comput. methods appl. mech. engrg.€¦ · bloco b sala b-101, cidade universitária, ilha do...

Comput. Methods Appl. Mech. Engrg. 201–204 (2012) 127–138

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Comput. Methods Appl. Mech. Engrg.

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Review

A finite element method based on capturing operator applied to wavepropagation modeling

Eduardo Gomes Dutra Do Carmo a, Cid Da Silva Garcia Monteiro b,⇑, Webe João Mansur ba COPPE/UFRJ – Post-Graduate Institute Alberto Luiz Coimbra of the Federal University of Rio de Janeiro, Nuclear Engineering Department, Centro de Tecnologia,Bloco G sala G-206, Cidade Universitária, Ilha do Fundão, 21945-970 Rio de Janeiro, RJ, Brazilb LAMEMO-COPPE/UFRJ – Post-Graduate Institute Alberto Luiz Coimbra of the Federal University of Rio de Janeiro, Civil Engineering Department, Centro de Tecnologia,Bloco B sala B-101, Cidade Universitária, Ilha do Fundão, 21945-970 Rio de Janeiro, RJ, Brazil

a r t i c l e i n f o

Article history:Received 19 November 2010Received in revised form 10 September 2011Accepted 3 October 2011Available online 8 October 2011

Keywords:Finite element methodDiscontinuity-capturing operatorsElastodynamicsStabilized methods

0045-7825/$ - see front matter � 2011 Elsevier B.V. Adoi:10.1016/j.cma.2011.10.006

⇑ Corresponding author. Tel.: +55 21 2562 7382; faE-mail addresses: [email protected] (E.G.

yahoo.com.br (C. Da Silva Garcia Monteiro), webe@co

a b s t r a c t

This paper presents a methodology for the development of discontinuity-capturing operators for generalelastodynamics. These operators are indicated for problems with sharp gradients in the space and in thetime. The development here presented is based on the methodology for obtaining discontinuity-captur-ing operators developed by Dutra do Carmo and Galeão (1986) for diffusion–convection problems and isinspired in the works presented by Hughes and Hulbert (1988 and 1990). It is shown that their operatorbelongs to the families of operators developed here. The formulation is applied to one-dimensional andtwo-dimensional problems. The results show that the method produces better results than classic meth-ods for the one dimensional case and presents robust performance for the two-dimensional case.

� 2011 Elsevier B.V. All rights reserved.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1272. General elastodynamic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1283. Space–time finite element formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1284. Discontinuity-capturing operators for elastodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1295. Families of discontinuity-capturing operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1306. Formulation for the one-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1307. Determination of the parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1318. Numerical examples for one-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

8.1. Example 1 – Homogenous one-dimensional elastic bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1318.2. Example 2 – Non-homogeneous one-dimensional elastic bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

9. Extension for the d-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13610. Numerical example for two-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

10.1. Example 3 – Transverse motion of quadrangular membrane under prescribed initial velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

11. Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

1. Introduction

Mathematical modeling of physical problems, static or dy-namic, linear or non-linear, generally results in a system of partial

ll rights reserved.

x: +55 21 2562 8464.Dutra Do Carmo), [email protected] (W.J. Mansur).

differential equations (PDE) that must be resolved analytically ornumerically to obtain the solution. In the specific case of dynamicproblems, algorithms using discretization in the time are rou-tinely used to find solutions dependent of the time. Among thevarious numerical methods used to solve such problems, the fi-nite difference method (FDM) has a high computational perfor-mance, while the finite element method (FEM) is versatile androbust.

http://dx.doi.org/10.1016/j.cma.2011.10.006mailto:[email protected]:csgm25@ yahoo.com.brmailto:csgm25@ yahoo.com.brmailto:[email protected]://dx.doi.org/10.1016/j.cma.2011.10.006http://www.sciencedirect.com/science/journal/00457825http://www.elsevier.com/locate/cma

128 E.G. Dutra Do Carmo et al. / Comput. Methods Appl. Mech. Engrg. 201–204 (2012) 127–138

The analysis of structural dynamics problems is usually car-ried out by the so called semidiscrete methods, where the FEMis used to model the spatial domain and the FDM is used to inte-grate on the time. Semidiscrete methods are effective for com-puting smooth responses, however, their performance to modelproblems with sharp gradients or discontinuities is notsatisfactory.

Alternatively to semidiscrete methods, the use of the FEM torepresent both space and time domains, was first proposed in[1–3]. Space–time finite elements can represent better the solutionof a problem than semidiscrete methods. However, both presentdifficulties in the presence of discontinuities as described in refer-ences [4,5].

It was developed in [6] a space–time finite element formulationfor elastodynamics where the solution and it’s derivative in thetime are discontinuous between of two consecutive time intervals.Capturing operators were included in this formulation to capturethe discontinuities.

