squares Method

When the boundary conditions are applied to the Eq. 8, then it can be rewritten as

[C]

∂uj

∂t

+ [Kj] uj = [C] pj (12)

where [Kj] = ν

([H] +

[βj

γj

][G]

). In a typical time element of length ∆t we approximate vectors uj

and qj as uj ≈ Φ1 (t) umj + Φ2 (t) u

m+1j where φ1 (t) = 1 − ξ, φ2 (t) = ξ with ξ = t − t

m are thelinear interpolation functions.We construct the error functional Πj as in [4]

Πj =

∫ tm

tm+1

rTrdt (13)

where r is the residual vector, which is obtained by substituting the approximate solution in Eq. 12.The LSM solution is obtained after minimizing the unknown uj from the system of equations as

[A]u

m+1j

= [B1]

u

mj

+ [B2]

p

mj

(14)

where

[A] =

(1

∆t2[C]T [C] +

1

2∆t

([C]T [Kj] + [Kj]

T [C])

+1

3[Kj]

T [Kj]

)


µ t=0.5 t=2.0∆t=0.01 ∆t=0.1 ∆t=0.5 ∆t=0.01 ∆t=0.1 ∆t = 0.5 ∆t=1.0

0.7 5.3 1.7(−3) 1.2(−2) 7.07 2.3(-3) 1.6(-2) 4.0(-2)0.8 1.2(−3) 2.0(−3) 1.3(−2) 1.1(-3) 2.8(-3) 1.9(-2) 4.6(-2)0.9 1.2(−3) 2.3(−3) 1.4(−2) 1.1(-3) 3.3(-3) 2.3(-2) 5.3(-2)

Table 1: Maximum absolute errors at small times with FDM for Problem 1

µ t=8.0 t=20.0∆t=0.01 ∆t=0.1 ∆t = 0.5 ∆t=1.0 ∆t=0.01 ∆t=0.1 ∆t = 0.5 ∆t=1.0

0.7 7.3 8.1(−4) 5.7(-3) 1.1(−2) 7.4 3.3(−6) 1.7(-5) 3.2(−5)0.8 4.5(−4) 8.7(−4) 6.1(-3) 1.3(−2) 6.0(−7) 3.6(−6) 1.8(-5) 3.5(−5)0.9 4.5(−4) 9.2(−4) 6.5(-3) 1.4(−2) 6.0(−7) 3.8(−6) 1.8(-5) 3.8(−5)

Table 2: Maximum absolute errors for steady state with FDM for Problem 1

[B1] =

(1

∆t2[C]T [C] −

1

2∆t

([C]T [Kj] + [Kj]

T [C])−

1

6[Kj]

T [Kj]

)

[B2] =

((1

∆t

)[C]T [C] +

1

2[Kj]

T [C]

).

The solution of the system of nonlinear reaction-diffusion equations is obtained from the solutionof the algebraic Eq.s 11 or 14 .

Note that in both time integration schemes the nonlinearity p (u) is approximated only at the timelevel t

m in order to obtain a linear system of equations at the end.

5. Numerical Results

Problem 1.We consider the nonlinear reaction-diffusion equation [2] with ν = 12, p1 (u1) =

u21 (1 − u1) in the square region

(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 for t ≥ 0 . The initial condition and the mixed type boundary condi-tions are taken appropriate with the exact solution u1(x, y, t) = 1

1+ep(x+y−pt) .

To measure the quality of the approximate solutions with both methods we use the maximumabsolute error which is defined bymax |uexact − unum| where uexact, unum denote the exact and numerical solutions for the problem.The results in terms of these maximum absolute errors for different time increments are presented inTables 1-2 at the point x = 1

2for several values of relaxation parameter. Table 3 shows maximum

absolute errors for the LSM solution at the same time levels with the same time increments, again atx = 1

2.

One can see from these tables that, the time increment ∆t = 0.1 is the right choice for boththe methods but an optimal value of relaxation parameter is required in FDM. This is of coursecomputationally expensive procedure. The LSM is preferred as a direct method although there is aone digit drop for small times and two digits drop for steady-state in the accuracy.

Problem 2. Next we solve the system of nonlinear reaction-diffusion equations with the nonlin-earities p1 (u1, u2) = 1 + u

21u2 −

32u1, p2 (u1, u2) = 1

2u1 − u

21u2 with ν = 1

500in the same unit square

∆t t=0.5 t=2.0 t=8.0 t=20.00.01 5.3(-1) 6.3(-1) 1.17 9.6(-1)0.1 2.8(-2) 6.9(-2) 6.5(-2) 1.6(-4)0.5 2.0(-2) 1.6(-1) 2.4(-1) 9.6(-4)1.0 1.3(-1) 2.0(-1) 1.4(-1)

Table 3: Maximum absolute errors with LSM for Problem 1


in problem 1, subject to the initial and Neumann type boundary conditions

(u1 (x, y, 0) , u2 (x, y, 0)) =

(1

2x

2−

1

3x

3,1

2y

2−

1

3y

3

)(x, y) ∈ Ω

(∂u1

∂n,∂u2

∂n

)= (0, 0) (x, y) ∈ Γ, t > 0.

Figure 1: Solution of Problem 2

For this problem we cannot compare the convergence of both methods since we do not have anyexact solution. But we see that both methods show the same expected behaviour with the referencesolution in [8], i.e. u1 → 1 and u2 → 1/2 as time increases. We have found the optimal value of therelaxation parameter in FDM as 0.3 for the solution. Fig. 1 shows that the behaviour of the solutionagrees with the behaviour of the solution in [8]. The solution is obtained with a considerable smallnumber of boundary elements (N=8) in the use of both FDM and LSM.

6. Conclusion

The system of nonlinear reaction-diffusion equations is solved by using the coupling of the methodDRBEM in space with both the FDM and the LSM in time.

The DRBEM results in a system of ODE’s in time and FDM(Euler) with a relaxation parametergives quite good accuracy without the need of very small time increment. The optimal value ofrelaxation parameter is time consuming. In the LSM the accuracy drops one or two digits in smalland large time values, respectively but it does not require very small time increment and relaxationparameter. The DRBEM requires very small number of boundary elements for obtaining a reasonableaccuracy when it is coupled either with FDM or LSM time integration schemes .

References

[1] M.M. Chawla, M.A. Al-Zanaidi, M.G. Al-Aslab, Computers and Mathematics with Applica-tions, 39,71-84(2000).

[2] M.M. Chawla, M.A. Al-Zanaidi, Computers and Mathematics with Applications, 42, 157-168(2001).

[3] K. M. Singh and M. S. Kalra, Engineering Analysis with Boundary Elements,18, 73-102(1993).

[4] K. M. Singh and M.S. Kalra, Computer Methods in Applied Mechancis and Engineering, 190,111-130(2000).


[5] K. M. Singh and M.S. Kalra, Communications in Numerical Methods in Engineering, 12, 425-431(1996).

[6] G. Meral, Proceedings of the International Conference BEM/MRM 27, 133-140(2005).

[7] G. Meral and M. Tezer-Sezgin, International Journal of Computer Mathematics, (in print).

[8] W.T. Ang, Engineering Anlaysis with Boundary Elements, 27, 897-903(2003).

[9] C.A. Brebbia and J. Dominguez, Boundary Elements an Introductory Course, ComputationalMechanics Publications McGraw-Hill book company(1992).

[10] P.W. Partridge, C.A. Brebbia and L.C. Wrobel, The dual reciprocity boundary element method,Computational Mechanics Publications Elsevier Applied Science(1992).


A Galerkin Boundary Element Method with the Laplace transformfor a heat conduction interface problem

Roman VodickaTechnical University of Kosice, Faculty of Civil Engineering,

Vysokoskolska 4, 042 00 Kosice, Slovakia

[email protected]

Keywords: boundary elements, domain decomposition, non-matching meshes, heat conduction, Laplacetransform.

Abstract. The solution of a heat conduction problem with domain decomposition by a Laplace-transformmethod based on a boundary element technique is treated. A series of the complex valued boundary value prob-lems for Helmholtz equation is solved in the frequency domain by the complex Symmetric Galerkin BoundaryElement Method and subsequently transformed back to the time domain by the inverse Laplace transform. Thealgorithm of the domain decomposition solution includes generally curved interfaces and independent meshingof each side of an interface producing non-matching boundary element meshes. The examples present obtainednumerical results and they are compared with analytical solutions.

Introduction

Time-dependent problems that are modeled by initial-boundary value problems (IBVP) can be treated byboundary integral equation (BIE) method. Such methods are widely and successfully being used also fornumerical modeling of problems in heat conduction. A nice survey of BIE applications for time dependentproblems is given in [2].

The present paper introduces an approach utilizing the Laplace transfrom. Such methods solve the problemsin frequency domain, usually for complex frequencies. For each fixed frequency, the heat conduction problemreduces to a BIE for a boundary value problem (BVP) of the modified Helmholtz equation. The transformationback to the time domain includes special methods for inversion of the Laplace transform guided by the choiceof the time dependence approximation.

The split of the space domain into several parts (due to physical properties of materials or for a paral-lelization of the solving process etc.) usually includes domain decomposition techniques to be used [6, 10].In the present approach, these methods due to interfaces are applied very naturally to the BIE system for theHelmholtz equation solved in the frequency domain as BIEs use unknowns directly on the boundary so thattechniques derived for elliptic problems can be applied. The Boundary Element Method (BEM) as a numericaltool for solving BIE is also used in this context [4, 5, 8].

The numerical implementation of a curved interface is very important – it is naturally included in the dis-cussed formulation together with the non-matching meshes along both sides of interfaces. As documented inthe aforementioned papers, the curved boundaries present a strong ability of each problem with an interfaceapproach. Various numerical implementations of the data transfer between two non-matching meshes via cal-culation of integrals over the discretized curved surfaces has been given in [1]. The present approach uses theimplementation of data transfer using an auxiliary interface mesh referred to as common-refinement mesh.

The heat conduction problem with an interface

Let us consider a body defined by a domain, Ω ⊂ R2 in a fixed cartesian coordinate system xi (i = 1, 2), with a

bounded Lipschitz boundary ∂Ω = Γ. Note, that Γ may include corners but not cracks and cusps. Let ΓS ⊂ Γdenote the smooth part of Γ, i.e. excluding corners, edges, points of curvature jumps, etc. Let n denote theoutward unit normal vector defined on ΓS . Although the developed formulation is valid in 3D space as well,for the sake of simplicity we confine ourselves only to the analysis in 2D continuum.


The presence of interfaces cause the domain Ω to be split into several parts. For the sake of simplicity, letus consider a split into two non-overlapping parts ΩA and ΩB whose respective boundaries we denote ΓA andΓB . There exists a common part of both boundaries ΓA and ΓB , let us denote this coupling boundary by Γc.

The initial-boundary value problem with an interface (IBVPwI) of a heat conduction problem for a temper-ature distribution u(x, t) without volume heat sources can be stated as

∂uη(x, t)

∂t− aη2∆uη(x, t) = 0, x ∈ Ωη, t > 0, η = A,B, (1a)

uη(x, 0) = uη0(x), x ∈ Ωη, (1b)

uη(x, t) = gη(x, t), x ∈ Γηu, qη(x, t) = −kη ∂uη(x, t)

∂n= hη(x, t), x ∈ Γη

q , t ≥ 0 (1c)

uA(x, t) − uB(x, t) = 0, qA(x, t) + qB(x, t) = 0, x ∈ Γc, t ≥ 0, (1d)

where the material parameter aη2 = kη

cηρη includes specific heat cη, density ρη and thermal conductivity kη. The

split of each boundary Γη due to the boundary and interface conditions can be written as: Γη = Γηu ∪ Γη

q ∪ Γc

(∅ = Γηu ∩ Γη

t = Γηu ∩ Γc = Γη

t ∩ Γc). Therefore, functions gη(x, t) and hη(x, t) introduce given boundaryconditions, while the function uη

0(x) defines the initial condition.The unilateral Laplace transform is applied to obtain the solution in the frequency domain, let the transform

temperature solution be

u(x, p) =

∫ ∞

0u(x, t)e−pt dt, p ≥ σ0, (2)

with σ0 being the abscissa of convergence for the Laplace transform. Moreover, let the boundary conditionsbeing defined by functions with separated variables, i.e. gη(x, t) = gη

x(x)gηt (t), hη(x, t) = hη

x(x)hηt (t). Then

the IBVPwI (1) is transformed into a domain BVP with an interface (BVPwI) for the modified Helmholtzequation with the parameter p

puη(x, p) − aη2∆uη(x, p) = uη0(x), x ∈ Ωη, (3a)

uη(x, p) = gηx(x)gη

t (p), x ∈ Γηu, qη(x, p) = hη

x(x)hηt (p), x ∈ Γη

q , (3b)

uA(x, p) − uB(x, p) = 0, qA(x, p) + qB(x, p) = 0,x ∈ Γc. (3c)

Naturally, the temperature solution u(x, t) is finally found by the inverse Laplace transform of u(x, p).

The boundary integral equation system

The problem (3) can be solved by a system of BIEs for a fixed parameter p. The fundamental solution of thegoverning differential equation and its necessary derivatives are known — they are given by the modified Besselfunctions of the second kind Ki(z), (i = 0, 1, 2). All necessary functions and derivatives can by introduced bythe following relations

Uη(x,y, p) = −(2πkη)−1K0(

√p

a‖x − y‖), Qη(x,y, p) = −kη ∂Uη(x,y, p)

∂nx,

Qη∗(x,y, p) = −kη ∂Uη(x,y, p)

∂ny, Dη(x,y, p) = kη2 ∂2Uη(x,y, p)

∂nx∂ny. (4)

The BIE system for BVPwI to be solved includes the boundary conditions (3b) and also the interface con-ditions (3c) in a weak form when solved by the Galerkin method. The formulation is based on a variationtechnique derived in [8]. In order to write the BIE system in a compact and transparent matrix form, weintroduce an operator notation:

ωηr Zη

rswηs =

∫Γη

r

∫Γη

s

ωη(y)Zη(x,y, p)wη(x, p) dΓ(x) dΓ(y), (5a)

ωηr Zη

rΩwη =

∫Γη

r

∫∫Ωη

ωη(y)Zη(x,y, p)wη(x) dΩ(x) dΓ(y), (5b)


where ω stands for a weighting function, w stands for u or q, r and s stand for u, q or c, Z stands for U , Q,Q∗ or D. Moreover, the inner integral can be regular, singular or, in the case of (5a), also Hadamard finite-partintegral. Hereinafter, indices u, t or c introduce a restriction of a function or of an operator to a part of boundaryΓ with the same index. The matrix form of the resulting system then reads

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

ϕAu

ϑAq

ϕAc

ϑAc

ϕBu

ϑBq

ϕBc

ϑBc

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

T⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

−UAuu QA

uq −UAuc QA

uc 0 0 0 0

QA∗qu −DA

qq QA∗qc −DA

qc 0 0 0 0−UA

cu QAcq −UA

cc − 1

2IAcc+QA

cc 0 0 0 IABcc

QA∗cu −DA

cq − 1

2IAcc+QA∗

cc −DAcc 0 0 0 0

0 0 0 0 −UBuu QB

uq −UBuc QB

uc

0 0 0 0 QB∗qu −DB

qq QB∗qc −DB

qc

0 0 0 0 −UBcu QB

cq −UBcc

1

2IBcc+QB

cc

0 0 IBAcc 0 QB∗

cu −DBcq

1

2IBcc+QB∗

cc −DBcc

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

qAu

uAq

qAc

uAc

qBu

uBq

qBc

uBc

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

ϕAu

ϑAq

ϕAc

ϑAc

ϕBu

ϑBq

ϕBc

ϑBc

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

T

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

UAuq − 1

2IAuu − QA

uu 0 0 − kA

aA2 UAuΩ

01

2IAqq − QA∗

qq DAuu 0 0 kA

aA2 QA∗uΩ

0

UAcq −TA

cu 0 0 − kA

aA2 UAcΩ 0

−QA∗cq DA

cu 0 0 kA

aA2 QA∗cΩ 0

0 0 UBuq − 1

2IBuu − QB

uu 0 − kB

aB2 UBuΩ

0 0 1

2IBqq − QB∗

qq DBuu 0 kB

aB2 QB∗uΩ

0 0 UBcq −QB

cu 0 − kB

aB2 UBuΩ

0 0 −QB∗cq DB

cu 0 kB

aB2 QB∗uΩ

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎜⎜⎝

hA

gA

hB

gB

uA0

uB0

⎞⎟⎟⎟⎟⎟⎟⎠

, (6)

where Irr is formal representation (5) of the identity (Iηrr) and projection (IAB

cc , IBAcc ) operators. Function ϕη

r

and ϑηr should be chosen to form the basis of the function spaces, where the functions q

ηr and u

ηr , respectively,

belong to. The found functions solve (3), while the solution of the original IBVPwI of heat conduction canbe found by inverse Laplace transform. Some details of the numeric procedure applied will be noted in whatfollows.

Notes on the numerical solution

Boundary element approximation. The numerical solution of BIE system (6) by a boundary element tech-nique includes discretization of the boundary Γη of each subdomain Ωη by W η

e boundary elements Γη k,k = 1, 2, . . . W η

e . For making the expressions simpler, let us omit the superscript index η in this paragraph.The present discretization utilizes conforming isoparametric elements. Let us consider an n-th order poly-

nomial parameterization of k-th boundary element Γk given by the relation x = Nk(ξ), ξ ∈ 〈0, 1〉. Then theapproximation of the functions u(x, p), q(x, p) can be written in the following form

u(x, p) =

Wu∑m=1

ϑm(x)um, q(x, p) =

Wq∑m=1

ϕm(x)qm, (7)

where the nodal shape functions ϑm(x), resp. ϕm(x) are equal to element shape functions Nkj(x) for somek, j such that x ∈ Γk, j ∈ 1, 2, . . . n + 1. Moreover, the functions ϑm should be continuous, while thefunctions ϕm do not have to be. Nevertheless, it is clear that all the functions defined by the parameterizationare smooth together with all their derivatives along an element. The relation between the parameterizationof boundary Nk(ξ) and the element shape functions Nkj(x) is Nkj(x) = Nkj(Nk(ξ)) = N j(ξ), wherethe polynomials N j(ξ) form the basis of the n-th order polynomials space. An example of the second ordercontinuous elements and their parameterization is shown on Figure 1. It can be seen that the support of afunction ϑm(x) can cover up to two neighbour elements: compare ϑm(x) and ϑm+1(x).

Calculation of integrals. The discretization of the BIE system (6) also includes calculation of the integrals (5)in the form∫

Γl

∫Γk

N li(y)Z(x,y, p)Nkj(x)dΓ(x)dΓ(y)=

∫ 1

0

∫ 1

0N i(υ)Z(Nk(ξ),N l(υ), p)N j(ξ)

∣∣N ′k(ξ)

∣∣∣∣N ′l(υ)

∣∣dξdυ (8)


0 1 ξ

1

12

N1(ξ)

N2(ξ)

N3(ξ)

N kj(x) = N j(ξ)Ω

Γq

ΓkΓk+1

ξk = 0

ξk = 1

ϑm+1(x) = Nk3(x)ϑm+1(x) = N (k+1)1(x)

ϑm(x) = Nk2(x)

Figure 1: Second order elements and their shape functions

for each particular integral kernel Z(x,y, p) and for each pair of boundary elements Γk and Γl. The idea ofthe calculation is to split the integral into a regular part and into a singular part if it occurs. The regular partsthen can be evaluated e.g. by a standard Gauss-Legendre quadrature rule, while the singular parts are treatedseparately according to the type of the singularity. The integrals with Z = U and Z = Q can contain onlylogarithmic singularities, the other case Z = D may result even in an hypersingular integral, see also [7].

In order to demonstrate the way of calculation, let us discuss the most singular case in more detail. Theintegral (8), with respect to (4) and to the notation r = |r|, r = x − y = Nk(ξ) − N l(υ) and c =

√p

aη , reads

∫ 1

0=

∫ 1

0N i(υ)

∂2K0(c r)

∂nx∂nyN j(ξ)

∣∣N ′k(ξ)

∣∣ ∣∣N ′l(υ)

∣∣ dξdυ =

∫ 1

0

∫ 1

0N i(υ)N j(ξ)

∣∣N ′k(ξ)

∣∣ ∣∣N ′l(υ)

∣∣×[

−K2(c r) − ln r I2(c r) +2

c2r2

](rT nx)(rT ny)

r2+

1

c

[K1(c r) − ln r I1(c r) − 1

c r

]nT

xny

r

dξdυ+

∫ 1

0

∫ 1

0N i(υ)N j(ξ)

∣∣N ′k(ξ)

∣∣ ∣∣N ′l(υ)

∣∣ ln r

[I2(c r)

(rT nx)(rT ny)

r2+

I1(c r)

c rnT

xny

]dξdυ−

∫ 1

0=

∫ 1

0N i(υ)N j(ξ)

∣∣N ′k(ξ)

∣∣ ∣∣N ′l(υ)

∣∣ 1

c2r2

[2(rT nx)(rT ny)

r2− nT

xny

]dξdυ = DR + DL − 1

c2DH , (9)

where Ii, i = 1, 2 are the modified Bessel functions of the first kind. The terms are reordered for the firstintegral DR to contain regular functions only, the second integral DL to contain a logarithmic singularity forr → 0 and the last term DH to be a hypersingular integral if r = 0 is a point of the integration domain.

The logarithmic singularity should be treated in such a way that a suitable weighted quadrature formulacould be used. Two cases should be distinguished for calculation of the integral DL: first, when the inte-grals in (8) are calculated with respect to the same elements, i.e. k = l and, second, when the elements areneighbouring in the mesh, having one common point.

In the former case, the distance function r renders

r = Nk(ξ) − Nk(υ) = N ′k(ξ)(ξ − υ) − 1

2N ′′k(ξ)(ξ − υ)2 + · · · = (ξ − υ)g(ξ, υ) (10)

for a sufficiently smooth non-vanishing vector function g(ξ, υ) if ξ = υ. Therefore DL can be written as

DL =

∫ 1

0

∫ 1

0ln r G(ξ, υ)dξdυ=

∫ 1

0

∫ 1

0ln |g(ξ, υ)|G(ξ, υ)dξdυ +

∫ 1

0

∫ 1

0ln |ξ−υ|G(ξ, υ)dξdυ=DLR+DLL, (11)

with G(ξ, υ) being the non-singular and smooth rest of the integrand and with the split of the result into theregular part DLR and the singular part DLL. To be able to correctly apply a quadrature rule which requires thelogarithmic function at the end point of the interval, following substitutions are required:

ξ ∈ 〈0, 1〉 ∧ υ ∈ 〈0, ξ〉 : ξ = γ, υ = γ(1 − δ), J = γ, γ ∈ 〈0, 1〉 ∧ δ ∈ 〈0, 1〉 (12a)

ξ ∈ 〈0, υ〉 ∧ υ ∈ 〈0, 1〉 : υ = γ, ξ = γ(1 − δ), J = γ, γ ∈ 〈0, 1〉 ∧ δ ∈ 〈0, 1〉 (12b)


to obtain

DLL =

∫ 1

0

∫ 1

0ln(γδ) G(γ, δ)γdγdδ =

∫ 1

0

∫ 1

0γ ln γ G(γ, δ)dγdδ +

∫ 1

0ln δ

[∫ 1

0γ G(γ, δ)dγ

]dδ. (13)

In the latter case, the relation to be satisfied for boundary parameterization is e.g. Nk(1) = N l(0). Theintegration domain is split into two triangles along the diagonal ξ = υ, with the following substitution to beapplied to the triangle with the singularity:

ξ ∈ 〈0, 1〉 ∧ υ ∈ 〈0, ξ〉 : ξ = γ(1 − δ) + 1, υ = γδ, J = γ, γ ∈ 〈0, 1〉 ∧ δ ∈ 〈0, 1〉. (14)

At the vicinity of the singularity, the distance function r can be expressed as

r = Nk(γ(1 − δ) + 1) − N l(γδ) = N ′k(1) [γ(δ − 1)] − N ′

l(0) (γδ) + · · · = γh(γ, δ), (15)

with a function h(γ, δ) sufficiently smooth and non-vanishing for γ = 0, as the Lipschitz condition supposedfor the boundary makes the expression N ′

k(1) (δ − 1) − N ′l(0) δ not to vanish. The integral DL obeys the

relation

DL =

∫ 1

0

∫ υ

0ln r G(ξ, υ)dξdυ +

∫ 1

0

∫ 1

0ln(γ|h(γ, δ)|) ˜G(γ, δ)γdγdδ =

DLR +

∫ 1

0

∫ 1

0γ ln |h(γ, δ)| ˜G(γ, δ)dγdδ +

∫ 1

0

∫ 1

0γ ln γ ˜G(γ, δ)dγdδ, (16)

where neither the first term DLR – the integral over the triangle without singularity, nor the second term containany singular function and only the last integral includes a logarithmic function with zero argument, thoughsmoothed by Jacobian γ.

The last term in (9), DH , requires exceptional treatment as it includes hypersingular integral. This singu-larity has been obtained in a limit procedure, where a point x at the boundary Γ(x) has been approached bymoving a point y from the interior of the domain Ω . Therefore, for the calculation of the integral, the boundaryΓ(y) can be shifted by a small ε inwards to Ω resulting in an integral, which does not contain any singularity.

It is also useful to calculate the integral DH introducing a complex function z(ξ, υ) =(Nk

1 (ξ) − N l1(υ)

)+

i(Nk

2 (ξ) − N l2(υ)

)because a simple calculation via integration by parts renders following result

DH = ∫ 1

0

∫ 1

0N i(υ)N j(ξ)

1

z2(ξ, υ)

∂z(ξ, υ)

∂ξ

∂z(ξ, υ)

∂υdξdυ =

[[− ln rN i(υ)N j(ξ)]ξ=1

ξ=0

]υ=1

υ=0+

[N j(ξ)

∫ 1

0ln r N i′(υ)dυ

]ξ=1

ξ=0

+

[N i(υ)

∫ 1

0ln r N j ′(ξ)dξ

]υ=1

υ=0

−∫ 1

0

∫ 1

0ln r N j ′(ξ)N i′(υ)dξdυ. (17)

The basic polynomial functions N j(ξ) and N i(υ) can be smoothly differentiated along an element, more-over they are (already parameterized) restrictions to particular boundary elements of continuous nodal shapefunctions ϑmj (x) and ϑmi(y), respectively. In the calculation, the integrals over the support of pertinent ϑ-functions, which vanish at the end points of their support, can be gathered together. Therefore, the three freeterms also vanish and only the last integral remains.

Finally, the aforementioned limit procedure should be performed. The integral, however, has remained onlywith logarithmic singularity, which can be treated as above.

The inverse Laplace transform. The algorithm of numerical inverse of the Laplace transform due to Weekshas been applied, see [9]. It is based on the fact that the Laplace transform of a function expressed by a serieswith respect to appropriately scaled orthonormal Laguerre functions can be easily modified to calculate the ap-proximation of its inverse Laplace transform by the quarter wave cosine fast Fourier transform. The numericalprocedure includes several parameters which should be carefully chosen to obtain a quickly convergent andaccurate results. The discussion of the choice of the parameters has been also presented in [3]. The param-eters which are necessary for calculation are: abscissa of convergence of Laplace transform σ0, abscissa of


evaluation of the inverse Laplace transform σ, scaling parameters of the Laguerre functions b and of the seriescoefficients evaluation r and also number N of functions used from the series, equal to the number of points infrequency domain for the inverse transform evaluation and the time interval determined by maximum requiredtime instance tmax.

An example

An example has been chosen to document the numerical behaviour of the proposed numerical method. Itincludes a simple square divided into two subdomains, see Figure 2, with the interface defined by a cubic splinepassing through the points E, Si and F , where the coordinates of Si are as follows: S1[0.5; 0.4], S2[1.0; 1.1],S3[1.5; 1.6].

0.4

0.2

x1

x2

2

O

E

FS3

S2

S1 ΩA

ΩB

Γc

ΓAu

ΓAq

ΓBq

ΓBq

ΓAq

Figure 2: A square domain with a curved interface with the pattern of element nods in the interface

The solution of the problem has been chosen so that it is analytically expressed by the series formula

u(x, t) = x21x

22

(1 − e−t

)+

∞∑m=1

[T0m(t) sin (2m+1)πx2

4 +

∞∑n=1

Tnm(t) sin (2m+1)πx2

4 cos nπx12

], (18a)

T0m(t) = 323π(2m+1) · λ0m−1−λ0me−t+e−λ0mt

λ0m(λ0m−1) + 32π2(2m+1)2

[(−1)m − 2

π(2m+1)

]

×[−8

3e−t−e−λ0mt

λ0m−1 + 4λ0m−1−λ0me−t+e−λ0mt

λ0m(λ0m−1)

], (18b)

Tnm(t) = 128(−1)n

π3n2(2m+1)· λnm−1−λnme−t+e−λnmt

λnm(λnm−1) − 1024π4n2(2m+1)2

[(−1)m − 2

π(2m+1)

]e−t−e−λnmt

λnm−1 , (18c)

λnm = n2π2

4 + (2m+1)2π2

16 . (18d)

It means that the vanishing initial conditions for simplicity (no need to compute the volume integrals in (6) dueto (5b)) are prescribed. The boundary conditions, which types are shown on Figure 2, read

u ((x1, 0), t) = 0, q ((2, x2), t) = −4x22

(1 − e−t

), (19a)

q ((0, x2), t) = 0, q ((x1, 2), t) = −4x21

(1 − e−t

). (19b)

The discretization by boundary elements has been made in such a way that the lengths of all elements areapproximately the same. Each side of the outer square contour is divided into 20 linear elements. The interfaceis meshed accordingly: the non-matching mesh taken contains 24 equally spaced elements along the face of thedomain ΩA, the coarser mesh, and 27 elements with respect to the other domain, finer mesh.

The parameters of the numerical inverse of the Laplace transform has been set to the following values:N = 16 points in the frequency domain, abscissa of the convergence σ0 set to zero, maximum time evaluationbeing unity, parameters r = 0.99, b = 16, and abscissa of evaluation σ being 8.

The results obtained in the interface has been focused on: first, the solution evolution in time and its erroralong the interface is shown on Figure 3 for the temperature u and on Figure 4 for the flux q, second, the


0.0 0.2 0.4 0.6 0.8 1.0t

0.00.5

1.01.5

2.0

x 10

2

4

6

8

10

uA

0.0 0.2 0.4 0.6 0.8 1.0t

0.00.5

1.01.5

2.0

x 1

0

0.004

0.008

0.012

0.016

0.02

|uAn−

uA

a |

Figure 3: Distribution of the temperature u and of its error from the mesh of the domain ΩA along the interfaceand with respect to the time t

0.0 0.2 0.4 0.6 0.8 1.0t

0.00.5

1.01.5

2.0

x 1

01234567

−qA

0.0 0.2 0.4 0.6 0.8 1.0t

0.00.5

1.01.5

2.0

x 1

00.050.1

0.150.2

0.250.3

|qAn−

qAa |

Figure 4: Distribution of the flux q and of its error from the mesh of the domain ΩA along the interface andwith respect to the time t

distribution of both functions (u and q) and their errors are evaluated at a specific time instance on Figure 5. Allgraphs use the x1-coordinate of interface points as the abscissa for the values of pertinent functions at a timeinstance.

Let us comment some observations. The time evolution graphs show nice relation between the error mag-nitudes and the high gradients of the solutions obtained along the coarser mesh of the interface, with respectto the domain ΩA. The errors are naturally worse for the fluxes q. The graphs contain the absolute errors, i.e.|un−ua|, and |qn−qa|, where the superscript ‘n’ denotes numerical solution obtained by the BIE system (6) andthe inverse Laplace transform and the superscript ‘a’ denotes the analytical solution obtained by (18a) truncatedwithin each sum to 20 terms, however, the maximum relative errors can be estimated from them: it is about0.002 for the temperatures u and about 0.04 for the fluxes q.

The same observation can be also done from Figure 5, which has been made for the maximum time value.Moreover, the pictures also include the results obtained along the finer mesh, so that both data can be comparedmutually. As the solutions and their errors are plotted in the same graph, the mutual relation between themagnitude of the error and the descent steepness is even more obvious. The interface mesh pattern shownon Figure 2 can help to understand the oscillating character in the error distributions: the errors cannot coincidebecause the meshes have few common points. Nevertheless, the errors are not significant and in the currentgraphs the results of both functions u and q, plotted for the domain ΩB , nicely fit with the analytical solution.


Conclusion

A test of a boundary element approach for solving interface heat transfer IBVP has been performed. The formu-lation utilizing the Laplace transform of the solution solves the problem in the frequency domain numericallyby a series of BVPs for Helmholtz equation. The numerical method used here includes the complex symmet-ric Galerkin BEM. The idea of the approach has been taken from a method previously derived by the authorsin [8] for problems of elasticity and it seems to work well with the presented example. The investigations andnumerical tests planned for the proposed method will give it more rigorous explanation for its applicability inwider range of the initial-boundary value problems with interfaces.

0.0

2.0

4.0

6.0

8.0

10.0

0.0 0.5 1.0 1.5 2.00.000

0.002

0.004

0.006

0.008

0.010

u

|un−

ua |

x1

|uAn − uAa||uBn − uBa|

uB

exact

0.0

2.0

4.0

6.0

8.0

0.0 0.5 1.0 1.5 2.0

0.000

0.050

0.100

0.150

0.200

q

|qn−

qa |

x1

|qAn − qAa||qBn − qBa|

qB

exact

Figure 5: Results along the interface for the time t = 1

Acknowledgement The author gratefully acknowledge the Scientific Grant Agency VEGA for supporting thiswork under the Grant No. 1/4198/07.

References

[1] Boer, A. de, Zuijlen, A.H. van, Bijl, H.: Review of coupling methods for non-matching meshes. Comput.Methods Appl. Mech. Engrg, 196, pp. 1515–1525, 2007.

[2] Costabel, M.: Time-dependent problems with boundary integral equation method. In: Encyclopedia ofComputational Mechanics, John Wiley & Sons, Eds. Stein, de Borst, Hughes, vol. 1, chap. 25, 2004.

[3] Garbow, B.S., Giunta, G., Lynnes, J.N., Murli, A.: Software for an implementation of Weeks’ method forthe inverse Laplace transform problem. ACM T. Math. Software, 14, pp. 163–170, 1988.

[4] Hsiao, G.C., Steinbach, O., Wendland, W.L.: Domain decomposition methods via boundary integralequations. J. Comp. Appl. Math., 125, pp. 521–537, 2000.

[5] Langer, U., Steinbach, O.: Boundary element tearing and interconnecting method. Computing, 71, pp.205 – 228, 2003.

[6] Puso, M.A.: A 3D mortar method for solid mechanics. Int. J. Num. Meth. Engrg., 59, pp. 315–336, 2004.

[7] Vodicka, R.: On evaluation of integrals arising in SGBEM solution of modified Helmholtz equation. In:VIII. vedecka konferencia Stavebnej fakulty v Kosiciach, TU v Kosiciach, Stavebna fakulta, pp. 91–96,2007.

[8] Vodicka, R., Mantic, V., Parıs, F.: Symmetric variational formulation of BIE for domain decompositionproblems in elasticity - an SGBEM approach for nonconforming discretizations of curved interfaces.CMES – Comp. Model. Eng., 17, pp. 173–203, 2007.

[9] Weeks, W.T.: Numerical inversion of Laplace transform using Laguerre functions. Jornal of the Associa-tion for Computing Machinery, 13, pp. 419–426, 1966.

[10] Wohlmuth, B.I.: Discretization Methods and Iterative Solvers Based on Domain Decomposition, LectureNotes in Computational Science and Engineering, vol.17, Springer, Berlin, 2001.


Assembled Plate Structures by the Boundary Element Method

D. D. Monnerat1, a, J. A. F. Santiago2, b and J. C. F. Telles2, c

1Exactum Consultoria e Projetos Ltda, Rua Sete de Setembro, 43

20050-003 Rio de Janeiro, RJ, Brazil

2Programa de Engenharia Civil - COPPE/UFRJ, Caixa Postal 68506

21941-972 Rio de Janeiro, RJ, Brazil

[email protected], [email protected], [email protected]

Keywords: Boundary element; Plates; Reissner’s plate theory.

Abstract. This paper deals with the analysis of assembled plate structures, subjected to arbitrary

loadings, for which two dimensional plane stress elasticity and shear deformable plate bending

theories are coupled. To this end, direct boundary element formulations, based upon Reissner’s

plate theory and 2-D elasticity, are presented for elastostatic problems.

The multi-region technique is employed to assemble the plates. Several plates sharing

common interface boundaries are accommodated, including inclined ones. After a standard

coordinate transformation, each region can be combined taking into account displacement

compatibility and equilibrium equations in order to obtain the final equation system.

1. Introduction

Structures composed of assembled plane elements with close or open cross sections have been

employed in several branch of engineering, such as civil, mechanic, naval, aeronautics, etc.; mainly

due to the advantage of attaining high flexural rigidity with low self weight.

Papers by Palermo [1] and Palermo et al. [2] discuss plate assembles with close and open cross

sections using Kirchhoff’s plate theory and two-dimensional elasticity. Dirgantara and Aliabadi [3]

and Baiz and Aliabadi [4] also analyzed assembled plate structures under arbitrary loadings using

Reissner’s plate theory and two regions sharing common interfaces.

In the present work, the direct boundary element formulation for the multi-region technique,

based upon Reissner´s plate and 2-D elasticity theories are presented for elastostatic problems,

considering isotropic materials, small deformations and small displacements. Several plates sharing

common interface boundaries are accommodated, including inclined ones, generalizing previous

analyses [5].

Several numerical examples are presented and results are compared with exact analytical and

finite element solutions to demonstrate the accuracy of the proposed formulation.

2. Boundary Integral Equations

The equations will be presented here in indicial notation. Here roman indices vary from 1 to 3 and

Greek indices vary from 1 to 2.

The integral equations adopted to represent displacement components can be written as

i) Reissner’s plate bending:

xdxqxwxwxdxpxw

xdxwxpwC

iijij

jijjij

,1

,,

,

*,2

*3

*

*

(1)


ii) Two dimensional plane stress elasticity:

,,, ** xdxtxuxdxuxtuC (2)

where the boundary integrals on the left-hand side are interpreted in sense of Cauchy’s principal

value.

In Equations 1 and. 2, are the rotations about the axes, are the displacements on the

plane and is the displacement in the direction. The terms are the bending moments

(j = 1, 2) and shear force (j = 3), which are given as

w x u

21xx 3w 3x jp

nMp and nQp3 , respectively. On

the other hand represent in plane tractions, given by t nNt , q is the distributed load and

h/10 , h being the thickness. The coefficients ijC depend on the boundary geometry at the

source point . The kernels and xwij ,* xpij ,* represent the fundamental solution for plate

bending, whereas and represent the fundamental solution for plane elasticity.

The complete expressions for them can be found in references [6-8].

xu ,* xt ,*

Equations 1 and 2 represent five integral equations per functional node of a structure under

bending and extensional effects in the local coordinate system. Three degrees of freedom come

from Reisser’s plate theory and two degrees of freedom from plane stress elasticity..

The domain integral of the equation 1 can be transformed in a boundary integral applying the

divergence theorem. Here q is considered constant (uniformly distributed load), hence Equation 1

can be written as

,,1

,,

,

*

2

*,

*

*

xdxnxwxqxdxpxw

xdxwxpwC

iijij

jijjij

(3)

where is the particular solution of the equation . The expression for can be found

in reference [8].

*i

*3

*, ii w *

,i

3. Assemble of the Equations System

Equations 2 and 3 represent the basic expressions for the solution of spatial assembled plate

problems using boundary element method (BEM).

