INTERNATIONAL
OURNAL FOR
NUMERICAL
METHODS
IN
ENGINEERING,
VOL.
29, 10031020 (1990)
ADVANCED DEVELOPMENT OF TIMEDOMAIN BEM FOR TWODIMENSIONAL SCALAR WAVE PROPAGATION
A.
S.
M.
ISRAIL. AND P.
K.
BANERJEE'
Department
of
Civil Engineering, Stale
UniDersity
of
New
York
at
Bufalo,
Bufalo,
New
York
14260,
U.S.A.
SUMMARY
An
improved formulation
for
solving
2D
transient scalar wave propagation problems by the Boundary Element Method (BEM) is presented. The kernels presented are simpler and better behaved than those that have appeared in the published literature. An appropriate set of temporal shape
functions
for
linear variation is used.
For
spatial variations isoparametric quadratic elements are
used.
All
of
these represent significant improvements over the present level
of
sophistication in the analysis
of
2D
transient scalar problems.
The
algorithm is implemented in a general purpose boundary element
code
known
as
GPBEST.
INTRODUCTION Scalar wave propagation problems are commonly encountered in various engineering disciplines. The propagation of acoustic waves, water waves, electromagnetic waves, etc. is all governed by the scalar wave equation. Kirchoff was the first to formulate the integral equation for scalar wave problems in terms of unknown potentials. However, the solution of such problems started only in the early
1960's
with the pioneering work of Friedman and Shaw.2 They solved
20
transient acoustic wave scattering problems by considering them as
3D
cylindrical ones with an arbitrary axis length using a timemarching scheme. Later, Banaugh and Goldsmith3 extended it to timeharmonic prob lems. Groenenboom4 recently reviewed some
of
these efforts and presented a general bounday element formulation for the solution of transient potential fluidflow problems in three dimen sions. It has been extended by Groenenboom
et
aL5
to incorporate coupling with the Finite Difference technique. Niwa
et
~1 ~
nd Fukui' derived the twodimensional boundary integrodif ferential equations directly from the threedimensional form of Love's integral identity and presented displacements due to the diffraction of an incident wave by a cylindrical cavity. Cole
et
af.8
mplemented a transient timestepping boundary element algorithm for scalar problems using a constant boundary element, which they found to
be
unstable, leading to a build up
of
errors. Nevertheless, this was a significant contribution towards a general
2D
imedomain formulation using a 2D timedependent fundamental solution. Later Mansur' used a similar timestepping scheme and incorporated both constant and linear elements
in
space together with constant and linear temporal variations. However, his transient kernel corresponding to the derivative
of
potential is obtained through complex manipulation of the Heaviside function appearing in the
Graduate
Student
Professor
00295981/90/05100318 09.00 990
by John Wiley Sons, Ltd.
Received
29
April
1989
Revised
28 July 1989
1004
A.
S.
M.
ISRAIL
AND P.
K.
BANERJEE
potential kernel. Also the resulting kernel, after convolution, is unable to produce the steadystate kernel in a single large time step, although this reduction to the steadystate value is a very important characteristic
of
the transient kernel. Moreover, he defined the temporal linear shape functions as a triangular function over two successive time steps, which is not at all justified. In the present paper, the potentialderivative kernel is derived by satisfying the causality condition and by taking the required derivatives in a consistent manner. The resulting kernel is explicit, simpler and, once the convolution integral is evaluated, it reduces to the steadystate counterpart for both constant and linear timevariations. The linear temporal shape functions are appropriately defined within one time step. The resulting formulation leads to very accurate results, as can be seen in the examples presented. Moreover, the algorithm presented incorporates isoparametric quadratic elements and has the capability
of
tackling problems with multiple regions by satisfying the continuity
of
potential and fluxes across common interfaces. The problem
of
socalled 'shadow zones' as discussed by Groenenboom4 and Antes and von EstorlT'O is also investigated in the context of
2D
scalar propagation problems. It has been shown that, contrary to their observation, the correct solution is reproduced even for concave surfaces since the formulation automatically obeys the causality and integrations are carried out to at least four digit precision. It
is
to be mentioned here that apparently the observation of 'shadow zones' has been recently* corrected by those researchers.'
'
TIMEDOMAIN BEM FORMULATION
I.
Governing equation
isotropic, homogeneous body is given by The differential equation governing transient scalar wave propagation through an elastic,
1
azp
c2
at2 V2p(x, )

(x, t)
+
b(x,
t)
=
0
where
p
is the potential,
x
the position vector,
t
the time,
b
the body source,
V2
the Laplacian operator and
c
the speed of wave propagation given by
@,
ith E the elastic modulus and
p
the mass density. For a wellposed problem, equation
1)
is accompanied by appropriate initial and boundary conditions.
2.
Boundary integral representation
The solution of equation
(1)
for
a
unit body source is known as the fundamental solution and is denoted by
G(x,
t;
t,
r).
It
represents the potential at a point
x
at time
t
due to a unit source applied at
t
at a preceding time
T.
The fundamental solution for twodimensional problems is well known.I3 The corresponding kernel for the potential derivative is obtained (as discussed later) by
F(x, t;
t,
r)
=
(aG/dn)(x,
;
t,
T).
The fundamental solution state, mentioned above,
can
be
com bined with the actual state through the use
of
the dynamic reciprocal identity, to yield the following integral eq~ation,'~
c(t)p(tt
t)
=
I
I
CW
;
t,
Wx,
)