Since then many works have been developed to represent dis-continuities or sharp gradients for the FD and FE methods. A gen-eral review can be seen in [7,8] and a very instructive reviewconcerning spurious oscillations for convection–diffusion equa-tions can be seen in [9].

In [10] it was presented another time discontinuous Galerkinmethod where time and space are decoupled.

The objective of the present work is to extend the methodologydeveloped in [11,12] to elastodynamics. This paper is organized asfollows.

In Section 2 we present the basic equations of the elastody-namics. Section 3 is devoted to the time discontinuous Galerkinformulation. In Sections 4 and 5 we present the general method-ology to obtain the discontinuity capturing operators. In Sections6 and 7 we show how to obtain the capturing operator for wavepropagation in one-dimensional spaces and determine the param-eters of the proposed operator. Section 8 presents two numericalexamples for one-dimensional case. In Section 9 we propose apossible extension of the discontinuity capturing operator to thed-dimensional case, and the Section 10 presents one numericalexample for two-dimensional case using the extension proposedin the previous section. Finally we present the conclusions inSection 11.

2. General elastodynamic equations

Let X � Rd, where d is space dimension, be an elastic linearsolid with boundary Lipshitz continuous C = oX = Cg [ Ch andbeing meas (Cg \ Ch) = 0, where meas(.) denotes Lebesgue posi-tive measure. Elastic wave propagation in solids is governed bythe following second order hyperbolic partial differentialequation

q€u�r:rðruÞ � f ¼ 0 on Q ¼ X� ½0; T�; ð1Þ

where the stress r(ru) is given by the generalized Hooke’s law:

rðruÞ ¼ C:ru; ð2Þ

where C is a fourth-order tensor whose components are the elasticcoefficients.

Therefore, the component ri j is given as follows

rij ¼Xdl¼1

Xdk¼1

Cijkl@uk@xl

; ð3Þ

with boundary conditions and initial conditions given below

u ¼ g on Cg � ½0; T�;n � rðruÞ ¼ h on Ch � ½0; T�;uðx;0Þ ¼ u0ðxÞ;_uðx;0Þ ¼ v0ðxÞ;

ð4Þ

where g and h represent respectively the prescribed boundaries dis-placement and traction, n denotes the unit outward vector normalto C, u denotes the displacement vector, _u denotes the differentia-tion of u with respect to the time variable t. T > 0, q denotes themass density and u0(x) and v0(x) represent respectively initial dis-placement and initial velocity.

3. Space–time finite element formulation

For n 2 {0, . . . ,N} consider the time interval In = [tn�1, tn], thetime step Dt = tn � tn�1, and the jump operator defined as

suðtnÞt ¼ u tþn� �

� u t�n� �

; ð5Þu tþn� �

¼ lime!0þ

u tn þ eð Þ; ð6Þ

u t�n� �

¼ lime!0�

u tn þ eð Þ: ð7Þ

The variational equation or weak form can be derived from aweighting residual form as given in [6,7], (see expression (8)), byconsidering the jump operator in the time for displacement andvelocity, as followsZ tn

tn�1

ZX

_Wðx; tÞq€uðx; tÞdXdt �Z tn

tn�1

ZX

_Wðx; tÞr:rðruðx; tÞÞdXdt

�Z tn

tn�1

ZX

_Wðx; tÞfðx; tÞdXdt þZ

X

_W x; tþn�1� �

qs _uðx; tÞtdX

þZ

XW x; tþn�1� �

r:rðsruðx; tn�1ÞtÞdX ¼ 0: ð8Þ

After applying integration by parts to reduce the order of the spatialoperator and by considering the divergence theorem, one obtainsZ tn

tn�1

ZX

_Wðx; tÞq€uðx; tÞdXdt þZ tn

tn�1

ZXr _Wðx; tÞrðruðx; tÞÞdXdt

�Z tn

tn�1

ZCh

_Wðx; tÞhðx; tÞdt �Z tn

tn�1

ZX

_Wðx; tÞfðx; tÞdXdt

þZ

X

_W x; tþn�1� �

q _u x; tþn�1� �

dX�Z

X

_W x; tþn�1� �

q _u x; t�n�1� �

dX� �þ

ZXrW x; tþn�1

� �r ru x; tþn�1

� �� dX

��Z

XrW x; tþn�1

� �r ru x; t�n�1

� �� dX�¼ 0: ð9Þ

In order to work out the approximate formulation through the finiteelement method, consider a usual partition of the domain X into neelements. For each Xe and each time interval In, let P

k Qen� �

be thespace of the polynomials of degree 6k in the local coordinateswhere Qen ¼ Xe � In. By considering k P 2 one has the set of theadmissible approximations