In general terms, the boundaries and interfaces of a structure are discretized in elements, for

which displacements and tractions are interpolated by means of functional node values. The integral

equations for plane stress elasticity and shear deformable plate bending are applied to all functional

nodes, in a corresponding local coordinate system, for every region, generating a linear equation

system of the following matrix form:

,0

0

0

0

1515551555 xx

P

s

x

P

S

x

P

S

x

P

S

b

0

p

t

G

G

w

u

H

H (4)


whereTS uu 21u ,

TP www 321w ,TS tt 21t and

Tppp 321p are the

displacements (u and w) and tractions (t and p) for plane stress elasticity and plate bending,

respectively, is the domain load vector and , , and are the element

influence matrices for plane stress elasticity (S) and plate bending (P), respectively.

Tq00b SG S

HP

GP

H

To solve the problem the equation system of each region must be referred to the same global

coordinate system. Therefore, the local equation system (referred to the individual plate) is

transformed to the global one by employing the coordinate transformation matrix as M

3

2

1

3

2

1

2

1

3

2

1

0

u

u

u

w

w

w

u

u

M ,

uMu

uMu

T

and

3

2

1

3

2

1

2

1

3

2

1

0 m

m

m

t

t

t

p

p

p

t

t

M ,

pMp

Mpp

T

(5)

where and T

uuu 321321uT

mmmttt 321321p represent the

displacements and tractions vectors, respectively, referred to the global coordinate system.

One can observe in equations 5 that a new rotation 3 (or bending moment ) about the x3m 3-axis

is added to the previous u (or p) vector. Therefore a new equation per functional node is needed,

namely a restrain equation, given by

,0333232131 mmm (6)

in which ij is the cosine of the angle between xi local and xj global axes.

After a standard coordinate transformation, the sub-regions can be combined taking into account

displacement compatibility and equilibrium of tractions along the interface boundaries, in order to

obtain the final global equation system. Notice that now six degree of freedom per functional node

are considered. These equations (compatibility and equilibrium equations) can be written as

follows:

i) Displacement compatibility

i

i

i

i

i

i

uuu

uuu

uuu

323

13

222

12

12

111

323

13

222

12

121

11

(7)

ii) Equilibrium of tractions


,0

0

0

0

0

0

323

13

222

12

121

11

323

13

222

12

121

11

i

i

i

i

i

i

mmm

mmm

mmm

ttt

ttt

ttt

(8)

where the index i is the sub-region (individual plate) sharing common interface boundaries.

After the boundary conditions are enforced, the global equation system can be solved in order to

produce the global unknowns (displacements and tractions) on external boundaries and interfaces of

the structure.

4. Numerical Examples

Several examples of 3-D assembled plate structures under flexural and extensional loads,

simultaneously, are analyzed. The results are compared with beam theory and finite element

solutions to demonstrate the accuracy of the proposed formulation.

4.1. Cantilever I Beam

In this example a cantilever beam with an I cross section (see figure 1) is studied. Dimensions

and properties are: L1 = L2 = 40 cm, L3 = 100 cm, t1 = 0.5 cm, t2 = 1 cm, E = 21000 kN/cm2 and =

0.3.

Figure 1: Geometrical dimensions of the beam

The beam is subjected to a linear distributed load q, varying from -100kN/cm to 100kN/cm, on

the web of the opposite end of the vertical support, as indicated in figure 2.


Figure 2: Cantilever beam subjected to linear distributed load

To analyze this example, 42 elements and 100 functional nodes have been employed over the

external boundaries and interfaces of the beam.

The obtained results are compared with beam theory and the finite element method. For beam

theory the solution is given by

2

2x

EI

Mxu , (9)

where M is the applied bending moment and I is the moment of inertia with respect to the neutral

axis.

The obtained results for the vertical displacements of the beam along the neutral axis, are

presented in figure 3.

-0,200

-0,175

-0,150

-0,125

-0,100

-0,075

-0,050

-0,025

0,000

0 10 20 30 40 50 60 70 80 90 100

x (cm)

w (

cm

)

Beam Theory BEM FEM

Figure 3: Comparison of vertical displacements

As seen in figure 3, the beam theory and both methods (BEM and FEM) are in close agreement,

confirming the validity of the proposed method.

4.2. L-shaped Plate Structure

In this second example three rectangular plates with different sizes and the same thickness were

assembled in order to form the L-shaped plate structure, showed in figure 4


The geometric constants are: L1 = L2 = 100 cm, L3 = L4 = 50 cm, t = 5 cm, = 120º. The

modulus of elasticity and Poisson´s ratio are taken to be 7000 kN/cm2 and 0.33, respectively.

Figure 4: Plate assembly geometry

The L-shaped plate structure is loaded by the uniformly distributed load qy = qz =0.05 KN/cm, in

the y and z directions, along the tip edge of the horizontal plate, as depicted in figure 5.

Figure 5: Cantilever plate subjected to the uniformly distributed loading in y and z directions

The problem is modelled with three sub-regions, each having 16 elements and 42 functional

nodes along the interfaces and external boundaries.

The results for vertical and horizontal displacements along the cross section at x = 50 cm are

shown in figure 6. These results are here compared with the finite element method (FEM).

In order to improve the comparison, figure 6 presents the deformed shape in expanded scale with

a factor of 10. As can be seen, the BEM result is in excellent agreement with the FEM.


0

10

20

30

40

50

60

70

80

90

100

110

120

130

0 10 20 30 40 50 60 70 80 90 100 110

v (cm)

w (

cm

)

Undeformed shape BEM (deformed) FEM (deformed)

Figure 6: Deformed structure

5. Conclusion

The presented analysis of structures composed of 3-D associations of plane panels subjected to

co-occurring bending and extension loads has been considered quite satisfactory, with accurate

results in comparison to existing alternative procedures. In addition, the discussed BEM

implementation has been found to lead to acceptable solutions even in case of rather coarse

discretization alternatives, employing a reduced number of elements. It can, therefore, be

recommended for such panel assembled structures existing in current engineering practice.

References

[1] L. Palermo Jr., M. Rachid and W.S. Venturini: Analysis of Thin Walled Structures using the

Boundary Element Method, Engineering Analysis with Boundary Elements. Vol. 9 (1992), pp.

359-363.

[2] L. Palermo Jr., Analysis of Thin Walled Structures as Assembled Plates by Boundary Element

Method (in Portuguese). Thesis of Doctor of Science (D.Sc.), Escola de Engenharia de São

Carlos / USP, São Carlos, SP, Brazil, (1989).

[3] T. Dirgantara and M.H. Aliabadi, Boundary Element Analysis of Assembled Plate Structures.

Commun. Numer. Meth. Engng. Vol.17 (2001), pp. 749-760.

[4] P.M. Baiz and M.H. Aliabadi, Local Buckling of Thin Walled Structures by BEM, Advances in

Boundary Element Techniques, pp. 39-44, (2007).


[5] D.D. Monneratt, Analysis of Assembled Plate Structures using the Boundary Element Method

(in Portuguese), Dissertation of Master of Science (M.Sc), COPPE / UFRJ, Rio de Janeiro, RJ,

Brazil, (2008).

[6] C.A. Brebbia, J.C.F Telles and L.C. Wrobel: Boundary Element Techniques: Theory and

Applications in Engineering, Springer-Verlag, Berlin, (1984).

[7] L.C. Wrobel and M.H. Aliabadi, The Boundary Element Method, Wiley, Chichester, (2002).

[8] F. Vander Weeën, "Application of the Boundary Integral Equation Method to Reissner's Plate

Model", International Journal for Numerical Methods in Engineering. Vol. 18 (1982), pp. 1-

10., (1982).


Development of a time-domain fast multipole BEM based on theoperational quadrature method in 2-D elastodynamics

Takahiro SAITOH1,a, Sohichi HIROSE 2,b and Takuo FUKUI3,c

1University of Fukui, 3-9-1, Bunkyo, Fukui-shi, Fukui, Japan2Tokyo Institute of Technology, 2-12-1,O-okayama, Meguro-ku, Tokyo, Japan

3University of Fukui, 3-9-1, Bunkyo, Fukui-shi, Fukui, [email protected], [email protected], [email protected]

Keywords: Operational Quadrature Method (OQM), Fast Multipole Method (FMM),Time-domain, Elastodynamics.

Abstract. This paper presents a new time-domain Fast Multipole Boundary Element Method in 2-Delastodynamics. In general, the use of direct time-domain BEM sometimes causes the instability oftime-stepping solutions and needs much computational time and memory. To overcome these diffi-culties, in this paper, the Operational Quadrature Method (OQM) developed by Lubich is applied toestablish the stability behavior of the time-stepping scheme. Moreover, the Fast Multipole Method(FMM) is adapted to improve the computational efficiency for large size problems. The proposedmethod is tested for large-size elastic wave scattering by many cavities.

Introduction

Since the Boundary Element Method (BEM) is known as a suitable numerical approach for wave anal-ysis, time-domain transient problems have been solved by many researchers using BEM by Mansurand Brebbia[1], and Hirose[2]. In general, transient problems can usually be solved for unknown time-dependent quantities by a direct time-domain BEM with a time-stepping scheme. However, the use ofdirect time-domain BEM sometimes causes two problems. The one is the instability problem of time-stepping procedure and the other one is computational efficiency problem for a large size problem.Recently, to overcome the former problem, the Operational Quadrature Method (OQM), proposed byLubich[3], has been used for the BEM formulation for some engineering problems such as 2-D scalarwave problem[4], poroelastic problem[5] and 2-D anisotropic problem[6]. In the formulation of BEMbased on OQM (OQBEM), the convolution integral is numerically approximated by a quadrature for-mula whose weights are determined by the Laplace transformed fundamental solution and a linearmultistep method. The computational complexity becomes O(LM2N) for the problem with M ele-ments, N time steps, and L expansion terms. On the other hand, the latter problem still remain becauseit is difficult to solve a large scale problem with the large number of M by using OQBEM.　 In this paper, we propose a new time-domain fast multipole BEM based on OQM in 2-D elastody-namics. The Fast Multipole Method (FMM), proposed by Greengard and Rokhlin[7], is applied to theOQBEM to resolve the computational efficiency problem. After the description of basic concept andformulation of proposed method in 2-D elastodynamics, numerical examples for elastic wave scatter-ing are demonstrated by using the proposed method. The computational efficiency of the proposedmethod is also confirmed.


Figure 1 A elastic wave scattering model.

Operational Quadrature Method

In this section, the operational quadrature method (OQM) is briefly described. The OperationalQuadrature Method (OQM), first proposed by Lubich, approximates the convolution f ∗ g(t) by adiscrete convolution using the Laplace transform of the time dependent function f(t − τ ). In general,the convolution integral is defined as follows:

f ∗ g(t) =

∫ t

0

f(t − τ )g(τ )dτ , t ≥ 0 (1)

where ∗ denotes the convolution. The convolution integral defined by Eq. 1 is approximated by OQMas follows:

f ∗ g(nt) ∑

j

ωn−j (t)g(jt) (2)

where time t was divided into N equal steps t. Moreover, ωj(t) denotes the quadrature weightswhich are determined by the coefficients of the following power series with complex variable z, namely

F (δ(ζ)

t) =

∞∑n=0

ωn(t)zn. (3)

In Eq. 3, F is the Laplace transform of the time dependent function f . The power series defined in Eq.3 can be calculated by Cauchy’s integral formula. Considering a polar coordinate transformation, theCauchy’s integral is approximated by a trapezoidal rule with L equal steps 2π/L as follows:

ωn(t) =1

2πi

∫|ζ|=ρ

F

(δ(ζ)

t

)ζ−n−1dζ ρ−n

L

L−1∑l=0

F

(δ(ζl)

t

)e

−2πinlL . (4)

where δ(ζ) is the quotient of the generating polynomials of a linear multistep method and ζl is givenby ζl = ρe2πil/L. In addition, ρ is the radius of a circle in the domain of analyticity of F .


Time-Domain BEM Formulation in 2-D Elastodynamics

We consider the 2-D elastic wave scattering by a scatterer D in an exterior elastic media D as shown inFig. 1. When the incident wave uin hits the boundary surface S of a scatterer D, scattered waves aregenerated by the interaction with the scatterer D. Assuming the zero initial conditions, i.e., ui(x, t =

0) = 0 and ∂ui(x, t = 0)/∂t = 0, the governing equations and boundary conditions are written asfollows:

µui,jj(x, t) + (λ + µ)uj,ij(x, t) = ρ∂2ui(x, t)

∂t2in D (5)

ui = ui on S1, ti = ti on S2, S2 = S \ S1 (6)

where ui and ti show the displacement and traction respectively, ρ is the density of elastic media D,and λ and µ indicate Lame constants. In Eq. 6, ui and ti are given boundary values. The time-domainboundary integral equation in 2-D elastodynamics can be expressed by

Cij(x)ui(x, t) = uini (x, t) +

∫S

Uij(x, y, t) ∗ tj(y, t)dSy −∫

S

Tij(x, y, t) ∗ uj(y, t)dSy. (7)

In Eq. 7, Uij(x, y, t) and Tij(x, y, t) denote the time-domain fundamental solution and its double layerkernel for 2-D elastodynamics and Cij is the free term[8]. Normally, Eq. 7 is discretized by using theappropriate interpolation functions for the unknown values and solved by a time-stepping algorithm.However, there are mainly two disadvantages of the conventional time-domain BEM. The first one isan instability encountered in the time-stepping procedure. The other is the difficulty in solving largescale problems.

Time-Domain FMBEM Based on OQM in 2-D Elastodynamics

To overcome the disadvantages of the conventional time-domain BEM, the Operational QuadratureMethod (OQM) and the Fast Multipole Method (FMM) are introduced.

BEM Formulation Based on OQM

In solving the system of the boundary integral equation (7) numerically, the boundary surface S isdiscritized into M elements due to a piecewise constant approximation of the unknown displacementui and traction ti. Taking the limit of x ∈ D → x ∈ S and applying Eq. 2 and Eq. 4 in OQM tothe convolution integrals in Eq. 7 yields the following discritized boundary integral equations for timeincrement t and n steps as follows:

1

2ui(x, n∆t) = uin

i (x, n∆t) +

M∑α=1

n∑k=1

[An−k

ij (x, yα)tαj (k∆t) − Bn−k

ij (x, yα)uαj (k∆t)

](8)

where Ami and Bm

i are the influence functions which are defined by


Amij (x, y) =

ρ−m

L

L−1∑l=0

∫S

Uij(x, y, sl)e− 2πiml

L dSy (9)

Bmij (x, y) =

ρ−m

L

L−1∑l=0

∫S

Tij(x, y, sl)e− 2πiml

L dSy. (10)

In Eq. 9 and Eq. 10, sl is given by sl = δ(ζl)/(t). The parameter ρ has to be ρ < 1 and is taken asρL =

√ε where ε shows the assumed error in the computation of Eq. 9 and Eq. 10. Uij(x, y, s) and

Tij(x, y, s) are Laplace domain fundamental solutions in 2-D elastodynamics as follows:

Uij(x, y, s) =1

2πµ

K0(sTr)δij − 1

s2T

[K0(sTr) − K0(sLr)],ij

(11)

Tij(x, y, s) = nj(y)ρ(c2L − 2c2

T )Uik,k(x, y, t) + ρc2T

(Uij,k(x, y, t) + Uik,j(x, y, t)

)nk(y) (12)

where cL and cT are the wave velocity of longitudinal and transversal waves respectively, r is givenby r = |x − y|, Kn is the modified Bessel function of the second kind in Eq. 11 and ni(y) is thecomponent of a outward unit normal vector with respect to y. Note that sL and sT are defined bysL = δ(z)/(cLt) and sT = δ(z)/(cTt) due to the simple expression. To determine δ(ζl), we usethe backward differential formula (BDF) of order two as follows:

δ(ζl) = (1 − ζl) +(1 − ζ2

l )

2. (13)

Note that Eq. 9 and Eq. 10 are identical to the discrete Fourier transform. Therefore, the calculationsof Eq. 9 and Eq. 10 can be evaluated by means of the FFT algorithm. After arranging Eq. 8 accordingto the boundary conditions, we can obtain

1

2ui(x, nt) +

M∑α=1

[B0

ij(x, yα)uαj (nt) − A0

ij(x, yα)tαj (nt)

]

=uini (x, nt) +

M∑α=1

n−1∑k=1

[An−k

ij (x, yα)tαj (kt) − Bn−k

ij (x, yα)uαj (kt)

]. (14)

For the n-th time step, all the quantities on the right-hand side are known. Therefore, the unknownvalues uα

i and tαi can be obtained by solving the above equation.

Unfortunately, we cannot solve a large scale problem with the large number of M by the time-domain BEM based on OQM because the required computational complexity and memory becomeO(LM2N) and O(M2L) in Eq. 14, respectively. Therefore, the time-domain BEM based on OQM isaccelerated by the Fast Multipole Method (FMM) in this research.

Time-Domain Fast Multipole BEM Formulation Based on OQM

The FMM proposed by Greengard and Rokhlin is a technique to reduce the computational time andmemory for a large scale problem. In recent years, Fast Multipole BEM, which is the coupling method


of BEM and FMM, has been developed to improve the computational efficiency for various large scaleproblems in many engineering fields, e.g., the 2-D scalar wave problem[9][10], and the 3-D sound andenvironmental vibration problems[11]. Since FMBEM algorithm has been described in detail in otherpublished papers (for example, see the paper of Nishimura[12]), we will summarize only the essentialformulas here.

Now, the fundamental solution in Eq. 11 is transformed into the following equation;

Uij(x, y, s) =1

µs2T

[ΦU

,i + e3ijΨU,j

](15)

where ΦU and ΨU are displacement potentials with respect to P and S-waves, which are defined by

ΦU =1

2πK0(sL|x − y|),k (16)

ΨU = e3kl1

2πK0(sT |x − y|),l. (17)

　 To apply FMM, we consider a point o near the source point y. Locations of field point x andsource point y are expressed as (r, θ) and (ρ, φ), respectively in polar coordinate system originated atthe point o. Using Graf’s addition theorem, we obtain the multipole expansions of the displacementpotentials as follows:

ΦU =1

2π

∞∑n=−∞

MUn Kn(sLr)einθ (18)

ΨU =1

2π

∞∑n=−∞

NUn Kn(sTr)einθ (19)

where the coefficients MUn and NU

n are called the multipole moments, which are given by

MUn = − ∂

∂yk

[In(sLρ)e−inφ

](20)

NUn = −e3kl

∂

∂yl[In(sT ρ)e−inφ]. (21)

In Eq. 20 and Eq. 21, In shows the modified Bessel function of the first kind. Multipole moments MTn

and NTn for Tij(x, y) is similarly obtained. Once the multipole moments are obtained, we can quickly

evaluate the matrix-vector products of the discritized integral equation (14) using the fast multipolealgorithm[7]. The translation formulas (M2M, M2L and L2L) are also derived from Graf’s additiontheorem of the fundamental solutions defined in Eq. 11 and Eq. 12.　 The modified Bessel function In(z) , in practice, tends exponentially to infinity for large argumentz. This fact sometimes causes the instability of the translation formulas when the cell size is large. Toresolve the problem, we introduced the scaling of the multipole and local expansion coefficients.

Numerical Examples

Time-domain BEM based on OQM (OQBEM) is applied to analyze the transient behaviors of a cavitywith radius a as shown in Fig. 2. The boundary of the cavity is supposed to be traction free. The


x1

2x

A

B

a

cavity

incident wave

C

Figure 2 A scattering model.

cLt/a

u1/u0

Pao and Mow-A

OQBEM-A

Pao and Mow-B

OQBEM-B

Pao and Mow-C

OQBEM-C

Figure 3 Analytic and numerical solutions u1/u0 atA, B and C. The solutions obtained by OQBEM areindicated by symbols and the analytical numericalresults of Pao and Mow are shown by solid lines.

The number of Elements

Co

mp

uti

ng

tim

ese

c(

) 108

107

106

105

104

103

101

102

101

102

103

104

105

OQBEM

FM OQBEM-

109

Figure 4 The comparision of CPU time betweenOQBEM and the FM-OQBEM.

incident wave

x2cavity

2a

3a

x1

Figure 5 A multiple scattering model.

number of elements is 64 and time increment is cLt/a = 0.0625. The parameters N and L are givenby N = L = 128. In addition, ρ is assumed to be ρ = 0.95609320 (ε = 10−10). The displacementcomponents of the incident wave are given by

uini (x, t) = u0δi1[(cLt − x1 − a)/a]H(cLt − x1 − a). (22)

Fig. 3 shows the displacement u1/u0 as a function of time at A, B and C on the boundary of thecavity. This problem has been analytically solved in the frequency domain by Pao and Mow[13]. Thetransient solution can be obtained by superposing the results in the frequency domain by means of thefast Fourier transform. The results by OQBEM are in good agreement with the analytical-numericalresults of Pao and Mow.

Fig. 4 shows the CPU time needed in order to solve scattering problems of the incident wavesby the cavity using time-domain BEM based on OQM (OQBEM) or fast multipole BEM based onOQM (FM-OQBEM). In this analysis, the number of elements is adjusted by changing the size ofthe element. We cannot solve the case that the number of elements is 512 or more with OQBEM


because of the restriction of the memory. We can see that FM-OQBEM is faster than OQBEM whenthe number of elements is several thousands or more as shown in Fig. 4.

Finally, we consider the scattering problem of an incident wave with wave length a/2 by 8 × 8

cavities with the radius a of the individual cavities and the cavity spacing 3a between two adjacentcavities along the x1 and x2 axis as shown in Fig. 5. The components of the incident wave are givenby

uini (x, t) = u0 (1 − cosπΘ) , 0 ≤ Θ = t − x1 + 11.5a

cL≤ 2π. (23)

The parameters are taken as N = L = 256, ρ = 0.95609320 (ε = 10−10) and cLt/a = 0.125. Thenumber of DOF in each time step is 8192. This problem cannot be solved by OQBEM. Therefore,the fast multipole method is applied to accelerate the matrix vector products of discritized boundaryintegral equation and to save the memory. Also, OpenMP with 8 threads is used to parallelize thisanalysis. Fig. 6 (a)-(d) show the time variations of the wave fields u1/u0 around cavities. We can seethat scattered waves are generated by the interaction of the incident wave and cavities.

Thus, time-domain fast multipole BEM based on OQM is very effective in both aspects of thecomputational time and required memory for a large scale problem.

Conclusions

In this paper, the time-domain fast multipole BEM formulation based on OQM was developed for 2-Delastodynamics. The convolution integrals were discritized by the operational quadrature method andthe fundamental solutions in Laplace domain were used for the calculations of influence functions.The fast multipole method was applied to accelerate the calculations of matrix-vector products for theretarded potential and to reduce the memory requirement. As numerical examples, scattering problemsof incident waves by cavities were demonstrated and the computational efficiency of the proposedmethod was confirmed. In near future, we will develop the time-domain fast multipole BEM based onOQM in 3-D elastodynamics.

Acknowledgement

This work is supported by the Japan Society of the Promotion of Science.

References

[1] W. J. Mansur and C. A. Brebbia Transient elastodynamics using a time-stepping technique, In;Boundary Elements, C. A. Brebbia, T. Futagami and M. Tanaka (Eds), 677-698 (1983).

[2] S. Hirose Boundary Integral equation method for transient analysis of 3-D cavities and inclusions,Engineering analysis with Boundary Elements, vol.8, No.3, 146-153 (1991).

[3] C. Lubich Convolution quadrature and discretized operational calculus I , Numer. Math.,52, 129-145 (1988).

[4] A. I. Abreu, J. A. M. Carrer and W. J. Mansur Scalar wave propagation in 2D: a BEM formulationbased on the operational quadrature method, Engineering analysis with Boundary Elements, 27,101-105 (2003).


x a2/ x a2/

x a2/x a2/

x a1/ x a1/

x a1/x a1/

(a) (b)

( )C (d)

Figure 6 Time variations of displacements u1/u0 around cavities. (a) cLt/a = 6.25 (b) cLt/a =

12.5 (c) cLt/a = 19.5 (d) cLt/a = 25.0.

[5] M. Schanz and V. Struckmeier Wave propagation in a simplified modelled poroelastic continuum:Fundamental solutions and a time domain boundary element formulation, Numer. Math.,64, 1816-1839 (2005).

[6] Ch. Zhang Transient elastodynamic antiplane crack analysis of anisotropic solids, Int. J. Solidsand Structures, vol. 37, 6107-6130 (2006).

[7] L. Greengard and V. Rokhlin A fast algorithm for particle simulations, Journal of ComputationalPhysics, 73, 325-348 (1987).

[8] S. Kobayashi Wave Analysis and Boundary Element Methods, Kyoto University Press (in Japanese),(2000).

[9] T. Fukui and J. Katsumoto Fast multipole algorithm for two dimensional Helmholtz equation andits application to boundary element method. Proc of the 14th Japan National Symposium on Bound-ary Element Methods (in Japanese), 81-86 (1997).

[10] T. Saitoh, S. Hirose, T. Fukui and T. ISHIDA, Development of a time-domain fast multipoleBEM based on the operational quadrature method in 2-D wave propagation problem, Advances inBoundary Element Techniques VIII, 355-360 (2006).

[11] T. Saitoh A study on effective 3-D numerical analysis of environmental vibration and noise in-duced by a moving train, doctral thesis in Tokyo Institute of Technology, (2006).

[12] N. Nishimura, Fast multipole accelerated boundary integral equation methods, Appl. Mech. Rev.,55, 299-324 (2002).

[13] Y.-H. Pao and C. C. Mow, Diffraction of Elastic Waves and Dynamics Stress Concentrations,Crane and Russak, New York, (1973).


Characteristic matrix in the bending plate analysis by SBEM

Panzeca T.1,a, Cucco F. 2,b, Salerno M. 1,c

1Diseg, Viale delle Scienze, 90128 Palermo, Italy

2Via E. Tricomi 8, 90127 Palermo, Italy


Keywords: bending plate, Symmetric Boundary Element Method, Kirchhoff shear force.

Abstract. This paper deals with the thin bending plate analysis by using the symmetric approach of

Boundary Element Method (SBEM).

A formulation is used in which the plate boundary is discretized into boundary elements and is

subjected to appropriate distributions of shear forces and couples, as well as of vertical

displacement and rotations.

These distributions are the causes and are modelled through appropriate shape functions, whereas

the generalized effects are obtained, according to the Galerkin approach, as weighting of the

displacements and the rotations, as well as of the shear forces and moments.

In the equations system the algebraic operator is a symmetric matrix whose coefficients are defined

as double integrals with high order singularities, all computed in closed form.

Introduction

The object of this paper is to consider some computational aspects regarding the thin bending plate

analysis with the SBEM (Bonnet et al. [1]). Respect to the collocation BEM, in which

nonsymmetric boundary integral formulations for bending plates have been studied by various

authors (Beskos [2], Aliabadi [3]), the SBEM approach shows few contributions about the plate

analysis (Tottenham [4], Frangi and Bonnet [5], Perez-Gavilan and Aliabadi [6]).

The computation of the solving equation coefficients presents considerable difficulties for the

presence of high order singularities in the kernels of the integrals and it is carry out either by means

of a derivative transfer technique, employing the integration by parts to reduce the singularity of the

fundamental solutions (Frangi and Bonnet [5]), or by means of an approach based on a limit process

(Frangi and Guiggiani [7] in the collocation context).

In the present paper a general computational methodology that makes easier the generation and the

check of the coefficients calculus is shown. In this methodology, already applied to the in-plane

loaded plate in Panzeca et al. [8], the bending plate is studied without considering the actual

constraint and boundary load conditions. For the plate an appropriate algebraic operator, so-called

characteristic matrix, which connects the kinematical and mechanical quantities along the boundary,

is introduced. This matrix is singular and its coefficients are valued by imposing in a sequential way

some distributions of causes on the boundary elements, and by computing the effects in the same

boundary through a weighting process of the response.

This approach is particularly useful when the matrix coefficients are determined; indeed the rigid

body technique allows to verify the effectiveness of the coefficients computation.

In the first section the peculiarities of the characteristic matrix are shown and an appropriate

rearrangement and employment of this matrix is explained in order to obtain the algebraic operators

of the mixed value elastic problems.

In the second section a technique to compute a coefficient of the characteristic matrix is illustrated.

The kernels of the integrals are defined as distributions in the Schwartz sense, specifically the

distribution definition is employed as the limit of a succession of functions. This approach makes it


possible to naturally cancel out the singularities of higher order in the cause integration, whereas the

lower order singularities are smoothed by the effect shape functions and eliminated by the outer

integration.

1. Symmetric characteristic operators

Let us consider the bending problem for a linearly elastic plate of domain and boundary ,

distinguished into constrained and free 1 2 . The plate is subjected to the following external

actions (Fig. 1a):

- body forces p normally applied to the middle surface in the domain ;

- displacements and rotations T

n nuu 0 imposed on the constrained boundary ;1

- forces and couples T

n nf c 0f given on the free boundary 2 .

The elastic response to the known external actions may be obtained in terms of boundary quantities,

defined along an element characterized by the outward normal :n

- shear forces and couples T

n n snf c cf on 1 ;

- displacement and rotationsT

n n snuu on 2 ;

To study the plate a general strategy is used, based on the introduction of a matrix, called

characteristic. The plate is embedded in an unlimited domain (Fig. 1b) having the same

Young’s modulus , Poisson’s ratio E and the same thickness h of the plate, and as a consequence

its boundary may be considered as boundary of or of the complementary domain \ .

cn

cn

nn

ff

p

1

u

u

a) b)

n

n

n

n

p

Fig. 1: a) A polygonal plate, b) the plate embedded in .

To derive the characteristic matrix no distinction is made between the constraint or free boundaries.

It involves that the entire boundary is subjected to a distribution of layered mechanical actions

and to a distribution of double layered kinematical discontinuities

f

u u .

It is known that the response in terms of kinematical u and mechanical quantities at every point

of the on the boundary

t

is given by the Somigliana Identities (SI). By imposing the boundary

conditions and on , which replace the classical Diriclet and Neumann conditions +u 0 0+t

u u and t f on , these SI may be written in compact form in the following way:

+PV

ˆ[ ] [ ] [p]1

u u f u u u u2

0 (1.a)

+PV

ˆ[ ] [ ] [p]1

t t f t t -u t2

0

(1.b)

where the following positions are valid:

uu PV ut uuˆ[ ] d , [ ] ( )d , [p] p du f G f u u G u u G (2a-c)


PV tu tt tuˆ[ ] d , [ ] ( )d [p] p d ,t f G f t u G u t G

U

(2d-f)

being the fundamental solutions matrices defined in Panzeca et al. [9].hk ( h,k = u, t)G

Let us operate the discretization of the boundary and introduce appropriate shape functions and

to model the layered mechanical and the double layered kinematical quantities: t

u

t u,f F u (3a,b)

where and are the vectors collecting nodal quantities. Linear shape functions are assumed for

, whereas quadratic ones in the bending rotation and hermitian ones in the displacement and

torsional rotation discontinuity are assumed for .

F U

t

u

Let us perform the weighing process in accordance with the Galerkin approach in the eqs. (1a,b), so

obtaining the following boundary integral equations:

+uu ut ut

1 ˆ( ) ( )2

W A F A U C U W 0 (4a)

+tu tt tu

1 ˆ( )2

P A F A U C F P 0 (4b)

where the following positions are set:

+

+ +

+ + +

+

+ T T + T 't t uu uu t uu t

T ' + T +ut t ut u ut t u

+ T + + T + T ' +u u tu tu u tu t

T +tu u t tt

ˆd , [ p d ]d , [ d ]d

1[ d ]d , [ d ],

2

ˆd , [ p d ]d , [ d ]d ,

1[ d ],

2

W u W G A G

A G C

P t P G A G

C A+

T ' +u tt u[ d ]dG

,

(5a-l)

In compact form one has:

ˆB X L 0 (6)

with

uu ut ut

tu tt tu

ˆ1 2ˆ( ), , , ,

ˆ1 2

A A 0 C F WB A + C A C X L

A A C 0 U P (7a-e)

The matrix is symmetric, whereas the matrix C , which includes the free terms, is emi-

symmetric. The matrix is unsymmetric and singular: the singularity depends on the circumstance

that the plate may be subjected to a rigid motion.

A

B

In the present paper all the coefficients of the matrix B have been computed in closed form.

The matrix B is used to solve the mixed value elastic problems. Indeed it allows to generate both

the pseudostiffness matrix and the load vectors due to the mechanical actions applied on the free


boundary and to the kinematical quantities imposed at the constrained boundary , as it has

been made by Panzeca et al. [8] in the in-plane loaded plate. 2 1

In order to get this aim, a rearrangement of the rows and columns of the characteristic matrix is

performed: indeed the vectors and ,F U W and P are redefined in the following way

T T TT T T T T T T T

1 2 1 2 1 2 1 2, , ,F = F F U = U U W = W W P = P PT

0 0

(8a-d)

Let us to introduce the Dirichlet and Neumann generalized conditions on the boundaries:

1W on , on (9a-b)1 2P 2

The rearrangement introduced in the eqs. (9a-b) allows to derive the following equations:

u2u2 u2u1 u2t 2 u2t12 2p2

u1u2 u1u1 u1t2 u1t11 1p1

t2u2 t2u1 t 2t 2 t2t12 2p2

t1u2 t1u1 t1t2 t1t11 1p1

= +

W B B B B WF

W B B B B WF

P B B B B PU

P B B B B PU

p

0

0

0

0

(10)

In compact form:

ˆ+ =K X L 0 (11)

where the following positions are made:

u1u1 u1t2 1

t 2 u 1 p

t2u1 t2t 2 2

1pu1u2 u1t1

t u p

2pt2u2 t2t1

ˆ= ; = ; = + (- ) + p ;

; ; ;

B B FK X L L F L U

B B U

WB BL L L

PB B

L

(12a-f)

In eq. (11) K is the pseudostiffness matrix, symmetric and non singular, whereas is the

generalized load vector.

L

2. Coefficients analytical computation in the Kirchhoff model.

In the Kirchhoff model the following Love-Kirchhoff hypotheses have been introduced:

- kinematical assumption: in the plate boundary, having normal and tangent , the torsional

rotation is the tangential derivative of the vertical displacement;

n s

- mechanical assumption: the distributed torsional moment along the plate boundary may be

replaced by an appropriate distribution of transversal shear forces, leading to the so-called

Kirchhoff shear, which is the sum of the shear force and the tangential derivative of the torsional

moment.

To get a symmetric formulation, a kinematical quantity, associated to the Kirchhoff shear force and

defined the Kirchhoff displacement discontinuity, has to be introduced.

The shape functions employed for the modelling process are assumed to be linear for the forces and

couples, quadratic for the rotation discontinuities, whereas they are assumed to be hermitian for the

Kirchhoff displacements discontinuities. The shape functions employed for the weighting process


are assumed to be linear for the generalized displacements and rotations, quadratic for the

generalized moments, whereas they are assumed to be hermitian for the generalized Kirchhoff

shear forces.

In the evaluation of the matrix coefficients the following steps are used:

- to impose a unitary value at the node according to the local node system (Fig. 2b),

- to model the cause along the boundary elements through appropriate shape functions according

to the local side system (Fig. 2e),

- to employ the Somigliana Identities,

- to perform the weighting process of the boundary quantities by means of appropriate shape

functions according to the Galerkin strategy.

In the case of a rectangular plate, with sides parallel to the Cartesian axes, let us suppose to

determine the generalized Kirchhoff shear force associated with node 1 as the effect of a

Kirchhoff displacement discontinuity imposed at the same node (Fig. 2) between the two

frontiers and .

KTKU 1

+

In the following schemes the use of the fundamental solutions is shown, with the objective to take

both the Kirchhoff kinematical and mechanical assumptions into account:

kinematical assumption sn

u

s '

n sn

11 12 13

tt n 21 22 23

sn 31 32 33

u

tt tt tttint egration by parts

m tt tt tt

m tt tt tt

G

n

1311 12

23tt n 21 22

33sn 31 32

u

tttt ttt

s '

ttm tt tt

s '

ttm tt tt

s '

G

mechanical assumptionK snm

t ts

n

1311 12

23tt n 21 22

33sn 31 32

u

tttt ttt

s '

ttm tt tt

s '

ttm tt tt

s '

G

n

K 1311

3212 K K

33 11 12tt 31 K

21 22

2321 22n

u

ttt tt

s ' tttt

tt tt ttstt

tt tts s '

tttt ttm

s '

G

As a consequence, the displacement discontinuity imposed at node 1 is transferred along the

boundary sides next to the node as the sum of two distributions: one concerning the vertical

displacement and the other one concerning the tangential derivative of the torsional rotation.

The generalized Kirchhoff shear force associated with the node 1 can be obtained as the sum of

the weighted shear force along the boundary sides a and b. On every side the weighted shear force

is obtained as the sum of two contributions and specifically the vertical force and the tangential

derivative of the torsional moment. At a point of each boundary side the Kirchhoff fundamental

solution , that has to be modelled by the cause shape functions and has to be weighted by the

effect shape function, may be expressed as the sum of four terms:

KT

K11tt


13 33K11 11 31

tt tttt tt tt

s ' s s '(13)

where is the Kirchhoff traction at point of normal n caused by:K11tt x

- a displacement discontinuity -u applied at a point 'x with normal n' (111 31tt tt s ).

- a tangential derivative of the torsional rotation sn- 1 applied at a point with tangent 's

(

'x

13 33tt s ' s tt s ' );

=?

b b

’

’a a

1 1

a)

d)

e)

f)

b) c)

x

n n’s s’

s s’

n n’

z

y

x’

z’

y’KT =-U=1KU

Distributions of shear forces andof tangential derivative of the torsional moment

’

’

BOUNDARY SIDES DISTRIBUTIONS

s n

n snC

C

F, W

nsn

, ,

x

yz

yx MM

T, U

yx ,

,

NODE QUANTITIES

Distributions of displacementdiscontinuities and of tangential derivativeof the torsional rotation

b

a

1b

a

1

Fig. 2: a) Weighted Kirchhoff equivalent shear force, b) node local system,

c) generalized Kirchhoff displacement discontinuity, d) distributions of the effects,

e) side local system, f) distributions of the causes.

The causes are imposed according to the node local system and their boundary effects are evaluated

according to the side local system.

The generalized shear force associated with node 1 as the effect of a Kirchhoff displacement

discontinuity imposed at the same node derives from the double integration of the product

between the fundamental solution and the shape functions

KTKU 1

K11tt '

u ( ')x and . The

fundamental solution is defined in the boundary elements a and b, where it is specified as the

following forms

u ( )x

K K K K11 11 11 11aa ab bb ba

tt , tt , tt , tt , in which the double indices represent the sides where

the effect and cause distributions are located.

Consequently one has:

Inner integrals:

- The Kirchhoff shear force at a point of the side a, caused by the kinematical distributions

associated to the Kirchhoff displacement discontinuity in the sides a and b:


31 1K KK

a 11 ua 11 ubaa ab0 0

E h 24 3(3 )t tt ( ') d ' tt ( ') d ' (

24 (1 ) 1) (14)

- The Kirchhoff shear force at a point of the side b, caused by the kinematical distributions

associated to the Kirchhoff displacement discontinuity in the sides a and b:

31 1K K Kb 11 ub 11 uabb ba0 0

E h 24 3( 5 )t tt ( ') d ' tt ( ') d ' (

24 (1 ) 1) (15)

where and are functions containing logarithmic singularity only. ( ) ( )

In eqs. (14) and (15), the fundamental solution K11 ii

tt shows a singularity of order 41 r in the

interval (0,1), instead the fundamental solution K11 ij

tt shows the same singularity only for

.' 0

Outer integral:

- Let us compute the primitives ,KaT K

bT of the Kirchhoff shear forces ,Kat

Kbt weighted through

hermitian shape functions:

K K Ka ua a b ub bT ( ) t d , T ( ) tKd , (16)

In these two primitives the singularity order decreases, remaining only the logarithmic one. The

two obtained expressions are functions of the same natural variable , therefore the

weighted Kirchhoff shear force associated to the node 1 is obtained as the sum of the two

primitives, defined in (0,1) interval:

(0,1)KT

3 21

K K Ka b 20

E h ( 1 )T = ( 3 2T + T

8(1 )Log[4]) (17)

3. Application.

The application concerns a square plate with two free sides and two simply supported sides, with

side and with the material elastic constants26E 1 10 , h 0.01 and , subjected to

uniform normal load of unit value (Fig. 3).