W
;
t,
~P(x,
1
dr dW
(2)
so
Suggested
by
one
of
the
reviewers
BEM
FOR
2D
SCALAR
WAVE
PROPAGATION
1005
where and
c(()
is known
as
the jump term and has the following values. (i)
1.0
for within the volume,
V
(ii)
0.5
for on the smooth boundary,
S
and (iii)
0
for outside
V.
In equation
(2),
the body is assumed to be initially at rest and no sources are present within the body. Solution of equation
(2)
is the essence of the Boundary Element Method.
3. Numerical implementation and solution procedure
Equation
(2)
is
an exact representation of the transient problem since
no
approximations have been introduced
yet.
However, for the solution of practical problems, suitable approximations are needed for both.spatia1 and temporal variations of the field variables.
As
will
be
seen later, the timefunctions involved are explicit and simple enough to perform the timeintegration analyti cally. However, numerical techniques are warranted for the spatialintegration. For temporal discretization, the time interval from
0
to
t
is divided into
N
equal
increments of duration
At,
i.e.
t
=
NAt.
Within each time step, the variables can be assumed to remain constant or vary linearly. Depending on such variations, the solution algorithm will
be
elaborated.
(i)
Constant time variation.
Since the fieldvariables remain constant during
a
time step, they can
be
taken out of the timeintegral,
so
that only the kernels have to
be
integrated. The timeintegrated kernel is denoted by GN+'
=
11:1
(x,
NAt;
r
r)d7
(3)
Combining this and a similar expression for the Fkernel in equation
(2),
one obtains
N
c(()pN(()
=
1
[G ' @(x)

FN'l pn(~)]dS(~)
(4)
n=
1
where
pN )
stands for the potential at time
t
=
NAt.
fieldvariables and they are given, in terms of the nodal quantities, by For the spatial discretization, quadratic variation is assumed over both the geometry and where
i
=
1,2
(for
2D),
a
=
1,2,3
3
nodes).
NJq)
are the shape functions and
q
the intrinsic coordinate
of
the elements. The Jacobian of transformation
is
1006
A.
S.
M.
ISRAIL AND
P.
K.
BANERJEE
After the spatial discretization, explained above, equation
(4)
takes the form where
M
represents the total number
of
boundary elements. Fkernel, the singular integral
of
the Fkernel
is
evaluated in the following way: Since the transient Fkernel has the same type of singularity
as
the corresponding steadystate where
F a S
nd
F
are the transient and steadystate Fkernels respectively. The first integral on the right side of equation (8) involving the steadystate kernel is singular and its evaluation using the uniform potential and the resulting zero flux solution is well known. However, for this process the body must have a closed boundary. Thus, for problems with an open boundary, the region of interest must be enclosed with special kinds
of
elements known as 'enclosing elements'. The steadystate kernel is integrated
only
during the first time step. More over, these elements do not have to be as fine as those used for modelling the problem. Thus, the additional computational effort warranted by these elements
is
not significant. Details of this can be found in papers by Ahmad and Banerjee and Henry and Banerjee.16 The second integral is nonsingular and its numerical treatment does not pose any special difficulty. In general the kernel functions are not continuous over the integration intervals and intelligent subsegmenta tion needs to
be
done
so
that these are evaluated to at least four digit precision. Moreover,
if
the wave has reached over part
of
an element the integration is carried out only over that portion of the element. Equation
(7)
is written for every boundary node and after the appropriate integration, can be cast into the following set of equations: where
{
p } an
{
q } are vectors of boundary nodal quantities, with the superscript referring to the time step index. At time
t,
only half of the boundary values are unknown. The remaining are known;
so
are the values at all previous times. Thus, putting the unknowns on the left, equation
(9)
can be rearranged as
N
1
[A']{XN}
=
[S']{
Y )

1
[GN'ln](@}

FN n](p })
(10)
n=l
which is put into the following simplified form: [A']{XN}
=
[B']{
Y }
+
{RN}
in which
{
X } and
{
Y },
respectively, are the vectors of the unknown and known variables and
{RN}
epresent the effects of past dynamic history on the current time node, N. Solution of equation
ll),
which involves only real quantities, can
be
carried out with any standard technique.
BEM
FOR
2.D
SCALAR
WAVE
PROPAGATION
1007
where
All ?)
and
MJT)
are the temporal interpolation functions corresponding to the local time nodes
1
and
2
(Figure
l),
and are expressed
by
?

nl
M1(T)
=
At
(13)
?,,I
<
T
<
tn
n

At
M2('T)
=
Here, the timeintegration involves the kernel and the temporal shape function products. After the analytical timeintegration, two sets of kernels are obtained, one corresponding to each time node. This is in contrast to one set of kernels for the constant variation case. The timeintegrated linear kernels are Expressions for the Fkernel are similar. The subscripts refer to the local time nodes. Following the procedure, similar to constant formulation, one obtains
N1
[A:]{XN)
=
[B:]{
Y }

1
[GY+'
+
Gyr]{qr)
r=l

F;+I
+
Fyn]{p*})
which can be simplified to
CA:I{XN}
=
CB:1{YN}
+
{RN}
Figure
1.
Local
time nodes at
a
time step