Sh;k ¼ uhjuh 2 C0 [Nn¼1Q n� ��

;uhe 2 Pk Qen� �

;uh ¼ g on Cg � In o

;

ð10Þ

and the space of the admissible variations

Vh;k ¼ WhjWh 2 C0 [Nn¼1Qn� ��

;Whe 2 Pk Qen� �

;Wh ¼ 0 on Cg � In o

;

ð11Þ

where uhe denotes the restriction of uh to Qen. The Time Discontinu-

ous Galerkin formulation associated to the variational problem

E.G. Dutra Do Carmo et al. / Comput. Methods Appl. Mech. Engrg. 201–204 (2012) 127–138 129

given by (9), consists of finding uh 2 Sh,k satisfying the variationalequation

ATDGðuh;WhÞn ¼ FTDGðWhÞn8Wh 2 Vh;k; ð12Þ

ATDGðuh;WhÞn ¼Xnee¼1

R tntn�1

RXe

_Whðx; tÞq€uhðx; tÞdXdtþR tntn�1

RXer _Whðx; tÞrðruhðx; tÞÞdXdtþR

Xe_Wh x; tþn�1� �

q _uh x; tþn�1� �

dXþRXerWh x; tþn�1

� �r ruh x; tþn�1

� �� dX

2666664

3777775;ð13Þ

FTDGðWhÞn ¼Xnee¼1

R tntn�1

RChe

_Whðx; tÞhðx; tÞdXdt�R tntn�1

RXe

_Whðx; tÞfðx; tÞdXdtþRXe

_Whðx; tþn�1Þq _uh x; t�n�1� �

dXþRXerWh x; tþn�1

� �r ruh x; t�n�1

� �� dX

26666664

37777775 if n > 1;ð14Þ

FTDGðWhÞ1 ¼Xnee¼1

R tntn�1

RChe

_Whðx; tÞhðx; tÞdXdt�R tntn�1

RXe

_Whðx; tÞfðx; tÞdXdtþRXe

_Wh x; tþn�1� �

qv0ðxÞdXþRXerWh x; tþn�1

� �rðru0ðxÞÞdX

26666664

37777775: ð15Þ

4. Discontinuity-capturing operators for elastodynamics

In this section we present a general methodology to developdiscontinuity-capturing operators for the general elastodynamicproblem presented in Section 2. We consider again, an elastic bodyoccupying a bounded region X contained in Rd, where d 2 {1,2,3},the stress components ri j being given as

rij ¼Xdl¼1

Xdk¼1

Cijkl@uk@xl

; ð16Þ

where i, j, k, l 2 {1,. . ., d} and Cijkl are the elastic coefficients.In order to extend to elastodynamics the methodology pre-

sented in [11] for diffusion–convection problem, we consider theequilibrium equations

q@2ui@t2�

Xdj¼1

Xdk¼1

Xdl¼1

Cijkl@2uk@xj@xl

þ @Cijkl@xj

@uk@xl

" #" #¼ fi ði ¼ 1; . . . ; dÞ:

ð17Þ

By following the methodology presented in [11], associated to oneuh fixed, we consider the approximate coefficients Chijkl;

@Chijkl@xi

and qhisatisfying

qhi@2uhi@t2�

Xdj¼1

Xdk¼1

Xdl¼1

dijklChijkl

@2uhk@xj@xl

þ djijkl@Chijkl@xj

@uhk@xl

" #" #� fi ¼ 0 ði ¼ 1; . . . ;dÞ; ð18Þ

where

dijkl ¼1; if Cijkl – 00; otherwise

�and djijkl ¼

1; if @Cijkl@xj

– 0

0; otherwise:

(ð19Þ

Therefore, given an approximate solution uh, the goal is to find the

approximate coefficients, Chijkl;@Chijkl@xi

and qhi satisfying (18) and as

close to Cijkl;@Cijkl@xi

and qi as possible. To achieve this goal, we con-sider the functional J⁄, defined as

J� ¼Xd

i

Xdj

Xdl

Xdk

Chijkl � Cijklh i2

2þ hi

2@Chijkl@xj� @Cijkl

@xj

" #28>:9>=>;

þ ci2

qhi � q 2

; ð20aÞ

where hi and ci are dimensional parameters so that the terms of theequation above can have the same dimension and are defined asfollows

ci ¼Ciiiiq

� �2; ð20bÞ

hi ¼ ðCouriÞ2ðhe;iÞ2; ð20cÞ

where Couri is the Courant number on the direction i, and he,i is thecharacteristic length on the direction i, both will be defined later onin this section.