0.3

x

zT, U

yM y,

yxM x,

Fig. 3: Plate subjected to a vertical load: constraint and load conditions.

The coefficients of the domain load vector are determined in closed form by means of a double

integration, the one regarding the cause through domain integrals, and the one regarding the effect

through boundary integrals.

In Table 1 the results obtained are compared with those obtained using the classic plate theory

(Timoshenko [10]).


ymxm

u

SGBEM

8 nodes

Classic

theory

(Timoshenko)

u (0,0) 2.19162 2.28708

u (1,0) 2.53317 2.63652

xm (0,0) 0.08331 0.1084

ym (0,0) 0.48137 0.4900

ym (0,1) 0.54752 0.5272

Table 1 - Displacements and moments in the plate.

References

[1] M. Bonnet, G. Maier, C. Polizzotto: Symmetric Galerkin boundary element method. Appl.

Mech. Rev. 51 (1998), p. 669-704.

[2] D.E. Beskos, (ed.). Boundary Element Analysis of plates and shells, Springer-Verlag, Berlin,

1991.

[3] M.H. Aliabadi (ed.). Plate bending analysis with boundary elements. In: Advances in Boundary

Elements. Computational Mechanics Publications, Southampton, 1998.

[4] H. Tottenham. The boundary element method for plates and shells. In: P.K. Banerjee, R.

Butterfield, (Eds.), Developments in Boundary Element Methods. vol. 1, Elsevier, Amsterdam,

1979.

[5] A. Frangi, M. Bonnet: A Galerkin symmetric and direct BIE method for Kirchhoff elastic

plates: formulation and implementation. Int. J. Num. Meth. Engng. 41 (1998), p. 337-369.

[6] J.J. Perez-Gavilan, M.H. Aliabadi: Symmetric Galerkin BEM for shear deformable plates. Int. J.

Num. Meth. Engng. 57 (2003), p. 1661-1693.

[7] A. Frangi, M. Guiggiani: Boundary element analysis of Kirchhoff plates with direct evaluation

of hypersingular integrals. Int. J. Num. Meth. Engng. 46 (1999), p. 1845-1863.

[8] T. Panzeca, F. Cucco, S. Terravecchia: Symmetric boundary element method versus Finite

Element Method. Comp. Meth. Appl. Mech. Engng. 191 (2002), p. 3347-3367.

[9] T. Panzeca, V. Milana, M. Salerno: A symmetric Galerkin BEM for plate bending analysis, in

press Eur. J. Mech. A/Solids, DOI 10.1016/j.euromechsol.2008.02.004, (2008).

[10]S.P. Timoshenko, S. Woinow-Sky-Krieger. Theory of plates and shells. McGraw-Hill Book

Company, 1959.


Computational aspects in thermoelasticity

by the Symmetric Boundary Element Method

Panzeca T.1,a, Terravecchia S.1,b and Zito L.1,c

1 Diseg Viale delle Scienze, 90128 Palermo, Italy.


Keywords: thermoelasticity, symmetric boundary element method, Galerkin approach, singular domain integral.

Abstract. Within the thermoelasticity field treated by the Symmetric Boundary Element Method

(SBEM) [1] some difficulties regard the removal of the strong singularities arising in the kernels of

the domain integrals. It happens because the differential operator necessary to obtain the stress is

applied to a singular displacements field.

When this method is applied to mechanics problems, several computational advantages happen if

the domain integrals are replaced by boundary integrals. These advantages are considerable

especially in the analysis phase because the evaluation of the domain load terms arising from a

weighted process may be easily computed in closed form.

The present approach acts in the displacements field, at first substituting the domain integrals by

boundary ones and successively applying the differential operator to a regular field in order to

obtain the thermal stress field..

Introduction

In the SBEM the stress state computation in the domain , due to a volumetric distortions

distribution, represents one of the topics more studied in the plastic analysis.

The evaluation of the stress field arises from the Somigliana Identities (S.I.) of the displacements.

The direct use of the differential operator to the singular field of the displacements produces a

domain integral which is interpreted as a Cauchy principal value and a “jump” term (Bui free term)

[2]. The evaluation of this latter integral is made through a regularization process.

Among the current methodologies, those based on the transformation of the singular domain

integrals regarding strain or stress fields into boundary ones, performed by Gauss theorem [3] or

Radial Integral Method (RIM technique) [4], are computationally more advantageous. The

advantages of these methodologies are more useful inside the SBEM because the load terms due to

domain actions, transferred on the boundary, permit to perform the Galerkin weighting on the

boundary without domain integrals.

In the present paper, the stress at an inner point is dealt with a different way in comparison with the

Huber [3] and Gao [4] approaches.

In the S.I. of displacements the domain integral is regularized and the singular integral is

transformed into boundary one through the RIM technique trasferring so the cause quantity on the

boundary and the effect point in the domain. A non-singular displacement field is obtained and the

differential operator may be directly applied. By using the Hooke law, the S.I. of stresses is found;

that latter is formally the same expression obtained by Huber [3], but in the present formulation the

jump term does not appear because the differential operator is applied to a regular displacements

field.

In the present formulation the inelastic domain actions are constant, and the proposed approach

permits to deal with inelastic actions, modelled either in the cells or in the body in a simple way.

By using Cauchy formula and through a limit approach the tractions are computed on the boundary;

these latter are utilized to obtain in a closed form the weighted values of a part of the load vector.


1. Stress field

The starting point of the elastic problem is the S.I. of the displacements. When we ignore the mass

forces this latter expression takes on the following form:

[ , , , ] ( )uu ut ud d du f u b G f G u G (1)

In the infinite domain the displacements are caused by mechanical f and kinematical u layered

discontinuities vectors, both collecting known and unknown functions, but also by the volumetric

distortion vector , like temperature variation, which in the present formulation are assumed

constant in . In eq.(1) the ukG are matrices of fundamental solutions symbolically introduced by

Maier and Polizzotto [1].

The compatibility condition is the following

[ , , ] [ , , ]t t

p p pf u D u f ux

(2)

where p is the volumetric distortion in P. The differential operator Dx

gives rise to the following

equation

( )u t x u pd d dG f G u D G . (3)

where the positions u uuG D Gx

and t utG D Gx

are assumed. In the domain integral, which

evaluates the effects of the volumetric distortions , the differential operator involves the presence

of hypersingularity in the displacement gradient.

Fig. 1: Body subjected to volumetric distortion; circle of esclusion having radius .

With reference to Fig. 1, generically the latter integral of eq.(3) may be written as follows

0 0lim limu u u pd d dD G D G D G

x x x

(4)

or in this way

0 0lim ( ) lim( )u u u pd d dD G D G D G

x x x

(5)

The expressions (4,5) give rise to formulations characterized by regularization techniques. The

solutions proposed by Gao [4] and Huber [3] are discussed in the following subsections. In the

eq.(5) the second limit is null.

1.1 Gao approach

The first integral of eq.(4) is meant as Cauchy Principal Value (CPV) and may be written in a

regularized form in the following way

0 0lim ( ) ( ) limu p pd d dD G G G

x

(6)


The singular integral is modified by the RIM technique into boundary integral

0lim ( ) t

p p pd Log r d dG G n r G (7)

not more singular.

The second integral of eq.(4), by using Gauss theorem and the relation (.) (.)D Dx x'

, is

transformed into an integral defined on the boundary of Fig. 1. Subsequently, by performing a

transformation of the Cartesian coordinates into polar ones, this integral is evaluated in closed form

so obtaining the matrix J of jump terms (Bui free term).

2

00

( )lim ' ' ( ) u

u ud d dJ D G N G Nx

(8)

The expressions of the regolarized strain and stress fields are obtained

( ) ( ) ( )u t p p pd d d dG f G u G G J I (9)

( ) ) ( )u t p p pd d d dE G f G u G G E J I (10)

1.2 Huber approach

The singular integral (5), transformed into polar coordinates, is modified through the Leibnitz

theorem so deriving the Bui free term. Subsequently, after a new transformation into Cartesian

coordinates, a regularization of the singular term is performed. One obtains

0 0lim ( ) ( ) lim 'u p u p u pd d d dD G G D G N G

x x

(11)

The Gauss theorem and the condition (.) (.)D Dx x'

, both utilized into the second integral of the

previous equation, transform the expression as follows

0lim ( ) ' ' 'u p u p u p u pd d d d dD G G N G N G N G

x

(12)

so obtaining the following regularized strain and stress fields

( ) ) 'u t p u pd d d dG f G u G N G (13)

( ) ( ) ( ' )u t p u pd d d dE G f G u G E N G I (14)

1.3 Present model

In the present model the strain expression [ ] given by eq.(5) is regularized and the singular term

is transferred on the boundary by using the RIM technique. Therefore the differential operator is

applied to a non-singular displacement field. The following expression of the strains [ ] is

obtained

0 0lim ( ) limu u p u pd d dD G D G D G

x x x

(15)

The first integral of the previous equation is regular and the differential operator may be applied

directly to the fundamental solution. The RIM technique is used in the second integral before the

application of the differential operator, so obtaining:


) t

u p u pd d dD G G D G n rx x

(16)

If now we apply the differential operator to the second integral of eq.(16) and consider the position

(.) (.)D Dx x'

, one obtains:

't t

u p p u pd d dD G n r G n r N Gx

(17)

Because the following position is valid for the circular integral

t dG n r 0 (18)

the strain and stress fields take on the following regularized form:

( ) ( ) ( ' )u t p u pd d d dG f G u G N G I (19)

( ) ( ) ( ' )u t p u pd d d dE G f G u G E N G I (20)

This latter expression is equal to those obtained by Huber approach (14).

Through a comparison of the expressions provided by Gao (10) and the Huber (14) approaches and

by the present model (20) the following equality is easily demonstrable:

Gao Huber, Present model

( ) 'p u pd dG E J E N G (21)

The expression (21) permits to assert that the Bui free term, present in Gao formulation, does not

appear in explicit form in the present formulation and in the Huber one.

When we apply the Cauchy formula in eq.(20), the S.I. of the traction on an element having the

slope defined by n is obtained, i.e.

( ) ( ) ( ' )T

tu tt tt p u pd d d dt G f G u G N E N G I (22)

In order to evaluate the traction on the discretized boundary element, it is necessary to make a limit

operation from the inner of the body. Indeed the first integral is meant as CPV to which the free

term / 2f must be added, whereas the fourth integral, where the Bui free term is present in implicit

form, gives rise to a term equivalent to the CPV and to a free term as shown in the following

section.

2. S.I. of the traction on the boundary

The fourth integral in eq.(22), below transcribed, is evaluated through a limit operation

'T

u pdN E N G . (23)

In Fig. 2 P represents the point, distant d from the boundary, where the traction has to be computed

on a slope characterized by a normal unit vector assumed parallel to the normal vector to the

generic boundary element. The limit operation gives rise to a coefficient obtainable in closed form.

Two jump terms are added to this coefficient: the first of which corresponds to an half of Bui free

term, i.e. (1/ 2) TN E J , the second one T

bN E J contains the arctangent function.

First jump term: when the point P approaches to the boundary ( 0d ) from the inner of the body,

the contribution (1/ 2) TN E J is obtained putting 1 / 2 , 2 / 2 . Some simple geometrical


considerations permit to observe that, whatever the body geometry may be, the same contribution

arises for each side, whereas the other half depends on the remaining boundary.

Second jump term: the additional term T

bN E J depends on the slope of the boundary element i

towards which the infinitesimal element approaches.

Fig. 2: Traction contribution on each boundary element

In detail:

0

1lim ' ' ( ) '

2i i i

T T T T T

u i u i b u id

d d dN E N G N E N G N E J J N E N G N E H . (23)

3. Solving system

In order to formulate the solving system, in the field of the Galerkin symmetric formulation, the S.I.

of the displacements and of the tractions, evaluated on the boundary, are necessary:

1( ) ( )

2

1( ) ' ( )

2

T

uu ut p u

T T

tu tt u p p

d d

d d d

u G f G u u G n r

t G f f G u N E N G N E H I

(24a,b)

written in the hypothesis of constant

These latter written in compact form on 1 and 2 take on the following form

1 2

1 2

1 1 1 2 2 1 2 1 1

2 1 1 2 2 2 2 1 2

( ) [ , , ] [ ]

( ) [ , , ] [ ]

uu ut

tu tt

d d

d d

u G f G u u f u u

t G f G u t f u t (25a,b)

Let us introduce the shape functions t and u in order to model the boundary layered forces and

distortions:

1 1 2 2 2 2 1 1, , , t t u uf = F f = F u = U u = U (26a-d)

where ( 1F , 2U ) and ( 2F , 1U ) are vectors collecting unknown and known nodal quantities,

respectively.

In order to obtain the solution for the examining body, the Dirichlet and Neumann conditions have

to be imposed, i.e.

1 1u u compatibility condition on 1 (27a)

2 2t f equilibrium condition on 2 . (27b)

In according with the Galerkin strategy, the previous boundary conditions may be written in the

following generalized (weighted) form, so obtaining:


1t 1 1 1( ) du u 0 global compatibility condition on 1 (28a)

2u 2 2 2( ) dt f 0 global equilibrium condition on 2 . (28b)

Introducing the relations (25a,b) and the modelling (26a-d) in these latter equations, one obtains:

1 1 1 2

1

T T

f uu f 1 1 1 f ut u 2 1 2

T

f 1 2 1 1 u 1 1

d d d d ( )

[ , ] [ ] d

G F G U

u F U u U 0

(29a)

2 1 2 2

2

T T

u tu f 1 2 1 u tt u 2 2 2

T

u 2 2 1 2 f 2 2

d d d d ( )

[ , ] [ ] d

G F G U

t F U t F 0

(29b)

whose, rearranged and rewritten in compact form as made by Panzeca et al. [5,6], provide the

following solving system:

uu ut 1 u 2 1 u

tu tt 2 t 2 1 t

[ , ] [ ]

[ , ] [ ]

A A F 0L F U L

A A U 0L F U L (30)

or also

A X + L = 0 . (31)

The vector X collects the unknown quantities defined at the boundary nodes, whereas the vector

L collects also the generalized displacements and tractions due to the volumetric distortions,

constant in .

4. Examples

The proposed approach is applied to two plane structures where the thermal loads, variables in the

domain, are considered zone-wise constant by using the substructuring approach developed by

some of the present authors [5,6] and implemented in the calculus code Karnak.sGbem [7]. The

analysis is made by using the displacement method. At the aim to obtain some results comparable

with the classical theories, the body force is not condidered.

4.1 Example 1

A plate having dimension (L = 2, H = 1) and unit thickness of Fig. 3, subjected to a quadratic

temperature variation in y direction, is analyzed.

The variation low is the following:

2

2 1 0T c y c y cx

(32)

where the constants are defined by

2 1 2 3 1 1 2 0 32

1 1c 2(T T 2T ) , c (T T ) , c T

H H, being 1 2 3T 20; T 40; T 0 . (33a-f)

The analytical solution of this problem, thought for a strain plane state, is the following:

2

ET ; T ; ( )

(1 ) (1 )

y

x x yy y yu y d y= = =x x

-H /2

(34a-c)


x

y

L

H

Fig. 3: Plate subjected to quadratic thermal load: subdivision in substructures.

The plate is subdivided into 10 substructures, each of those is subjected to a constant thermal

variation obtained by eq.(32); for every substructure the following mechanical and physical

characteristics have been adopted: E 10000 daN / cmq, 0.3; = 0.00001. A discretization

having constant step 0.1 p cm has been introduced. In the barycentre of every substructure the

stresses x and the vertical displacements yu have been evaluated by using the strategy developed

in the present paper. The results obtained are compared with the analytical solution (see Tab.1).

y sx analytical sx uy analytical uy

0.45 2.07692 2.07692 1.84751 e-5 1.819 e -5

0.35 1.76923 1.76923 2.59857 e-5 2.569 e -5

0.25 1.37363 1.37362 3.21429 e-5 3.182 e -5

0.15 0.89011 0.89011 3.65857 e-5 3.624 e -5

0.05 0.318631 0.318681 3.89714 e -5 3.859 e -5

-0.05 - 0.340659 - 0.340659 3.89571 e -5 3.855 e -5

-0.15 - 1.08791 - 1.08791 3.62000 e -5 3.576 e -5

-0.25 - 1.92308 - 1.92307 3.03571 e -5 2.989 e -5

-0.35 - 2.84615 - 2.84615 2.10857 e -5 2.059 e -5

-0.45 - 3.85714 - 3.85714 0.80428 e -5 0.752 e -5

Table 1: Analytical solution x and yu compared with the present approach.

4.2 Example 2

As second example a beam having dimension (L = 300 cm, H = 50 cm) and thickness s = 30 cm of

Fig. 4 is analyzed. In the middle section of the beam a concentrated inelastic distortion is applied,

consisting in a relative rotation tk , where tk is the curvature due to a linear thermal

variation and is the thermal gradient.

In the example the concentrated distortion is simulated by introducing in the middle section of

the beam a strip long 1 cm, subdivided into 20 substructures, each of whose is subjected to the

constant thermal variation Tx

.

The data of the examining problem are: E 300000 daN / cmq , 0.12 , = 0.000012 , T =+25s

x ,

T =-25i

x , 1, tk . A discretization having constant step 2.5 p cm has been

adopted.

The analytical solution, in the case of mono-dimensional solid, foresees a built-in moment

M (EI / L) 3750daN cm and a linear stress distribution along the height (M / I) yx in

every section.


x

y

L

H

y

x

1 cm

+25°

-25°

a)

b) c)

Fig. 4: Beam subjected to concentrated distortion ; a) subdivision into substructures; b)

temperature distribution.

y sx analytical sx

-1.25 0.015 0.014663

-3.75 0.045 0.045395

-6.25 0.075 0.076141

-8.75 0.105 0.106874

-11.25 0.135 0.137534

-13.75 0.165 0.168024

-16.25 0.195 0.198180

-18.75 0.225 0.227784

-21.25 0.255 0.256727

-23.75 0.285 0.285236

M analytical M computed

-3750 -3751

Table 2: Analytical solution x compared with the present approach.

References

[1] Maier G., Polizzotto C., (1987), “A Galerkin approach to boundary element elastoplastic

analysis”, Comp. Meth. Appl. Mech. Engng., 60, 175-194.

[2] Bui H. D., (1978), “ Some remarks about the formulation of three-dimensional

thermoelastoplastic problems by integral equation”. Int. J. of Solids and Structures., 14, 935-

939.

[3] Huber O., Dallner P., Parteymuller P., Kuhn G., (1996), “Evaluation of the stress tensor in 3D

elastoplasticity by direct solving of hypersingular integrals”. Int. J. Num. Meth. Engrg., 39,

2555-2573.

[4] Gao X. W., (2003), “Boundary element analysis in thermoelasticity with and without internal

cells”. Int. J. Num. Meth. Engrg., 57, 975-990.

[5] Panzeca T., Salerno M., Terravecchia S., (2002), “Domain decomposition in the symmetric

boundary element method analysis”. Comp. Mech., 28, 191-201.

[6] Panzeca T., Cucco F., Terravecchia S., (2002), “Symmetric boundary element method versus

Finite element method”. Comp. Meth. Appl. Mech. Engng., 191, 3347-3367.

[7] Cucco F., Panzeca T., Terravecchia S., (2002), The program Karnak.sGbem Release 1.0.


Natural convection flow of micropolar fluids in a square cavity byDRBEM

Sevin Gümgüm1,a, Münevver Tezer-Sezgin2,b

1Department of Mathematics, Izmir University of Economics, Izmir, Turkey2Department of Mathematics, Middle East Technical University, Ankara, Turkey

ae-mail:[email protected], be-mail:[email protected].

Keywords: DRBEM, natural convection, micropolar fluids

Abstract. The main purpose of this study is to present the use of the Dual Reciprocity Bound-

ary Element Method (DRBEM) in the analysis of the unsteady natural convection flow of

micropolar fluids in a differentially heated square cavity. For the resulting system of ordinary

differential equations in time, the finite difference method (FDM) is made use of. The results

are reported for the configuration in which the cavity is heated from the vertical walls while

the horizontal walls are insulated. Solutions are obtained for several values of microstructure

parameter and Rayleigh number (Ra). The heat transfer rate (average Nusselt number) of

micropolar fluids is found to be smaller than that of the Newtonian fluid. Numerical results

are given in terms of streamlines, isotherms, vorticity contours as well as a table containing

Nusselt number values for several Ra.

Introduction

The micropolar fluid model, introduced by Eringen [1], is a generalization of the well-

established Navier-Stokes model in the sense that it takes into account the microstructure of

the fluid. This model introduces a new kinematic variable called microrotation which describes

rotation of particles. The theory is expected to describe successfully the non-Newtonian be-

havior of certain real fluids, such as liquid crystals, colloidal fluids, and liquids with polymer

additives. Physically, micropolar fluids may be represented by fluids consisting of dumb-bell

molecules or short rigid cylindrical elements.

Natural convection heat transfer in enclosures has been of considerable research interest

in recent years due to the coupling of fluid flow and energy transport. Most of the previous

studies on natural convection in enclosures have been related to Newtonian fluids. The study

of Lo et al [2] represented a numerical algorithm which has been implemented to analyze

natural convection in a differentially heated cavity. In their study, higher-order polynomial

approximation used for discretizing the partial derivatives in the DQ method has been used to

obtain accurate numerical results while solving the velocity-vorticity form of the Navier-Stokes

equations. The penalty finite element method was applied to solve natural convection flows in

a square cavity with non-uniformly heated walls by Roy and Basak [3]

Hsu and Chen [4] numerically investigated the natural convection of a micropolar fluid in

an enclosure heated from below using the cubic spline collocation method. They studied the

effects of microstructure on the convective heat transfer and found that heat transfer rate of

micropolar fluids was smaller than that of the Newtonian fluid. In another work, Hsu et al [5]

studied natural convection of micropolar fluids in an enclosure with isolated heat sources. They

observed that the heat transfer rate is sensitive to the microrotation boundary coonditions and

the average Nusselt number is lower for a micropolar fluid, as compared to a Newtonian fluid.

In a recent study, Aydin and Pop [6], numerically investigated the steady natural convective

heat transfer of micropolar fluids in a square cavity with differentially heated walls using the

finite difference method. They found that the average Nusselt number increases with increasing

Rayleigh number and an increase in the material parameter reduces the heat transfer.

In this study, the dual reciprocity boundary element method is employed to discretize the

spatial derivatives in the stream function-vorticity form of the Navier-Stokes equations, energy


and the microrotaion equations. DRBEM idea is applied to the Laplace operator in each equa-

tion by using the fundamental solution of Laplace equation and keeping all the other terms as

nonhomogeneity. The resulting matrices contain integrals of logarithmic function or its normal

derivative. The DRBEM reduces all calculations to the evaluation of the boundary integrals

only. This fact might be advantageous in geometrically involved situations that are frequently

encountered in fluid flow problems. DRBEM application of unsteady natural convection flow of

micropolar fluids gives rise to a system of initial value problems in time. The finite difference

scheme is made use of solving this system.

Governing Equations

The non-dimensional unsteady equations of motion, energy and microrotation can be written

as follows [6]

∇2ψ = −w

(1 + K)∇2w − K∇2N +Ra

Pr

∂T

∂x=

∂w

∂t+ u

∂ w

∂x+ v

∂ w

∂y

1

Pr∇2T =

∂T

∂t+ u

∂ T

∂x+ v

∂ T

∂y

(1 +K

2)∇2N − 2KN + Kw =

∂N

∂t+ u

∂ N

∂x+ v

∂ N

∂y

(1)

where (x, y) ∈ Ω ⊂ R2, t > 0. Ra, Pr and K are the Rayleigh number, Prandtl number,

and material parameter respectively. ψ and w are the stream function and the vorticity with

u = ∂ψ∂y

, u = −∂ψ∂x

and w = ( ∂v∂x

− ∂u∂y

). The initial and the boundary conditions are taken as

w = T = N = 0 when t = 0

x = 0 : 0 ≤ y ≤ 1, u = v = 0, T = 0.5, N = 0

x = 1 : 0 ≤ y ≤ 1, u = v = 0, T = −0.5, N = 0

y = 0, 1 : 0 ≤ x ≤ 1, u = v = 0, ∂T/∂y = 0, N = 0 .

(2)

The vorticity boundary conditions are derived from the Taylor series expansion of the stream

function equation.

For K = 0, stream function, vorticity transport and energy equations describe the classical

problem of natural convection of a Newtonian fluid in a differentially heated square cavity, first

considered by Vahl Davis [8].

The heat transfer coefficient in terms of the local Nusselt number, Nu, and the average

Nusselt number, Nuav at the vertical walls are defined by

Nu = −(∂T

∂x

)|x=0,1 , Nuav = −

∫ 1

0

Nudy . (3)

Application of DRBEM

The equations in (1) are weighted through the domain Ω as in [7], by the fundamental

solution

u∗ =1

2πln

1

r


of Laplace equation in two dimensions in which r is the distance between the source and the

fixed points.

Applying Green’s second identity, we have the following integral equations for each source

point i:

ciψi +

∫Γ

(ψq∗ψ − ψ∗ψq)dΓ =

∫Ω

(−w)ψ∗dΩ

(1 + K)ciwi + (1 + K)∫

Γ(wq

∗w − w∗wq)dΓ =∫

Ω(∂w

∂t+ u∂w

∂x+ v ∂w

∂y+ K∇2N − Ra

Pr∂T∂x

)w∗dΩ

1

PrciTi +

∫Γ

1

Pr(Tq

∗T − T ∗Tq)dΓ =

∫Ω

(∂T

∂t+ u

∂T

∂x+ v

∂T

∂y)T ∗dΩ

(1 + K2)ciNi + (1 + K

2)∫

Γ(Nq

∗N − N∗Nq)dΓ =∫

Ω(∂N

∂t+ u∂N

∂x+ v ∂N

∂y+ 2KN − Kw)N∗dΩ

(4)where the subscript ′q′ indicates the normal derivative of the related function and ci = θi/2πwith the internal angle θi at the source point i.

Expanding the nonhomogeneties in each equation in terms of the radial basis functions fj’s

−w =

N+L∑j=1

αjfj(x, y)

∂w

∂t+ u

∂w

∂x+ v

∂w

∂y+ K∇2N − Ra

Pr

∂T

∂x=

N+L∑j=1

αj(t)fj(x, y)

∂T

∂t+ u

∂T

∂x+ v

∂T

∂y=

N+L∑j=1

αj(t)fj(x, y)

∂N

∂t+ u

∂N

∂x+ v

∂N

∂y+ 2KN − Kw =

N+L∑j=1

αj(t)fj(x, y)

(5)

where αj, αj, αj and αj are undetermined coefficients. The numbers of boundary and selected

internal nodes are denoted by N and L , respectively.

The radial basis (coordinate) functions fj are linked to the particular solutions of each

equation with the Laplace operator.

Substituting these expansions in Eq. (4) and the application of Green’s second identitiy to

the right hand sides will result in matrix vector equations for each unknown ψ, w, T and N .

Hψ − Gψq = (Hψ − Gψq)α

(1 + K)(Hw − Gwq) = (Hw − Gwq)α

1

Pr(HT − GTq) = (HT − GTq)α

(1 +

K

2

)(HN − GNq) = (HN − GNq)α

(6)

where G and H are (N + L) × (N + L) matrices defined by

Hij = ciδij +1

2π

∫Γj

∂

∂n

(ln(

1

r)

)dΓj , Gij =

1

2π

∫Γj

ln(1

r) dΓj.


The matrices ψ, w, T and N are constructed by taking the corresponding particular solutions

as columns.

Evaluation of the right hand sides of each equation in (5) at all boundary and interior (N+L)

points gives

Hψ − Gψq = (Hψ − Gψq)F−1 −w

(1 + K)(Hw − Gwq) = (Hw − Gwq)F−1

∂w

∂t+ u

∂w

∂x+ v

∂w

∂y+ K∇2N − Ra

Pr

∂T

∂x

1

Pr(HT − GTq) = (HT − GTq)F

−1

∂T

∂t+ u

∂T

∂x+ v

∂T

∂y

(1 +

K

2

)(HN − GNq) = (HN − GNq)F

−1

∂N

∂t+ u

∂N

∂x+ v

∂N

∂y+ 2KN − Kw

(7)where F is the (N + L) × (N + L) matrix containing coordinate functions fj’s as columns.

Derivatives of w, T and N are approximated by the DRBEM idea

∂w

∂x=

∂F

∂xF−1w ,

∂w

∂y=

∂F

∂yF−1w

∂T

∂x=

∂F

∂xF−1T ,

∂T

∂y=

∂F

∂yF−1T

∂N

∂x=

∂F

∂xF−1N ,

∂N

∂y=

∂F

∂yF−1N

∂2N

∂x2=

∂2F

∂x2F−1N ,

∂2N

∂y2=

∂2F

∂y2F−1N .

(8)

Substituting convection terms back into Eq. (7), and finally rearranging, we end up with

the following system of ordinary differential equations for w, T and N respectively

w − Hw + Gwq − SF = 0

T − HtT + GtTq = 0

N − HnN + GnNq − SF1 = 0

(9)

and a linear system of equations for ψ

Hψ − Gψq = −Sw


with S = (Hψ − Gψq)F−1 and the matrices SF , H, G, Hn, Gn, Ht and Gt are

SF =Ra

Pr

∂T

∂x− K∇N, SF1 = Kw

H = S−1 (1 + K) H −(u∂F

∂xF−1 + v

∂F

∂yF−1

)

Hn = S−1 (1 +K

2) H −

(u∂F

∂xF−1 + v

∂F

∂yF−1

)− 2K

Ht = S−1 1

PrH −

(u∂F

∂xF−1 + v

∂F

∂yF−1

)

G = S−1 (1 + K) G , Gn = S−1 (1 +K

2) G, Gt = S−1 1

PrG .

(10)

For the derivatives of w, T and N in Eq. (9) implicit central differences are used assuming

the previous two time level solutions are known.

Results and Discussion

A numerical model was developed to validate the accuracy for the solutions of 2D unsteady

natural convection flow of micropolar fluids in a square cavity given in Eq. (1) and (2). The

no-slip boundary conditions of the velocities are assumed. The horizontal walls are adiabatic,

while the vertical walls are isothermally heated. Solutions are obtained by using N = 120

boundary elements and L = 400 interior nodes.

Computations are carried out for Ra = 103, 104 and 105 with the time increments ∆t = 0.5,

0.01 and 0.003 respectively. The material parameter K is taken as 0, 0.5, 1 and 2. An increase

in Rayleigh number results in intensified circulation inside the cavity, and thinner thermal

boundary layers for all the variables, stream function, vorticity and isotherms near the heated

and cooled walls. For Ra = 103 the vortex at the center was in circular pattern. With the

increase in Rayleigh number the vortex changes its shape to elliptical form. Since the viscous

forces are dominating when Ra = 103, there is not enough convective motion of the fluid within

the cavity. The isotherms are almost vertical in this case. As the Rayleigh number increases,

the isotherms undergo an inversion at the central region of the cavity. These behaviors can be

seen from Fig. 1 and Fig. 2.

The effect of varying Ra on the average Nusselt number at the heated wall is shown in Table

1 for some values of K and a fixed value of Prandtl number, Pr = 0.71. It shows that for a

fixed value of Ra, an increase in K reduces the heat transfer. In addition, the Newtonian fluid

(K = 0) is found to have higher average heat transfer rates than a micropolar fluid (K = 0).

This is because an increase in the vortex viscosity would result in an increase in the total

viscosity of the fluid flow, thus decreasing the heat transfer. The results are in good agreement

with the results given in [6].


Figure 1: Streamlines, vorticity contours, isotherms for Ra = 103, K = 0.5, K = 1 and K = 2


Figure 2: Streamlines, vorticity contours, isotherms for Ra = 105, K = 0.5, K = 1 and K = 2


K Ra = 103 Ra = 103 Ra = 104 Ra = 104 Ra = 105 Ra = 105

Present [6] Present [6] Present [6]0 1.118 1.118 2.217 2.234 4.476 4.4860.5 1.057 1.057 1.925 1.947 4.023 4.0331 1.031 1.034 1.735 1.771 3.703 3.7292 1.012 1.016 1.529 1.545 3.302 3.314

Table 1: The effect of K on the average Nusselt number Nuav for different values of Ra

Conclusion

The unsteady natural convective heat transfer of micropolar fluids in a differentially heated

square cavity is computationally studied using the DRBEM. Time derivative is discretized

using implicit central difference scheme. The results are obtained for all variables, stream

function, vorticity and temperature also for a Newtonian fluid for comparison. Simulations are

performed to investigate the effects of the Rayleigh number, Ra, and the material parameter,

K, on the momentum and heat transfer. As the Rayleigh number increases boundary layer

formation starts and the average Nusselt number increases. However, an increase in the

material parameter reduces the average Nusselt number.

References

[1 ] AC. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966); 1-18.

[2 ] D.C Lo , D.L Young and C.C Tsai, High resolution od 2D natural convection in a cavity

by the DQ method, JCAM, 203 (2007); 219-236.

[3 ] S. Roy and T. Basak, Finite element analysis of natural convection flows in a square

cavity with non-uniformly heated walls, Int. J. Engrg. Sci., 43 (2005); 668-680.

[4 ] T.H Hsu and C.K. Chen, Natural convection of micropolar fluids in a rectangular enclo-

sure, Int. J. Engrg. Sci., 34(4) (1996); 407-415.

[5 ] T.H. Hsu, P.T. Hsu and S.Y. Tsai, Natural convection flow of micropolar fluids in an

enclosure with heat sources, Int. J. Heat Mass Transfer, 40, No.17 (1997); 4239-4249.

[6 ] O. Aydin and I. Pop, Natural convection in a differentially heated enclosure filled with

a micropolar fluid, Int. J. of Thermal Sciences, 46 (2007); 963-969.

[7 ] P.W. Partridge, C.A. Brebbia and L.C. Wrobel The Dual Reciprocity Boundary Element

Method, Comp. Mech. Pub. Southampton and Elsevier Sci., London, (1992).

[8 ] G. Vahl Davis, Natural convection in a square cavity: A benchmark solution, Int. J.

Numer. Meth. Fluids, 3 (1983); 249-264.


Two-dimensional Thermo-Poro- mechanic fundamental solution for unsaturated soils

Pooneh Maghoul a, Behrouz Gatmiri c,b, Denis Duhamel a

a Université Paris-Est, Institut Navier, LAMI, Ecole des Ponts, Paris, France Email: [email protected]

b Université Paris-Est, Institut Navier, CERMES, Ecole des Ponts, Paris, France Email: [email protected]

c Department of Civil Engineering, University of Tehran, Tehran, Iran

Keywords: Boundary element method; fundamental solution; time domain; unsaturated soil; porous media; thermo-poro-elastic behaviour

Abstract. In this article, after a brief discussion on the unsaturated soils’ governing differential equations

including the equilibrium, air, moisture and heat transfer equations, the closed form transient thermal

fundamental solution of the governing differential equations for an unsaturated two-dimensional deformable

porous medium with linear elastic behaviour for a symmetric polar domain have been introduced. The

derived fundamental solution has been verified mathematically by comparison with the previously

introduced corresponding fundamental solution.

Introduction

There are numerous media encountered in engineering practice whose behaviour is not consistent with the

principles and concepts of classical saturated soil mechanics. An unsaturated porous medium can be

represented as a three-phase (gas, liquid, and solid) or three-component (water, dry air, and solid) system.

The liquid phase is considered to be pure water containing dissolved air and the gas phase is assumed to be a

binary mixture of water vapour and “dry” air [1, 2]. The air in an unsaturated soil may be in an occluded

form when the degree of saturation is relatively high. At a lower degree of saturation, the gas phase is

continuous. Commonly it is the presence of more than two phases that results in a medium that is difficult to

deal with in engineering applications. The phenomenon of coupled heat and moisture transfer in a

deformable partly saturated porous medium is important in relation to several problems, including

underground storage of hot and cold materials, disposal of high-level nuclear waste and so on. Thermally-

induced moisture movements can lead to changes in both the thermal and isothermal properties of the soil

which can subsequently affect the functioning of the soil for its intended purpose.

In order to model unsaturated soil behaviour, firstly the governing partial differential equations should be

derived and solved. Because of the complicated forms of governing partial differential equations the different

numerical methods are presented for solving them. Among them, the boundary element method as the most

efficient is going to be employed for more complicated and coupled ones regarding the behaviour and

consequently the governing differential equations. As in this method, during formulation boundary integral

equations, the applied mathematics concept of the Green functions has been employed. This type of

fundamental solutions for the governing partial differential equations should be first derived. Indeed,

attempting to solve numerically the boundary value problems for unsaturated soils using boundary element

method leads one to search for the associated Green functions [3]. The comprehensive state-of-the-art review

by [4, 5, and 6] provides clearly presented information on the fundamental solution applied in the saturated

soil. For unsaturated soils, the first Green functions for the nonlinear governing differential equations for

static and quasi-static poroelastic media for both two and three-dimensional problems have been derived by

[6, 7]. The thermo-poro-elastic Green functions for the nonlinear governing differential equations for static

and for both two and three-dimensional problems have been derived by [8]. The present research is an

attempt to derive these Green functions for two-dimensional deformable quasi-static unsaturated soil.

Following some reasonable and necessary simplifications, the fundamental solutions will be introduced in

both frequency and time domains.


Governing Equations

For an unsaturated material influenced by heat effects, the governing partial differential equations considered

are of four main groups: equilibrium equations, moisture transfer equations, air transfer equations and heat

diffusion equations [9].

Solid Skeleton. The equilibrium equation and the constitutive law for the soil’s solid skeleton including the

effects of suction and temperature [10]:

0,

,ij ij a a i i

jP P b (1)

ij ij a s ij a w T ijd P Dd d P P d T (2)

Where 1.s sD D and 1

.T TD D . With 1

s sD m and 1

T TD m in which 1

s

a w

e

e P P and

1T

e

e T and 1 1 0m .

The linear elastic matrix D is written as:

ijkl ij kl ik jl il jkD (3)

Considering the strain-deformation relations:

1

2, ,ij i j j i

u u (4)

One can conclude:

1 0, , , , ,i ij j ii s a j s w j T j j

u u P P T b (5)

Continuity and transfer equations for moisture. This part will be divided into vapour transfer and liquid

transfer formulation as follows:

Liquid phase transfer. According to [11] the unsaturated flow equation can be written as

/ .w w w

U q K z (6)

The capillary potential varies with moisture content and temperature. The capillary potential in a

reference temperature in terms of suction will take the following form:

( ) /r g w wP P (7)

The variation of capillary potential according to the temperature is considered by the introduction of surface

tension )(T .

( , ) ( ). ( ) /r rT T (8)

Where ( )T is surface tension of the water and r and ( )r are respectively, the surface tension of the

water and the capillary potential in a reference temperature.

Substituting from eqs (7, 8) and after manipulating some mathematical operations a new suction-based

formulation of water movement equation can be found.

/w w Tw Pw g w wU q D T D P P D z (9)

Where TwD is thermal liquid diffusivity( ) ( )r

w

r

d TK

dT, PwD is isothermal liquid diffusivity

( )

.w

r w

TK and

wD is gravitational diffusivity, wK .

