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Journal ArticleDOI

A comparative study on three boundary element method approaches to problems in elastodynamics

01 Jan 1983-International Journal for Numerical Methods in Engineering (John Wiley & Sons, Ltd)-Vol. 19, Iss: 1, pp 73-91

AbstractIn this study, three boundary element method approaches are compared by solving a typical problem in linear elastodynamics. The first approach formulates and solves the problem in the real time domain in conjunction with a time-stepping algorithm. The other two approaches formulate and solve the problem in the Laplace and the Fourier transformed domains, which necessitates a numerical inversion of the results to the time domain. The results obtained compare favourably with available analytic solutions. A detailed tabulation of the computer time and memory requirements of each approach is presented.

Summary (3 min read)

INTRODUCTION

  • An approach in which the integral equation formulation is applied to the Laplace transform of the governing equations of motion for the two-dimensional case.
  • All of the aforementioned cases share some common traits.
  • Following that, the results drawn from applying the three aforementioned BEM approaches to the same problem, which is shown in Figure 1 , are discussed in detail.

STATEMENT OF THE PROBLEM AND DEFINITIONS

  • Let V denote the area occupied by a given body and S denote its bounding surface.
  • Associated with each surface point is an outward pointing unit normal vector n.
  • Under the assumption of small displacement theory and linear elastic isotropic and homo~eneous material behaviour, the equations of motion of the body are (1) where u;(x, t) is the displacement vector and[; is the body force vector per unit mass.
  • The propagation velocities of the pressure and shear waves in the body are given, respectively, as ci=(A+2M)/p; c~=IJ.IP (2) where A and IJ. are the Lame constants and p is the mass density.
  • Furthermore, commas indicate space differentiation, dots indicate time differentiation and repeated indices imply the summation convention.

The Laplace transform

  • A Laplace transform of the equations of motion (1), which are hyperbolic partial differential equations, results in the following system of elliptic partial differential equations: ( EQUATION which is more amenable to numerical solutions.
  • It should be noted in passing that the Navier equations of equilibrium are also of elliptic nature, i.e. they are similar in form to equation (7).
  • This series of solutions must finally be numerically inverted back to the original time domain.
  • Also, the body forces are taken equal to zero as well.
  • These assumptions are made for the sake of convenience and no additional conceptual difficulty in the solution procedure is encountered if these assumptions are relaxed.

INTEGRAL EQUATION FORMULATION

  • It is well known (DeHoop/ 1 Wheeler and Sternberg 22 and Kupradze 4 ) that a system of partial differential equations along with the appropriate boundary and initial conditions may be cast in an integral equation form.
  • This holds true for either the system of equations (1)- (5) or the system of equations ( 7)-( 9), depending on whether one wishes to establish an integral equation formulation in either the time domain or in a transformed domain.
  • The basis for an integral equation formulation in elastodynamics is the dynamic extension of Betti's reciprocal theorem.
  • The obvious choice for one of the two elastodynamic states is the unknown state the authors are seeking to find.
  • For more details, an appropriate reference is Banerjee and Butterfield.

Time domain integral equation formulations

  • In (11), the prime symbol used as a superscript denotes three-dimensional quantities with the pertinent subscripts ranging from 1 to 3.
  • Finally, F ;i represents the resulting tractions due to the same unit impulse and can be obtained from G ~i through the use of the constitutive equations and surface normals.
  • Along with details of the solution procedure.
  • The approach delineated there is essentially a three-dimensional formulation applied to an infinitely long cylindrical body whose longitudinal axis coincides with the Z axis of the cartesian co-ordinate system .
  • The problem, of course, remains a plane strain one.

Laplace transform domain integral e'quation formulation

  • In the Laplace transform domain, the Green's function can be found in Cruse and Rizzo 8 and Nowacki 24 and the singular integral equations are of the form EQUATION.
  • The explanatory notes following equation ( 11) apply here as well.
  • It should be reminded at this point that bars over a quantity denote the Laplace transform of that quantity.
  • The solution procedure in the Laplace transform domain is elaborated in detail in Manolis and Beskos 10 and only a few remarks will be made here.

Fourier transform domain integral equation formulation

  • For this case, the fast Fourier transform algorithm of Cooley and Tukey, 26 which has been extensively used by numerous investigators in the past is employed.
  • Since the upper and lower limits in (18) are replaced by 0 and !1 (the maximum frequency of interest), the well-known problem associated with the finite Fourier transform, namely the approximation a non-periodic motion by a periodic one, arises.
  • In the particular example used here, this numerical difficulty was not encountered.
  • Some details on this subject can be found in Kobayashi and Nishimura.

NUMERICAL EXAMPLE

  • This section describes the detailed solution of a numerical example that serves as the vehicle for a comparison study of the three BEM approaches.
  • It can be observed from Figures 4 and 6 that the results obtained from the three BEM formulations agree reasonably well with the results of analytic-numerical approaches.
  • The true behaviour at this location is probably better represented by the results obtained by Baron and Matthews, because when the P-wave pulse first reaches the cavity, the points around(}= oo act as a rigid wall and as a consequence the applied stress iiy should double.
  • Also, it should be mentioned that all computations were performed on a CDC Cyber 176 computing machine.