Our objective is to minimize the functional J⁄ subjected to therestriction given by (18). The problem of minimization of the func-tional J⁄ satisfying the restrictions (18) is equivalent to minimizethe functional J, defined as

J¼Xd

i

Pdj

Pdl

Pdk

Chijkl�Cijkl½ �2

2 þhi2

@Chijkl@xj� @Cijkl

@xj

� �2( )þ ci2 qhi �q

2þki �

Pdj

Pdk

Pdl

Chijkldijkl@2uh

k@xj@xl

� �þ djijkl

@Chijkl@xj

� �@uh

k@xl

� �� þqhi

@2uhi

@t2� fi

" #2666664

3777775;ð21Þ

where ki are Lagrange Multipliers.By minimizing the J functional with respect to Chijkl;

@Chijkl@xi

;qhi andki, for one uh fixed, and by performing subsequently some manip-ulations and by considering the vector

UiðuhÞ ¼ Ui;1ðuhÞ;Ui;2ðuhÞ; 1ffiffiffifficip @2uhi@t2

!; ð22aÞ

where

Ui;1ðuhÞ ¼ di111@2uh1@x21

; . . . ; dijkl@2uhk@xj@xl

; . . . ; diddd@2uhd@x2d

!; ð22bÞ

Ui;2ðuhÞ ¼ d1i111ffiffiffiffihip @u

h1

@x1; . . . ;

djijklffiffiffiffihip @u

hk

@xl; . . . ;

ddidddffiffiffiffihip @u

hd

@xd

!; ð22cÞ

one can obtain the Lagrange multipliers in the compact format

ki ¼RiðuhÞkUiðuhÞk2

ði ¼ 1; . . . ;dÞ; ð23aÞ

where

RiðuhÞ ¼ �Xn

j

Xnk

Xnl

Cijkl@2uhk@xj@xl

!þ @Cijkl

@xj

@uhk@xl

" #" #þ q @

2uhi@t2

� fi ði ¼ 1; . . . ; dÞ: ð23bÞ

By defining the vector of disturbance or error vector

Vh;ip ðuhÞ ¼ Vh;i;1p ;V

h;i;2p ;

ffiffiffiffici

pq� qhi� ��

ð24aÞ


where

Vh;i;1p ðuhÞ ¼ Chi111 � Ci111

� �; . . . ; Chijkl � Cijkl

� �; . . . ; Chiddd � Ciddd

� �� ð24bÞ

Vh;i;2p ðuhÞ ¼ffiffiffiffihi

p @Chi111@x1

� @Ci111@x1

!; . . . ;

ffiffiffiffihi

p @Chijkl@xj� @Cijkl

@xj

!; . . . ;

ffiffiffiffihi

p @Chiddd@xd

� @Ciddd@xd

!!; ð24cÞ

with (i = 1, . . . ,d), and from Eqs. (22a-c), (23a-b) and (24a-c) weobtain

Vh;ip ðuhÞ ¼RiðuhÞUiðuhÞ

UiðuhÞ�� 2 ði ¼ 1; . . . ;dÞ; ð25Þ

bVh;ip ðuhÞ ¼ Vh;ip ðuhÞVh;ip ðuhÞ�� ¼ Riðu

hÞUiðuhÞjRiðuhÞj UiðuhÞ

�� ði ¼ 1; . . . ;dÞ; ð26ÞAssociated to Ui(uh) we consider the vector Ui,loc(uh) in the localcoordinates,

Ui;locðuhÞ ¼ di111@2uh1@n21

. . . ; dijkl@2uhk@nj@nl

; . . . ; diddd@2uhd@n2d

;@2uhi@n20

!;

ði ¼ 1; . . . ;dÞ; ð27Þ

where (n1, . . . ,nd) are the dimensionless coordinates of the elementrelated to the physical or global coordinates and n0 is a dimension-less coordinate related to the time.

In order to obtain the Petrov–Galerkin disturbance necessaryfor building a discontinuity-capturing operator family, we usethe expressions (24a)–(24c), (25)–(27) for introducing thefunctions

hlocðuhÞ ¼ Ui;locðuhÞkUiðuhÞk

" #1=2; ð28Þ

siðuh;aiÞ ¼1

kUiðuhÞkRiðuhÞ

qðciÞ1=2

�� h iai ; if kUiðuhÞk > 00; if kUiðuhÞk ¼ 0

8

Lx

A (Lx, Δt/2)

A (Lx, Δt)

Δt

Δxx

t

…

Fig. 2. Space–time slab associated to the bar.

A f(t)

x0

E, ρ, c

Lx=1.0m

y

Fig. 1. Homogeneous elastic bar used to determine the discontinuity capturingparameters.


dimensionless functions dependent of the Courant number. In thenext section we present how the parameters for the one-dimen-sional elastic bar are determined.