Vapour transfer. The equation of vapour diffusion in porous media according to [12] theory is given as

. . .vap vap

q D v n (10)

In order to find vap

as a function of temperature and moisture content, local thermodynamic equilibrium

should be assumed. Under this assumption the following thermodynamic relationships can be introduced:


0.

vaph (11)

exp.

( ).

gh

R T(12)

Where0is the density of saturated water vapour and h is is the relative humidity.

In this equation 0 is a function of T only and h is a function of only. Then:

0

0vap

d dhh T

dT d

(13)

Substituting Eq (13) in Eq (10) and considering the hypotheses presented in [13]:

/vap w Tv Pv g wq D T D P P (14)

Where

0

0

( ). . . . ,

,

,

a

Tv w

T dDv n h n

D T dT

n

and

0

( ). . . ,

.

,

vap

w r wPv

D g Tv n n

RTD

n

Total moisture transfer. The total moisture movement in unsaturated soil due to temperature gradient

and its resulting moisture content gradient is equal to the sum of the flows which take place in both phases,

vapour and liquid. Thus

/ w T P g w wq V U D T D P P D z (15)

Where TD is the thermal water diffusivity, and is equal toTvap Tw

D D , and PD is isothermal water diffusivity

and is equal toPvap Pw

D D .

Moisture mass conservation. The conservation law for moisture mass is written:

1 0. . . .( ) ( . )w r vap r w

n S n S div V Ut

(16)

It seems reasonable to dispense with the variations of aK and wK due to the variations of rS and consequently

of g wP P for simplicity, since deriving the considered Green functions will become too difficult, at least

with usual methods, due to the nonlinearity of or existence of non-constant coefficients in the governing

differential equations. In this manner, the effects of r

S have been considered in air and water coefficients of

permeability by assuming aK and wK as a multi-linear function of g wP P for each finite domain [6].

However, Eq (16) may be written as:

2 2 21.( ) . r

w r vap r w vap w T w P g w P w

SnS S n D T D P D P

t t (17)

Considering that, the porosity equals the volumetric strain:

,kk k kn u (18)

And the definition of the degree of saturation in the linear form:

01 1( ) ( ) ,

r s g w sS b P P d T T (19)

The governing equation for the moisture becomes:

2 2 2

1 21

,( )

( ) . . . .g wk k

w r vap r w vap w vap w T w P g w P w

P Pu TS S n g n g D T D P D P

t t t

(20)

Where1

/r g wg S P P and2

/rg S T .

Continuity and transfer equations for air. Considering the generalized Darcy’s law, the air flow equation

can be given as:

/ . /g g g g g g

V q K P z (21)

Considering that g

P is a function of temperature, this equation can be written as


. . . /g g

g g g g

g

K PV T K P z

T(22)

Using the thermodynamic state equations for gases, the 1

.g

g

P

Tcan be replaced by

1

273 .

g g atm

pg

g g

P P P

T T(23)

This yields

. . .g

g g pg g

g

PV K T K z (24)

With

1.

. . .( )dg

g r

g

gK c e S (25)

c and d are constants.

With the same approach presented for the moisture equations, the mass conservation of air can be written as:

0div vt

(26)

,

1 2

2 2 2

( ). 1 1 . 1 . 1

. . . . . 0

k k a w

a r a a

a

a Pw w Pw a a Pg Tw

a

u P P TH S n H g n H g

t t t

kH D P H D P k H D T

(27)

Heat flow. Total flow of latent and sensitive heat in an unsaturated porous medium is given based on Philip

and De Vries’ theory as:

0. [ . . . . . . ] . . . .

mw w mv w v mg g g w fg v v fg gq T C U C V C V T T h V h V (28)

Energy Conservation Equation. The differential equation for heat flow is a description, in mathematical

terms, of the law of conservation of energy. The energy conservation equation in a porous medium can be

expressed by

0( )div qt

(29)

In which q is heat flux and is the volumetric bulk heat content of medium which can be defined by

0. . .

T v fgc T T n h (30)

Where Tc is the volumetric heat capacity of unsaturated mixture and can be written as:

1( ). . . . ( ). ( ).T s ms w mw v mv g mg

c n C C n C n C (31)

Combining the above equations yield the general differential equation of energy conservation as:

2 2 2

1 2 1 2 1 2 2 3 4 5 6

,.

gk k w

g w

Pu P Tg g g T P P

t t t t

(32)

Where

1 01 1 1. . . .( ) .( ). . .( )ms s mw r w mv r v mg r g v fg rC C S C S C S T T h S

2 0. . . . . . . .mw w mv v mg g v fgC n C n C n T T h n

3

4 0

1 1 1. .( ) . . .( ) . .( ).

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

ms s mw r w mv r v mg r g

m mw w Tw mv w Tv mg g Pg g w fg Tv v fg g Pg

C n C S C S C n S

C D C D C K T T h D h K


5 0

6 0

. . . . . . . . . .

. . . . . .

g

mw w Pw mv w Pv mg g g w fg Pv v fg

g

mw w Pw mv w Pv w fg Pv

KC D C D C K T T h D h

C D C D T T h D

Set of governing equations. The governing partial differential equations based on the linearization

assumptions considered may be summarized and simplified as:

1 20 0( ,( )( , ) ). . ( , ) , ,

tU x x t x SU x t t t (33)

Where1 2

0 0 0( , )T

x t b b ,1 2

( , )T

w gU x t u u P P T , 2( , ) ,x x y D and the components

of1are: 1

0i j 1

0kl 1 63

.i i

a1 733

a1 834

a1 935

a 1 134.

i ia

1 1443

a1 1544

a1 1645

a 1 205.

i ia1 2153

a1 2254

a1 2355

a

And the components of 2 are: 2 1 2. . .iji ij ja a2 33

.i i

a2 44

.i i

a2 55

.i i

a

20

l k 2 1033a

2 1134a

2 1235a

2 1743a

2 1844a

2 1945a

2 2453a

2 2554a

2 2655a

Where 1 2 3 5, , , , , ,i j k l and is Laplace operator.

Laplace transform domain fundamental solution

The objective of this section is to derive the fundamental solution associated with equation (33) which is the

response of the medium to unit point excitation (continuous unit line excitation in 2D). The general solution

procedure developed by Kupradze [14] is used in this study for the derivation of the fundamental solution

[5]. For a continuous unit line force in the i th direction suddenly applied at the origin, i.e.

( , ) ( ). ( )x t x H t where ( )H t is the Heaviside step function, the Laplace transform of which is 1/ ( )p x .

Then, one can rewrite equation (33) in the following form:

2

1 2

10. . ( , ) . ( , ) I ( ) , ,D x p D x p Rp x

px (34)

1( , ) ( , ) ( ) 0,D Ix p x p x

p(35)

Where I denotes the unit matrix of order 5, 5 5

D = ijD is the transformed fundamental solution matrix and

1 2( , ) . ( ) ( )px p x x is the differential operator matrix with the components as follows:

1 2

2( , ) . .i ij ij jx p a a 3 3( , )i ix p a

4 4( , )i ix p a 5 5( , )i ix p a

3 6( , ) . jj px p a 4 13( , ) .j jx p ap

5 20( , ) .j jx p ap 2

33 7 10( , ) . .px p a a

2

34 8 11( , ) . .px p a a 2

35 9 12( , ) . .px p a a

43 14 17

2( , ) . .x p a ap 44 15 18

2( , ) . .x p a ap

45 16 19

2( , ) . .x p a ap 53 21 24

2( , ) . .x p a ap

54 22 25

2( , ) . .x p a ap 55 23 26

2( , ) . .x p a ap

(36)

/ , 1, 2.x and 2 is the Laplacian operator.

The first stage is to determine ( , )x p , adjoint differential operators of ( , )x p , which is defined by

( , ) ( , ) det ( , )ik kj ijx p x p x p (37)

In which the determinant of ( , )x p is given by

1 2 3

4 6 102

4

3 8det ( , ) . . . . . . .p p px p D D D D (38)


Where 1 2 3, ,D D D and 4D are constants including above ija coefficients. Built from the cofactors of ( , )x p ,

the element of differential operator ( , )x p can be expressed as:

1 2 3

2 4 6 8 2 4

4 5

3 2 3

8

62

6 7

*( , ) . . . . . . .. . . . . . . .. . . . ..

i j i j iij i jj i jp p p pD x p B B B B pB B pB B

3 9 10

2 6

11

2 4*( , ) . . . . . . . .

i i iiD x p B Bp p B

4 12 13 1

2 4 6

4

2*( , ) . . . . . . .

i i iiD x p B Bp p B

2 2 4

5 15 16 17

6*( , ) . . . . .. . .

i ii ipD x p B pB B 3

3 18 19 2

2

0

62 4*( , ) . . . . . . . . .

ii i iD x p B Bp p pB

33 21 22

4 62

3

8

2

*( , ) . . . . .D x p B pB Bp

34 24 25

4 62

6

8

2

*( , ) . . . . .D x p B pB Bp

35 27 28

4 62

9

8

2

*( , ) . . . . .D x p B pB Bp 3

4 30 31 3

2

2

62 4*( , ) . . . . . .. . .


43 33 34

4 62

5

8

3

*( , ) . . . . .D x p B pB Bp

44 36 37

4 62

8

8

3

*( , ) . . . . .D x p B pB Bp

45 39 40

4 62

1

8

4

*( , ) . . . . .D x p B pB Bp 3

5 42 43 4

2

4

62 4*( , ) . . . . . . . . .


53 45 46

4 62

7

8

4

*( , ) . . . . .D x p B pB Bp

54 48 49

4 62

0

8

5

*( , ) . . . . .D x p B pB Bp

55 51 52

4 62

3

8

5

*( , ) . . . . .D x p B pB Bp (39)

For the second stage, we assume that ,r p is a scalar solution to the equation

1det ( , ) , 0p r p

px x (40)

Which gives

1( , ) ( , ) ( ) 0*

Ix p xp

x p (41)

Consequently, we get

,*D px (42)

Equation (42) enables us to determine the twenty five functions ijD by applying the differential operator

,* px to the single unknown function ,r p . Now, equations (38) and (40) may be combined to yield

3 2 1

4

4 4 4

2 36 4 2 4 0. . . . . . .p p p pD D D

D xD D D

(43)

This relation, with the introduction of as 4

4. .p D , leads to the following differential equation:

2 2 2

1 2

2 2 2

30x (44)

After manipulating some mathematical operations, one may obtain the ,r p function as follows:

0 1 0 2 0 3

2 2 2 2 4 2 2 2 2 4 2 2 2 2 44 2 1 3 1 1 1 2 3 2 2 1 3 2 3 3

. . .1

2 . .

K r

p

K r K r

D(45)

In which:

1 1 2 2 3 3, ,m mp mp p (46)

And the im coefficients in eq (46) are:

3 3 3

1 2 3

4 4 4

1 3 1 3

3 2 3 2 2 3 2

. ., . , .

D D Di im h t m h t h t m h t h t

D D D

4 5 6

2 1 3 1 1 2 3 2 1 3 2 3

1 1 1, ,m m m

m m m m m m m m m m m m

(47)

Where

3 2 3 2 2 33 3 32 2 13 3

4 4 4 4 4 4

3 9 9 27 2 54, , ( ) / , . ( ) /D D DD D D

h r q r t r q r q rD D D D D D

(48)

Finally, by applying the differential operator ,* px to ,r p and by definition of the i interim functions,

we get the 2D fundamental solution of (33) as:

11 12 13 11 3 4 42. . . .1 C C C C

11 12 13 15 7 8 46. . . .2 C C C C

(49)


21 22 23 25 7 8 46. . . .3 C C C C

2

3 3

2. .. .( , ) . .

i j iji j

i j ij 1 2 3

x x rr x xD x p .

r r

3 9 1022 23 2411( , ) . . . .i

i

xD x p C C C

r

22 234 1 13 1 242 4( , ) . . . .i

i

xD x p C C C

r

22 235 1 16 1 245 7( , ) . . . .i

i

xD x p C C C

r

27 253 18 19 20 26( , ) . . . .i

i

xD x p C C C

r

27 254 30 31 32 26( , ) . . . .i

i

xD x p C C C

r

27 255 42 43 44 26( , ) . . . .i

i

xD x p C C C

r

12 13 1433 21 22 23( , ) . . .D x p C C C (50)

12 13 1434 24 25 26( , ) . . .D x p C C C

12 13 1435 27 28 29( , ) . . .D x p C C C

12 13 1443 33 34 35( , ) . . .D x p C C C

12 13 1444 36 37 38( , ) . . .D x p C C C

12 13 1445 39 40 41( , ) . . .D x p C C C

12 13 1453 45 46 47( , ) . . .D x p C C C

12 13 1454 48 49 50( , ) . . .D x p C C C

12 13 1455 51 52 53( , ) . . .D x p C C C

In which theij

C coefficients are constants and the kl interim functions are:

0 1 0 2 0 3

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

2 1 3 1 1 1 2 3 2 2 1 3 2 3 3

0 1 0 2 0 3

2 2 2 2 2

2

2 2 2 2 2 2 2

2 1 3 1 1 2 3 2 1 3 2 3

2 2

1 0 1 2 0 2

2 2 2

11

12

1 2

2 1 3

3

1

. . ..

. . ..

. . . .

K r K r K r

K r K r

p

pK r

K r K 2

3 0 3

2 2 2 2 2 2 2 2

1 2 3 2 1 3 2 3

4 4 4

1 0 1 2 0 2 3 0 3

2 2 2 2 2 2 2 2 2 2 2 2

2 1 3 1 1 2 3 2 1 3 2 3

1 1 1 2 1 3

2 2 2 2

14

21 3 2 2 2 2 3

2 1 3 1 1 1 2 3

2

2 2 1

1

. .

. . . . . ..

. . ..

r K r

K r K r K r

K r K r K

p

pr

2 2 2 2 3

3 2 3 3

1 1 1 2 1 3

2 2 2 2 2 2 2 2 2 2 2 222

23

2 1 3 1 1 1 2 3 2 2 1 3 2 3 3

1 1 1 2 1 2 3 1 3

2 2 2 2 2 2 2 2 2 2 2 2

2 1 3 1 1 2 3 2 1 3 2 3

. . ..

. . . . . .

K r K r K r

K r K r K r

p

3 3 3

1 1 1 2 1 2 3 1 3

2 2 2 2 2 2 2 2 2 2 2 2

2 1 3 1 1 2 3 2 1 3 2

2

3

4

1 . . . . . ..

K r K r K r

p

1 1 1 2 1 2 3 1 3

2 2 2 2 2 2 2 2 2 2 2 2

2 1 3 1 1 2 3 2 1 3

2

2 3

3 3 3

1 1 1 2 1 2 3 1 3

2 2 2 2 2 2 2 2 2 2 2 2

5

2

2 1 3 1 1 2 3 2 1 3 2 3

6

. . . . . ..

. . . . . .

K r K r K r

K r K r K

p

r

1 1 1 2 1 3

2 2 2 2 2 2 2 2 2 2 2 2

2 1 3 1 1 1 2 3 2 2 1

2

3 2 3 3

7

2. . .

.pK r K r K r

(51)


Transient fundamental solution

To obtain the time domain fundamental solution, one needs to evaluate the analytical inversion of ijD . First,

it is necessary to find out the inverse transform ,kl

r t of the functions containing the modified Bessel

functions ,kl

r t . By the use of the following formulas [15]:

20 1

02 4

1L

. . .. , ,

j j

j

K p

p

r m

t

m rm r t

0

2 2 20

04 4 4

1L

. . . . .. , exp . ,

..

j j j j

j

K m r m rt

t t

m r m rp

p pm r

1

(52)

21 1

4

1L

. . .. , exp

..

j j

j

j

K m r m rm r

m

pt

p r t2

Expressions of the interim functions are obtained:

1 5 64

0 1 0 2 0 3

1 2 3

1

4 0 1 5

11 11

12 12

13

0 2 6 0 3

1

4 1 0 1 5 2 0 2 613

1

3 0 3

. , . , . ,

. , . , . ,

. . . , . . . , . ,, .

,

,

.

m mmm r m r m r

m m m

m m r m m r m m r

m m m r m

t t t

m m

t t t

t t

r t

r t

r r m r tt m m

L

L

L

4 14

1 2 2 2

4 1 0 1 5 2 0 2 6 3 0 3. ., . , . . . , . . . ,m m m r m mr m r m m trt t mt L

1 5

21 21

64

1 1 1 2 1 3

1 1 2 2 3 3

. , . ,, . ,m mm

m r m r m rm m m m m m

tr t t tL

1 5 64

1 1 122 22 2 1 3

1 2 3

. , . , ., ,t tm mm

m r m r mr tm m m

trL (53)

1

4 1 1 1 5 2 1 2 6 3 12 23 33. . . , . . . , . . ,, .t tm m m tr m m m r m m rr t mL

1

4 1 1 1 1 5 2 2 1 2 64 3 3 12 34 2. . . , . . . , . . . ,, m m m m r m m m m rt t m mr tt m m rL

1

4 1 2 1 5 2 2 2 6 3 22 25 35. . . , . . . , . . ,, .t tm m m tr m m m r m m rr t mL

1

4 1 1 2 1 5 2 2 2 2 66 3 3 22 36 2. . . , . . . , . . . ,, m m m m r m m m m rt t m mr tt m m rL

1 5 64

2 1 227 27 2 2 3

1 2 3

. , . , ., ,t tm mm

m r m r mr tm m m

trL

Finally, the fundamental solutions are obtained: 2

3 3

2. .. .( , ) . .

i j iji j

i j ij 1 2 3

x x rr x xD x p .

r r

3 9 1022 23 2411( , ) . . . .i

i

xD x p C C C

r

22 234 1 13 1 242 4( , ) . . . .i

i

xD x p C C C

r

22 235 1 16 1 245 7( , ) . . . .i

i

xD x p C C C

r

27 253 18 19 20 26( , ) . . . .i

i

xD x p C C C

r

27 254 30 31 32 26( , ) . . . .i

i

xD x p C C C

r

27 255 42 43 44 26( , ) . . . .i

i

xD x p C C C

r

12 13 1433 21 22 23( , ) . . .D x p C C C

12 13 1434 24 25 26( , ) . . .D x p C C C

12 13 1435 27 28 29( , ) . . .D x p C C C

12 13 1443 33 34 35( , ) . . .D x p C C C

12 13 1444 36 37 38( , ) . . .D x p C C C

12 13 1445 39 40 41( , ) . . .D x p C C C

12 13 1453 45 46 47( , ) . . .D x p C C C


12 13 1454 48 49 50( , ) . . .D x p C C C

12 13 1455 51 52 53( , ) . . .D x p C C C (54)

Where

11 121 2 3 413 14. . . .1 B B B B

11 125 6 7 813 14. . . .2 B B B B (55)

21 225 6 7 823 24. . . .3 B B B B

For instance, the derived Green functions are shown through Figs. 1 to 2:

-0.00001

-5´10-6

0

5´10-6

0.00001

x

-0.00001

-5´10-6

0

5´10-6

0.00001

y

-0.0005

0

0.0005

0.001

z

-5´10-6

0

5´10-6x

0.002

0.004

0.006

0.008

0.01

r

200

400

600

800

1000

t

-2

-1.5

-1

-0.5

0

z

002

0.004

0.006

0.008r

Fig 1: Green function D11

Solid skeleton displacment in direction one due to a unit point

load in direction one.

Fig 2: Green function D34

Variations of water pressure due to a unit increment in air pressure.

Verification

Firstly, the new transient fundamental solution is compared to the steady THHM fundamental solution [8] by

substituting the coefficients of terms in which the time variation is present, by zero: 2

2 2 2 2 212

11 12 11 12 12 2

8 6

4

12 1 2

4 2( , ) . . . , , , ,

. ..

i

i j

j

ij ij ij

x xFD x p F F Ln r F j

rpF F i

D

2

2 13

3 13

4

6 1

4 2( , ) . .

. .i i i

FD x p F x L

pn r

D

2

2 14

4 14

4

6 1

4 2( , ) . .

. .i i i

FD x p F x L

pn r

D

2

2 15

5 15

4

6 1

4 2( , ) . .

. .i i i

FD x p F x L

pn r

D

0 3 5 1 2( , ) , , , , ,i j

D x p i j 2

2 21

3 1

4

8

3 22

( , ) .. .

FD x p F Ln r

Dp

2

2 22

3 2

4

8

4 22

( , ) .. .

FD x p F Ln r

Dp

2

2 23

3 3

4

8

5 22

( , ) .. .

FD x p F Ln r

Dp (56)

2

2 31

4 1

4

8

3 32

( , ) .. .

FD x p F Ln r

Dp

2

2 32

4 2

4

8

4 32

( , ) .. .

FD x p F Ln r

Dp

2

2 33

4 3

4

8

5 32

( , ) .. .

FD x p F Ln r

Dp

2

2 41

5 1

4

8

3 42

( , ) .. .

FD x p F Ln r

Dp

2

2 42

5 2

4

8

4 42

( , ) .. .

FD x p F Ln r

Dp

2

2 43

5 3

4

8

5 42

( , ) .. .

FD x p F Ln r

Dp

Where 8

4

125 12

3 538 944, , . . .r Ln r

Dpand 2

ijF are the constant coefficients.

Secondly, if the coefficients representing the thermal behaviour of the phenomenon and Henry’s coefficient

approach to zero the steady fundamental solution (eq. 56) will approach the corresponding isothermal

solutions [7]:


2

2

2 3 2

8 2

( ) ( ). ln . . ( ). .( , )

. .( ).

ij i j

i j

r r x xD x p

r

3

1 2

8 2

. . . ln( , )

. .( ).

w s i

i

w

x rD x p

p K

4

11 2

8 2

.( ).( , ) .

.ln

. .( ).

g s i

i

gp

xD x p r

K

3 40

i iD D (57)

332

. ln( , )

. .

w

w

Dp

rx p

K

442

. ln( , )

. .

g

g

Dp

rx p

K

34 430D D

5 50 1 4, , , ,

i jD D i j

Also, it is evident that while s approaches zero, the fundamental solution in eq (57) approaches elastostatic

fundamental solution [16, 17]: 2

2

2 3 2

8 2

( ) ( ). ln . . ( ). .( , )

. .( ).

ij i j

i j

r r x xD x p

r

41 2

8 2

.( , ) . ln

. .( ). .

g i

i

g g

xD x p r

K

332

. ln( , )

. .

w

w w

rD x p

K44

2

. ln( , )

. . .

g

g g

rD x p

Kp3

0( , )i

D x p3 4

0i i

D D34 43

0D D

5 50 1 4, , , ,

i jD D i j (58)

References

[1] D.W Pollock Water resources research, 22 (5), 765-775 (1986).

[2] S.Olivella, J.Carrera, A.Gens, E.E.Alonso Transport Porous Media, 15, 271–293(1994).

[3] E.Jabbari, B.Gatmiri, 7th International Conference on Boundary Element Techniques (B.Gatmiri,

A.Sellier, M.H.Aliabadi), EC:, Paris, 247-248 (2006).

[4] B.Gatmiri, M.Kamalian International Journal of Geomechanics 2(4), 381–398 (2002).

[5] B.Gatmiri, K.V.Nguyen Communications in Numerical Methods in Engineering 21 (3), 119–132

(2005).

[6] B.Gatmiri, E.Jabbari International Journal of Solids and Structures 42, 5971–5990 (2005).

[7] B.Gatmiri, E.Jabbari 5th International Conference on Boundary Element Techniques (M.H.Aliabadi,

V.M.A.Leitão), EC:, Lisbon, 217-221 (2004).

[8] E.Jabbari, B.Gatmiri International Journal of Computer Modelling in Engineering and Sciences 18(1),

31-43 (2007).

[9] B.Gatmiri, P.Delage, M.Cerrolaza Advances in Engineering Software 29(1), 29-43 (1998).

[10] B.Gatmiri, P.Delage, 1th International Conference on Unsaturated Soils (E.E.Alonso, P.Delage), EC:,

Paris, 1049-1056 (1995).

[11] L.A.Richards J. Physics 1, 318-333 (1931). [12] J.R.Philip, D.A.de Vries Trans. Am. Geophys 38, 222-232 (1957).

[13] J.Ewen, H.R.Thomas Géotechnique, 39(3), 455-470 (1989).

[14] V.D.Kupradze et al. Three-dimensional Problems of the Mathematical Theory of Elasticity and

Thermoelasticity, North-Holland, Netherlands (1979).

[15] M.Abramowitz, I.A.Stegun Handbook of Mathematical Functions, National Bureau of Standards,

Washington, D.C. (1965).

[16] P.K.Banerjee The Boundary Element Methods in Engineering, McGraw-Hill Book Company,

England (1994).

[17] G.Beer Programming the Boundary Element Method, John Wiley and Sons, England (2001).


A Three-Step MDBEM for Nonhomogeneous Elastic Solids

X.W. Gao1, a, J. Hong1,b and Ch. Zhang2,c

1 Department of Engineering Mechanics, Southeast University, Nanjing 210096, PR China

2 Department of Civil Engineering, University of Siegen, D-57068 Siegen, Germany

a [email protected], b [email protected], c [email protected]

Keywords: Boundary element method; Nonhomogeneous elastic solids; Multi-domain technique; Functionally graded materials (FGMs).

Abstract. In this paper, a three-step boundary element method (BEM) is presented for solving

boundary value problems in two-dimensional (2D) and three-dimensional (3D) nonhomogeneous and

linear elastic solids by using the multi-domain boundary element method (MDBEM). Fundamental

solutions for homogeneous and linear elastic solids are adopted in the MDBEM. Boundary-domain

integral equations expressed in terms of normalized displacements and tractions are formulated for

each sub-domain. The first step is the elimination of internal variables, and the second one is the

elimination of boundary unknowns used only by individual sub-domains, and the last step is the

establishment of a system of linear algebraic equations according to the continuity/discontinuity

conditions of the displacements and the tractions at common nodes on the interfaces. Discontinuous

elements are utilized to model the traction discontinuity at corner nodes. Numerical examples are

presented to demonstrate the accuracy and the efficiency of the present three-step MDBEM.

Introduction

The material properties of continuously nonhomogeneous solids such as functionally graded

materials (FGMs) are dependent on spatial positions [1]. Although the boundary element method

(BEM) has been successfully developed and applied to homogeneous and linear elastic solids since

many years, its extension and applications to continuously nonhomogeneous and linear elastic solids

are not straight-forward. The main reason is the fact that the required fundamental solutions or

Green's functions for general nonhomogeneous and linear elastic solids are either mathematically too

complicated or not available, which makes an easy and efficient numerical implementation difficult.

One remedy for this difficulty is the use of fundamental solutions for homogeneous and linear elastic

solids in the BEM formulation, which involves domain-integrals in the boundary-domain integral

equations. Although the domain integrals can be converted to global boundary integrals [2] or local

ones [3], the resulting system of algebraic equations involves unknown quantities consisting of both

boundary unknowns and internal displacements. The numerical solution of such a system with mixed

boundary and internal variables is in general very time consuming or even unfeasible for large-scale

problems using the single-domain BEM. An efficient numerical way to solve boundary value

problems in nonhomogeneous and linear elastic solids is the multi-domain boundary element method

(MDBEM) [4,5], where the considered domain is divided into several sub-domains. There exist two

different efficient solution techniques in the MDBEM. One is the variable condensation technique

and the other is the iterative technique. The first technique results in a small system of equations by

eliminating some variables in an intermediate step, while the latter solves the large system of

algebraic equations by using a fast iterative scheme such as the Krylov’s iteration method [6].

In this paper, a three-step BEM is presented for solving boundary value problems in

two-dimensional (2D) and three-dimensional (3D) nonhomogeneous and linear elastic solids by

using the MDBEM [5]. Fundamental solutions for homogeneous and linear elastic solids are adopted

in the MDBEM. Boundary-domain integral equations with normalized displacements and tractions

are formulated for sub-domains. The first step of the method is the elimination of internal variables

for each sub-domain. The second step is the elimination of boundary unknowns defined over nodes

used only by the sub-domain itself. And the third step is the establishment of a system of linear


algebraic equations according to the continuity/discontinuity conditions of the displacements and the

tractions at common nodes on the interfaces. Discontinuous elements are implemented to properly

model the traction discontinuity at corner-nodes. The present three-step MDBEM has two important

features, namely, only interfacial displacements are unknowns in the final system of algebraic

equations, and the coefficient matrix is blocked and sparse. Consequently, large-scale 2D and 3D

boundary value problems in nonhomogeneous and linear elastic solids can be dealt with efficiently.

Integral equations for functionally graded materials (FGMs)

In isotropic, continuously nonhomogeneous, and linear elastic solids, such as functionally graded

materials, the shear modulus is a function of spatial coordinates, while, for most cases, the

Poisson’s ratio can be regarded as a constant. Under this assumption, integral equations for

boundary and internal nodes can be derived by using Gauss’ divergence theorem as follows [5,7]

( ) ( , ) ( ) ( , ) ( ) ( , ) ( )p p p p

i ij j ij j ij jcu U t d T u d V u dx x x x x x x x x x , (1)

where c=1 for internal points and c=1/2 for smooth boundary points, ijU and ijT are the Kelvin

displacement and traction fundamental solutions [8] for an isotropic, homogeneous and linear elastic

solid with 1 , and

, , , , , , , ,

1[(1 2 ) ] (1 2 )( )

4 (1 )ij k k ij i j i j j iV r r r r r

r, (2)

where =2 for 2D and =3 for 3D problems and = -1. In Eqs. (1) and (2), ( )ju x and are the

normalized displacements and shear modulus defined by

( ) ( ) ( )i iu ux x x , ( ) log ( )x x . (3)

From Eq. (1), it can be seen that no displacement gradients are involved in the boundary-domain

integral equations. This is attributed to the use of the normalized quantities defined in Eq. (3).

Comparison of Eq. (1) to the conventional boundary integral equations for isotropic, homogeneous

and linear elastic solids [8] shows that there is a domain integral appearing in the integral equations.

This domain integral is converted into an equivalent boundary integral using the radial integral

method (RIM) [2]. Since the unknown variables ju are included in the domain integral, some internal

points are required to be placed inside the domain to improve the computational accuracy in the

approximation of ju in terms of the radial basis functions (RBFs) in the application of RIM [2,7].

Consequently, the resulting system of algebraic equations includes both boundary unknowns and

internal normalized displacements as the system unknowns. To solve such a system using a single

domain technique, the computational time and the required memory storage would be huge for

complicated 3D problems. Therefore, a robust MDBEM solution technique is desired for solving

such large-scale problems.

Three-step MDBEM for isotropic, nonhomogeneous and linear elastic solids

As shown in Fig. 1, the domain of concern is divided into a number of sub-domains. The nodes used

for each sub-domain are classified into three types: “self nodes”, “internal nodes”, and “common

nodes”.

To efficiently exploit the MDBEM technique, the order of the three types of nodes is arranged in

such a way that the self nodes are numbered first, followed by the common nodes and finally the

internal nodes. To model the traction discontinuity at a corner or an edge, discontinuous element [5]

is used and more than one nodes are defined at an internal corner at which different sub-domains meet


and at a boundary corner at which at least one of the components is specified with the displacement

boundary condition (see Fig. 1).

common nodeinternal node

self node

tt

uu

jj tt

i

j

k

uu

Fig. 1. Definition of three types of nodes

After using the node arrangement strategy described above and applying RIM [2] to transform the

domain integral into a boundary integral, the boundary-domain integral equations (1) can be

converted into a system of algebraic equations for each sub-domain, which can be written in the

matrix form as

bs s bc c bi i bs s bc cH u H u H u G t G t (4)

for boundary nodes, and

is s ic c ii i is s ic cH u H u H u G t G t (5)

for internal nodes.

In Eqs. (4) and (5), the subscript b denotes quantities for boundary nodes consisting of self nodes

and common nodes, and the subscripts s, i and c represent quantities for self, internal and common

nodes, respectively. Also, su , cu , iu , st and ct are displacement and traction vectors corresponding

to the three types of nodes. It is noted that for piecewise homogeneous solids, the matrix iiH is an

identity matrix and biH is a zero matrix.

After invoking all specified displacement and traction boundary conditions in Eqs. (4) and (5), the

following equation set can be obtained for each sub-domain

bs s bc c bi i b bc cA x H u H u y G t , (6)

is s ic c ii i i ic cA x H u H u y G t , (7)

where sx is the vector consisting of unknown displacements and unknown tractions over the self

nodes, and by and iy are the known vectors formed by multiplying all given boundary displacements

and tractions with their corresponding matrix elements. To solve Eqs. (6) and (7) for the unknown

vectors sx , cu , ct and iu , a three-step solution technique is applied, which is described in the

following.


Step 1: Eliminating internal unknowns for each sub-domain

The matrix iiH in Eq. (7) is a square matrix and well-posed, so eliminating the internal displacements

iu from Eqs. (6) and (7) it follows for each sub-domain

bs s bc c b bc cA x H u y G t , (8)

where

1

1

1

1

( ) ,

( ) ,

( ) ,

( ) .

bs bs bi ii is

bc bc bi ii ic

bc bc bi ii ic

b b bi ii i

A A H H A

H H H H H

G G H H G

y y H H y

(9)

It is noted again that for piecewise homogeneous solids, the matrix iiH is an identity matrix and biH

is a zero matrix, and therefore it is unnecessary to form Eq. (8) since the matrices to be formed in Eq.

(9) reduce to the original matrices.

Step 2: Eliminating boundary unknowns for each sub-domain

Since for each sub-domain the boundary nodes are composed of the self nodes and the common

nodes, Eq. (8) can be divided into two sets of equations for self nodes and common nodes, i.e.,

ss s sc c s sc cA x H u y G t , (10)

cs s cc c c cc cA x H u y G t . (11)

All matrices in Eqs. (10) and (11) are sub-matrices of the corresponding matrices in Eq. (8). Now,

elimination of sx from Eqs. (10) and (11) yields

ˆˆ ˆcc c c cc cH u y G t , (12)

where

1

1

1

ˆ ( ) ,

ˆ ( ) ,

ˆ ( ) .

cc cc cs ss sc

cc cc cs ss sc

c c cs ss s

H H A A H

G G A A G

y y A A y

(13)

Step 3: Assembling the system of equations from all sub-domain’s contributions

Equations (10) and (11) can be applied to every sub-domain. For the n-th sub-domain, the traction

vector ct for the common nodes can be expressed as

( ) ( ) 1 ( ) ( ) ( )ˆ ˆ ˆ( ) ( )n n n n n

c cc cc c ct G H u y . (14)

Assembling all sub-domain’s contributions for the global common nodes and applying the traction

equilibrium condition ( ) 0n

c

n

t results in the final system of algebraic equations as

cc c cK U Y , (15)

where


( ) 1 ( ) ( )ˆ ˆ( )n n n

cc cc cc

n

K G H Q , (16)

( ) 1 ( )ˆ ˆ( )n n

c cc c

n

Y G y , (17)

where ( )nQ is the location matrix consisting of 0 and 1, which relates the local displacement vector ( )n

cu to the global one cU .

Solving Eq. (15) for cU , we can obtain the displacements at all common interface nodes, and then

substituting it back into Eq. (14) we can compute the tractions at each domain’s common nodes.

Using these results, the boundary unknowns at self nodes of each sub-domain can be calculated by

applying Eq. (10). Equation (15) shows that the number of degrees of freedom of the system is only

the number of degrees of freedom of the common interface nodes, which is much smaller than those

of all boundary and internal nodes. It is noted that for piecewise homogeneous solids, the present

three-step solution technique reduces to a two-step solution technique consisting of the last two steps

as described above.

Numerical example

The numerical example considered here is a multi-planar tubular DX-Joint as depicted in Fig. 2. This

example has been analyzed in reference [8]. The DX-Joint consists of a large diameter tube (chord)

intersected orthogonally by two smaller diameter tubes (braces). The outer radii of the chord and

braces are 228.6mm and 28.58mm, respectively, while the inner radii are one-half of these values. For

simplicity, only a short section of the tubes is analyzed and the eight-fold symmetry is exploited (Fig.

3). The half-lengths of the tubes are given by 900mmxL and 660mmy zL L . A constant

Poisson’s ratio = 0.3 is used and a tensile load F=0.234GPa is applied to each end of the DX-Joint.

F

F

F

F

F

F

Fig. 2. Multi-planar tubular DX-Joint subjected to axial loads


The computational domain is divided into three sub-domains as shown in Fig. 3. The BEM mesh

consists of 1660 linear quadrilateral boundary elements with 1606 boundary nodes (including 56

interface elements and 60 interface nodes) and 329 internal nodes.

xy

z

1

3

2

Fig. 3. BEM mesh of the DX-Joint

Case 1: Homogeneous DX-Joint

For the purpose of the validation of the present MDBEM, numerical calculations are first carried out

for the homogeneous case by setting the Young’s modulus E = 200GPa for all the three sub-domains.

Figure 4 shows the distribution of the displacement ux along the middle line of the outer surface of the

chord. For comparison, the results using the single-domain code BEMECH listed in [8] are also given.

It can be seen that the two sets of numerical results are in very good agreement.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 150 300 450 600 750 900

x (mm)

Dis

pla

cem

en

t (m

m) BEMECH

Present

Fig. 4. Displacement for homogenous DX-Joint with E = 200GPa


Case 2: Functionally graded DX-Joint

The second computation is performed by assuming 1

x

LE E x , yy

L eEE2 and zz

LeEE3 for

the sub-domains 1 , 2 and 3 , respectively. The parameters , and are determined by

x

L

x

E E

L,

( / )y

L

y

log E E

L,

( / )z

L

z

log E E

L,

where E=200GPa, 10x

LE E , 5y

LE E and 10z

LE E . Figure 5 plots the distribution of the

displacement ux along the middle line of the outer surface of 1 , and Fig. 6 shows the displacements

uy and uz along the middle lines of the outer surfaces of 2 and 3 , respectively. Figure 7 illustrates

the axial stresses yy and zz along the middle lines of the domains 2 and 3 , respectively.

From Figs. 6 and 7 it can be seen that the axial displacements along domains 2 and 3 are quite

different, since the variation of the Young’s modulus is different for the two domains. On the other

hand, Fig. 7 shows the same pattern for the axial stresses of the two domains 2 and 3. This indicates

that the distribution of the axial stresses mainly depends on the applied loads in the considered case.

0

0.08

0.16

0.24

0.32

0 150 300 450 600 750 900

x (mm)

Dis

pla

cem

en

t (m

m)

ux (over surface of domain 1)

Fig. 5. Variation of ux over the outer surface of 1

0.0

0.1

0.1

0.2

0.2

0.3

0.3

0.4

0.4

0.5

200 300 400 500 600 700

y or z over surface 2 or 3 (mm)

Dis

pacem

en

ts (

mm

)

uy (over surface of domain 2)

uz (over surface of domain 3)

Fig. 6. Variations of uy and uz over the outer surfaces of 2 and 3


0.20

0.28

0.36

0.44

0.52

0.60

0.68

200 255 310 365 420 475 530 585 640

y or z over surface 2 or 3 (mm)

Str

esses

Sigma-yy (over domain 2)

Sigma-zz (over domain 3)

Fig. 7. Variations of axial stresses along middle lines of 2 and 3

Summary

A three-step MDBEM is presented in this paper for the numerical solution of boundary value

problems in 2D and 3D isotropic, continuously nonhomogeneous and linear elastic solids.

Fundamental solutions for isotropic, homogeneous and linear elastic solids are implemented in the

present MDBEM. Boundary-domain integral equations expressed in terms of normalized

displacements and tractions are formulated for each sub-domain. The domain-integrals are

transformed to boundary integrals by using the radial integration method (RIM). Through a two-step

elimination procedure, a system of linear algebraic equations for the displacements at common nodes

is established. Discontinuous elements are adopted to model the traction discontinuity at corner

nodes. The number of unknowns in the present three-step MDBEM is much smaller than that of the

classical single-domain BEM. Numerical examples show that the present three-step MDBEM is

efficient and suitable for solving large-scale problems in isotropic, continuously nonhomogenous and

linear elastic solids.

