-
Comput. Methods Appl. Mech. Engrg. 201–204 (2012) 127–138
Contents lists available at SciVerse ScienceDirect
Comput. Methods Appl. Mech. Engrg.
journal homepage: www.elsevier .com/locate /cma
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@
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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
-
130 E.G. Dutra Do Carmo et al. / Comput. Methods Appl. Mech.
Engrg. 201–204 (2012) 127–138
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.
E.G. Dutra Do Carmo et al. / Comput. Methods Appl. Mech. Engrg.
201–204 (2012) 127–138 131
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.
-
132 E.G. Dutra Do Carmo et al. / Comput. Methods Appl. Mech.
Engrg. 201–204 (2012) 127–138
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.
E.G. Dutra Do Carmo et al. / Comput. Methods Appl. Mech. Engrg.
201–204 (2012) 127–138 133
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
-
134 E.G. Dutra Do Carmo et al. / Comput. Methods Appl. Mech.
Engrg. 201–204 (2012) 127–138
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.
E.G. Dutra Do Carmo et al. / Comput. Methods Appl. Mech. Engrg.
201–204 (2012) 127–138 135
-
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.
136 E.G. Dutra Do Carmo et al. / Comput. Methods Appl. Mech.
Engrg. 201–204 (2012) 127–138
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
-
E.G. Dutra Do Carmo et al. / Comput. Methods Appl. Mech. Engrg.
201–204 (2012) 127–138 137
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.
138 E.G. Dutra Do Carmo et al. / Comput. Methods Appl. Mech.
Engrg. 201–204 (2012) 127–138
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).
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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