7. Determination of the parameters

The parameters �a; lþðCourÞ and l�(Cour) are determined vianumerical experiments. The problem chosen to determine theseparameters is a one-dimensional elastic bar. The bar has one endfixed and the other loaded by an axial force as shown in Fig. 1,and has the following properties: the length is Lx = 1 m, the squarecross section area is 0.01 m2, the mass density is q = 1 kg/m3 andYoung’s modulus is E = 1 N/m2. The exact solution for this problemcan be found in [13].

The developed formulation uses space–time elements unliketraditional semidiscrete finite element formulations. The space–time slab was discretized using 50 quadrilateral elements of 9nodes, as shown in Fig. 2.

The variables used to determine the functions l+(Cour),l�(Cour) and the exponent �a are the velocity distribution along

Fig. 3. Behavior of the function l+/

the time at the point A (free end of the bar) and the stress distribu-tion along the bar at a specific time.

The experiment for the determination of the parametersl+(Cour), l�(Cour) and the exponent �a is limited to Cour 2 [0.2,4.0], with the Courant number step D Cour = 0.05. The exponent�a was determined as follows: the functions l+/�(Cour) were fixedequal to unity, and for each Courant number in the range above,the values 0.25, 0.50 and 0.75 for the exponent �a were tested.The best value found was 0.50 which corresponds to the smallestmean square error between exact and numerical solutions. Afterdetermining the �a value, the next step was to determine thel+/�(Cour) functions. By using �a ¼ 0:5, for each Courant numberfixed, was determined as being the best value for these functionsthose that gave the smallest mean square error between exactand numerical solutions. The final results are presented in theFig. 3. We notice that a fast interpolation scheme to obtain thefunctions l+/�(Cour) at any point inside the range can be easilydetermined.

8. Numerical examples for one-dimensional case

In this section we present two numerical examples for a one-dimensional case. The operator obtained with the methodologyhere presented will be denoted by TDG + DC. The results are com-pared with exact solution, TDG method and with the method pre-sented in [8], which will be denoted by TDG + HH.

8.1. Example 1 – Homogenous one-dimensional elastic bar

The first example presented here considers a homogenous one-dimensional elastic bar with one end fixed and the other loaded byan axial force, which represents a typical one-dimensional wavepropagation problem. In this example both stresses and velocitiespresent discontinuity on space and time. The properties of thebar are: length Lx = 4 m, square cross section 0.04 m2, mass densityq = 1 kg/m3 and Young’s modulus E = 1 N/m2. A 200 � 1 mesh ofquadratic Lagrangian space–time elements was used in eachtime-step. The force applied at the free end of the bar at initial timeis a Heaviside of intensity 10�2N.

The results for velocities and stresses obtained for Courantnumbers 0.57, 1.03 and 2.03, which correspond respectively to

�(Cour) with Courant number.


the following time steps 1.14 � 10�2 s, 2.06 � 10�2 s and4.06 � 10�2 s, are depicted in Fig. 4. Figs. 4a, 4c and 4e show thebehavior of the velocity at the tip of the bar along the time, whileFigs. 4b, 4d and 4f show the behavior of the stress distribution at aspecific time along the bar. It should be noted that only a smallinterval of the time and space are presented in those figures. Theexact solution of this example can be found in [13].

Fig. 4a. Velocity at the tip of the bar alon

Fig. 4b. Stress distribution along th

Fig. 4c. Velocity at the tip of the bar alon

Some remarks can be done about the behavior of the methodsresults. In all tests, the TDG method presented an overshoot andan undershoot close to the discontinuity, because this method doesnot control the second derivative. The other methods do notpresent these oscillations. For all ranges of Courant numbers, theproposed method (TDG + DC) is less dissipative than the TDG + HHmethod. The difference between the two methods is more

g the time for Courant number 0.57.

e bar for Courant number 0.57.

g the time for Courant number 1.03.

Fig. 4d. Stress distribution along the bar for Courant number 1.03.

Fig. 4e. Velocity at the tip of the bar along the time for Courant number 2.03.

Fig. 4f. Stress distribution along the bar for Courant number 2.03.


pronounced for Courant numbers close to 1. We can observe fromthe results presented that the TDG + DC method does not presentovershoots and undershoots. It should be noted that the methodsTDG + DC and TDG + HH are non-linear; for the numerical tests,presented here, the number of iterations to achieve convergenceof the proposed method was equal to two.

8.2. Example 2 – Non-homogeneous one-dimensional elastic bar

The second example analyzed considers a one-dimensionalelastic bar consisting of two different homogenous domains. Thematerial properties are: L1 = 2 m, E1 = 4 N/m2; c1 = 2.0 m/s andL2 = 2 m, E2 = 1 N/m2; c2 = 1.0 m/s, and the square cross section is


0.04 m2 along the entire bar. A 200 � 1 mesh of quadratic Lagrang-ian elements was used in each time-step. The force applied at thefree end of the bar has intensity of 10�2N and short duration (onlyone time step) as depicted in Fig. 5. Fig. 6 shows results obtainedfor Courant numbers equal to 0.57, 1.03 and 2.03, which corre-spond respectively to the following time steps 1.14 � 10�2 s,2.06 � 10�2 s and 4.06 � 10�2 s.