Acknowledgement

Support by the German Research Foundation (DFG) under the project number ZH 15/10-1 is

gratefully acknowledged.

References

[1] X.W. Gao, Ch. Zhang, J. Sladek and V. Sladek: Compos. Sci. Tech. Vol. 68 (2008), p. 1209.

[2] X.W. Gao: Eng. Anal. Bound. Elem. Vol. 26 (2002), p. 905.

[3] J. Sladek, V. Sladek and Ch. Zhang: Building Research Journal, Vol. 53 (2005), p. 71.

[4] S. Ahmad and P.K. Banerjee: Int. J. Numer. Meth. Engng., Vol. 26 (1988), p. 891.

[5] X.W. Gao, L. Guo, and Ch. Zhang: Eng. Anal. Bound. Elem. Vol. 31 (2007), p. 965.

[6] K. Davey, S. Bounds, I. Rosindale and M.T.A. Rasgado: Comp. & Struct. Vol. 80 (2002), p. 643.

[7] X.W. Gao, Ch. Zhang and L. Guo: Eng. Anal. Bound. Elem. Vol. 31 (2007), p. 974.

[8] X.W. Gao and T.G. Davies: Boundary Element Programming in Mechanics. Cambridge

University Press, 2002.


DBEM for Fracture Analysis of Sti ened Curved Panels

(Plates and Shallow Shells Assemblies)

P.M. Baiz and M.H. AliabadiDepartment of Aeronautical Engineering, Imperial College London

South Kensington campus, London SW7 2AZ

Keywords: Fracture Mechanics, Shear Deformable, Plate and Shallow Shell Assemblies.

Abstract. This paper presents applications where the DBEM formulation presented by Dirgantaraand Aliabadi [3] is combined with the multi region BEM presented recently by Baiz and Aliabadi [2],for the analysis of cracked shear deformable plates and shallow shell assemblies. Stress intensity factorsare obtained using the CTOD technique. Several examples are solved to demonstrate the capabilitiesof the proposed technique. Comparing DBEM with FEM models, it was clear that good accuracy ande ciency can be achieved with the present multi region DBEM approach.

Introduction. Cracks are present in most structural members either as a result of the manufac-turing process or due to a localized damage during service life. These cracks may grow by fatigue,corrosion or creep, decreasing the strength and leading to the failure of the structure. The dualboundary element method (DBEM) is based on the use of di erent equations on each crack surface(displacement and traction integral equations). During the past years, the dual boundary elementmethod has emerged as a robust numerical method for fracture mechanics problems [1]. Applicationsof the dual boundary element method to fracture mechanics of shear deformable plates have beenreported independently by Rashed, Aliabadi and Brebbia [6] and Ahmadi-Brooghani and Wearing [8]while DBEM for shear deformable shallow shells have been derived by Dirgantara and Aliabadi [3].

Multi-region BEM (Plate and Shallow Shells). Lets consider M assembled cylindricalshallow shells or plates joined at J as shown in Figure 1a. The global coordinate system is given by-x1-x2-x3, and the local coordinate systems for each region by -x1 -x2 -x3 (m = 1;M).

The plates or shallows shells have an uniform thickness h, Young’s modulus E, Poisson’s ratioº. As shown in Figure 1c, w represent rotations of the middle surface, w3 denotes the out-of-planedisplacement, and u represent in-plane displacements. And generalized tractions are denoted as: pdue to the stress couples, p3 due to shear stress resultant and t due to membrane stress resultants.Shallow shells are defined using a curvilinear coordinate system. This means that contrary to flatplates which have a fix normal (local coordinate system) through the whole plate, the local coordinatesystem of a shallow shell changes with the curvature (see Figure 1b).

In the simple case of two shallow shells or plates with the same axis orientation at the junction line(see Figure 1c), the continuity and equilibrium equations along the joint can be written as follows:

u = u +1;X=1

t = 0 (1)

w = w +1;X=1

p = 0

Because two or more angled plates or shallow shells joined together are considered, an approachsimilar to that proposed in [10] is developed. To simplify this approach, the local coordinate systemsof each region is assumed to be defined such that the x2 directions are all aligned with the globaldirection x2, following the implementation for plate assemblies presented by Wen et. al. [9] or Di


x1

x3x2

x1

x3

x2

m

m

m

m

m

Jn

c

c

x1

x3m

m

u1

w3

R

D

Jn

x1

x3

cShallow Shell Base Plane

x2

a) b)

x1

x3x2

x1

x3

x2

m

m mx1

x3

x2m+1

x1

x3x2

x1

x3x2

mm

m

m+1

m+1m+1

m+1

m+1

u1

u2

w1

w2

w3

u1

u2

w1

w2

w3m+1

m+1

m+1

m+1

m+1

m

m

m

m

m

t1

t2

p1

p2

p3

m

m

m

mm

t1

t2

p1

p2

p3

m+1m+1

m+1

m+1

m+1

Displacements

Tractionsc)

Figure 1: a) Plate and Shallow Shell Assembly, b) Local Coordinate System of Shallow Shell, c) SimpleAssemblies.

Pisa [4]. Based on the above simplification, w3 and u1 displacements for any given shallow shell ata junction line (J ) can be presented as shown in Figure 1b. Therefore, compatibility equations foreach pair of adjacent shallow shells (e.g. m = 1 and m = 2) could be written as follows:

u11(n111n

111 + n131n

113) + w13(n

113n

111 + n133n

113) = u21(n

211n

211 + n231n

213) +w23(n

213n

211 + n233n

213)

u11(n111n

131 + n131n

133) + w13(n

113n

131 + n133n

133) = u21(n

211n

231 + n231n

233) +w23(n

213n

231 + n233n

233)

u12 = u22

w11 = w21

w12 = 0

w22 = 0 (2)

where n are the components of the rotation matrix of the shallow shell base plane m from localto global coordinates [10], and n are the components of the rotation matrix from the curvilinearcoordinate system to the shallow shell base plane m. The components of n are given by:

n11 = cos(® ); n12 = 0; n13 = cos(90 + ® )

n21 = 0; n22 = 1; n23 = 0

n31 = cos(90¡ ® ); n32 = 0; n33 = cos(® ) (3)

where ® is measured with respect to the shallow shell base plane, as shown in Figure 1b.Equations in (2) result in a system of 5M¡4 compatibility conditions, and have to be supplemented


with 4 equilibrium conditions as follows:

X=1

[t1 (n11 n11 + n31 n13) + p3 (n13 n11 + n33 n13)] = 0

X=1

[t1 (n11 n31 + n31 n33) + p3 (n13 n31 + n33 n33)] = 0

X=1

t2 = 0

X=1

p1 = 0 (4)

to produce the required 5M equations. This approach relies on the assumption that the plate orshallow shell flexural rigidity in its own plane is so large that it is possible to ignore its associateddeformation, in another words, there is no drilling rotation.

Boundary Integral Formulation. The dual boundary element method is based on the use of

two independent equations, the displacement and traction boundary integral equations, at each pair

of coincident source points on the surfaces that define a crack. The displacement integral equations

for collocation points on one crack surface (x+ 2+), can be written as follows [3]:

1

2(x+) +

1

2(x ) +¡

Z(x+ x) (x) (x) =

Z(x+ x) (x) (x)

¡

Z3

¡x+X¢ 1¡

2

·(X) + (X) +

2

1¡(X)

¸(X)

¡

Z3

¡x+X¢

((1¡ ) + ) 3 (X) (X)

+

Z3(x

+X) 3(X) (X) (5)

and,1

2(x+) +

1

2(x ) +¡

Z(x+ x) (x) (x)

+

Z(x+ x) [ (1¡ ) + ] 3(x) (x) (x)

¡

Z(x+ X) [ (1¡ ) + ] 3 (X) (X)

=

Z(x+ x) (x) (x) +

Z(x+ X) (X) (X) (6)

In order to avoid an ill-conditioned system, the traction integral equations are used for collocations

on the other crack surface (x 2 ) [3]:

1

2(x )¡

1

2(x+) + (x ) =

Z(x x) (x) (x) + (x )¡

Z3(x x) 3(x) (x)

= (x )¡

Z(x x) (x) (x) + (x )

Z3(x x) 3(x) (x)

¡ (x )

Z1¡

2

µ(X) + (X) +

2

1¡(X)

¶3(x X) (X)


¡ (x )

Z((1¡ ) + ) 3 (X) 3(x X) (X)

+ (x )

Z3(x X) 3 (X) (7)

1

23(x )¡

1

23(x

+) + (x )¡

Z3 (x x) (x) (x) + (x ) =

Z3 3(x x) 3(x) (x)

= (x )

Z3 (x x) (x) (x) + (x )¡

Z3 3(x x) 3(x) (x)

¡ (x )

Z1¡

2

µ(X) + (X) +

2

1¡(X)

¶3 3(x X) (X)

¡ (x )

Z((1¡ ) + ) 3 (X) 3 3(x X) (X)

+ (x )

Z3 3(x X) 3 (X) (8)

and1

2(x )¡

1

2(x+) + (x ) =

Z(x x) (x) (x)

+ (x )¡

Z(x x) [ (1¡ ) + ] 3(x) (x) (x)

¡ (x )

Z(x X) [ (1¡ ) + ] 3 (X) (X)

= (x )¡

Z(x x) (x) (x) + (x )

Z(x X) (X)

+1

2(x ) [(1¡ ) + ] 3(x ) (9)

Equations (5-6) and (7-9) represent displacement and traction integral equations on the cracksurfaces, respectively; and together with the displacement integral equations (see equations 3.1 and3.2 in [3]) for collocation on the rest of the boundary ¡ , form the dual boundary integral formulationin shallow shell problems. It is worth notice that as the source points x+ and x are coincident, extrafree terms appear in equations (5-9) for collocation on both crack surfaces.

Solution Strategy. The implementation of the dual boundary element formulation requiresthat boundary ¡ to be discretized. In the case of shallow shell regions several uniformly distributeddomain points are required for the application of the dual reciprocity method (DRM). In the case ofthe boundary: continuous, semi discontinuous and discontinuous quadratic isoparametric boundaryelements are used to describe the geometry of each region (plate or shallow shell).

The detailed modelling strategy is similar to the one described by Portela Aliabadi and Rooke [5]and can be summarized as follows [3]:

² The crack boundaries are discretized with discontinuous quadratic elements (each node of onecrack surface is coincident with another node on the opposite crack surface).

² Continuous quadratic elements are applied along the remaining boundary of the structure, exceptat the intersection between a crack and an edge or at corners, where semi-discontinuous elementsare required in order to avoid a common node at intersections.

² The traction integral equations are used for collocation on one crack surface (x 2 ).


-6.00E-04

-4.00E-04

-2.00E-04

0.00E+00

2.00E-04

1.00E+02 1.50E+02 2.00E+02

ABAQUS

DBEM

u /

h2

x /h1 1

1

x1

x3

x2

h3

h1

h2

0.075 m 0.035 m

0.040 m

E=71.016 GPav=0.33

q=0.006 MPa

h =0.005 m

h =0.0025 m

h =0.0075 m

b=0.75 m

1

2

3

Figure 2: Crack Opening Displacement for Curved Sti ened Panel.

² The displacement integral equations are used for collocation on the opposite crack surface (x+ 2

+).

² In the non-crack boundaries (x0 2 ) the common displacement integral equations are em-ployed.

² For shallow shell, several uniformly distributed DRM points are used in the domain.

This simple strategy is very robust, making the DBEM an e ective tool for the modeling of generaledge or embedded crack problems.

Numerical Example. A curved sti ened panel with a centre crack as shown in Figure 2 isanalyzed. The material properties and dimensions are also presented in Figure 2. The curved panelis simply supported along all the boundary and is subjected to an uniform internal pressure q. Inorder to validate the DBEM formulation in shallow shell and plate assemblies, results are comparedwith FEM solutions. The FEM half model has a total of 5922 elements and 18133 nodes. The DBEMmodel contains 7 shallow shells and 6 flat plates, with a total of 248 elements and 214 DRM points(14 elements per crack side). Figure 2 presents the in-plane displacement u2 along the symmetry lineof the central cracked shallow shell (crack opening displacement). From Figure 2 is evident the goodagreement between both numerical solutions (DBEM and FEM).

Conclusion. In this work, applications of the DBEM for the fracture mechanics analysis ofshear deformable plate and shallow shells assemblies was presented. A multi-region technique wasused to model plate and shallow shell assembled structures subjected to arbitrary loading. Additionalequations were obtained by imposing compatibility and equilibrium equations along the interfaceboundaries. The DBEM shallow shell formulation was developed by coupling boundary element for-mulations of shear deformable plate bending and two dimensional plane stress elasticity; as a result,domain integrals appear in the formulation and are treated with the Dual Reciprocity Technique.


Traction integral equations were applied on one crack surface and the usual displacement integralequations on the other crack surface and non-crack boundaries. Special crack tip elements are usedto model accurately the displacement field. These displacements were used for the evaluation of SIFusing the CTOD technique. Comparing DBEM with FEM models, it was clear that good accuracyand e ciency can be achieved with the present multi region DBEM approach.

References

[1] Aliabadi, M.H., The Boundary Element Method, vol II: application to solids and structures, Chich-ester, Wiley (2001).

[2] P.M. Baiz, M.H. Aliabadi, Local Buckling of Thin-Walled Structures (Plate and Shallow ShellAssemblies) by the Boundary Element Method. Submitted to International Journal of Solids andStructures.

[3] Dirgantara, T., Aliabadi, M.H., Dual boundary element formulation for fracture mechanic analysisof shear deformable shells, International Journal of Solids and Structures, 28, 7769-7800 (2001).

[4] Di Pisa, C., Boundary Element Analysis of Multi-layered Panels and Structures, PhD Thesis,Department of Engineering, Queen Mary University of London (2005).

[5] Portela, A., Aliabadi, M.H. and Rooke, D.P., The dual boundary element method: e ectiveimplementation for crack problems, International Journal for Numerical Methods in Engineering,33, 1269-1287 (1992).

[6] Rashed, Y. F., Aliabadi, M. H. and Brebbia, C. A., Hyper-singular boundary element formulationfor Reissner plates, International Journal of Solids and Structures, 35, 2229-2249 (1998).

[7] Reissner, E., On a Variational Theorem in Elasticity, Journal of Mathematics and Physics, 29,90-95 (1950).

[8] Wearing, J.L., Ahmadi-Brooghani, S.Y., Fracture analysis of plate bending problems using bound-ary element method, in Plate Bending Analysis with Boundary Elements, Advanced in BoundaryElement Series, M. H. Aliabadi (Ed.), Computational Mechanics Publications, Southampton(1998).

[9] Wen, P.H., Aliabadi, M.H., Young, A., Crack growth analysis for multi-layered airframe structuresby boundary element method, Engineering Fracture Mechanics, 71, 619-631 (2004).

[10] Zienkiewicz, O.C., Taylor, R., The Finite Element Method, Vol 2: Solid Mechanics, B-H, Oxford,(2000).


An Incremental Technique to Evaluate the Stress Intensity Factors by the Element-Free Method

P.H. Wen1 and M.H. Aliabadi

2

1 Department of Engineering, Queen Mary, University of London, London, UK, E1 4NS

2 Department of Aeronautics, Imperial College, London, UK, SW7 2BY

Abstract

In this paper an incremental technique was developed to evaluate stress intensity factors

accurately by the use of the element-free Galerkin method based on the variation of potential

energy. The stiffness matrix is evaluated with a domain integral by the use of radial basis

function interpolation without elements in the domain. The Laplace transformation technique

and the Durbin inversion method are used to obtain the time domain physical values. The

applications of the proposed incremental technique to two-dimensional fracture mechanics have

been presented. Comparisons have been made with benchmark analytical solutions and

boundary element method.

Key words: Element-free Galerkin method, stress intensity factor, Laplace transformation

method, Fracture mechanics, moving least square interpolation.

1. Introduction

Although the FEM and BEM have been very successfully established and applied in

engineering as numerical tools, the development of new advanced methods nowadays is still

attractive in computational mechanics. Meshless approximations have received much interest

since Nayroles et al[1]

proposed the diffuse element method. Later, Belyschko et al[2]

and Liu

et al[3]

proposed element-free Galerkin method and reproducing kernel particle methods,

respectively. One key feature of these methods is that they do not require a structured grid and

are hence meshless. Recently, Atluri and his colleagues presented a family of Meshless

methods, based on the Local weak Petrov-Galerkin formulation (MLPGs) for arbitrary partial

differential equations [4]

with moving least-square (MLS) approximation. MLPG is reported to

provide a rational basis for constructing meshless methods with a greater degree of flexibility.

However, Galerkin-base meshless methods, except MLGP presented by Atluri[5]

still include

several awkward implementation features such as numerical integrations in the local domain. A

comprehensive review of meshless methods (MLPG) can be found in the book [6]

by Atluri.

A variety of local interpolation schemes that interpolate the randomly scattered points is

currently available. The moving least square and radial basis function interpolations are two

popular approximation techniques recently. With comparisons of these two techniques, the

moving least-square approximation is generally considered to be one of the best schemes with a

reasonable accuracy, particularly for static elasticity demonstrated by Wen et al[7]

. In this paper,

the mesh free Garlerkin method is presented with the radial basis function interpolation and an

incremental technique has been developed to calculate the stress intensity factors with high

accuracy. In addition, the enriched radial basis function and high density of node distribution


near the crack front are not needed in this approach. The accuracy of proposed method has been

demonstrated through benchmark examples.

2. Variation of potential energy and MLS

Based on the variation of potential energy, the element free Galerkin method is

developed on the basis of finite element method by the use of radial base function interpolation

in this paper to evaluate static and dynamic stress intensity factor with an incremental

technique. For a linear two dimensional elasticity, the equilibrium equations can be written as

iijij uf, (1)

where ij denotes the stress tensor, if the body force, is the mass density, 22 / tuu ii the

acceleration. Considering the variation of the total potential energy, with respect to each nodal

displacement, and the relations DuBuu andˆ,ˆ yields a linear algebraic equation

system in a matrix form as ˆˆ

222222 NNNNNN fuMuK (2)

where N is the total number of node in the domain . The stiffness and mass matrices are:

)(),(),(,)(),()(),( TTyyxyxMyyxByDyxBK dd (3)

in which Nii ,...2,1xx , and nodal force vector is defined by

)()(),()()(),( TTyytyxyybyxf dd (4)

where ),(),...,,(),,(),( )()(2211 yy xyxyxyxy nn andT)(21 ˆ,...,ˆ,ˆˆ y

un

iiii uuu are vector of

shape functions and nodal values of displacement. The collocation points ,, )(

2

)(

1

kk

k xxx

),(,...,2,1 ynk k are the shape functions and n(y) the total number of nodes in the local

domain named as supported domain as shown in Figure 1. For a two dimensional plane stress

case, we can rearrange the above equation in a matrix form as T)(

2

)(

1

2

2

2

1

1

2

1

1

T

21ˆ,ˆ...,ˆ,ˆ,ˆ,ˆˆ;ˆ),(,)( yy

uuxyyunn uuuuuuuu (5)

For convenience of analysis, the tilde (^) is removed in the following discussion. Applying the

Laplace transform to the equation (2) yields ~~2fuMK s (6)

where s is the Laplace parameter. We assume that the displacements u(y) at the point y can be

approximated in terms of the nodal values in a local domain (see Figure 1) as

i

n

k

k

ikki uu uxyxyyy

ˆ),(ˆ),()()(

1

(7)

where ),(),...,,(),,(),( )()(2211 yy xyxyxyxy nn , 2,1,ˆ,...,ˆ,ˆˆT)(21 iuuu n

iiii

yu and

)(ˆ xiu is the nodal values at point )(,...,2,1,, )(

2

)(

1 yx nkxx kk

k . For the two dimensional

plane stress case, we can rearrange the above relation as follows


Figure 1. Sub-domain y for MLS/RBF interpolation of the field point y and support domains.

.ˆ,ˆ...,ˆ,ˆ,ˆ,ˆˆ

,0...00

0...00

0

0),(

,ˆ),(,)(

T)(

2

)(

1

2

2

2

1

1

2

1

1

)(21

)(21

T

21

yy

y

y

u

xy

uxyyu

nn

n

n

uuuuuu

uu

(8)

3. Radial bases function

The distribution of function u in the sub-domain y over a number of randomly distributed

notes )(,...,2,1, yx nii can be interpolated, at the point y, by

)(),(),()( T)(

1

yaxyRxyyyn

i

iii aRu (9)

where ),(),...,,(),,(),( )(21

T xyxyxyxyR ynRRR is the set of radial basis functions centred at

the point y,)(

1

yn

kka are the unknown coefficients to be determined. The radial basis function

has been selected to be the following multi-quadrics 22),( kk cR xyxy (10)

with a free parameter c and in this paper, we select c=h (h is specified length in each example).

From the interpolation strategy in Eq. (9) for RBF, a linear system for the unknowns

coefficients a is obtained by

uaR0 (11)

It is apparent that the interpolation of field variable is satisfied exactly at each node. As the

RBFs are positive definite, the matrix 0R is assured to be invertible. Therefore, we can obtain

the vector of unknowns from Eq. (11)

)()(1

0 xuxRa (12)

So that the approximation u(y) can be represented, at domain point y, as

field point y

node xi

sub-domain y


- 4 -)(

1

1

0

T ),()()(),()(y

uxyxuxRxyRyn

k

kk uu

(13)

where the nodal shape function are defined by

)(),(),( 1

0

TxRxyRxy (14)

It is worth noticing that the shape function depends uniquely on the distribution of scattered

nodes within the support domain and has the Kronecker Delta property. As the inverse matrix

of coefficient )(1

0 xR is a function only of distributed node xi in the support domain, it is much

simpler to evaluate the partial derivatives of shape function. From Eq. (13), we have )(

1

,. ),()(y

uxyyn

i

iki,kk uu (15)

4. Incremental technique

To derive the integral for stress intensity factor for dynamic problem, one static

reference problem has to be determined first. Let t and u be the traction and displacement

boundaries respectively. If there is an increment of crack surface a at crack tip, an increment

of displacement ku in the domain and on the traction boundary and kt on the displacement

boundary will occur. The stress intensity factor in the static case can be written as2/1

0

)1(ut

dua

tdt

a

uK k

kk

kI (16)

The numerical results of static case can be used directly to the dynamic problem. The

relationship between the stress intensity factors for the reference problem and the real dynamic

problem can be written as

ut

dua

usdu

a

tdt

a

u

KK k

kk

kk

k

I

I~~~

)1(

~ 2

0 (17)

In order to evaluate the stress intensity factor in the time domain, the Durbin’s inverse method

is employed in this paper K

k

t

T

tki

T

kff

T

etf

0

2exp

2~Re)(

~

2

12)( (18)

where )(~

ksf is the transformed variables in the Laplace transform domain when the parameter

Tiksk /2 . The selection of parameters and T affects the accuracy slightly. In the

computations, we have chosen 0/5 t and 20/ 0tT in the following examples, where

10 / cht , h is the height of cracked sheet and )21)(1(/)1(1 Ec . Two numerical

examples are given to demonstrate the accuracy and efficiency of the proposed technique.


(a) (b)

Figure 2. Square plate with a central crack (h=b) under tension 0 : (a) a quarter of the plate; (b)

normalized stress intensity factor, where a/a=10-n

.

(a) (b)

Figure 3. Square plate with a central crack subjected to dynamic tension )(0 tH : (a) h=b; (b)

h=2b.

5. Examples

5.1 A central crack in rectangular sheet under uniform static load 0

A square plate of width 2b and height 2h containing a centred crack of 2a subjected to a

uniform shear load 0 on the top and the bottom is analysed. Due to the symmetry, a quarter of

plate is considered as shown in Figure 2 (a). Here Poisson’s ratio =0.3. A set of 11×11

( 121totalN ) uniformly distributed nodes is used and the integration is performed by dividing

the square into 10×10 cells with 4×4 Gauss points. The support domain is selected as a circle of

radius yd centered at field point y, which is determined such that the minimum number of

nodes in the sub domain 0)( Nn y , here the number 0N is selected to be 10 for all following

examples. Figure 2(b) shows the convergence of the normalized stress intensity factor

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 2 4 6 8 10 12 14 16

This method

BEM

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 2 4 6 8 10 12 14 16

This paper

BEM

KI/

0a

c1t/h

KI/

0a

c1t/h

h

y1

y2

0

a

b

crack tip K

I/0

a

1.00

1.10

1.20

1.30

1.40

1.50

0 1 2 3 4 5

Increment technique

handbook

n

1.325 [8]


aK I 0

0 / against the length of the increment of crack surface a . Excellent agreement with

Reference[8]

can be achieved when 310/ aa .

5.2 A Single central crack in rectangular plate under tensionConsider a rectangular plate of width 2b and length 2h with a centrally located crack of

length 2a. It is loaded dynamically in the direction perpendicular to the crack by a uniform

tension )(0 tH on the top and the bottom. Due to the symmetry, a quarter of plate is considered

as shown in Figure 2(a). Poisson’s ratio =0.3 and Young’s modulus is unit. Two geometries of

rectangular plate are considered in this example, i.e. h=b and h=2b. To demonstrate the

accuracy of mesh free method, the results given by Wen [9] using the indirect boundary

element method (fictitious load method) are plotted for comparison. Normalize dynamic stress

intensity factors aK I 0/ by these two techniques are shown in the Figures 3(a) and 3(b).

Apparently before the arrival time of dilatation wave traveling from the top of plate, the stress

intensity factor should remain to be zero. The agreement between the solutions is considered to

be good.

6. References

[1] B. Nayroles, G. Touzot & P. Villon, Generalizing the finite element method: diffuse

approximation and diffuse elements, Computational Mechanics, 10, 307-318, 1992.

[2] T. Belytschko, Y.Y. Lu & L. Gu, Element-free Galerkin method, Int. J. Numerical Methods

in Engineering, 37, 229-256, 1994.

[3] W.K. Liu, S. Jun & Y. Zhang, Reproducing kernel particle methods, Int. J. Numerical

Methods in Engineering, 20, 1081-1106, 1995.

[4] S.N. Atluri & T. Zhu, A new meshless local Peyrov-Galerkin (MLPG) approach to

nonlinear problems in computational modelling and simulation, Comput Model Simul Engng, 3,

187-196, 1998.

[5] S.N. Atluri & T. Zhu, The meshlesss local Peyrov-Galerkin (MLPG) approach for sovling

problems in elasto-statics, Comput Mech, 25, 169-179, 1999.

[6] S.N. Atluri, The Meshless Method (MLPG) for Domain and BIE Discretizations, Forsyth,

GA, USA, Tech Science Press, 2004.

[7] P.H. Wen and M.H. Aliabadi, An Improved Meshless Collocation Method for Elastostatic

and Elastodynamic Problems, Communications in Numerical Methods in Engineering, 2007 (to

appear).

[8] D.P. Rooke and D.J. Cartwright, A Compendium of Stress Intensity Factors, HMSO,

London, 1976.

[9] P.H. Wen, Dynamic Fracture Mechanics: Displacement Discontinuity Method, Computa-

tional Mechanics Publications, Southampton, 1996.


Stress analysis of composite laminated plates by the boundary

element method

F. L. Torsani, A. R. Gouveia, E. L. Albuquerque, and P. Sollero

Faculty of Mechanical Engineering, State University of Campinas

13083-970, Campinas, Brazil, [torsani,adriana,ederlima,sollero]@fem.unicamp.br

Keywords: Plates, boundary element method, laminated composites, and stress analysis.

Abstract. This paper presents a boundary element analysis of stresses in laminate composite platesfollowing Kirchhoff hypothesis. Stress integral equations are derived from the transversal displacementintegral equation. All derivatives of anisotropic thin plate fundamental solutions are computed ana-lytically. Stresses are computed in each lamina at internal points of the plate. A numerical exampleis presented in order to assess the proposed method. Results are compared with solutions found inliterature, showing good agreement.

Introduction

The material anisotropy presents two different hands. On one hand, it turns the material analysisextremely hard due the large number of variables necessary to represent its mechanical properties.On the other hand, the use of anisotropic materials allows the designer to control their mechanicalproperties along each direction, increasing the material strength without increasing the weight. Withthe demand by optimization of natural resources and the large offer of computational resources, thedesigner target are changing from simple analysis to optimized performance.

In this way, the use of high performance composite materials is an interesting option because theyallow the control of their mechanical properties either by the choice of their components, matrix andreinforcement, or by the component order inside the material. Together with the demand by highperformance composite materials, it has also increased the demand by reliable and accurate numericalprocedures for this materials analysis.

The complexity of the anisotropic material analysis is evident in literature. It can be noted thatthe number of references in which the boundary element method is applied for anisotropic materials issignificantly smaller than those treating isotropic materials. However, in the last ten years, importantadvances on boundary element techniques applied to anisotropic materials were published in theliterature. For example, plane elasticity problems were analyzed by [1, 2], [3], and [4, 5, 6, 7], out ofplane elasticity problems by [8], tri-dimensional problems by [9, 10, 11], and Kirchhoff plates by [12].

Boundary element formulations have been applied to plate bending anisotropic problems consid-ering Kirchhoff as well as shear deformable plate theories. [13] presented a boundary element analysisof plate bending problems using fundamental solutions proposed by [14] based on Kirchhoff platebending assumptions. [15] proposed a formulation in which the singularities were avoided by placingsource points outside the domain. [16] presented an analytical treatment for singular and hypersingu-lar integrals of the formulation proposed by [13]. Shear deformable plates have been analyzed usingthe boundary element method by [17, 18] with the fundamental solution proposed by [19].

In this work the calculation of internal point stresses of anisotropic plates using the boundaryelements method is presented. Stress integral equations are derived from the transversal displacementintegral equation. Stresses are computed in every lamina at internal points of the plate. Results arecompared with solutions found in literature, showing good agreement.


Boundary integral equation

As shown by [12], the boundary integral equation for transversal displacements w in a boundary pointof an anisotropic plate can be written as:

1

2w(Q) +

∫Γ

[V ∗

n (Q,P )w(P ) − m∗

n(Q,P )∂w(P )

∂n

]dΓ(P ) +

Nc∑i=1

R∗

ci(Q,P )wci

(P ) −

∫Γ

[Vn(P )w∗(Q,P ) − mn(P )

∂w∗

∂n(Q,P )

]dΓ(P ) +

Nc∑i=1

Rci(P )w∗

ci(Q,P ) +

∫Ω

g(P )w∗(Q,P )dΩ. (1)

where ∂∂n

is the derivative in the direction of the outward vector n that is normal to the boundary Γ;mn and Vn are, respectively, the normal bending moment and the Kirchhoff equivalent shear force onthe boundary Γ; Rc is the thin-plate reaction of corners; wc is the transverse displacement of corners;P is the field point; Q is the source point; and an asterisk denotes a fundamental solution.

Fundamental solutions required at equation (1) are given by [12].

Strain and displacement in laminate composite plates

Laminates are fabricated such that they act as an integral structural element. To assure this condition,the bond between two laminae in a laminate should be infinitesimally thin and not shear deformableto avoid the laminae slip over each other, and to allow displacement continuity along the bond [20].Thus, we could consider that strains are continuous along its thickness. However, as each laminae iscompounded by different materials, stresses present discontinuities along laminate interfaces.

In Kirchhoff plates, strains are given by:

εx = −z∂2w

∂x2,

εy = −z∂2w

∂x2,

γxy = −2z∂2w

∂x∂y. (2)

So, in order to obtain strains, second order derivatives of integral equations (1) need to be com-puted. For internal points, these derivatives are given by:

∂2w(Q)

∂x2=

∫Γ

[∂2V ∗

n

∂x2(Q,P )w(P ) −

∂2m∗

n

∂x2(Q,P )

∂w(P )

∂n

]dΓ(P ) +

Nc∑i=1

∂2R∗

ci

∂x2(Q,P )wci

(P ) −

∫Γ

[Vn(P )

∂2w∗

∂x2(Q,P ) − mn(P )

∂3w∗

∂n∂x2(Q,P )

]dΓ(P ) +

Nc∑i=1

Rci(P )

∂2w∗

ci

∂x2(Q,P ) +

∫Ω

g(P )∂2w∗

∂x2(Q,P )dΩ, (3)


∂2w(Q)

∂y2=

∫Γ

[∂2V ∗

n

∂y2(Q,P )w(P ) −

∂2m∗

n

∂y2(Q,P )

∂w(P )

∂n

]dΓ(P ) +

Nc∑i=1

∂2R∗

ci

∂y2(Q,P )wci

(P ) −

∫Γ

[Vn(P )

∂2w∗

∂y2(Q,P ) − mn(P )

∂3w∗

∂n∂y2(Q,P )

]dΓ(P ) +

Nc∑i=1

Rci(P )

∂2w∗

ci

∂y2(Q,P ) +

∫Ω

g(P )∂2w∗

∂y2(Q,P )dΩ, (4)

and

∂2w(Q)

∂x∂y=

∫Γ

[∂2V ∗

n

∂x∂y(Q,P )w(P ) −

∂2m∗

n

∂x∂y(Q,P )

∂w(P )

∂n

]dΓ(P ) +

Nc∑i=1

∂2R∗

ci

∂x∂y(Q,P )wci

(P ) −

∫Γ

[Vn(P )

∂2w∗

∂x∂y(Q,P ) − mn(P )

∂3w∗

∂n∂x∂y(Q,P )

]dΓ(P ) +

Nc∑i=1

Rci(P )

∂2w∗

ci

∂x∂y(Q,P ) +

∫Ω

g(P )∂2w∗

∂x∂y(Q,P )dΩ. (5)

As presented by [20], stresses at each laminae can be evaluated from strain given by equation (2)as following:

⎧⎪⎨⎪⎩

σx

σy

τxy

⎫⎪⎬⎪⎭ =

⎡⎢⎣

Q11 Q12 Q16

Q12 Q22 Q26

Q16 Q26 Q66

⎤⎥⎦

⎧⎪⎨⎪⎩

εx

εy

γxy

⎫⎪⎬⎪⎭ , (6)

where matrix[Q

]is given by:

[Q

]= [T ]−1 [Q] [T ] . (7)

The transformation matrix [T ] is given by:

[T ] =

⎡⎢⎣

cos2 θ sin2 θ 2 sin θ cos θsin2 θ cos2 θ −2 sin θ cos θ

− sin θ cos θ sin θ cos θ cos2 θ − sin2 θ

⎤⎥⎦ , (8)

where θ is the angle between the fibre orientation and the direction of axis x.The stiffness matrix [Q] is given, in terms of engineering constants, by:

[Q] =

⎡⎢⎣

EL

1−νLT νTL

νLT ET

1−νLT νTL

0νLT ET

1−νLT νTL

ET

1−νLT νTL

0

0 0 GLT

⎤⎥⎦ , (9)

where EL is the elastic modulus in the parallel to the fibre direction, ET is elastic modulus in thetransversal to the fibre direction, GLT is the shear modulus in the plane of the laminae, and νLT isthe principal Poisson ratio in the plane of the laminae.

Numerical results

To validate the procedures implemented, a nine-layered, symmetrical angle-ply laminate with stackingsequence [+θ/− θ/+ θ/− θ/+ θ/− θ/+ θ/− θ/+ θ] with 0 ≤ θ ≤ 45o was chosen. The plate is squarewith edge length a = 1 m. All edges are simply-supported and all layers have the same thickness.The total thickness is equal to h = 0.01 m and material properties are: EL = 207 GPa, ET = 5.2


GPa, GLT = 3.1 GPa, and νLT = 0.25. Figures 1 and 2 show the effect of the variation of θ on thedisplacement and bending stress resultants, respectively, at the centre of the plate. They are comparedwith finite element results obtained by [21]. It is worth to say that the finite element formulationconsiders the effect of the shear deformation. As it can be seen, in both cases the agreement betweenthe boundary element thin plate and the finite element shear deformable plate is very good.

0 5 10 15 20 25 30 35 40 452

2.5

3

3.5

4

4.5

1000×

(wE

Th

3/qa

4)

θ (degrees)

w, this workw, Reference [21]

Figure 1: Effect of the orientation θ in the transversal displacement response at the centre of the plate.

0 5 10 15 20 25 30 35 40 450

2

4

6

8

10

12

14

100×

(Mx/qa

2),

100×

(My/qa

2)

θ (degrees)

Mx, this workMy, this workMx, Reference [21]My, Reference [21]

Figure 2: Effect of the orientation θ in the moment response at the centre of the plate.

Figure 3 shows the stress distribution (σx) along the thickness of the plate. It can be seen thatstress are discontinuous at the interface and vary linearly along each laminae.


0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

2.5(1

0×

σx×

h2)/

q

z/h

Figure 3: Stress distribution (σx) along the thickness for θ = 45o at the centre point of the plate.

Conclusions

This paper presented a boundary integral formulation for the computation of stress in internal pointsof anisotropic thin plates. An integral equation for the second displacement derivative is developedand all derivatives of the fundamental solution are computed analytically. The obtained results are ingood agreement when compared with finite element thick plate results.

Acknowledgment

The authors would like to thank the State of Sao Paulo Research Foundation (FAPESP) for financialsupport for this work (grant number: 03/09498-0).

References

[1] P. Sollero and M. H. Aliabadi, Fracture mechanics analysis of anisotropic plates by the boundaryelement method. International Journal of Fracture, 64: 269-284, 1993.

[2] P. Sollero and M. H. Aliabadi, Anisotropic analysis of composite laminates using the dual bound-ary elements methods. Composite Structures, 31:229-234, 1995.

[3] A. Deb, Boundary elements analysis of anisotropic bodies under thermo mechanical body forceloadings. Computers and Structures, 58:715-726, 1996.

[4] E. L. Albuquerque, P. Sollero and M. H. Aliabadi, The boundary element method applied totime dependent problems in anisotropic materials. International Journal of Solids and Structures,39:1405-1422, 2002.

[5] E. L. Albuquerque, P. Sollero and P. Fedelinski, Dual reciprocity boundary element methodin Laplace domain applied to anisotropic dynamic crack problems. Computers and Structures,81:1703-1713, 2003.

[6] E. L. Albuquerque, P. Sollero and P. Fedelinski, Free vibration analysis of anisotropic materialstructures using the boundary element method. Engineering Analysis with Boundary Elements,27:977-985, 2003.


[7] E. L. Albuquerque, P. Sollero and M. H. Aliabadi, Dual boundary element method for anisotropicdynamic fracture mechanics. International Journal for Numerical Method in Engineering, 59:1187-1205, 2004.

[8] Ch. Zhang, Transient elastodynamics antiplane crack analysis of anisotropic solids. International

Journal of Solids and Structures, 37:6107-6130, 2000.

[9] M. Kogl and L. Gaul, A boundary element method for transient piezoelectric analysis. Engineering

Analysis with Boundary Elements, 24:591-598, 2000.

[10] M. Kogl and L. Gaul, A 3-d boundary element method for dynamic analysis of anisotropic elasticsolids. CMES-Computer Modeling in Engineering and Science, 1:27-43, 2000.

[11] M. Kogl and L. Gaul, Free vibration analysis of anisotropic solids with the boundary elementmethod. Engineering Analysis with Boundary Elements, 27:107-114, 2003.

[12] E. L. Albuquerque, P. Sollero, W. S. Venturini and M. H. Aliabadi, Boundary element analysis ofanisotropic kirchhoff plates. International Journal of Solids and Structures, 43:4029-4046, 2006.

[13] G. Shi and G. Bezine, A general boundary integral formulation for the anisotropic plate bendingproblems. Journal of Composite Materials, 22:694-716, 1988.

[14] B. C. Wu and N. J. Altiero, A new numerical method for the analysis of anisotropic thin platebending problems. Computer Methods in Applied Mechanics and Engineering, 25:343-353, 1981.

[15] C. Rajamohan and J. Raamachandran, Bending of anisotropic plates by charge simulationmethod. Advances in Engineering Software. 30:369-373, 1999.

[16] W. P. Paiva, P. Sollero and E. L. Albuquerque, Treatment of hypersingularities in boundaryelement anisotropic plate bending problems. Latin American Journal of Solids and Structures,1:49-73, 2003.

[17] J. Wang and K. Schweizerhof, Study on free vibration of moderately thick orthotropic laminatedshallow shells by boundary-domain elements. Applied Mathematical Modelling, 20:579-584, 1996.

[18] J. Wang and K. Schweizerhof, Free vibration of laminated anisotropic shallow shells includingtransverse shear deformation by the boundary-domain element method. Computers and Struc-

tures, 62:151-156, 1997.

[19] J. Wang and K. Schweizerhof, The fundamental solution of moderately thick laminated anisotropicshallow shells. International Journal of Engineering and Science, 33:995-1004, 1995.

[20] B. D. Agarwal and L. J. Broutman, Analysis and performance of fiber composites. 2nd Edition,John Wiley & Sons Inc, New York, 1990.

[21] H. V. Lakshminarayana and S. S. Murthy, 1984. A shear-flexible triangular finite element modelfor laminated composite plates. International Journal for Numerical Methods in Engineering, 20:591–623, 1984.


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Homogenization of nonlinear composites using Hashin-Shtrikman principles and BEM

P.Prochazka1,a and Z. Sharif Khodaei2,b

1Society of Science, Research and Advisory, Czech Association of Civil Engineers 2Czech Technical University in Prague, Faculty of Civil Engineering, Department of Mechanics,

Prague, Czech Republic [email protected]

[email protected]

Keywords: Homogenization of composites, nonlinear problem, Hashin-Strikman principle, boundary elements, eigenparameters, Lippmann-Swinger equation

Abstract. Boundary element methods suffer from one disadvantage, which is description of

inhomogeneous and plastic structures. It appears that one useful trick can bridge this issue. This is

generated by generalized Hashin-Strikman variational principles, in which the eigenparameters are

involved. Such an approach leads to separation of phases in the composites. In very many cases one

phase behaves linearly (it is mostly the fiber) and the other (matrix in prevailing cases) nonlinearly.

Using an equivalent integral formulation of the H-S principles yields exclusion of fiber influence in

the problem. This is an impact of identification of material properties of the fiber with that of

comparative medium. New unknown strains or stresses in the matrix occur instead of linear

mechanical properties defined in the fibers. When considering nonlinear material behavior of the

matrix, these strains or stresses play a role of new quantities appearing in iteration steps needed for

identification of plastic behavior.

1. Introduction

The approach for calculating plastic deformations (or alternatively relaxation stresses) selected in

this paper starts with the idea of Hashin-Strikman variational principles, [1], which lead to

variational bounds of linear composites. Extending the principles by introducing eigenparameters

into the formulation, [2], new free parameters occur in the postulation of the problem and they will

serve plastic deformation or relaxation stresses. They were successfully used in optimization of

prestress in laminated composites in [3], for example. Survey of access to the homogenization

techniques of periodic composites using numerical procedures can be found in [4].

In this paper basic relationships are derived and the approach is briefly described. An example is

presented in the end of this paper. The plastic behavior is described in [5] for the Mises hypothesis.

2. Basic considerations

Before we tackle the formulation of the problem of finding overall material properties of a

composite nonlinear structure using boundary element method a useful approach derived in a

similar way as Hashin-Shtrikman variational principles will be mentioned. For this reason the H-S

principles will be briefly mentioned in the sequel. Only primary principle is considered for

application to lower bounds (lower estimate of the overall properties – deformation method) while

the dual principle can be applied to force method of finding upper estimate of the effective material

properties.


Let a domain 3R with its boundary 0, pupu , be given, describing a shape

of the body under consideration. Moreover, let us assume that along the boundary u of the body

displacements 3,2,1, iui , be prescribed and along the boundary p tractions be given. Influence

of volume weight forces is neglected.

The approach is split into two steps:

1st step: displacements 00

iuu , tractions 00ipp , strains 00

ij and stresses 00ij

( 3,2,1, ji ) are calculated for homogenized, isotropic comparative medium and they are

considered to be known in the next.

Statical equations for the stress field and geometric boundary conditions are valid:

0

0

j

ij

x , ii uu0 on u , ii pp0 on p (1)

and also homogeneous and isotropic Hooke’s law holds valid with material stiffness matrix 0L :

000klijklij L in (2)

2nd

step: quantities for inhomogeneous anisotropic medium are to be determined, i.e. displacements

u , tractions p , strains and stresses are unknown. In this step geometrically same body is

assumed with prescribed boundary displacements u and given tractions p from the first step.

Hooke’s law is now valid for material stiffness matrix of the whole body (no more isotropic

homogeneous) involving also eigenstress field :

ijklijklij L in , ii uu on u , ii pp on p (3)

Periodic unit cell is considered, for which it holds:

0)),(()()(),()( ijijijijijijijiji EEuxEu xuxxxx (4)

where ijE are components of the overall (macroscopic) strain tensor, iu are components of fluctuating

displacement vector. In periodic structure it holds for the unit cell: u is same at each boundary point in

the direction of invariance of the layer, while tractions are there antisymmetric. Similarly to the

classical H-S principle symmetric polarization tensor is introduced, defined as:

ijklijklij L0 (5)

Inasmuch as is statically admissible, it also holds:

0)( 0

j

ijklijkl

x

L (6)

Next, subtracting (5) and (3), considering (4), provides:

00 ,in0][ LLLijklijklklijklij ELL (7)


3. Lippmann-Swinger equation for plasticity

In this section integral formulation to the H-S variational problem is created. Multiplying (7)

successively by test functions 3,2,1,ii , integrating the result over the domain , applying two

times Green’s theorem and setting for i the kernel displacement components yield:

xxxxxxxxx ),(d)(),()(d)(),()(d)(),()( ***kliklkikkiki uppuu (8)

where

jlmkijlmj

m

lkijlmik

j

ikikl nLn

x

uLp

x

u),(

),(),(,

),(),( *0

*0*

**

x

x

x

x

x

Here and in what follows geometrically and materially symmetric unit cell is considered, so that the

boundary displacements u vanish.

Positioning on the boundary and taking into account the boundary conditions (4), the

integral equation equivalent to (6) is then:

xxxxx ),(d)(),()(d),(0 **kliklkik pu (9)

Differentiating (8) with respect to j provides the expression

xxxxx ),(d)(),()(d),()( **klijklkijkij ph (10)

and the volume integral is taken in Hadamard’s sense. Equations (9) and (10) can be expressed in a

comprehensive form: there is an operator )(G which relates the fluctuating strain and polarization

tensor as:

))(()( xx G (11)

Let us substitute for )(xkl from (7) into the latter equation and split the integration over fiber

and matrix, which provides:

µ

xx

xxxx

),(d)(][),(

)(d))()(]()[,()(d),()(

0*

0**

klijklklijklijkl

klijklijklijklijklkijkij

ELL

ELLph

f

m

(12)

as there is no eigenstrain (plastic strain) inside of a fiber. If we set fijklijkl LL0 , the latter equation

simplifies as:


µ

xx

xxxx

),(d),(

)(d))()(]()[,()(d),()(

*

**

ijklklfijkl

klfijklkl

fkl

mklijklkijkij

EL

ELLLphm

(13)

Next, equation (13) can be written in increment form (note that no differentiation is carried out, it is not

permitted here for the singular integrals involved in such a process do not admit it):

mijkl

fijkl

mijklijklkijkij LLph

m

µ xxxx ),(d))()(]()[,()(d),()( ** (14)

and the points of observer belong now only to the matrix.

For completeness let us write integral equations which are equivalent to (9):

mklkl

fijkl

mkliklkik LLpu

m

µ xxxx ),(d))()(]()[,()(d),(0 ** (15)

Equations (9) and (13) establish a simultaneous system for computation of elastic state in

inhomogeneous material composed form fiber and matrix due to unit strain impulses. The latter two

equations (14) and (15) create a simultaneous system for improvement of boundary tractions and

fluctuating strains due to plasticity.

In conclusion, the strain components at any point of the domain are dependent on boundary

tractions and strains and eigenstrains in matrix only, and the boundary tractions are dependent on

the strains and eigenstrains in matrix. The latter equations lead us to an approach, which is

described in more details in the following section.

4. Calculation of plasticity effect

First, let us consider geometry of a composite unit cell. The solution of such a cell in periodic structure is

concisely described in [4], for example. Applying appropriate unit displacements to the boundary of the cell,

its elastic material properties are identified with that of fiber, null superscript quantities are straightforwardly

obtained, and even on much more complicated geometry than ours. The values of displacements, strains and

stresses are also speculated at starting levels.

Figure 1: Unit cell used in the study

Dividing the external boundary of the unit cell into subsurfaces in 3D or subintervals in 2D as

)()()(),()()(),()()(111

k

M

kijkijk

M

kijkijj

N

jiji QQIpp µµ (16)


where are the characteristic functions being equal to one for inside the subregion iI or i and zero

otherwise, iI is a boundary subregion, i is a subregion in the domain, NM , are respectively numbers of

domain subregions and boundary subregions. If we put jj

ijiij ppp ,)()( is centered in jI ,

kijkijijk )()( and k

kijkijijk ,)()( µµµ is identical with the center of gravity of the subregion

k , all quantities are uniformly distributed inside their subregions (elements). Obviously, iI is a boundary

element while k is an internal cell. Applying these approximations, putting first 0ijµ overall, from (12)

and (13) strains, tractions and stresses follow. Then stresses are obtained from (5). Testing them for plastic

rules, distribution of ijµ is specified and (14) with (15) delivers the increments of strains using only the

domain of the matrix. This is the greatest advantage of the above described approach, which consists in

concentration of the problem of improvement of boundary tractions and fluctuating strain to the domain of

matrix only, while fiber is no longer involved into computation.

5. Example

A plane square unit cell is considered with fiber volume ratio equal to 0.6 according to Fig. 2, so

that only first quarter is assessed. The following elastic material properties of phases are assumed:

Young's modulus of fiber Ef = 210 GPa, Poisson's ratio f = 0.16; on the matrix E

m = 17 GPa, and

m = 0.3. The Mises plasticity is considered with plastic coefficient MPa200k . Displacements in

the planar principal directions are drawn in Fig. 2 for unit displacement excitation in x-direction.

Figure 2: Distribution of elastic and plastic displacements in the first quarter of the unit cell


6. Conclusions

In this paper useful treatment of plastic behavior on a unit cell is described when using boundary

element method. This splendid numerical method suffers from one important problem:

inhomogeneous material properties (so is plasticity, for example) are hardly involved in the

computation. It appears that for particular problems there is a way on how to bridge this problem.

Extended Hashin-Shtrikman variational principles are used for equivalent formulation, which

covers polarization tensor. A special choice of elastic material properties (identification of

comparative medium with fiber) leads us to elimination of fibers from the nonlinear computations

and basic simplification of iterative procedure.

From the pictures describing distribution of displacements in the first quarter of the cell at elastic

and plastic states it is seen that principal redistribution of extreme displacements is attained. This

result is obtained in very short computational time, as only two iterations are necessary to get

significant values of the displacements. The continuation of iterations is no more necessary, is

insignificant. Moreover, in each iteration step only multiplication of vectors and matrices is

required; there is no solution of equations. Only in the elastic state standard linear algebraic

equations are solved.

Acknowledgments: The financial support of this work provided for the first author by grants No.

GACR 103/08/1197 and CZE MSM 6840770001 of the Grant Agency of the Czech Republic are

gratefully acknowledged.

References

[1] Z. Hashin, S. Shtrikman: On some variational principles in anisotropic and nonhomgeneous

elasticity, J Mech Phys Solids 10(3), (1962), p. 35-42.

[2] P.P. Procházka, J. Šejnoha: Extended Hashin-Shtrikman variational principles, Applications of

Mathematics 49 (4), (2004), p. 357-372.

[3] G.J. Dvorak, P.P. Procházka: Thick-walled composite cylinders with optimal fiber prestress,

Composites Part B 27B, (1996), p. 643-649.

[4] P.M. Suquet: Homogenization techniques for composite media, Lecture Notes in Physics 272,

Berlin, Springer (1985), p. 194-278.

[5] M. Duvant, J.-P. Lions: Variational inequalities in mechanics, Dunod, Paris (1978).


Transient Dynamic Analysis of Interface Cracks in 2-D Anisotropic Elastic Solids by a Time-Domain BEM

Stefanie Beyer1, a, Chuanzeng Zhang2, b, Sohichi Hirose3, c,

Jan Sladek4, d and Vladimir Sladek4, e

1 Energy Sector, Siemens AG, D-02826 Görlitz, Germany

2 Department of Civil Engineering, University of Siegen, D-57068 Siegen, Germany

3 Department of Mechanical and Environmental Informatics, Tokyo Institute of Technology, Tokyo 152-8552, Japan

4 Institute of Construction and Architecture, Slovak Academy of Sciences, 84503 Bratislava, Slovakia

a [email protected], b [email protected], c [email protected],

d [email protected], e [email protected]

Keywords: Interface cracks; Layered anisotropic elastic solids; Dynamic stress intensity factors; Time-domain BEM; Multi-domain technique.

Abstract. In this paper, transient elastodynamic analysis of an interface crack in two-dimensional

(2-D), layered, piecewise homogenous, anisotropic and linear elastic solids subjected to an impact

loading is presented. A time-domain boundary element method (BEM) is developed for this

purpose. Displacement boundary integral equations (BIEs) in conjunction with a multi-domain

technique are applied in the present time-domain BEM. Collocation method is used for both the

spatial and the temporal discretizations. Numerical examples for computing the complex dynamic

stress intensity factor are presented and discussed.

Introduction

Interface cracks and interface debonding are the most distinct failure mechanisms in composite

materials because of the mismatch in the material properties. To characterize the crack-tip stress

and deformation fields, stress intensity factors (SIFs) and energy release rates are often used in

linear elastic fracture mechanics. Because of the mathematical complexity of the interface crack

problems, only very few investigations on interface cracks in generally anisotropic and linear

elastic solids under impact loading conditions can be found in literature. Although the time-domain

boundary element method (BEM) can be utilized in principle for this purpose, its numerical

implementation and applications to dynamic analysis of interface cracks in homogeneous,

generally anisotropic and linear elastic solids have been reported in literature only very recently.

The main reason for this deficiency is due to the fact that the required elastodynamic fundamental

solutions in the time-domain BEM do not have explicit closed-form analytical expressions and they

have a very complex mathematical structure.

This paper presents a transient dynamic analysis of interface cracks in two-dimensional (2-D),

layered, anisotropic and linear elastic solids. A time-domain BEM is developed for this purpose. A

multi-domain BEM is applied, which divides the inhomogeneous layered anisotropic solid into

homogeneous and anisotropic layers. Time-domain elastodynamic fundamental solutions for

homogeneous, anisotropic and linear elastic solids and displacement boundary integral equations

are used for each layer. For both temporal and spatial discretizations of the boundary integral

equations, collocation methods are adopted. By using the continuity/discontinuity conditions of the

displacements and the stresses on the interfaces and the crack-faces and initial conditions, an


explicit time-stepping scheme is obtained for computing the discrete unknown boundary data

including the crack-opening-displacements (CODs). An efficient technique for computing the

complex dynamic SIFs from the numerically calculated CODs is presented and discussed. To

demonstrate the accuracy and the efficiency of the present time-domain BEM, numerical examples

are presented and discussed.

Initial boundary value problem and time-domain BIEs

We consider a layered, anisotropic and linear elastic solid with an interface crack as shown in Fig.

1. In the absence of body forces, the layered solid satisfies the equations of motion

,ij j iu , (1)

Hooke’s law

,ij ijkl k lE u , (2)

the initial conditions

( , ) ( , ) 0i iu t u tx x for 0t , (3)

the boundary conditions

( , ) 0if tx , (1) (2)

c c cx , (4)

*( , ) ( , )i if t f tx x , fx , (5)

*( , ) ( , )i iu t u tx x , ux , (6)

an the continuity/discontinuity conditions on the interface int and the crack-faces (1,2)

c

(1) (2)( , ) ( , )i iu t u tx x , (1) (2)( , ) ( , )ij ijt tx x , intx , (7)

(1) (1) (2) (2)( , ) ( , ) ( , )i i c i cu t u t u tx x x , cx .(8)

In Eqs. (1)-(8), iu , ij and i ij jf n represent the displacement, the stress and the traction

components, jn is the outward normal vector, is the mass density, ijklE is the elasticity tensor,

(1) (2)

c c c denotes the crack-faces, ex f u stands for the external boundary with f

and u being the boundary parts with prescribed tractions *

if and displacements *

iu , int is the

interface, and iu is the crack-opening-displacements (CODs), respectively. A comma after a

quantity represents spatial derivatives while a dot over a quantity denotes time differentiation.

Greek indices take the values 1 and 2, while Latin indices take the values 1, 2 and 3. Unless

otherwise stated, the conventional summation rule over repeated indices is implied.

Fig. 1: An interface crack in a layered anisotropic and linear elastic solid

2Domain

2Material

(1)

c

(2)

c int

int

ex

1Domain

Material 1


To each sub-domain of the cracked solid, the following time-domain displacement BIEs are

applied

( , ) -G G

ij j ij j ij j yc u t u f t u dsx , (1,2)

ex int cx , (9)

where ijc is the free-term depending on the smoothness of the boundary, ( , ; , )G

iju tx y and

( , ; , )G

ijt tx y are the elastodynamic displacement and traction fundamental solutions for

homogeneous, anisotropic and linear elastic solids, x and y represent the observation and the

source points, and denotes Riemann convolution

0

( ) ( )

t

f g f t g d . (10)

For smooth boundaries 0.5ij ijc , where ij is the Kronecker-delta function.

The elastodynamic fundamental solutions for homogenous, anisotropic and linear elastic solids

derived by Wang and Achenbach [1] are implemented in the present time-domain BEM. Note here

that the elastodynamic fundamental solutions for homogenous, anisotropic and linear elastic solids

cannot be given in closed forms in contrast to homogeneous, isotropic and linear elastic solids. In

2-D case, they can be represented by line-integrals over a unit circle. It should be also remarked

here that the displacement BIEs (9) have a strong singularity in the sense of Cauchy-principal value

integrals.

Numerical solution procedure

To solve the strongly singular displacement BIEs (7), a numerical solution procedure is developed.

Collocation method is applied for both the temporal and the spatial discretizations. The

displacements and the tractions are approximated by

( ) ( )

1 1

( , ) ( ) ( )M P

pp

i m u u i mm p

u uy y , (11)

( ) ( )

1 1

( , ) ( ) ( )M P

pp

i m f f i mm p

f fy y , (12)

where ( ) ( )m y is the spatial shape functions, ( ) ( )p is the temporal shape function, ( ) p

i mu and

( ) p

i mf are discrete values at the m-th collocation point and p-th time-step. Also, M is the total

element number and the time is divided into P equal time-steps, i.e., P t . In this analysis,

constant spatial and linear temporal shape functions are adopted for simplicity, i.e.,

1,( ) ( )

0,

m

m u m f

m

y

y y

y

, (13)

( ) ( )

1( ) ( ) ( 1) ( 1) 2

( 1) ( 1) ,

p p

u f p t H p t p t H p tt

p t H p t

(14)

where [ ]H is the Heaviside step function. In Fig. 2, the used spatial and temporal shape functions

are depicted.


Fig. 2: Spatial and temporal shape functions

By substituting Eqs. (11) and (12) into the time-domain BIEs (9), applying the discretized BIEs

to each sub-domain and invoking the initial conditions (3), a system of linear algebraic equations

can be obtained as

-11 1 - 1 - 1

1

NN N N n n N n n

n

A u B f B f A u , (15)

where nA and n

B are the system matrices, Nu is the vector containing the boundary

displacements, and Nf is the traction vector for the external boundary and the crack-faces. By

considering the boundary conditions (4)-(6) and the continuity conditions (7), equation (12) can be

rearranged as

-1-1

1 1 - 1 - 1

1

NN N N n n N n n

n

x C D y B f A u , (16)

in which Nx represents the vector with unknown boundary data, while N

y denotes the vector with

known boundary data. The explicit time-stepping scheme (16) is applied for computing the

unknown boundary data time-step by time-step.

By using constant spatial and linear temporal shape-functions (13) and (14), time and spatial

integrations can be performed analytically. Strongly singular integrals are computed analytically by

a special regularization technique. Only the line-integrals over the unit circle in the elastodynamic

fundamental solutions have to be computed numerically by using standard Gaussian quadrature

formula. More details on the numerical implementation of the time-domain BEM can be found in

the recent work of Beyer et al. [2] and Beyer [3].

Computation of the complex dynamic stress intensity factor

For an interface crack, the displacement and the stress fields near the crack-tip can be characterized

by a complex stress intensity factor which is defined as

21 KiKK , (17)

where 1K and 2K are the real and the imaginary parts of the complex stress intensity factor. The

amplitude and the phase angle of the complex dynamic stress intensity factor can be obtained by

using the following equations

tp )1(tp )1(

( ) ( )p

p t

( ) ( )m y

1m 2m 3m

1l 2l 3l

y


2 2

1 2

2 22

2 1 2 2 1 2 2 1

1 2 1 2

( ) ( ) ( ) ( )

( , ) ( , ) ( , ) ( , )1 4 2,

4cosh

KK t t K t K t

t u t t u t d u t d u t

d t t d

(18)

2

1 2 12

1 1 2 1 2

( , ) / ( , )( )tan ( )

( ) ( , ) / ( , )

d d u t u tK tt

K t t u t u t t.

(19)

In Eqs. (18) and (19), is a small distance to the crack-tip, is the bi-material constant, and the

constants d , d , t ( 1,2 ) can be found in [4].

Since the stress field near the tip of an interface crack shows a very complicated oscillating

singularity, no special crack-tip elements are implemented in the present time-domain BEM for

simplicity. For this reason, the local behavior of the crack-opening-displacements (CODs) cannot

be described properly by using the present time-domain BEM. To minimize the numerical error in

the computation of the complex dynamic stress intensity factor from the CODs, a least-squares

technique based on the minimization of the quadratic deviations of the displacements on the crack-

faces is applied.

Numerical examples

As first numerical example, we consider an interface crack of length 2a in a rectangular plate

consisting of two homogeneous, anisotropic and linear elastic materials as depicted in Fig. 3. The

plate is subjected to an impact tensile loading of the form 0 ( )H t , where 0 is the loading

amplitude and ( )H t is the Heaviside step function.The geometry of the cracked plate is defined by

2w = 20,0mm, 2h = 40,0mm and 2a = 4,8mm. The outer boundary of the plate is discretized by 80

constant elements with 20 elements for each side, and the crack is discretized by 15 constant

elements. A time-step 0.3t s is applied. Plane strain condition is assumed in the numerical

calculations.

Two different Graphite-epoxy composites are considered in the first example, which have the

following material constants

(1)

122.77 3.88 18.88 0 0 0

16.34 4.79 0 0 0

28.3 0 0 0

3.91 0 0

21.83 0

6.94

ijC GPa

sym

, (1)

= 1600kg/m3

(2)

65.41 4.29 26.17 0 0 0

16.34 4.39 0 0 0

54.99 0 0 0

5.27 0 0

33.96 0

5.58

ijC GPa

sym

, (2)= 1600kg/m

3

Numerical results for the normalized amplitude of the complex dynamic stress intensity factor

( ) / st

IK t K ( 0

st

IK a ) are presented in Fig. 4. A comparison with the numerical results of


t

(t)

0

Wünsche [5] shows a very good agreement. Wünsche used a time-domain BEM based on a

collocation method for the temporal discretization and a Galerkin-method for the spatial

discretization. Figure 4 shows that the normalized amplitude of the complex dynamic stress

intensity factor is zero before the wave arrival time at the crack-tip (1)/ 6.25Lt h c s , where

(1)

Lc

is the velocity of the quasi-longitudinal wave of the domain 1 . After the wave arrival, ( ) / st

IK t K

increases rapidly and it reaches its maximum value ( ) / 3.82st

IK t K at 9.9t s . Then it

decreases to a local minimum and thereafter it increases again. A second peak is obtained at

18.3t s .

0 4 8 12 16 20

t [ s]

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0K~

(t)/

KIst

WÜNSCHE [5]

TDBEM

Fig. 3: An inner interface crack Fig. 4: Normalized amplitude of the complex

dynamic stress intensity factor

0 5 10 15 20 25

t [ s]

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

K~(t

)/K

Ist

a = 3 mm

a = 4 mm

a = 4 mm

a = 6 mm

a = 7 mm

a = 8 mm

0 5 10 15 20 25 30 35

t [ s]

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

K~(t

)/K

Ist

a = 9 mm

a = 10 mm

a = 11 mm

a = 12 mm

a = 13 mm

a = 14 mm

Fig. 5: Effects of the crack-length on the normalized amplitude of the complex dynamic stress

intensity factor

a2h2

w2

1

2


By using the same temporal and spatial discretizations, the effects of the crack-length on the

normalized amplitude of the complex dynamic stress intensity factor are investigated. The

corresponding numerical results are presented in Fig. 5. Figure 5 reveals that the crack-length has

significant influences on ( ) / st

IK t K . Both the maximum value and the corresponding time instant

are dependent on the crack-length. For small crack-length the first peak of ( ) / st

IK t K is also its

maximum value, while for large crack-length the second peak becomes its maximum.

In the second example, we consider an edge interface crack of length a in a rectangular plate

consisting of two different anisotropic and linear elastic materials as depicted in Fig. 6. The

geometry of the cracked plate is defined by 2w = 20,0mm, 2h = 40,0mm and a = 4,8mm. The plate

is subjected to an impact tensile loading of the form 0 ( )H t . The outer boundary of the plate

is discretized by 80 constant elements with 20 elements for each side, and the crack is discretized

by 15 constant elements. A time-step 0.4t s is chosen. Plane strain condition is assumed in

the numerical calculations. The same material combination as in the first example is investigated.

Figure 7 shows the normalized amplitude of the complex dynamic stress intensity factor versus

the time. Here again, a comparison of the present numerical results with that of Wünsche [5] shows

again a good agreement. The ( ) / st

IK t K -curve shows a more smooth increase with time after the

wave arrival time (1)/ 6.25Lt h c s than in the first example for a central interface crack. The

maximum value of ( ) / st

IK t K is attained at about 13.6t s and a second peak is observed at

about 19.2t s .

0 4 8 12 16 20

t [ s]

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

K~(t

)/K

Ist

WÜNSCHE [5]

TDBEM

Fig. 6: An inner interface crack Fig. 7: Normalized amplitude of the complex

dynamic stress intensity factor

Finally, the effects of the length of the edge interface crack on the normalized amplitude of the

complex dynamic stress intensity factor are investigated by using the same temporal and spatial

discretizations. Figure 8 presents the corresponding numerical results for ( ) / st

IK t K . Similar to the

first example for a central interface crack, the crack-length may affect the behaviour of the

( ) / st

IK t K -curve considerably. The numerical results given in Fig. 8 confirm again that both the

maximum value and the associated time depend strongly on the crack-length. From Fig. 8 it can be

concluded that in comparison to the ( ) / st

IK t K -curve for a central interface crack as shown in Fig.

ah2

w2

1

2t

(t)

0


5, the variations of the ( ) / st

IK t K -curve with time are even more complex in the case of an edge

interface crack.

0 5 10 15 20 25 30 35 40 45 50

t [ s]

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

K~(t

)/K

Ist

a = 3,0mm

a = 4,0mm

a = 5,0mm

0 5 10 15 20 25 30 35 40 45 50

t [ s]

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

K~(t

)/K

Ist

a = 6,0mm

a = 7,0mm

a = 8,0mm

Fig. 8: Effects of the crack-length on the normalized amplitude of the complex dynamic stress

intensity factor

Summary

This paper presents a time-domain BEM for transient elastodynamic analysis of an interface crack

in 2-D, layered, piecewise homogeneous, anisotropic and linear elastic solids. Time-domain

displacement BIEs in conjunction with a multi-domain technique are applied in the present time-

domain BEM. For both the temporal and the spatial discretizations, a collocation method is

adopted. Constant spatial shape functions and linear temporal shape functions are used for

simplicity, which allow us to perform the temporal and the spatial integrations analytically. Only

the line-integrals appearing in the elastodynamic fundamental solutions have to be computed

numerically. An explicit time-stepping scheme is obtained for computing the unknown boundary

data numerically. An efficient least-squares technique is applied for accurately compute the

complex dynamic stress intensity factor from the CODs. Numerical results for the complex

dynamic stress intensity factor are presented and compared with available reference solutions.

Acknowledgement

This work is supported by the German Research Foundation (DFG) under the project numbers ZH

15/5-1 and ZH 15/5-2, which is gratefully acknowledged.

References

[1] C.-Y. Wang and J.D. Achenbach: Geophys. J. Int. Vol. 118 (1994), p. 384.

[2] S. Beyer, Ch. Zhang, S. Hirose, J. Sladek and V. Sladek: Structural Durability & Health

Monitoring Vol. 3 (2007), p. 177.

[3] S. Beyer: PhD Thesis (in German), TU Bergakademie Freiberg, Germany, 2008.

[4] S.B. Cho, K.R. Lee, Y.S. Choy and R. Yuuki: Engineering Fracture Mechanics Vol. 43 (1992),

p. 603.

[5] M. Wünsche: PhD Thesis (in German), TU Bergakademie Freiberg, Germany, 2008.


Modelling of topographic irregularities for seismic site response

F. J. Cara1, B. Benito2, I. Del Rey1, E. Alarcón1.

1ETSI Industriales Universidad Politécnica de Madrid

José Gutierrez Abascal, 2 28006, Madrid, Spain [email protected]

2 ETSI Topografía, Geodesia y Cartografía Universidad Politécnica de Madrid

Carretera de Valencia, km 7 28031, Madrid, Spain

Keywords: BEM, wave propagation, layered media, irregular interfaces, wave scattering, site effects

Abstract. Seismic evaluation methodology is applied to an existing viaduct in the south of Spain, near

Granada, which is a medium seismicity region. The influence of both geology and topography in the spatial

variability of ground motion are studied as well as seismic hazard analysis and ground motion

characterization. Artificial hazard-consistent ground motion records are synthesised applying seismic hazard

analysis and site effects are estimated through a diffraction study. Direct BEM is used to calculate the valley

displacement response to vertically propagating SV waves and transfer functions are generated allowing the

transformation of free field motion to motion at each support. A closed formulae is used to estimate these

transfer function. Finally, the results obtained are compared.

Introduction

Performance Based Engineering is the reassurement of the classical line which tries to use the most advanced

and comprehensive procedures to give confidence to the designer and quantify structural damages in terms

that both owner and society can understand the involved risks. Among others, this new paradigm includes

hazard analysis and site effects for important bridges.

The hazard analysis will produce two main results. The first is the characterization of the hazard itself. The

second one is the identification of most-likeable scenarios, allowing selection or generation of records

compatible with them. Seismic hazard analysis is beyond the scope of this paper, although a brief summary

is presented below.

Then site effects will be investigated, assessing the need of considering multiple support excitations. Direct

Boundary Element Method (DBEM) will be used to compute transfer functions allowing the transformation

of free field motion to motion at the foundation of each pier.

The influence of topographic details in earthquake accelerations has been recognized for a long time. As the

analytical solution is limited to very simple geometric types, all research on realistic cases is based on

numerical techniques that can be the so-called Indirect Boundary Element method (IBEM) [1], a pure Direct

Boundary Element Method (DBEM) [2,3], or a mixture of Finite and Boundary elements [4].

The boundary element method is especially well suited for the analysis of the seismic response of valleys of

complicated topography and stratigraphy. Infinite regions are naturally represented, and the radiation of

waves towards infinity is automatically included in the model which is based on an integral representation

valid for internal and external regions.

The main focus of this work is related to the evaluation of the different accelerations at the pier foundations

of bridges, and generating semi analytically recommendations that can be applied to the general response

spectra.


Description of the viaduct and the valley

The bridge is part of the Spanish Highway network, which means that the Spanish code considers it as an

important infrastructure with a design life of 100 years.

The bridge has two-decks, supported by u-shaped piers in their tops. The total length is 305 m distributed in

six spans: 45,50 m + 4x53,50 m + 45,50 m. The pier heights are (from left to right) 27, 64 m, 74, 79 m and

50 m.

The valley geometry can be seen in Figure 1. The left hillside has a slope of approximately 30º and the right

one is about 40º. The central part is almost horizontal around the river bed.

Figure 1. Valley geometry.

The geology is shown in Figure 2. There is an erosive contact between the Pliocene and the Tertiary rocks

that produces a shallow layer of conglomerates on a dome of schists. Inside this one a dolomitic inclusion

has been detected near pier number 4. Both abutments are founded on conglomerates.

Figure 2. Valley geology.

S waves velocity properties, the density and the shear modulus G of the different materials are shown in

Table 1.

Cs (m/s) ( kg/m3 ) G ( GPa)

Conglomerates 1000 2200 2,20

Schists 1800 2400 7,76

Dolomites 2400 2700 15,55

Table 1. Material properties.


Seismic hazard analysis

Seismic hazard assessment at the bridge site is computed following the standard zonified probabilistic

approach, where earthquake occurrence is modelled as a poissonian process and earthquake recurrence is

characterized by a doubly-truncated Gutenberg-Richter relation [5].

There is a relatively poor knowledge on potential fault sources in the study area. However, geomorphologic

and neotectonic analyses provide rough estimates on present-day activity and maximum possible magnitudes

of some faults. According to these studies, it is possible to relate a magnitude Mw 5.5-6.5, epicentral

distance Repi 30-40km event with the Ventas de Zafarraya and Pinos-Puente principal faults.

That scenario was used to performance simulations of hazard-consistent acceleration-time histories.

Site modelling

Structural analysis of the bridge showed that one of the limit conditions was related to the longitudinal bridge

behaviour. So it was decided to conduct a bidimensional study in the plane containing the bridge vertical and

longitudinal directions, and to study for the valley response to the incidence of vertically propagating SV

waves.

The objective is to obtain, in the frequency domain, the transfer functions between the displacements at

every pier foundations and a reference point in a fictitious outcrop of schists far from the site that will be

supposed to be subjected to the seismic hazard defined in the previous point.

The use of a DBEM is especially appropriate for this kind of analysis. In this case 4 subregions have been

discretized. The boundary element mesh has been interrupted 2000 m from the centre of the valley where the

behaviour is similar to a monodimensional column of stratified soil.

The total number of elements is 640, with parabolic interpolation of displacements and tractions. At the

surface interface, the element size is equal to 15 meters and, at internal interfaces, the element size has been

chosen taking into account the maximum shear wave velocity of the two materials separated by the interface.

Figure 3. Time domain displacement of the valley to incident vertical SV waves. The incident time signal is

a Ricker wavelet. The stations are located along the free surface of the half-space, at a horizontal

dimensionless horizontal coordinate x=a.


Synthetic seismograms were computed using the FFT algorithm for a Ricker wavelet. Time series were

obtained from the transfer functions estimated at receivers placed along the free surface. Figure 3 shows a

sample of the synthetic seismograms, computed at the surface of the model defined in Figure 2, for the

vertical incidence of SV waves (Ricker wavelet’s central frequency is fp=1.0cs/2a = 2.90 Hz).

The response spectra of the horizontal component are plotted in Figure 4b at every support of the bridge. The

response spectra have been calculated from a simulated acceleration-time history and are presented in terms

of pseudo-acceleration as a function of time. Figure 4b shows the comparison between the spectra generated

at every support combining the free field motion and the transfer functions from Figure 4a and the spectra

computed directly from the free field acceleration time history. In all cases, the differences are noticeable,

and at some periods are as high as the 50%. These results show that it is mandatory to conduct a diffraction

study.

(a) (b)

Figure 4. (a) Horizontal displacement transfer functions obtained for BEM model. (b) Elastic response

spectra generated at every support taking into account BEM transfer functions and free field response

spectrum.

Closed form solution

Bridge designers not always have the time or the specific techniques to develop complex diffraction

problems. In addition, the uncertainties involved in the quantification of seismic action suggest the

possibility of using the closed form solution associated to the transfer function of a layered media to estimate

the relative displacement between piers. In addition, main interest of designers is centred around response

spectra method, so that this section tries to compare the relative differences that can be found when the

approach is applied to this complicated layer media.

Consider a soil deposit consisting of 2 horizontal layers resting on a semi-infinite media. Assuming that an

incident harmonic SV vertical wave is propagating in the semi-infinite media, satisfaction of the

requirements of equilibrium and compatibility at each interface give rise to a system of simultaneous

equations, which allows the amplitudes for the reflected and refracted waves to be expressed in terms of the

amplitude of the incident SV wave, so that, one can define a transfer function

1 2 31 1 s 1 2 2 s 2 3 s 3H f ( H , ,c , ,H , ,c , , ,c , , ) relating the horizontal displacement amplitude at the free

boundary to the incident waves (where H is layer height, is material density, cs is SV-wave propagation

velocity and is the material damping ratio, for superficial layer (1), intermediate layer (2) and semi-infinite

media (3); is the angular frequency of the incoming waves). Applying Haskell propagator methodology

[6]:


1 1 1 1 2 2 2 2ik H ik H ik H ik H1 1 2 1 2

2 2H ·

1 e 1 e 1 e 1 e (1)

1 1 2 21 2

2 2 3 3

G k G k,

G k G k (2)

1 1 1 1

1 1 1 1

ik H ik H1 1

1 ik H ik H1 1

1 e 1 e

1 e 1 e (3)

n

n 1

2s n

k

c 1 i2

(4)

Figure 5 shows the transfer functions obtained with the closed form solution, where , cs are the same from

Table 1, is equal to 0.05 in all cases, H2 is equal to 200 m and using for H1 the depth of the stratum under

each pier.

(a) (b)

Figure 5. (a) Horizontal displacement transfer functions obtained for closed formulae. (b) Elastic response

spectra generated at every support taking into account BEM transfer functions, closed formulae transfer

functions and free field response spectrum.

Figure 5b shows the comparisons between the spectra generated through those transfer function at every

support and those from Figure 4b. For all practical purposes, these response spectra are similar to those

obtained with BEM, and therefore, for practical design the approach seems valid.

Summary

Principal seismic codes define the seismic actions by means of elastic response spectrum. Therefore, bridge

designers use elastic response spectrum in bridge projects and, at least, for the general proportion of

structural members, they are mainly interested for simle procedures.

This paper presents how BEM can be used to obtain acceleration-time histories at every bridge pier

foundations taking into account site effects, valleys of irregular geometry and complicated strata with

irregular interfaces. Elastic response spectra obtained with these transfer functions have been compared with

free field elastic response spectrum. The results show that free file spectrum introduces significant errors and

justified the consideration of site effects. Finally, it has been shown that close form solutions produce

acceptable results from the engineering practical view point, especially if, as usual, before been applied those

spectra are smoothed.


Acknowledgement

This work has been produced as a part of the research founded by Spanish Ministero de Fomento, within the

National Plan of Scientific Research, Development and Technological Innovation 2004-2007, with number

of project 80007/A04.

The authors gratefully acknowledge the kindness of Prof. R. Gallego who provided the boundary element

code LiPoSo [5].

References

[1] Sánchez-Sesma FJ, Campillo M. Diffraction of P, SV, and Rayleigh waves by topographic features:

a boundary integral formulation. Bull Seismol Soc Am 1991; 81: 2234–53.