Figs. 6a, 6c and 6e show the time history of the displacement atthe tip of the bar, while the Figs. 6b, 6d and 6f correspond to the

Lx1 = 2.0m

Material 2

0

Material 1

Lx2 = 2.0m

Fig. 5. Two mater

Fig. 6a. Displacement at the tip of the bar a

Fig. 6b. Stress distribution along th

behavior of the stress distribution along the bar at a specific time.It should be noted that only a short interval of the time and spaceare presented in those figures.

Again, the proposed discontinuity-capturing operator does notpresent oscillations and we observe that for all ranges of Courantnumber, the proposed method (TDG + DC) is less dissipative thanthe TDG + HH method, and the difference between the two meth-ods is more pronounced for Courant numbers close to 1. Again,the convergence of the method was achieved with two iterations.

f(t)

t(s)

10-2N

0 Δt

A f(t)

x

ial elastic bar.

long the time for Courant number 0.57.

e bar for Courant number 0.57.

Fig. 6c. Displacement at the tip of the bar along the time for Courant number 1.03.

Fig. 6d. Stress distribution along the bar for Courant number 1.03.

Fig. 6e. Displacement at the tip of the bar along the time for Courant number 2.03.


Fig. 6f. Stress distribution along the bar for Courant number 2.03.

Lx

Ly

1.0 x0.5 0.6 0.4

1.0

y

0.5

0.6

0.4

A

0.0

Fig. 7. Square membrane under prescribed initial velocity, over the area A, superiorview.


9. Extension for the d-dimensional case

In this section we propose an extension of the discontinuity-capturing operator for d-dimensional problems. However, theextension for the d-dimensional problem of the capturing operatoris not trivial. The results obtained from a one-dimensional experi-ment, suggests that the stabilization parameter of the discontinu-ity-capturing operator is a function of Courant number. However,for problems of dimension greater than one, there are many possi-ble choices to evaluate the Courant number for each direction.Based on numerical experiments, an optimal or quasi-optimalparameter was obtained for one-dimensional problems. However,numerical tests with quadrilateral elements using this parameteron all directions and with only one Courant number suggested:(a) Appropriated Courant numbers must be evaluated on eachdirection and, b) one reduction factor must multiply the one-dimensional stabilization parameter for each direction in d-dimen-sional problems, otherwise, excessive dissipation can appear.

Numerical experiments with quadrilateral elements have indi-cated that an appropriate expression for the Courant number onthe i-direction can be similar to that given in Section 4 and adoptedas follows

Couri ¼jCiiii jq

� �1=2Dt

he;i; ð40Þ

where he,i is as given in Section 4, Eq. (31). The same numericalexperiments also have indicated that an appropriate reduction fac-tor must possess information concerning the distortion of theelement.

By noting that the Jacobean matrix possesses intrinsically thisinformation, we propose the following expression for the reductionfactor on the i-direction

Fred;i ¼ Fred;0ðHe;JÞikHe;Jk

; ð41Þ

with He,J given by

He;J ¼ Jhe;1

:

he;d

0B@1CA; ð42Þ

where J is the Jacobean matrix, (He,J)i is the i-order component ofvector HJ and Fred,0 6 1 is a dimensionless number.

Extensive numerical tests with meshes of quadrilateral ele-ments were made to Fred,0 = 0.25, Fred,0 = 0.50 and Fred,0 = 0.75. De-spite the low sensitivity for this range of values of Fred,0, wasobserved that the best results for all ranges of Courant numberwere obtained with Fred,0 = 0.50.

Preliminary tests with various Courant numbers using triangu-lar meshes obtained from regular meshes with quadrilateral ele-ments, making each quadrilateral into two triangles, have beenmade with Fred, 0 = 0.25, Fred,0 = 0.50 and Fred,0 = 0.75. Again, therewas little sensitivity and the best results were obtained for Fred,0 = 0.50. However it would be good to repeat these tests for trian-gular unstructured meshes in order to confirm this result.