[2] Dominguez, J. Boundary elements in dynamics. ElSevier & CMP 1993.

[3] Alvarez-Rubio S., Benito J.J., Sanchez-Sesma F.J. and Alarcon E. The use of direct boundary

element method for gaining insight into complex seismic site response. Computers & Structures 83; 821-835.

2005.

[4] Faccioli E, Paolucci R, Vanini M. TRISEE 3D site effects and soilfoundation interaction in

earthquake and vibration risk evaluation. In: European Commission, Directorate General XII for Science,

Research and Development. 1994–1998 Environments and Climate Programme-Climate and Natural

Hazards Unit

[5] Gaspar-Escribano J.M., Benito, B. Ground-Motion Characterization of Low-to-Moderate Seismicity

Zones and Implications for Seismic Design: Lessons from Recent Mw 4.8 Damaging Earthquakes in

Southeast Spain. Bulletin of the Seismological Society of America, Apr 2007; 97: 531 - 544.

[6] Kramer, S.L. Geotechnical Earthquake Engeneering. Prentice Hall. New Jersey. 1996.

[5] Gallego R. BEM program LiPoSo. Internal report. ETSI Caminos, Canales y Puertos. Universidad de

Granada.


A fast 3D BEM for anisotropic elasticity based on hierarchical matrices

I. Benedetti1,a, A. Milazzo1,b, M.H. Aliabadi2,c

1 Dipartimento di Tecnologie ed Infrastrutture Aeronautiche, Viale delle Scienze, Edificio 8, 90128 Palermo - Italy

2 Department of Aeronautics, Imperial College London, South Kensington Campus, Roderic Hill Building, Exhibition Road, SW72AZ, London, UK


Keywords: Fast BEM solvers, hierarchical matrices, anisotropic elasticity.

Abstract. In this paper a fast solver for three-dimensional anisotropic elasticity BEM problems is developed.

The technique is based on the use of hierarchical matrices for the representation of the collocation matrix and

uses a preconditioned GMRES for the solution of the algebraic system of equations. The preconditioner is

built exploiting the hierarchical arithmetic and taking full advantage of the hierarchical format. The

application of hierarchical matrices to the BEM solution of anisotropic elasticity problems has been

numerically demonstrated highlighting both accuracy and efficiency leading to almost linear computational

complexity.

Introduction.

The use of composite materials in many engineering applications enables improved design for structures,

equipment and devices. The performance of such inherently anisotropic materials must be carefully

evaluated to meet increasing requirements in critical engineering applications. Much effort has been devoted

to experimental studies for composite materials characterization. On the other hand numerical modeling and

analysis have recently gathered significant momentum.

Computational methods such as the finite difference method (FDM), the finite element method (FEM)

and the boundary element method (BEM) have been widely exploited to carry out numerical analyses of

structural problems involving both isotropic and anisotropic materials. The boundary element method is

particularly appealing for many structural applications, although its extensive industrial usage, especially

when large scale computations are involved, is hindered by some limitations, mainly related to the features of

the solution matrix. Such matrix is generally fully populated, thus resulting in increased memory storage

requirements as well as increased solution time with respect to other numerical methods for problems of the

same order.

Moreover, the analysis of three-dimensional anisotropic elastic solids in the framework of the BEM

requires some additional considerations. The lack of anisotropic Green’s functions for the construction of the

boundary integral representation [1] results in the use of either the integral expression of the fundamental

solutions [2-5] or explicit expressions with complex calculations [6-14]. Due to the form of the 3D

fundamental solutions, BEM techniques for anisotropic elasticity applications resulted in slower

computations with respect to the isotropic case, for which analytical closed form fundamental solutions are

known.

Many investigations have been carried out to overcome such limitations. In particular, fast multipole

methods (FMMs) have been developed to solve efficiently boundary element formulations for elasticity

problems [15]. Although FMMs are very effective, they require the knowledge of the kernel expansion in

advance in order to carry out the integration and this is particularly complex for anisotropic elasticity

problems, for which analytic closed form expressions of the kernels are not available.

In the present paper the Fast BEM based on hierarchical matrices and their algebra proposed in reference

[17] for three-dimensional isotropic elasticity is extended to anisotropic applications. The main step is the

construction of the approximation of suitable blocks of the boundary element matrix based on the

computation of only few entries of the original blocks. This approximation, in conjunction with the use of

Krylov subspace iterative solvers, leads to relevant numerical advantages, namely reduced memory storage

requirements and reduced computational time for the solution. The effectiveness of the technique for the

analysis of anisotropic solids is numerical demonstrated in the reported applications.


The boundary element model

The boundary integral equation governing the behavior of a three-dimensional anisotropic body with

boundary , in absence of applied body forces, is given by

0 0 0 0, ,ij j ij j ij jc u T u d U t dx x x x x x x x (1)

where ju and jt are the boundary displacement and tractions. ijU and ijT are the anisotropic fundamental

solution kernels, whose expression is given in references [10] where different approaches for their

computation are discussed. After standard BEM discretization [1] eq. (1) leads to a linear system of the form

Hu G t (2)

where u and t are the vectors collecting the components of the nodal values of displacement and boundary

tractions, respectively. The solution of system (2), after forcing the boundary conditions in terms of

prescribed nodal values, provides the values of the unknown displacement and tractions on the body

boundary. The system of algebraic equations presents a coefficient matrix which is fully populated and

neither symmetric nor definite. This results in increased memory requirements as well as increased assembly

and solution time with respect to other numerical methods for problems of numerical comparable size.

Moreover, in the case of anisotropic elasticity, all the matrix entries need to be computed integrating kernels

with complex expression, due to the lack of closed form Green’s functions for three-dimensional anisotropy.

The use of hierarchical matrices for the approximation and solution of BEM systems of equations arising in

isotropic elasticity applications has been proposed in [16,17]. Such technique seems very appealing for

anisotropic BEM problems where the computation and integration of the integral equation kernels is very

involved. In the following the basic principles and the practical steps needed to generate the BEM matrix

approximation using hierarchical matrices are summarized. The reader is referred to the references [16,17]

for more details.

The construction of the fast BEM solver is based on a hierarchical representation of the collocation

matrix. Such representation is built by representing the matrix as a collection of sub blocks, some of which

admit a special approximated and compressed format. Such blocks, referred to as low rank blocks, can be

approximated by computing only some of the entries of the original blocks (18) through adaptive algorithms

known as Adaptive Cross Approximation (ACA) (19,20). Low rank blocks represent the numerical

interaction, through asymptotic smooth kernels, between sets of collocation points and clusters of integration

elements which are sufficiently far apart from each other. The distance between clusters of elements enters a

certain admissibility condition, based on some selected geometrical criterion, for the existence of a low rank

approximant. The blocks that do not satisfy such condition are called full rank blocks and they need to be

computed and stored entirely. The low rank representation of the collocation matrix allows to reduce

memory storage requirements as well as to speed up operations involving the matrix.

The process leading to the subdivision of the matrix into low and full rank blocks is based on geometrical

considerations on the boundary mesh of the analyzed body, as schematically illustrated in fig. (1). Each

block in the collocation matrix is related to two sets of boundary elements, the one containing the collocation

points corresponding to the matrix row indices and the one grouping the elements over which the integration

is carried out, that contains the nodes corresponding to the matrix columns. If these two sets of boundary

elements are separated, then the block will be represented and stored in low rank format and it is called

admissible, while it will be entirely generated and stored in full rank format otherwise. The admissibility of a

candidate block is based on the inequality

, ( , ) xo x xo xmin diam diam dist (3)

where 0 is a parameter influencing the number of admissible blocks on one hand and the convergence

speed of the adaptive approximation of low rank blocks on the other hand [21].

The matrix block-wise subdivision and classification is based on a previous hierarchical partition of the

matrix index set aimed at grouping subsets of indices corresponding to contiguous nodes and elements on the

basis of some computationally efficient geometrical criterion. The partition is stored in a binary tree of index

subsets, or cluster tree, that constitutes the basis for the following construction of the hierarchical block

subdivision that will be stored in a quaternary block tree. A possible algorithm, leading to geometrically

balanced trees, is given in reference [22] and considers for simplicity boxes framing the considered clusters.


Figure 1. Schematic of the boundary subdivision.

As the admissible blocks have been located, their approximation is generated through ACA algorithms

which allow to reach adaptively the a priori selected accuracy c . Additionally, to optimize memory storage

requirements and reduce the overall computational complexity, the low rank blocks are recompressed

without accuracy penalties, taking advantage of the reduced Singular Value Decomposition (SVD) [23].

Moreover, since the initial matrix partition is generally not optimal, once the blocks have been generated and

recompressed, the entire structure of the hierarchical block tree can be modified through a coarsening

procedure, which reduces the storage requirements and speeds up the solution maintaining the preset

accuracy [24].

As an almost optimal representation is obtained, the solution of the system can be tackled either directly,

through hierarchical matrix inversion [25], or indirectly, through iterative methods [26]. In both cases, the

efficiency of the solution relies on the use of a special arithmetic, i.e. a set of algorithms that implement the

operations on matrices represented in hierarchical format, such as addition, matrix-vector multiplication,

matrix-matrix multiplication, inversion and hierarchical LU decomposition. A collection of algorithms that

implement many of such operations is given in [21] while the hierarchical LU decomposition is discussed in

[26].

The use of iterative methods takes full advantages of the hierarchical representation exploiting the

efficiency of the low-rank matrix-vector multiplication. The convergence of iterative solvers can be

improved, or sometimes obtained from a non convergent scheme, by using suitable preconditioners. In the

present approach an LU preconditioner matrix is built in hierarchical format starting from a coarse

approximation with accuracy p of the collocation matrix [26]. An iterative GMRES algorithm is finally

used in conjunction with such precoditioner for solving the system.

Numerical experiments and discussion

The hierarchical computational scheme described in [17] has been modified to perform analyses on

anisotropic bodies. In particular the anisotropic fundamental solutions are computed by using the technique

proposed by Wilson and Cruse in reference [5]. In this technique the dependence of the fundamental solution

kernels on the source point to field point direction is obtained by interpolation from a database containing the

so called modulation functions for the fundamental solution displacement and displacement derivatives. The

database is actually constituted by tables containing the modulation functions for different source point to

field point directions, described in terms of two angles 1 and 2 in spherical coordinates.

All the computations have been performed using an Intel® CoreTM Duo processor T9300 (2.5 GHz) with 2

GB of RAM. In order to compare the obtained results with those obtained for isotropic bodies, the same

configuration proposed in reference [17] with anisotropic material properties has been analyzed. The

analyzed mechanical element loaded anti-symmetrically at the holes and clamped at the center cylinders is

shown in fig. 2 with the features of the considered meshes.


Nr of Nodes Nr of Elements

Mesh A 412 138

Mesh B 1150 384

Mesh C 3094 1032

Figure 2. Geometry and mesh data.

A first set of analyses has been carried out on the mesh C to investigate the effect of the preconditioner

accuracy p on the convergence of the hierarchical GMRES iterative solution. For this purpose, the accuracy

of the collocation matrix has been set to 510c , the admissibility parameter has been chosen as 2 ,

the minimal block size has been set to 36minn [17] and the GMRES relative accuracy has been set to 810 .

The results obtained are shown in Table 1 where the preconditioner percentage of storage, the solution times

and the solution speed up ratios with respect to standard anisotropic BEM are given.

A second set of analyses has been performed to study the influence of the admissibility parameter on

the solution. The results obtained are presented in Table 2 in terms of percentage storage before and after

coarsening, speed up ratios and accuracy with respect to the standard BEM solution. The storage memory

requested by the fast hierarchical BEM is independent from the admissibility parameters. The same is not

true for the solution time and consequently for the speed up of the solution which are affected by the chosen

value of . Therefore, as expected, there is an optimum value of which set the best block partition of the

matrix and minimize the time requested for the solution. For a detailed discussion abut the influence of

see reference [17].

Table 3 reports memory requirements before and after coarsening, assembly time and speed up ratio,

solution time and speed up ratio and the accuracy of the final solution at different values of the collocation

matrix requested accuracy with the other parameter set at the values shown in the table. The memory

requirements, assembly times and solution times decrease when the preset accuracy decreases, as the average

rank of the approximation is reduced.

Table 1. Effect of the preconditioner matrix accuracy (mesh C, 510c , 2 , 36minn )

pPreconditioner

Storage %

Preconditioner

Time (s)

Precond.

LU time

GMRES

time

Total Solution

time (s)

Solution

Speed up

Total

Speed up 610 33.60 616.91 616.23 1.90 750.88 0.61 0.63 510 33.60 444.13 443.45 1.62 578.58 0.47 0.51 410 25.10 336.83 278.62 2.18 476.28 0.37 0.42310 17.26 186.2 128.04 2.96 326.6 0.25 0.33

210 10.49 90.3 57.57 5.46 230.6 0.19 0.28110 4.51 39.67 19.34 369.37 542.69 0.43 0.47

Table 2. Storage, times and speed up ratios for different (mesh C, 510c , 210p , 36minn )

Storage %

before coars.

Storage %

after coars.

Assembly

time (s)

Solution

time (s)

Assembly

Speed up

Solution

Speed up

Total

Speed up

L2

norm

0.5 73.37 33.55 183.68 313.86 0.67 0.23 0.31 4101.2

2 48.27 33.60 191.92 230.65 0.70 0.19 0.28 4103.4

2 44.23 33.67 193.15 210.50 0.71 0.16 0.26 4105.8

4 36.68 33.31 215.09 163.49 0.78 0.13 0.25 4108.4

6 36.14 32.88 223.37 173.75 0.81 0.13 0.26 4100.7

Table 3. Effect of hierarchical matrix accuracy (mesh C, 4 , 210p , 36minn )

cStorage %

before coars.

Storage %

after coars.

Assembly

time (s)

Solution

time (s)

Assembly

Speed up

Solution

Speed up

Total

Speed up

L2

norm210 26.54 10.45 142.74 104.21 0.52 0.08 0.16

1107.5


310 29.27 17.22 158.63 136.86 0.58 0.10 0.18 1103.1

410 32.67 24.94 178.48 156.47 0.65 0.13 0.23 3106.9

510 36.68 33.31 215.09 163.49 0.78 0.13 0.25 4108.4

610 45.52 45.04 955.80 451.60 3.47 0.36 0.93 5104.3

Table 4. Effect of fundamental solution description accuracy (mesh C, 510c , 4 , 210p , 36minn )

Storage %

before coars.

Storage %

after

coars.

Assembl

y time (s)

Solution

time (s)

Assembl

y

Speed up

Solution

Speed up

Total

Speed up

L2

Norm

0.5 42.93 41.16 526.27 254.00 1.96 0.21 0.52 3101.1

0.1 37.93 34.81 368.75 201.77 1.36 0.16 0.37 4106.4

5.0 36.68 33.31 215.09 163.49 0.78 0.13 0.25 4108.4

Regarding the collocation matrix hierarchical approximation, an interesting issue which need to be

investigated is the effect of the degree of accuracy in the interpolation of the modulation functions of the

fundamental solutions. In particular, with reference to the approach employed in the present paper, this effect

has been studied by performing analyses with three different accuracy of the tables describing the

modulation functions. Each tables is characterized by a different number of entries for the source point to

field point direction, obtained by setting the angular separation trough the different directions to a fixed

value 1 2 . The results are shown in Table 4, where it is evidenced that the degree of accuracy in

the description of the kernels affects the performance of the ACA block approximation. High accuracy in the

kernel description determines a lower average rank in the ACA approximation of admissible blocks, with

better resulting performances of the method.

Finally, the performances of the method are highlighted in Fig. 3, where the memory usage and the

solution time obtained by analyzing three different meshes are shown. The variation of memory usage and

solution time with respect to the degrees of freedom clearly shows that, also for anisotropic bodies, the

presented approach requires almost linear computational complexity. Its efficiency improves with the

problem dimension and it appears very appealing for large scale systems.

0 2000 4000 6000 8000 10000

DoF

0

400

800

1200

1600

So

luti

on

Tim

e [s

ec]

0 2000 4000 6000 8000 10000

DoF

0

0.2

0.4

0.6

0.8

Mem

ory

Usa

ge [

GB

] Standard BEM

Fast Hierarchical BEM

Standard BEM

Fast Hierarchical BEM

Figure 3. Memory usage and solution time with respect to problem DoF.

Conclusions

A Fast solver for 3D anisotropic elasticity BEM problems based on hierarchical matrices has been

developed. The performed tests demonstrated the applicability of the technique to the analysis of anisotropic

bodies. The performances of the method, namely relevant memory storage and solution time savings,

previously demonstrated for isotropic BEM systems, make the technique appealing also for anisotropic

problems. Almost linear computational complexity at increasing degrees of freedom has been evidenced.

Such features make the technique very appealing for large scale applications.


References

[1] M.H. Aliabadi, The Boundary Element Method: Applications in Solids and Structures, vol. 2. John

Wiley & Sons Ltd (2002).

[2] D.M. Barnett, Phys. Stat. Sol. (b), 49, 741-748 (1972).

[3] L.J. Gray, A. Griffith,L. Johnson, P.A. Wawrzynek, Electron. J. Bound. Elem , 1,. 68-94 (2003).

[4] S.M. Vogel, F.J. Rizzo, J. Elast., 3, 203-216 (1973).

[5] R.B. Wilson, T.A. Cruse, Int. J. Numer. Meth. Eng., 12, 1383-1397 (1978).

[6] T. Chen, F.Z. Lin, Comput. Mech. ,15,. 485-496 (1995).

[7] T.C.T. Ting, V.G. Lee, Q. J. Mech. Appl. Math., 50, 407-426 (1997).

[8] G. Nakamura, K. Tanuma, Q. J. Mech. Appl. Math., 50, 179-194, (1997).

[9] E. Pan, F.G. Yuan, Int. J. Numer. Meth. Eng., 48, 211-237 (2000).

[10] N.A. Schclar, Anisotropic analysis using boundary elements. Comp. Mech. Publ. (1994).

[11] M.A. Sales, L.J. Gray, Comput. Struct., 69, 247-254 (1998).

[12] F. Tonon, E. Pan, B. Amadei, Comput. Struct. ,79, 469-482 (2001).

[13] V.G. Lee, Mech. Res. Commun., 30, 241-249 (2003).

[14] C.Y. Wang, M. Denda, Int. J. Solids Structures., 44 7073-7091 (2007).

[15] V. Popov, H. Power, Eng. An. Bound. Elem., 25, 7–18 (2001).

[16] M. Bebendorf, R. Grzhibovkiskis, Math. Meth. Appl. Sciences, 29, 1721-1747 (2006).

[17] I. Benedetti, M.H. Aliabadi, G.Davì, Int. J. Solids Structures, 45, 2355-2376 (2008).

[18] E. E. Tyrtyshnikov, Calcolo, 33, 47-57, (1996).

[19] M. Bebendorf, Numerische Mathematik, 86, 565-589, (2000).

[20] M. Bebendorf, S. Rjasanow, Computing, 70, 1-24, (2003).

[21] S. Börm, L. Grasedyck and W. Hackbusch, Eng. An. Bound. Elem., 27, 405–422, (2003).

[22] K. Giebermann, Computing, 67, 183-207, (2001).

[23] M. Bebendorf, Effiziente numerische Lösung von Randintegralgleichungen unter Verwendung von

Niedrigrang-Matrizen, Ph.D. Thesis, Universität Saarbrücken, 2000. dissertation.de, Verlag im

Internet, ISBN 3-89825-183-7, (2001)

[24] L. Grasedyck, Computing, 74, 205-223, (2005).

[25] L. Grasedyck, W. Hackbush, Computing, 70, 295-334, (2003).

[26] M. Bebendorf, Computing, 74, 225-247, (2005).


A BEM approach in nonlinear acoustics

V. Mallardo1 and M. H. Aliabadi2

1 Department of Architecture, University of Ferrara, Italy, [email protected] at the Imperial College London as research associate

2 Department of Aeronautics, Imperial College London, UK, [email protected]

Keywords: Nonlinear acoustics, integral equations, dual reciprocity.

Abstract. The present paper deals with a novel application of the Boundary Element Method (BEM)to two-dimensional (2D) nonlinear acoustics. The acoustic waves are supposed to be of finite-amplitudeand the analysis is performed in the frequency domain. By applying the perturbation technique,the governing differential equations are transformed into a system of two Helmholtz equations, onehomogeneous and the other one inhomogeneous. The Dual Reciprocity Boundary Element Method(DRBEM) is used to transfer the domain integral to the boundary. The procedure is validated bycomparison with an ad-hoc analytical solution and tested for different basis functions.

Introduction

The interaction of an acoustic signal with matter is said to be linear if the response of the materialand the strength of the ouput signal vary linearly with the strength of the input signal. For instance,in one dimension (1D) the equation of motion reduces to the linear wave equation only if the acousticMach number M is negligible in comparison with unity. Furthermore, any wave will become distorted,no matter how small M is, if it can propagate a sufficient distance (see [1]).

For high input signal strengths, or for materials with some special properties, some nonlineareffects may occur. Some of these effects are increasingly used for nondestructive characterization ofmaterials and damage detection in industrial products. A review of the nonlinear acoustic applicationsfor material characterization can be found in [2]. The nonlinear phenomena which are linked to finite-amplitude waves, i.e. when the acoustic Mach number is not negligible in comparison with the unity,are involved in the mechanisms which determine a great number of practical applications. For instancein solids, plastic and metal welding, machining and cutting, material forming. In fluids, particlefiltration, defoaming, drying.

The BEM is a numerical approach for solving field problems based on the boundary integralequation (BIE) formulations. The BEM has been used to solve exterior and interior linear acousticproblems for many years (see for instance [3]) because of its boundary only discretisation and automat-ically satisfaction of the radiation condition at infinity. A review on the applications in elastostatics,thermoelasticity, elastoplasticity, contact and fracture mechanics can be found in [4].

The interest of extending such a tool to realistic modeling of the nonlinear acoustic field generatedby finite-amplitude acoustic waves is evident. This is the main purpose of the present work which islimited to the 2D analysis.

The governing differential equations are threaten by a perturbative approach, then transformedinto homogeneous/inhomogeneous integral equations and finally numerically solved by coupling theconventional BEM with the DRBEM. Some numerical examples are presented in order to demonstratethe efficiency of the proposed procedure.

The governing equations

The equations which govern the general motion of an unviscous fluid are mass conservation, momentumconservation and thermodynamic state (see for instance [1]):

ρ0 − ρ

ρ= ∇ · u (1a)

1


ρ0∂2u∂t2

+ ∇P = 0 (1b)

P = A

(ρ

ρ0

)γ

− Q (1c)

where ρ and ρ0 are the actual and the ”no-perturbation” mass density, respectively, u is the fluiddisplacement vector, P is the thermodynamic pressure, γ = cp/cv is the ratio of the specific heats atconstant pressure (cp) and constant volume (cv) and Q is a constant to be determined from experi-mental data.

Dissipation mechanisms are neglected. By Taylor expanding the density of the fluid ρ up to thesecond order term of the acoustic disturbance p = P − P0, the following governing wave equation canbe obtained:

∇2P − P,tt

c20

= − β

ρ0c40

(p2),tt (2)

where comma derivative notation is adopted, c0 is the sound speed in linear acoustics and β = (1+γ)/2is referred to as the coefficient of nonlinearity. For instance, the water at 20C has β = 3.5.

By adopting the following approximation:

p = pl + pc (3)

where bar indicates the dependance on the time variable, pl represents the first order approximation ofthe acoustic disturbance and pc furnishes its second order correction, and by assuming time-harmonicwaves, i.e. pl = ple

iωt and pc = pce2iωt, the differential Eq.(2) furnishes the following nonlinear system

of differential equations (see [6] for details):

∇2pl + k2l pl = 0 (4a)

∇2pc + k2cpc =

4ω2β

ρ0c40

p2l (4b)

where kl = ω/c0 and kc = 2kl. Third or higher order terms are neglected. This means that finite butof moderate amplitude waves are considered.

In conclusion, the total acoustic pressure is obtained by solving the above system of differentialequations in terms of pl and pc, i.e.:

p(x, t) = pleiωt + pce

2iωt (5)

The system of Eqs.(4) can be solved analytically only in very simple cases. For complex geometriesnumerical techniques must be involved.

The numerical implementation

Finite element methods (FEM) for time-harmonic acoustics governed by the Helmholtz equation havebeen an active research area for nearly 40 years. The BEM has demonstrated to be efficient especiallyfor scattering problems: in no other field is the BEM used so intensively by industry. Furthermore, re-cent advances such as the fast multipole and the panel clustering methods have tremendously improvedits efficiency both for pulsating and for internal problems.

The nonlinear acoustic boundary integral equations (NABIEs) can be derived by applying theweighted residual technique together with the Green’s identities (see [3]) to the Eqs.(4) to give:

c(ξ)pl(ξ) + −∫

Γq∗(ξ,x)pl(x)dΓ(x) −

∫Γ

p∗(ξ,x)ql(x)dΓ(x) = 0 (6a)

c(ξ)pc(ξ) + −∫

Γq∗(ξ,x)pc(x)dΓ(x) −

∫Γ

p∗(ξ,x)qc(x)dΓ(x) =∫

Ωp∗(ξ,x)b(pl(x))dΩ(x) (6b)

2


where

b(pl(x)) =4ω2β

ρ0c40

p2l (x) = cβ p2

l (x) (7)

The integral on the left hand side is to be interpreted in the sense of Cauchy principal value and the freeterm c(ξ) is equal to 0.5 if the tangent line at ξ is continuous. The symbols ξ and x denote the sourceand the field points, respectively, Γ is the boundary of the domain Ω under analysis. The fundamentalsolutions p∗, q∗ are given in any BEM book (see for instance [5]). In 2D they are expressed in termsof the modified zero and first order Bessel functions of the second kind.

In the conventional BEM the boundary is divided into NE elements (quadratic in the presentpaper) and the Eq.(6a) is collocated in each boundary node to furnish a discrete system of equationsin terms of the acoustic either pressure or flux in the boundary nodes. The final system is solved byany numerical technique after applying the boundary conditions.

Such a procedure cannot be applied directly to Eq.(6b) if a boundary-only formulation is required.The domain integral can be transformed into the sum of boundary integrals by the DRBEM (see [7]).The keypoint is the approximation of the right hand side of Eq.(6b), i.e. b = b(pl(x)), by a finite seriesof basis functions for which a particular solution is available. It can be written as:

b(pl(x)) ≈N+L∑j=1

fjαj =N+L∑j=1

f(x,ηj)αj (8)

where fj is function of the distance between the field point x and the dual collocation point ηj . The αj

coefficients are unknown and they are determined by collocating Eq.(8) at N +L (N on the boundaryand L in the domain) arbitrary points.

Most applications concern elastostatics and elastodynamics as well as the Poisson equation. Sofar no applications have been proposed concerning the Helmholtz equation even if much effort hasbeen made to determine various particular solutions [8-9]. In the present paper various approximatingfunctions fj are compared, i.e. the well trodden 1+r along with the 1+r2, the thin plate spline (TPS)r2Logr and the augmented thin plate spline (ATPS) r2Logr+αN+L+1 +αN+L+2x1 +αN+L+3x2. Thecorresponding particular solutions can be found in [6], [8-9].

On the basis of the above considerations, the NABIEs governing the propagation of finite butmoderate amplitude acoustic waves can be written:

c(ξ)pl(ξ) + −∫


∫Γ

p∗(ξ,x)ql(x)dΓ(x) = 0 ξ ∈ ∂Ω (9a)

c(ξ)pl(ξ) + −∫


∫Γ

p∗(ξ,x)ql(x)dΓ(x) = 0 ξ ∈ Ω (9b)

c(ξ)pc(ξ) + −∫

Γq∗(ξ,x)pc(x)dΓ(x) −

∫Γ

p∗(ξ,x)qc(x)dΓ(x) =∫

Ωp∗(ξ,x)b(pl(x))dΩ(x) ξ ∈ ∂Ω (9c)

The Eq.(9b) is included in the system of equations when some (let’s say L) internal points areconsidered to better evaluate αj in the application of the Dual Reciprocity (DR) approach to Eq.(9c).

In the numerical procedure the modified Bessel functions needs to be computed at the integrationpoints. Their accurate computation is very important. Two different series expansions, as suggestedin [5], are adopted, i.e. one for small arguments and the other for large arguments.

The numerical scheme is a standard collocation one. The boundary is discretised into quadratic,isoparametric elements. The discrete system of equations is solved by the LU decomposition.

Numerical results

In order to demonstrate the efficiency of the proposed procedure, some numerical examples are ac-quainted. They all refer to the wave propagation inside the cylinder of radius R = 1 depicted inFig. 1(a) and with the following parameters: ρ0 = 100, c0 = 100, β = 3.5, all in compatible units.

3


First of all the numerical solution is compared to the analytical one. In such a simple geometry, infact, an analytical solution can be obtained in terms of a power series. The comparison is performedfor klR = 1 and klR = 10. p0 = 1000 is imposed on the whole boundary.

θA

B

C

D

(a) (b) (c)

Figure 1: (a) Geometry. Adopted internal points: (b) klR = 1, (c) klR = 10.

Fig. 2 convey the behavior of pl, see Fig. 2(a), and pc, see Fig. 2(b), in the case of klR = 1.The agreement is excellent. Twelve quadratic boundary elements and 25 internal points, as shown inFig. 1(b), are sufficient to obtain a relative error of less than 0.5%, but it must be underlined that 9internal points would furnish an error of less than 2%.

0 0.2 0.4 0.6 0.8 1

Distance from the center

1000

1100

1200

1300

1400

Pre

ssure

pl

Analytical solution

BEM solution

(a)

0 0.2 0.4 0.6 0.8 1


-18

-16

-14

-12

-10

-8

-6

-4

-2

0

Pre

ssure

pc

Analytical solution

BEM solution

(b)

Figure 2: Comparison between analytical and numerical results (a) pl and (b) pc. klR = 1

Fig. 3 draw the behavior of the pressure for higher frequency, i.e. in the case klR = 10. Afiner boundary discretisation and more internal points are necessary due to the lower value of thewavelength. In fact, 144 boundary elements and 841 internal points, as depicted in Fig. 1(c), arenecessary to converge to the analytical solution. The very good agreement can be noticed in this casetoo.

In order to acquaint the efficiency of the DR approach, the procedure is tested with reference totwo special expressions of the source term for which an analytical solution is easily obtained. For bothk = 1. The first equation is:

∇2p(x) + p(x) = x1 (10)

with solution:pA1 = sinx1 + sinx2 + x1 (11)

The second differential equation is:

∇2p(x) + p(x) = 4x21 + 4x2

2 + 12x1x2 + 3x31x2 + 2x2

1x22 − x1x

32 (12)

4


with solution:pA2(x) = 3x3

1x2 + 2x21x

22 − x1x

32 (13)

Seventy-two boundary elements are used in order to cancel any error source due to the boundarydiscretisation.

The value of the flux corresponding to both the analytical solutions at the boundary nodes illus-trated in Fig. 1(a) is reported in Table 1 and in Table 2.

0 0.2 0.4 0.6 0.8 1


-5000

-4000

-3000

-2000

-1000

0

1000

2000

Pre

ssure

pl

Analytical solution

BEM solution

(a)

0 0.2 0.4 0.6 0.8 1


-40

-20

0

20

40

60

80

Pre

ssure

pc

Analytical solution

BEM solution

(b)

Figure 3: Comparison between analytical and numerical results (a) pl and (b) pc. klR = 10

The last column of each table furnishes the highest value, among the four points A,..,D, of therelative error for each basis function. Concerning the solution A1, the error results to be less than 2%for all the basis functions, but it is reduced to 0.2% for the one which best fits, i.e. ATPS. The erroris slightly higher with reference to the analytical solution A2. The last column does not consider theerror at the node A where the exact value is zero.

A B C D err(%)Analytic 1.540 1.836 1.782 1.295

1 + r 1.560 1.855 1.798 1.304 1.31 + r2 1.542 1.838 1.784 1.297 0.2TPS 1.512 1.810 1.763 1.285 1.9

ATPS 1.541 1.838 1.784 1.297 0.2

Table 1: Flux at some boundary points for different basis function. First analytical solution. 0 internalpoints

A B C D err(%)Analytic 0. 4.41 4. 0.414

1 + r 4.9e-2 4.55 4.14 0.524 211 + r2 857. 1450. 2090. 1370. –TPS -1.99e-2 4.45 4.03 0.417 0.8

ATPS 1.82e-2 4.49 4.07 0.456 9.2

Table 2: Flux at some boundary points for different basis function. Second analytical solution. 25internal points

It must be underlined that 1 + r2 fails in furnishing an acceptable solution for the analyticalsolution A2. The reason is well reported in literature and it is related to the ill-conditioned feature of

5


the matrix involved in Eq.(8). Furthermore, the convergence criterion of the radial basis function r2n

in 2D is not supported by a mathematical proof and, hence, its use is not recommended.

Conclusions

A numerical method for studying the nonlinear 2D propagation of high-intensity acoustic waves hasbeen presented. A perturbation theory up to the second-order approximation has been first applied.The nonlinearity provokes a domain integral which has been transformed into boundary integrals bythe DRBEM. The efficiency of the procedure has been verified in a case for which an analytical solutioncan be obtained. Different basis functions have been tested for some special cases and the results havebeen compared and discussed.

Finite-amplitude waves are directly involved in the mechanisms that determine a great number ofpratical applications in industrial processing. Numerical models are clearly advantageous over otherapproaches as giving solutions for a large number of different cases and for any irregular geometry orspecial boundary conditions. Thus, the method above presented can be an excellent tool to determinethe acoustic field in the industrial processing system and to assist the experimental tests. The mainadvantage over the FE approach stands in the possibility not to discretise the domain under analysis,but to limit the discretisation to the boundary only. Furthermore, it furnishes very accurate resultsin terms of both pressures and fluxes.

References

[1] R. T. Beyer. Nonlinear acoustics. In Physical Acoustics Edited by W.P. Mason, Academic, NewYork, 1965.

[2] Y. Zheng, R. G. Maev, I. Y. Solodov. Nonlinear acoustic applications for material characterization:A review. Can. J. Phys., 77:927–967, 1999.

[3] L. C. Wrobel. The Boundary Element Method Volume 1: Applications in Thermo-Fluids andAcoustics. Wiley, Chichester, West Sussex, 2002.

[4] M. H. Aliabadi. The Boundary Element Method Volume 2: Applications in Solids and Structures.Wiley, Chichester, West Sussex, 2002.

[5] J. Dominguez. Boundary Elements in Dynamics. Computational Mechanics Publications, Southamp-ton, Boston, 1993.

[6] V. Mallardo, M. H. Aliabadi. The Dual Reciprocity Boundary Element Method (DRBEM) innonlinear acoustic wave propagation. Computer and Experimental Simulations in Engineeringand Science CESES, in print, 2008.

[7] P. W. Partridge, C. A. Brebbia, L. C. Wrobel. The Dual Reciprocity Boundary Element Method.Computational Mechanics Publications, Southampton, London & New York, 1992.

[8] S. Zhu. Particular solutions associated with the Helmholtz operators used in DRBEM. Bound.Elem. Abstracts, 4(6):231–233, 1993.

[9] A. H. D. Cheng. Particular solutions of Laplacian, Helmholtz-type, and polyharmonic opera-tors involving higher order radial basis functions. Engineering Analysis with Boundary Element,24:531–538, 2000.

6


TIME CONVOLUTED DYNAMIC KERNELS FOR 3D SATURATED POROELASTIC MEDIA WITH INCOMPRESSIBLE CONSTITUENTS

M. Jiryaei Sharahi1, M. Kamalian2 and M.K. Jafari3

1 International Institute of Earthquake Engineering and Seismology (IIEES), Tehran, Iran, [email protected]



Keywords: boundary element, dynamic poroelasticity, time convoluted dynamic kernels

Abstract. This paper presents the explicit and simple analytical time domain convoluted kernels that appear

in the discretized governing BIE of the three-dimensional well known u-p formulation of saturated porous

media with incompressibile fluid and solid particles. At first, the corresponding boundary integral equations

are obtained for the governing differential equations which are established in terms of solid displacements

and fluid pressure. Subsequently, the analytical time domain convoluted kernels that appear in the BIE are

derived. Finally, a set of numerical results are presented which demonstrate the accuracies and some salient

features of the proposed solutions.

INTRODUCTION

The dynamic analysis of saturated porous media, is of interest in various fields, such as geophysics,

acoustics, soil dynamics and many earthquake engineering problems. From a macroscopical point of view,

saturated soil is a two-phase medium constituted of solid skeleton and fluid. Dynamic behaviours of each

phase as well as that of the whole mixture are governed by the basic principles of continuum mechanics. In

phenomena with medium speeds, such as earthquake problems, it is reasonable to neglect the fluid inertial

effects, and to reduce the complete dynamic governing differential equations to the simple commonly called

u-p formulation [1,2,3]. The governing differential equations could be further simplified by neglecting the

compressibility of the solid particles and fluid, which could be reasonably assumed incompressible compared

to the soil skeleton [4,5].

The BEM is one of the most efficient numerical mehods for solving wave propagation problems in elastic

media, because of its efficiency in dealing with semi-infinite or infinite domain problems that has long been

recognized. Predeleanu [6], Manolis & Beskos [7] and Wiebe & Antes [8] were among the firsts who

developed boundary integral equations and fundamental solutions governing the dynamics of poroelastic

media, in terms of solid skeleton displacement and fluid displacements components. Later, Cheng et al. [9],

Dominguez [10], Chen & Dargush [11] and recently Schanz [12] developed another forms of boundary

integral equations and fundamental solutions of dynamic poroelasticity in terms of less independent

variables. But their algorithms were based on transformed domain fundamental solutions.

Obviously, time domain BEM for modeling the transient behaviour of media is preferred than the

transformed domain BEM, because formulating the numerical procedure entirely in time domain and

combining it with the FEM, provides the basis for solving nonlinear wave propagation problems.

Proper displacement and traction fundamental solutions are one of the key ingredients required for solving

wave propagation problems in saturated porous media by the BEM.Considering the independent practical

variables of solid skeleton displacement and fluid pressure, Kaynia [13] was the first who presented


approximate transient 3D displacement fundamental solutions for the special case of short-time. Chen [14,

15]proposed another approximate transient 2D and 3D displacement solutions for the special case of short

time as well as the general case, which were too complicated to be applied in BE algorithms. Gatmiri &

Kamalian [16] showed that Chen's approximation could not be used in the simplified case of u-p

formulation. They derived another approximate transient 2D displacement fundamental solutions for the u-p

formulation which were still too complicated to be used in BE algorithms. Later Gatmiri & Nguyen [5]

proposed much less complicated transient 2D fundamental solutions for the u-p formulation of saturated

porous media consisted of incompressible constituents. Recently Kamalian et al[17,18] derived the transient

displacement and traction fundamental solutions for the simplified u-p formulation of 3D poroelastic media

with incompressible constituents.

Presentation of the analytical time domain convoluted dynamic kernels that appear in the discretized BIE for

the u-p formulation of 3D saturated porous media with incompressible constituents, constitutes the main

essence of this paper. Some numerical results are plotted to show the accuracies and some salient features of

the proposed solutions.

GOVERNING EQUATIONS

The governing equations of dynamic poroelasticity were first derived by Biot [19] using the concept of

variational formulation. These equations were later recast by using other theories such as the theory of

mixtures (Prevost [2]) and the principles of continuum mechanics (Zienkiewicz and Shiomi [1], Gatmiri [3],

etc.). Following the procedure outlined by Zienkiewicz and Shiomi [1], one can write the equations

describing, respectively, the conservation of total momentum, the constitutive equation of the solid skeleton,

the flow conservation for the fluid phase and the generalized Darcy’s law as follows:

Equilibrium equation:

ifiijij uf, (1)

Constitutive relation:

ijijjiiiij puuu )( ,,, (2)

Flow conservation for fluid phase:

Q

puw iiii ,, (3)

Generalized Darcy’s law:

iifii wmuwk

p1

, (4)

where

nm f , fs nn)1( , sf KnKnQ )(1 , sKK1

ui represents the displacement of the solid skeleton, p denotes the excessive fluid pore pressure and wi

represents the average displacements of the fluid relative to the solid. ij represents the total stress, the elastic

constants and µ denote the drained Lame constants and =k/ is the permeability coefficient, with and k


denoting the fluid dynamic viscosity and the intrinsic permeability of the solid skeleton, respectively. s is

the solid density, f denotes the fluid density, represents the density of solid-fluid mixture, m denotes the

mass parameter and n is the porosity. In addition, and Q are material parameters which describe the

relative compressibility of the constituents. Ks and Kf denote the bulk modulus of the solid grains and the

fluid while K represents the bulk modulus of the solid skeleton. Finally, fi and denote the body force and

the rate of fluid injection into the media, respectively.

Omiting all terms of fluid acceleration in equation (1) as well as all dynamic terms in equation (4) and

eliminating wi from equations (3) and (4), the well known governing u-p formulation of a poroelastic media

with incompressible solid particles and fluid, in which all coefficients 1/Ks, 1/Kf as well as 1/Q tend towards

zero, could be easily obtained in the Laplace transform domain as follows:

0~~~~)(~

,

2

,, iiijijjji fpusuu (5)

0~~~,, iiii upk (6)

the tilde denotes the Laplace transform and s demonstrates the Laplace transform parameter. In equations (5)

and (6), the contributions due to initial conditions are neglected.

BOUNDARY INTEGRAL EQUATIONS

The governing boundary integral equations will be deduced starting from the equilibrium equation and using

the well known weighted residual method. Weighting equation (5) by the displacement type function u'i,

integrating over the body , using integrations by parts twice and finally grouping the corresponding terms

together, one finds the following expression:

0~~~~~~~~~~~~dppdufufdutut iiiiiiiiiiii (7)

Also weighting equation (6) by the pressure type function p', integrating over the body , using

integrations by parts twice and finally grouping the corresponding terms together, one finds the

following expression:

0~~~~~~~~~~~~ dp-psdp-pdppppk iiiin,n, (8)

Eliminating the common term from equations (8) and (9) and returning to the time domain, one obtains the

governing boundary integral equation as follows:

0~~~~f dp-pduufdqp-qpdutut iiiiiiii (9)

where

n

pkq (10)

ti and q denote, respectively, the traction vector on and the flux normal to the boundary ( ).


By Assuming zero body forces (fi and ) and assigning proper unit Heaviside point forces and supplementary

impulse scalar sources to f'i and as

)()()()(),( 321 tHxxxtxf (11)

)()()()(),( 321 txxxtx (12)

we can obtain the Somigliana type integral equations:

dFpqGduFGttxuxc jjiijijiiij ).**()**(),().( 4400 (13)

dFpqGduFGttxpxc iiii ).**()**(),().( 44444400 (14)

x is the source point; x0 is field point; c (x0) is a matrix of constants, depend only upon the local geometry of

the boundary at x0; , =1, 2, 3, 4. G and F are the time domain displacement and traction fundamental

solutions are derived by Kamalian et al.[16, 17] as

dd

ij

d

ijijij

ijv

rtHtre

v

Ctre

v

Btre

Ate

AG ,,,

232216

s

ij

s

ij

s

ijij

v

rtHD

v

Ctre

v

Btre

A254 ,,

2(15)

ter

r

v

rtHtrg

rv

rtre

r

rG

i

dd

ii

i 82

,

2

,

72

,

44

,4

2,

4(16)

ter

r

v

rtHtre

rv

rtre

r

rG

i

dd

ii

i 92

,

2

,

12

,

44

,4

2,

4(17)

ttrev

rtH

krG

d

2exp,4

1744 (18)

mimjmijijkkjij nGGnGGF )()( ,,4, (19)

mimmiikki nGGnGGF )()( ,4,444,44 (20)

mmjj nGkF )( ,44 (21)

mm nGkF )( ,4444 (22)

where


k2

2

(23)

2dv

(24)

sv(25)

3,2,1, ji , ii xxr 2,

r

xr i

i, (26)

)3(4

1,,3 ijjiij rr

rA

(27)

)3(4

1,,2 ijjiij rr

rB

(28)

)(4

1,, jiij rr

rC

(29)

ijijr

D4

1(30)

Functions ei(r,t) and gi(r,t) are given in Appendix.

TIME DOMAIN CONVOLUTED DYNAMIC KERNELS

Implementation of boundary integral equations needs approximation in temporal variations of the field

variables. For temporal integration the time axis is divided into N equal steps and the field variables are

assumed to remain constant during a time step, so that they can be taken out of the convolution integral, thus

the time integration involves only the kernels and is expressed by

dGtdGtN

n

nN

ij

n

iiji

1

1* (31)

dtGGtn

tn

ij

nN

ij

1

1(32)

then by tNT :

tnN

tnNij

nN

ij TGG11

(33)

by similar way


tnN

tnNj

nN

j TGG1

4

1

4

(34)

tnN

tnNi

nN

i TGG1

4

1

4(35)

tnN

tnN

nN TGG1

44

1

44(36)

tnN

tnNij

nN

ij TFF11

(37)

tnN

tnNi

nN

i TFF1

4

1

4(38)

tnN

tnNmmj

nN

j nTkGF1

,4

1

4 (39)

tnN

tnNmm

nN nTkGF1

,44

1

44 (40)

with the temporal discretization described above, equations (13) and (14) transforms into:

N

nn

n

i

nNnN

i

nN

j

nN

ij

n

n

i

nNnN

i

nN

j

nN

ij

N

N

iij dp

u

FF

FF

q

t

GG

GG

pc

uc

11

44

1

4

1

4

1

1

44

1

4

1

4

1

(41)

BEHAVIOUR OF TRANSIENT KENRNELS AT LARGE TIME STEP

One of the important properties of the transient kernels is that at a very large time step the convoluted

kernels should reduce to the corresponding steady state kernels.

At the first time step N=1 so that TT :

tGG ijtijt limlim 1

ij

ss

ij

ji

ji

G

r

xx

r

r

xx

43116

1

22

1

8

1

2

3

(42)

Similarly

0lim 1

4it G (43)

0lim 1

4it F(44)

44

1

444

1lim G

rkG ss

t

(45)


442

,1

444

lim Fr

rnF ssmm

t (46)

r

x

kG

j

jt28

lim 1

4(47)

r

rrnF

miimm

jt

,,1

4)2(8

lim(48)

where Gssand Fss

are steady state fundamental solutions. as can be seen, all the convoluted transient

kernels reduce to corresponding elastostatic kernels at a very large time step.

NUMERICAL RESULTS

A set of numerical results are presented in this section to demonstrate the accuracy and some salient features

of the proposed transient convoluted kernels. A saturated soft soil with incompressible solid grains and pore

water was considered in which the material properties were defined in the metric system as follows: =12.5

MPa, µ=8.33 MPa, =2120 kg/m3, =1, =1×10-7 m4/Ns. The point force (or fluid source) is applied at the

coordinate (0,0,0) at time t=0 and the receiver is located at coordinate (0.01,0.02,0.03). Figures 1-4 depict the

presented analytical closed form kernels of 1

11G ,1

14G ,1

41G , and1

44G components respectively. As can be

seen, dynamic kernels gradually move toward static kernels at a very large time step. It is also interesting to

note the arrival times of the pressure (vp= ), diffusive (vd=117.3 m/s) and shear (vs=62.7 m/s) waves,

which could be detected by sudden changes appearing in the dynamic kernels. The notable initial values of

the pressure components 1

14G and 1

44G differ from zero, because the pressure wave with its wave

propagation velocity of infinity arrives immediately and affects the media's response.

CONCLUSION

Boundary integral equation, temporal discretization of BIE and analytical closed form expressions for

transient dynamic kernels are presented for the well known u-p formulation of 3D saturated porous media

with incompressible constituents, in terms of the practical variables of solid skeleton displacement and fluid

pressure. A set of numerical results are presented that demonstrate some salient features of the dynamic

kernels. The derived kernels could be simply implemented in time domain BEM for modeling the transient

behaviour of saturated porous media and provides the basis to develop more effective numerical hybrid

BE/FE methods for solving 3D nonlinear wave propagation problems in the near future.


-0.1

-0.05

0

0.05

0.1

0.15

0.2

0 0.5 1 1.5 2 2.5 3 3.5

Time (ms)

KE

RN

EL

G 11

(µ

m)

DynamicStatic

Figure 1. dynamic and static kernels of 1

11G at (0.01,0.02,0.03)

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0 0.5 1 1.5 2 2.5 3 3.5

Time (ms)

KE

RN

EL

G 41

(pa

)

DynamicStatic


41G at (0.01,0.02,0.03)

0

2

4

6

8

10

12

0 0.5 1 1.5 2 2.5 3 3.5

Time (ms)

KE

RN

EL

G 14

(m)

DynamicStatic


14G at (0.01,0.02,0.03)


-0.1

4.9

9.9

14.9

19.9

24.9

0 0.5 1 1.5 2 2.5 3 3.5

Time (ms)

KE

RN

EL

G 44

(Mp

a)

DynamicStatic


44G at (0.01,0.02,0.03)

APPENDIX: FUNCTIONS trei

, AND t,rgi

drgt

ttret

vr d

,2

2exp

2

1

2

1, 11

drgv

tret

vrd

d

,1

, 22

drgv

tret

vrd

d

,1

, 123

trgv

trgrv

ttre

dd

,,, 23124

tt

ttre 2exp12

1

2

1,

2

2

5

drgttret

vr d

,2exp1, 16

drgttret

vr d

,2exp, 17

d

d

d

d

v

rtvrtI

vrt

vrttrg

22

122

1 exp,

22

02 exp, dvrtIttrg


REFERENCES

1. Zienkiewicz OC and Shiomi T. Dynamic behavior of saturated porous media, the generalized Biot

formulation and it’s numerical solution. Int. J. Numer. Anal. Methods Geomech 1984; 8: 71-96.

2. Prevost JH. Dynamics of porous media. Geotechnical Modeling And Applications. S M Sayed, ed. Gulf

Publishing Company 1987; 76-146.

3. Gatmiri B. A simplified finite element analysis of wave-induced effective stresses and pore pressures in

permeable sea beds. Geotechnique 1989; 40: 15-30.

4. Gatmiri B and Nguyen KV. Time 2D fundamental solutions for saturated porous media with

incompressible fluid. Commun. Numer. Meth. Engng. 2004; 21(3): 119-132.

5. Schanz M., Pryl D., “Dynamic fundamental solutions for compressible and incompressible modeled

poroelastic continua”, International Journal of Solids and Structures, 41: 4047-4073, 2004.

6. Predeleanu M., “Development of Boundary Element Method to Dynamic Problems for Porous Media”,

Appl. Math. Modelling, 8: 378-382, 1984.

7. Manolis G.D. & Beskos D.E., “Integral Formulation and Fundamental Solutions of Dynamic

Poroelasticity and Thermoelasticity”, Acta Mechanica, 76: 89-104, 1989.

8. Wiebe T.H. and Antes H., “A Time Domain Integral Formulation of Dynamic Poroelasticity”, Acta

Mechanica 90: 125-137, 1991.

9. Cheng A.H.D. and Badmus T., “Integral Equation for Dynamic Poroelasticity in Frequency Domain with

BEM Solution”, Journal of Engineering Mechanics-ASCE, 117(5): 1136-1157, 1991.

10. Dominguez J., “Boundary Element Aproach for Dynamic Poroelasticity Problems”, J. of Numer. Meth.

Engng , 35: 307-324, 1992.

11. Dargush G F and Banerjee P K. A time domain boundary element method for poroelasticity. Int. J.

Numer. Methods In Eng. 1989; 28: 2434-2449.

12. Schanz, M., “poroelastodynamic Boundary Element Formulation”,Wave Propagation in Viscoelastic and

Poroelastic Continua: A Boundary Element Approach, Springer-Verlag publication, 77-98, 2001.

13. Kaynia A.M., “Transient greens functions of fluid-saturated porous media”, Computers & Structures, 44:

19-27, 1992.

14. Chen J. Time Domain Fundamental Solution To Biot’s Equations Of Dynamic Poroelasticity. PartI:

Two-Dimensional Solution. Int. J. of Solids & Structures 1994; 31(10): 1447-1490.

15. Chen J. Time Domain Fundamental Solution To Biot’s Equations Of Dynamic Poroelasticity. PartII:

Three-Dimensional Solution. Int. J. of Solids & Structures 1994; 31(2): 169-202.

16. Gatmiri B., Kamalian M., “On the Fundamental Solution of Dynamic Poroelastic Boundary Integral

Equations in Time Domain”, ASCE; The International Journal of Geomechanics, 2(4): 381-398, 2002.

17. Kamalian M., Gatmiri B. and Jiryaee Sharahi M., “Time domain 3D fundamental solutions for saturated

porelastic media with incompressible constituents”, Proc. of the 7th International Conference on Boundary

Element Techniques (BeTeq2006-Paris), 2006.

18. Kamalian M. and Jiryaee Sharahi M., “Traction transient fundamental solutions for 3D saturated

porelastic media with incompressible constituents”, Proc. of the 4th International Conference on

Earthquake Geotechnical Engineering (4ICEGE-Thessaloniki), 2007.

19. Biot M.A., “Theory of propagation of elastic waves in a fluid-saturated porous solid: I. Low-frequency

range, II. higher frequency range”, J. Acoust. Soc. Am., 28: 168-191, 1956.


Stability analysis of composite plates by the boundary

element method

E. L. Albuquerque1, P. M. Baiz2, and M. H. Aliabadi3

1 Faculty of Mechanical Engineering, State University of Campinas

Campinas, Brazil, [email protected]

Currently at Imperial College London as an academic visitor.

2 Department of Aeronautics, Imperial College London

London, UK, [email protected]

3 Department of Aeronautics, Imperial College London

London, UK, [email protected]

Keywords: Stability of structures, linear buckling, composite plates, radial integration method.

Abstract. This paper presents a boundary element formulation for the stability analysis of symmetriclaminate composite plates where only the boundary is discretized. Body forces are written as a sumof approximation functions multiplied by coefficients. Domain integrals which arise in the formulationare transformed into boundary integrals by the radial integration method. Plate buckling equationsare written as a standard eigenvalue problem. The accuracy of the proposed formulation is assessedby comparison with results from literature. Buckling coefficients and buckling modes are obtainedusing this formulation.

Introduction

Demand by an accurate stability analysis of anisotropic materials has increase with the increasing useof composite materials in engineering projects. In general, composites panels are very light structuresthat present high stiffness and strength. However, due to their slenderness, buckling is one of the mainconcern during their design.

The boundary element method (BEM) has provided a powerful solution to the field of plate buck-ling. Syngellakis and Elzein [1] presented a boundary element solution of the plate buckling based onKirchhoff theory under any combination of loadings and support conditions. Nerantzaki and Katside-lakis [2] developed a boundary element method for buckling analysis of plates with variable thickness.Elastic buckling analysis of plates using boundary elements can also be found in [3]. Buckling analysisof shear deformable isotropic plates was presented in [4]. To the best of author’s knowledge, the onlywork that presents a boundary element formulation applied to non-isotropic plates is due to [5] whopresented an orthotropic formulation with a domain discretization.

In this paper, a boundary element formulation for the stability analysis of general anisotropic plateswith no domain discretization is presented. Classical plate bending and plane elasticity formulationsare used and the domain integrals due to body forces are transformed into boundary integrals usingthe radial integration method. Numerical results are presented to assess the accuracy of the method.Buckling coefficients computed using the proposed formulation are in good agreement with resultsavailable in literature.

Boundary integral equations

In the absence of body forces, the governing equation of the anisotropic thin plate buckling is givenby:

Nij,j = 0, (1)


D11u3,1111 + 4D16u3,1112 + 2(D12 + D66)u3,1122 + 4D26u3,1222 + D22u3,2222 = Niju3,ij, (2)

where i, j, k = 1, 2; uk is the displacement in directions x1 and x2, u3 stands for the displacement inthe normal direction of the plate surface; Nij are the in-plane stress components, D11, D22, D66, D12,D16, and D26 are the anisotropic thin plate stiffness constants.

The boundary integral equation for in-plane displacements, obtained by applying reciprocity andGreen theorems at equation (1), is given by [6]:

cijuj(Q) +

∫Γ

t∗ik(Q,P )uk(P )dΓ(P ) =

∫Γ

u∗

ik(Q,P )tk(P )dΓ(P ) (3)

where ti = Nijnj is the traction in the boundary of the plate in the plane x1−x2, and nj is the normalat the boundary point; P is the field point; Q is the source point; and asterisks denote fundamentalsolutions. The anisotropic plane elasticity fundamental solutions can be found, for example, in [7].The constant cij is introduced in order to take into account the possibility that the point Q can beplaced in the domain, on the boundary, or outside the domain.

The in-plane stress resultants at a point Q ∈ Ω are written as:

cikNkj(Q) +

∫Γ

S∗

ikj(Q,P )uk(P )dΓ(P ) =

∫Γ

D∗

ijk(Q,P )tk(P )dΓ(P ) (4)

where Dikj and Sikj are linear combinations of the plane-elasticity fundamental solutions.The integral equation for the plate buckling formulation, obtained by applying reciprocity and

Green theorems at equation (2), is given by:

Ku3(Q) +

∫Γ

[V ∗

n (Q,P )w(P ) − m∗

n(Q,P )∂w(P )

∂n

]dΓ(P ) +

Nc∑i=1

R∗

ci(Q,P )u3ci

(P )

=Nc∑i=1

Rci(P )u∗

3ci(Q,P ) +

∫Γ

[Vn(P )u∗

3(Q,P ) − mn(P )∂u∗

3

∂n(Q,P )

]dΓ(P )

+λ

[∫Ω

Niju∗

3,ij dΩ +

∫Γ

(tiu

∗

3u3,i − tiu3u∗

3,i

)dΓ

], (5)

where ∂()∂n

is the derivative in the direction of the outward vector n that is normal to the boundary Γ;mn and Vn are, respectively, the normal bending moment and the Kirchhoff equivalent shear force onthe boundary Γ; Rc is the thin-plate reaction of corners; u∗

3ciis the transverse displacement of corners;

λ is the critical load factor; the constant K is introduced in order to take into account the possibilitythat the point Q can be placed in the domain, on the boundary, or outside the domain. As in theprevious equation, an asterisk denotes a fundamental solution. Fundamental solutions for anisotropicthin plates can be found, for example, in [8].

A second integral equation is necessary in order to obtain the thin plate buckling boundary elementformulation. This equation is given by:

K∂u3

∂m(Q) +

∫Γ

[∂V ∗

n

∂m(Q,P )w(P ) −

∂M∗

n

∂m(Q,P )

∂w(P )

∂n

]dΓ(P ) +

Nc∑i=1

∂R∗

ci

∂m(Q,P )u3ci

(P )

=Nc∑i=1

Rci(P )

∂u∗

3ci

∂m(Q,P ) +

∫Γ

[Vn(P )

∂u∗

3(Q,P )

∂m− mn(P )

∂2u∗

3

∂n∂m(Q,P )

]dΓ(P )

+λ

[∫Ω

u3Nij

∂u∗

3,ij

∂mdΩ +

∫Γ

(tiu

∗

3

∂u3,i

∂m− tiu3

∂u∗

3,i

∂m

)dΓ

], (6)


where ∂()∂m

is the derivative in the direction of the outward vector m that is normal to the boundaryΓ at the source point Q.

As can be seen in equations (5) and (6), domain integrals arise in the formulation owing to thecontribution of in-plane stresses to the out of plane direction. In order to transform these integralsinto boundary integrals, consider that a body force b is approximated over the domain Ω as a sum ofM products between approximation functions fm and unknown coefficients γm, that is:

b(P ) ∼=M∑

m=1

γmfm. (7)

The approximation function used in this work is:

fm = 1 + R, (8)

Equation (7) can be written in a matrix form, considering all boundary and domain source points,as:

b = Fγ (9)

Thus, γ can be computed as:

γ = F−1b (10)

Body forces of integral equations (5) and (6) depend on displacements. So, using equation (10)and following the procedure presented by Albuquerque et al. [9], domain integrals that come fromthese body forces can be transformed into boundary integrals.

Matrix Equations

After the discretization of equations (5) and (6) into boundary elements and collocation of the sourcepoints in all boundary nodes, a linear system is generated. It is worth notice that the only loadsconsidered in the linear buckling equations are that related to the in-plane stress Nij and tractions tithat are multiplied by the critical load factor λ. This means that all the known values of u3, ∂u3/∂n,Mn, Vn, wci, Rci (boundary conditions) are set to zero.

Dividing the boundary into Γ1 and Γ2 (Figure 1), this linear system can be written as:

Γ1: u3 =∂u3

∂n= 0

Γ2: Vn = Mn=0

Ω

Figure 1: Domain with constrained and free degrees of freedom.

[H11 H12

H21 H22

] w1

w2

−

[G11 G12

G21 G22

] V1

V2

= λ

[M11 M12

M21 M22

] w1

w2

, (11)

where Γ1 stands for stands for the part of the boundary where displacements and rotations are zeroand Γ2 stands for the part of the boundary where bending moment and tractions are zero. Indices 1and 2 stand for boundaries Γ1 and Γ2, respectively. Matrices H, G, and M are influence matrices ofthe boundary element method due to integral terms of equations (5) and (6).


As w1 = 0 and V2 = 0, equation (11) can be written as:

H12w2 − G11V1 = λM12w2,

H22w2 − G21V1 = λM22w2 (12)

or

Hw2 = λMw2, (13)

where H and M are given by:

H = H22 − G21G−111 H12,

M = M22 − G21G−111 M12. (14)

The matrix equation (13) can be rewritten as an eigen vector problem

Aw2 =1

λw2, (15)

whereA = H−1M. (16)

Provided that A is non-symmetric, eigenvalues and eigenvectors of equation (15) can be foundusing standard numerical procedures for non symmetric matrices.

Numerical results

The numerical results are presented in terms of the dimensionless parameter Kcr which is given by:

Kcr =Ncra

2

D22(17)

where Ncr is the critical load (Ncr = λ× the applied load) and a is the edge length of the square plate.Consider a square graphite/epoxy plate under different boundary conditions. The thickness of the

plate is h = 0.01 m. The material properties are: elastic moduli E1 = 181 GPa and E2 = 10.3 GPa,Poisson ratio ν12 = 0.28, and shear modulus G12 = 7.17 GPa.

The mesh used has 28 quadratic discontinuous boundary elements of the same length (7 per edge)and 49 (7 × 7) uniformly distributed internal points.

The plate is under uniformly uniaxial compression and the critical load parameter Kcr is computedconsidering all edges simply-supported (SSSS), all edges clamped (CCCC), and two edges clampedand two edges simply supported (CSCS). In the last case, the two edges where the load is applied aresimply supported and the two remaining edges are clamped. The results are shown in Table 1 togetherwith results obtained by [5] using a boundary element formulation with domain discretization and theanalytical solution presented by [10].

As it can be seen, there is a good agreement between the results obtained in this work and thosepresented in literature. Critical buckling modes for each case are shown in figures 2, 3, and 4.

Conclusions

This paper presented a boundary element formulation for the stability analysis of symmetric laminatedcomposite plates where domain integrals are transformed into boundary integrals by the radial inte-gration method. As the radial integration method doesn’t demand particular solutions, it is easier toimplement than the dual reciprocity boundary element method. Results obtained with the proposedformulation are in good agreement with results presented in literature.


Figure 2: Critical buckling mode of cases 1, 3 and 5.

Figure 3: Critical buckling mode of case 2.

Figure 4: Critical buckling mode of cases 4 and 6.


Table 1: Critical load parameter Kcr for a graphite/epoxy plate with different boundary conditions.

Case Boundary conditions Loadings This work Reference [5] Reference [10]

1 SSSS N1 = 0 130.82 – 129.78

2 SSSS N2 = 0 71.53 71.36 69.46

3 CCCC N1 = 0 493.70 481.21 –

4 CCCC N2 = 0 168.27 168.16 –

5 CSCS N1 = 0 161.47 163.24 162.03

6 CSCS N2 = 0 146.47 143.89 141.33

Acknowledgment

The first author would like to thank the CNPq (The National Council for Scientific and TechnologicalDevelopment, Brazil), AFOSR (Air Force Office of Scientific Research, USA), and FAPESP (the Stateof Sao Paulo Research Foundation, Brazil) for financial support for this work.

References

[1] S. Syngellakis and E. Elzein. Plate buckling loads by the boundary element method. International

Journal for Numerical Methods in Engineering, 37:1763–1778, 1994.

[2] M. S. Nerantzaki and J. T. Katsikadelis. Buckling of plates with variable thickness an analogequation solution. Engineering Analysis with Boundary Element, 18:149–154, 1996.

[3] J. Lin, R. C. Duffield, and H. Shih. Buckling analysis of elastic plates by boundary elementmethod. Engineering Analysis with Boundary Element, 23:131–137, 1999.

[4] J. Purbolaksono and M. H. Aliabadi. Buckling analysis of shear deformable plates by boundaryelement method. International Journal for Numerical Methods in Engineering, 62:537–563, 2005.

[5] G. Shi. Flexural vibration and buckling analysis of orthotropic plates by the boundary elementmethod. J. of Solids and Structures, 26:1351–1370, 1990.

[6] M. H. Aliabadi. Boundary element method, the application in solids and structures. John Wileyand Sons Ltd, New York, 2002.

[7] P. Sollero and M. H. Aliabadi. Fracture mechanics analysis of anisotropic plates by the boundaryelement method. Int. J. of Fracture, 64:269–284, 1993.

[8] E. L. Albuquerque, P. Sollero, W. Venturini, and M. H. Aliabadi. Boundary element analysis ofanisotropic kirchhoff plates. International Journal of Solids and Structures, 43:4029–4046, 2006.

[9] E. L. Albuquerque, P. Sollero, and W. P. Paiva. The radial integration method applied to dynamicproblems of anisotropic plates. Communications in Numerical Methods in Engineering, 23:805–818, 2007.

[10] S. G. Lekhnitskii. Anisotropic plates. Gordon and Breach, New York, 1968.


BEM model of mode I crack propagation along a weak interface applied to the interlaminar fracture toughness test of composites

L. Távara, V. Manti , E. Graciani, J. Cañas, F. París

Grupo de Elasticidad y Resistencia de Materiales, Escuela Técnica Superior de Ingenieros, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092 Sevilla, España

[email protected], [email protected], [email protected], [email protected], [email protected]

Keywords: composites, interlaminar fracture, weak interface, linear elastic-brittle law, BEM

Abstract. A numerical study of damage propagation in composite laminates is presented. Interlaminar

fracture toughness (GIc) test of two unidirectional carbon fiber laminates bonded by an adhesive layer is

studied. Displacement control is used in the numerical test simulation to ensure stable crack propagation. The

adhesive layer is modelled in the 2D Boundary Element Method (BEM) code developed as a weak interface

by means of a continuous distribution of springs governed by a linear elastic-brittle law. In this law, the

normal stresses across the interface are proportional to the relative normal displacements (opening) up to a

certain maximum stress value. It is shown that this approach provides a good representation of the actual

adhesive behaviour. An important feature of the BEM approach developed is that the parameters governing

the springs are independent of the boundary element mesh, i.e. distances between springs and element types

used. This fact allows us to perform an easy mesh refinement if required. It is shown that the local properties

of the numerical solution obtained near the crack tip agree with the predictions obtained with the weak

interface theory. The present model permits the study of both crack propagation and crack initiation. An

excellent agreement is observed between the load – displacement diagrams obtained in the BEM analysis and

in the laboratory tests. The computational procedure developed can be used to estimate the maximum

allowed load of a structure including similar adhesive bonded joints of laminates.

Introduction

Traditionally, the methods that simulated crack propagation were based on Linear Elastic Fracture

Mechanics (LEFM) assuming the presence of a crack, which made difficult the study of damage and/or crack

initiation occurring in the first step of fracture process. Recently, other models have been intensively

developed as cohesive crack model which assumes hypotheses different to those adopted in LEFM avoiding

the presence of a stress singularity at the crack tip. These models are suitable to study both crack initiation

and crack propagation, and also to estimate the fracture energy and the maximum allowable load of a

structure.

In many practical situations, the behavior of adhesive joints can be described modeling the thin adhesive

layer as a continuum spring distribution [1] with an appropriate stiffness parameter. This interface model is

usually called weak interface or elastic interface [2,3]. In the present work linear elastic – brittle constitutive

law of these springs is adopted in order to allow an easy modeling of crack propagation along a weak

interface. In the present work the above described weak interface behavior has been implemented in a 2D

BEM code [4,5], whose original version allowed modeling of plane elastic problems, including several linear

elastic anisotropic solids with strong interfaces or contact zones between them. The new feature incorporated

to this code is the incorporation of the possibility of defining weak interfaces between the elastic solids

where required. Another feature of the code is that the equilibrium and compatibility conditions, along

contact zones and strong or weak interfaces, are imposed using a weak formulation allowing an easy use of

non-conforming discretizations [4,5,6].

The understanding of the adhesive layer behavior is very important in the quality evaluation of this kind

of joints, and particularly in determining the parameters that characterize its resistance to fracture and failure.

These parameters can then be used in design and quality control of the productive process. The quality of an

adhesive joint between composite laminates is usually evaluated by an interlaminar fracture test, where an

estimation of the critical interlaminar fracture energy (GIc) is obtained. An extensive experimental study and

a numerical study by Finite Element Method of this test and of different adhesives were recently carried out

by the present authors and their co-workers [7,8].


Weak interface

According to Lenci and co-workers [2,3], a weak interface is considered as a model of a thin linear elastic

adhesive layer between two surfaces. In the present wok, adhesive damage and/or rupture are modeled as a

free separation of both surfaces. Thus, the springs that simulate an adhesive layer are governed by the

following linear elastic-brittle law, shown also in Fig. 1:

cck if0andif , (1)

where is the normal stress in a spring, is the relative opening of the extremes of the spring (separation

between surfaces), k is a stiffness parameter, c and c are, respectively, the critical normal stress and the

critical relative normal displacement leading to the spring rupture. Fracture mechanics is indirectly involved

through the area under the linear law line in Fig. 1 given by GIc value, .21

cG cIc

Figure 1. Linear elastic-brittle law of a spring.

According to the weak interface theory [2,3], interface tractions are bounded at the tip of a crack situated

along a weak interface, whereas these tractions are singular (unbounded) at the tip of an interface crack

situated along a perfect interface (called also strong interface, where no relative displacements of bonded

surfaces are allowed). Thus, during crack growth along a weak interface these tractions are kept bounded. It

appears that local normal tractions in the zone close to the interface crack tip follow the law [2]:

1)ln(10 (2)

where and are constants and x = a , where x is the distance from the crack tip to a point (in the bonded

part of the interface) where these tractions are evaluated and a is a characteristic length, usually the crack

length or semilength.

Mode I weak interface implementation in the 2D BEM code

Incremental formulation. The numerical solution of the non-linear problem formulated is based on a

gradual application, by means of a load factor , 0 1, of the loads and displacements imposed. The

solution procedure is given by a series of lineal stages, “load steps”. At the beginning of each load step an

actual adhesively bonded zone is defined, which defines the actual linear system of equations. By solving

this system the corresponding elastic solution is obtained. This solution fulfills all the conditions of the weak

interface formulation up to a certain maximum value of the load factor associated to this load step. A

further increment of the load factor leads to rupture of some springs.

Thus, the solution of the problem will be divided into a number M (a priori unknown) of load steps

where the values of the problem variables vary linearly:

)x(),x( m (3)

with m-1 m, m = 1,..., M, and 0=0, and where ),x( is the value of any problem variable at a point x

after fraction of load is applied, )x(m being the variable value obtained in the solution of the linear

system corresponding to the m-th load step.

c

c

GIc


This procedure can be repeated as many times as necessary to reach the equilibrium after the whole load

is applied. Nevertheless, changing the conditions node by node can make the crack propagation to be very

smooth (especially for fine meshes), in opposite to the experimental evidence for some industrial adhesives

that show crack growing by small jumps. That is why for a specific case of adhesive, like that simulated in

the present work, the end of a load step can be defined by a situation where a fixed number of consecutive

nodes (number 20 is chosen in the present study) do not fulfill the condition (7).

Laboratory test

Test description. The test used in the aeronautical industry to evaluate the interlaminar fracture toughness in

composite-composite joints is performed following AITM 1.0005 [9] and/or I+D-E 290 [10] standards. The

specimen used is the Double Cantilever Beam (DCB) shown in Fig. 3(a). The DCB specimen is formed by

two laminates joined by a thin adhesive layer. The laminates are processed according to EN 2565 standard,

and the specimens are cut after the panel has been cured. The specimen is fixed to the grips of the universal

testing machine through small tabs bonded to laminates as shown in Fig. 3(b). During crack propagation the

load (P) and the displacement (d) of the wedge grips are continuously registered.

L = 250 ± 5 mm L1 = 25 ± 1 mm w = 25.0 ± 0.2 mm t = 3.0 ± 0.2 mm

w

0º

t

l1

L

P

d

Figure 3. (a) Scheme of the DCB specimen, (b) Test configuration.

Adhesive type. In a study of experimental results obtained from GIc tests for different kinds of adhesive [7],

it was observed that the adhesives FM 300K0.5 and EA 9695 K.05 present falls in the experimental load –

displacement curve. This behavior was explained by the presence of a polyester support in these adhesives.

Evaluation of the adhesive model parameters. As the parameters of the adhesive model adopted here are a

priori unknown, they are adjusted by fitting the experimental and numerical load-displacement curves.

Experimental results provide estimations of GIc values, the crack length for some load values and the load –

displacement curves. With these data, and using equation (9), a first estimation of the critical displacement

( c) and the slope k is obtained. After a comparison between the numerical and experimental results these

values can be adjusted better.

2

2c

Ic

kG (9)

Other trial and error methods to obtain an estimation of Kadh value are presented in [7], the estimated

values of Kadh obtained therein and of k obtained here are in a good agreement according to (4).

Numerical Results

In the present numerical study, a 2D model has been solved using the BEM code described above, where the

plane strain and linear elastic behavior hypotheses have been assumed. The laminate considered is a

8552/AS4 carbon fiber – epoxy composite (0º plies), with the following orthotropic properties: Ex=135GPa,

Ey=10GPa, Ez=10GPa, Gxy=5GPa, Gxz=5GPa, xy=0.3, yz=0.4 and xz=0.3. The adhesive used is EA 9695


K.05, an epoxy adhesive with a polyester support. The estimated properties of the adhesive spring model are:

k=1514GPa/m and c =1514MPa.

A maximum displacement of 25 mm was progressively applied in the direction normal to the specimen

boundary at a 15 mm distance from the specimen extreme where the initial crack is situated.

The normal stresses along the bonded zone obtained in the last load step, are shown in Fig. 4(a). The

initial longitude of the adhesive layer 225 mm is discretized by 468 or 936 springs placed between the nodes

of the conforming boundary element meshes on A and B sides of the weak interface.

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.025 0.05 0.075

Distance to the crack tip (m)

No

rma

l s

tre

ss

(M

Pa

)

468 nodes1

936 nodes

0.8

0.6

0.4

Figure 4. (a) Normal stresses near the crack tip, (b) Fitting of the normalized local stress solution by an

analytic expression (x = a ) (10).

It is noteworthy that the local stress solution near the crack tip agrees very well with the predictions of

the weak interface theory (2). In Fig. 4(b), the normalized stresses / c represented as a function of (the

initial adhesive layer being modeled by 468 springs) are compared with the curve of expression (10),

obtained from (2) by applying the least square method.

ln89711.00658.1c

. (10)

Comparison between the experimental and numerical load - displacement diagrams

As can be observed in Fig. 5, the numerical results obtained provide a good approximation of the

experimental results. Therefore, the use of the weak interface formulation seems to be a promising approach

to model composite adhesive joints.

0

50

100

150

200

250

300

350

0.000 0.010 0.020 0.030 0.040 0.050

Displacement d (m)

Lo

ad

P (

N)

numerical

experimental

Figure 5. Comparison between the experimental and numerical load - displacement diagrams, and a detail of

the polyester support of the adhesive used.

0.01 0.02 0.03 0.04 0.05 0.06

0.2

0.07

/ c

P

a

Y

X


Conclusions

As shown by the numerical results presented, the weak interface formulation modeled by a spring

distribution, correctly describes the behavior of adhesive joints in the aeronautical industry.

An analytic expression for the local solution of normal tractions at the crack tip, deduced in the weak

interface theory, has been successfully compared with the present numerical solution. Noteworthy is the

bounded character of stresses along the weak interface, the maximum value of stresses being achieved at the

crack tip.

The spring constitutive law introduced and included in the incremental algorithm of the BEM code has

the advantage of being independent of the number of springs used in the interface.

It has been proved that the real behavior of an adhesive layer with a polyester support that joins two

unidirectional laminates can be approximated very well by means of BEM and a distribution of springs

which follow a linear elastic-brittle constitutive law, by adjusting the parameters of the discrete model (k, c,

and the number springs that breaks in a load step). This fact will allow predicting the real behavior of

structures that include similar adhesive joints by the model developed here.

From laboratory test and fractographic analysis it has been concluded that the jumps appearing in the

experimental load – displacement curve are caused by the polyester support of the adhesive resin. The results

obtained in this work can also be considered as a starting point for a study of adhesives including different

kinds of adhesive support.

Acknowledgements

The authors acknowledge the support of the Junta de Andalucía (Projects of Excellence TEP-1207 and TEP

02045) and also the support by the Spanish Ministry of Education and Science through Projects TRA2005-

06764 and TRA2006-08077.

References

[1] Erdogan F., Fracture mechanics of interfaces, In: Damage and Failure of Interfaces, Rossmanith ed.,

Balkema, Rotterdam, (1997).

[2] Lenci S., Analysis of a crack at a weak interface, International Journal of Fracture, 108: 275-290, (2001).

[3] Geymonat G., Krasucki F., Lenci S., Mathematical analysis of a bonded joint with a soft thin adhesive,

Mathematics and Mechanics of Solids, 4: 201-225, (1999).

[4] Graciani E., Formulación e implementación del método de los elementos de contorno para problemas

axisimétricos con contacto. Aplicación a la caracterización de la interfase fibra matriz en materiales compuestos, Tesis Doctoral, ETSI – Universidad de Sevilla, (2006).

[5] Graciani, E., Mantic, V., París, F. y Blázquez, A., Weak formulation of axi-symmetric frictionless contact

problems with boundary elements. Application to interface cracks, Computer and Structures, 83: 836-855,

(2005).

[6] Blázquez A., París F., Manti V., BEM solution of two-dimensional contact problems by weak application of contact conditions with nonconforming discretizations, International Journal of Solids and Structures,

35: 3259-3278, (1998).

[7] Jiménez M. E., Modelización del ensayo de tenacidad a la fractura interlaminar en materiales

compuestos, Proyecto de Fin de Carrera, ETSI – Universidad de Sevilla, (2006).

[8]Jiménez M. E., Cañas J., Manti V., Ortiz J. E., Estudio numérico y experimental del ensayo de tenacidad a fractura interlaminar de uniones adhesivas composite-composite, MATCOMP ‘07, Valladolid, (2007).

[9] AITM 1-0005, Determination of interlaminar fracture toughness energy. Mode I. Issue 2, AIRBUS,

(1994).

[10] I+D-E-290, Ensayo de tenacidad a la fractura interlaminar sobre estratificados de fibra de carbono,

CASA, (1988).


solution of nonlinear reaction-diffusion equation by using...

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