By using the functions l+/�(Couri) obtained for the one-dimen-sional case, one can define the factors Fþ=�1 ðCour1Þ; . . . ; F

þ=�d ðCourdÞ

to each direction x1, . . .xi, . . .xd as follows

Fþ=�i ðCouriÞ ¼ Fred;i � lþ=�ðCouriÞ: ð43Þ

By considering the cross factors Fþ=�mk and Fþ=�t given by the expres-

sions that follow

Fþ=�mk ¼ Fþ=�m ðCourmÞ � F

þ=�k ðCourkÞ

� �1=2; ð44Þ


and

Fþ=�t ¼ Max Fþ=�1 ðCour1Þ; . . . ; F

þ=�d ðCourdÞ

n o; ð45Þ

then for each i 2 {1, . . . ,d} a function Wi(Wh) can be defined for thed-dimensional case as follows

WiðWhÞ ¼Wi;�ðWhÞ; if Couri < 1Wi;þðWhÞ; if Couri P 1

(; ð46Þ

Wi;�ðWhÞ¼ W1i;�ðWhÞ;W2i;�ðW

hÞ;F�t �ðCouriÞ4 1ffiffiffifficip @

2Whi@t2

!; ð47aÞ

W1i;�ðWhÞ¼ F�11di111

@2Wh1@x21

;...;F�jl dijkl@2Whk@xj@xl

;...;F�dddiddd@2Whd@x2d

;

!ð47bÞ

W2i;�ðWhÞ¼ F�11

d1i111ffiffiffiffihip @W

h1

@x1;...;F�jl

djijklffiffiffiffihip @W

hk

@xl;...;F�dd

ddidddffiffiffiffihip @W

hd

@xd;

!ð47cÞ

Wi;þðWhÞ¼ W1i;þðWhÞ;W2i;þðW

hÞ;Fþt1ffiffiffifficip @

2Whi@t2

!; ð48aÞ

W1i;þðWhÞ¼ðCouriÞ�4 Fþ11di111

@2Wh1@x21

; .. . ;Fþjl dijkl@2Whk@xj@xl

; .. .;Fþdddiddd@2Whd@x2d

;

!ð48bÞ

W2i;þðWhÞ¼ðCouriÞ�4 Fþ11

d1i111ffiffiffiffihip @W

h1

@x1; .. . ;Fþjl

djijklffiffiffiffihip @W

hk

@xl; .. .;Fþdd

ddidddffiffiffiffihip @W

hd

@xd;

!ð48cÞ

where Whi is the component of ith order of the vector Wh.

Velocity for Couran

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

0.00 0.05 0.10 0.15

T

Velo

city

(m/s

)

Fig. 8a. Velocity for Cou

Velocity for Cour

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

0.00 0.05 0.10 0.15

Tim

Velo

city

(m/s

)

Fig. 8b. Velocity for Cou

10. Numerical example for two-dimensional case

In this section we present one numerical example for a two-dimensional case. The operator obtained with the methodologyhere presented will be denoted by TDG + DC. The results are com-pared with the exact solution found in [13], classical Newmark’smethod and TDG method.

10.1. Example 3 – Transverse motion of quadrangular membraneunder prescribed initial velocity

The last example considers the transverse motion of a squaremembrane with initial velocity 1 m/s applied transversely overthe shaded area A (0.2 m x 0.2 m) and zero displacements pre-scribed over all boundary and zero initial displacement prescribedover the domain (see Fig. 7). The length of each side of the mem-brane is equal to 1 m and the wave propagation velocity is 1 m/s.The variable chosen to verify accuracy is the velocity time historyat the center of the membrane. Due to the symmetry of the prob-lem, only the one quarter of the membrane needs to be discret-ized. Each space–time slab of the one quarter was discretizedwith 1600 hexahedral elements of twenty-seven nodes. In thisexample, the methods considered were classical Newmark withparameters (d = 0.50 and a = 0.25) as given in [14], TDG andTDG + DC. The results are compared with the exact solution. TheCourant numbers considered were 0.50, 1.00 and 1.75 whichcorrespond respectively to the time steps 6.25 � 10�3 s,1.25 � 10�2 s, and 2.19 � 10�2 s. The results are presented in theFig. 8.

t number 0.50

0.20 0.25 0.30 0.35 0.40

ime (s)

TDGTDG+DCExact solutionNewmark

rant number 0.50.

ant number 1.00

0.20 0.25 0.30 0.35 0.40

e (s)

TDGTDG+DCExact solutionNewmark

rant number 1.00.

Velocity for Courant number 1.75

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

Time (s)

Velo

city

(m/s

)

TDG

TDG+DC

Exact solution

Newmark

Fig. 8c. Velocity for Courant number 1.75.


This example uses the extension proposed to 2D problems inprevious section to the discontinuity-capturing operator. As wellas the one-dimensional examples, this case presents discontinuityin the velocity. As can be seen in Fig. 8, the (TDG + DC) method hasdissipation slightly higher that the TDG method but without spuri-ous oscillations, overshoots and undershoots.

Remark. As can be observed in the results presented, the New-mark’s method has spurious oscillations in the presence of highgradients (on velocities and stresses) while the TDG method doesnot exhibit strong spurious oscillations but only presents over-shoots and undershoots, and has equivalent dissipation or less thanthe Newmark’s method. These observations apply to other tradi-tional methods as well as to all methods of the Newmark’s family.Therefore, the TDG method can be the basis for evaluating theperformance of the (TDG + DC) method presented in this paper.

11. Conclusions

In this paper we present a general methodology to obtain fam-ilies of discontinuity capturing operators for elastodynamics. Thismethodology is based on the work developed in [12] to diffu-sion–convection problems, and inspired in the operator presentedin [7]. The operators are indicated for problems with discontinu-ities. The methodology was applied to problems with discontinu-ities in the time and in the space and for all ranges of Courantnumbers presented the operator proposed here was less diffusivethan the operator presented in [7]. The tests show that the pro-posed method (TDG + DC) does not present overshoots and under-shoots, because the discontinuity-capturing operator controls thesecond order derivatives of the displacement. This difference ismore pronounced for Courant numbers close to 1.

In addition it is proposed an extension for d-dimensional prob-lems. It should be noted that the proposed extension presented inSection 9 is not definitive, but a possible extension for d-dimen-sional case. Robust tests with distorted meshes including thosewith triangular elements must be done, in such a way to validateor suggest modifications on the proposed extension.

The TDG + DC and TDG + HH are non-linear methods, and thenumerical simulations with the proposed discontinuity-capturingoperator give a clear indication that two iterations are sufficient

to achieve convergence of the solution, more iterations lead to noimprovement.

Acknowledgment

The authors gratefully acknowledge the financial support ofCAPES (Coordenação de Aperfeiçoamento de Pessoal de Nı́vel Supe-rior) and the Brazilian Research Funding Agencies (CNPq).

References

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[2] J.H. Argyris, D.W. Scharpf, Finite elements in time and space, NuclearEngineering and Design 10 (1969) 456–464.

[3] I. Fried, Finite element analysis of time-dependent phenomena, AIAA Journal13 (1969) 1154–1157.

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[5] P. Lesaint, P.A. Raviart, On a finite element method for solving the neutrontransport equation, in: C. de Boor (Ed.), Mathematical Aspects of FiniteElements in Partial Differential Equations, Academic Press, New York, 1974, pp.89–123.

[6] T.J.R. Hughes, G.M. Hulbert, Space–time finite element methods forelastodynamics: formulations and error estimates, Computer Methods inApplied Mechanics and Engineering 66 (1988) 339–363.

[7] G.M. Hulbert, T.J.R. Hughes, Space–time finite element methods for second-order hyperbolic equations, Computer Methods in Applied Mechanics andEngineering 84 (1990) 327–348.

[8] G.M. Hulbert, Discontinuity-capturing operators for elastodynamics, ComputerMethods in Applied Mechanics and Engineering 96 (1992) 409–426.

[9] V. Jonh, P. Knobloch, On spurious oscillations at layers diminishing (SOLD)methods for convection–diffusion equations: Part I – A review, ComputerMethods in Applied Mechanics and Engineering 196 (2007) 2197–2215.

[10] X.D. Li, N.E. Wiberg, Implementation and adaptivity of a space–time finiteelement method for structural dynamics, Computer Methods in AppliedMechanics and Engineering 156 (1998) 211–229.

[11] E.G. Dutra do Carmo, A.C. Galeão, A consistent formulation of the finiteelement method to solve convective–diffusive transport problems, BrazilianJournal of Mechanical Sciences 4 (1986) 309–340.

[12] A.C. Galeão, E.G. Dutra do Carmo, A consistent approximate upwind petrov–galerkin method for convection-dominated problems, Computer Methods inApplied Mechanics and Engineering 68 (1988) 83–95.

[13] W.J. Mansur, A Time-Stepping Technique to Solve Wave Propagation ProblemsUsing the Boundary Element Method, Ph. D. Thesis, University ofSouthampton, England, 1983.

[14] K.J. Bathe, Finite Element Procedures, Prentice-Hall, Englewood Cliffs, NewJersey, 1996.

A finite element method based on capturing operator applied to wave propagation modeling1 Introduction2 General elastodynamic equations3 Space–time finite element formulation4 Discontinuity-capturing operators for elastodynamics5 Families of discontinuity-capturing operators6 Formulation for the one-dimensional case7 Determination of the parameters8 Numerical examples for one-dimensional case8.1 Example 1 – Homogenous one-dimensional elastic bar8.2 Example 2 – Non-homogeneous one-dimensional elastic bar

9 Extension for the d-dimensional case10 Numerical example for two-dimensional case10.1 Example 3 – Transverse motion of quadrangular membrane under prescribed initial velocity

11 ConclusionsAcknowledgmentReferences

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