CONCLUSIONS

  • On the basis of the preceding presentation, the following conclusions can be drawn: 1. Three general BEM formulations for solving problems in transient elastodynamics involving bodies of arbitrary shape under conditions of plane stress or plane strain are presented.
  • All three formulations are viable alternatives for the solution of general elastodynamic problems.
  • Integral transform formulations tend to reach a plateau corresponding to the solution of the static equivalent of the problem, while time domain formulations tend to diverge after a very large number of time steps.
  • To circumvent this economic difficulty, one may employ, in the beginning stages of an analysis/design procedure, simplified methods of analysis and resort to the powerful BEM approaches at the end of the procedure.
  • Some further advantages of the time domain BEM are that it is more suitable for extension to nonlinear material behaviour through the use of incremental and iteration schemes and that the extension to three-dimensional problems is straightforward.

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A comparative study on three boundary element
method approaches to problems in elastodynamics
George D. Manolis
To cite this version:
George D. Manolis. A comparative study on three boundary element method approaches to problems
in elastodynamics. International Journal for Numerical Methods in Engineering, Wiley, 1983, 19 (1),
pp.73 - 91. �10.1002/nme.1620190109�. �hal-01382196�

A COMPARATIVE
STUDY
ON THREE
BOUNDARY
ELEMENT METHOD APPROACHES TO PROBLEMS
IN ELASTODYNAMICS
GEORGE
D.
MANOLIS
Department
of
Civil Engineering, State University
of
New
York
at
Buffalo, Buffalo,
New
York,
U.S.A.
SUMMARY
In this study,
three
boundary
element
method
approaches
are
compared
by
solving a typical problem
in linear elastodynamics.
The
first approach formulates and solves
the
problem in
the
real time domain
in conjunction with a time-stepping algorithm.
The
other
two approaches formulate
and
solve
the
problem in
the
Laplace
and
the
Fourier transformed domains, which necessitates a numerical inversion
of
the
results
to
the
time domain.
The
results obtained compare favourably with available analytic
solutions. A detailed tabulation of the
computer
time and memory requirements
of
each approach is
presented.
INTRODUCTION
Perhaps
surprisingly,
the
use
of
integral equation formulations in
the
analysis of transient
phenomena
in solids
and
fluids has a long history, going back
over
one
hundred
years
to
the
Helmholtz-Kirchhoff integral formula, which is
the
mathematical interpretation of Huygens'
principle.
1
In a large
number
of these
phenomena,
part
of
the
boundary
is
at
infinity, which
makes the use of integral
equation
methods
particularly attractive, since they
are
capable of
accounting for
the
presence
of
infinite domains. Morse and Feshbach,
2
Eringen
and
Suhubi
3
and
Kupradze
4
are
but
a few of the references
that
contain details
of
the classical work done
in elastodynamics
and
related topics.
Although as
mentioned
above the basic integral equation formulations for wave propagation
problems have
been
known
for
quite
some
time, their adaptation for constructing numerical
algorithms for use in the solution of
boundary
value problems is new. Some of the earliest
such developments were by
Friedman
and
Shaw
5
and
Chen
and
Schweikert,
6
in acoustics,
and
by
Banaugh
and
Goldsmith,
7
in steady-state elastic wave propagation. Since then, many
other
investigators have formulated
and
used numerical implementations known as integral equation
methods (IEM),
boundary
integral
equation
methods
(BIEM)
or
boundary
element methods
(BEM) in various fields of engineering.
In
the field of elastodynamics, Cruse
and
Rizzo
8
and
Cruse
9
derived a
BIEM
approach in conjunction with
the
Laplace transform
and
employed
it
to
solve a half-plane wave propagation problem. A modified version of their approach was
used later
by
Manolis
and
Beskos
10
to
investigate stress concentrations
around
cylindrical
openings of arbitrary
shape
due
to
the
scattering of elastic waves of a general transient nature.
The
determination of
the
steady-state solution
by
BEM
approaches and the reconstitution of
the
transient response by
Fourier
synthesis was
done
by Niwa
et
al.
11
'
12
in the course of their
investigations of the transient stresses produced
around
cavities during the passage of travelling

waves. A time
domain
formulation
and
solution for elastic wave scattering problems was
done
by
Cole
et
al.
13
for
the
anti-plane strain case
and
by
Niwa et al.
14
for
the
general two-dimensional
plane
stress/plane
strain
cases.
It
therefore
appears
that
a study comparing
the
various
BEM
approaches
used
to
solve
the
propagation, diffraction
and
scattering
problems
arising in
the
field
of
elastodynamics would
be
of value
to
the
engineering
community.
In
this work,
three
possible
BEM
approaches
are
investigated
and
compared.
All
three
approaches
are used
to
study
the
classical
problem
of
the
scattering
of
a
pressure
wave pulse by a circular cylindrical cavity in a linear elastic,
homogeneous
and
isotropic medium. This is a two-dimensional
problem
of
the
plane
strain
kind.
Analytic-numerical
solutions for
the
displacement, velocity
and
stress fields
at
the
circumference
of
the
cavity
were
obtained
by
Baron
and
Matthews,
15
Baron
and
Parnes
16
and
by
Pao
and
Mow.
17
The
former
two pairs
of
investigators
used
an
integral transform,
separation
on
the
circumferential angle (8)
method.
The
latter
investigators (Pao
and
Mow) used
the
method
of wave function expansion coupled with
the
Fourier
synthesis technique. Miklowitz
18
used a special technique to bring
out
hitherto
undetected
late-time singular, non-decaying
Rayleigh waves
on
the
cavity walls
due
to impulsive loads.
It
is interesting to
note
that
this
classical
problem
has
been
resolved recently
by
other
analytical
means
as, for instance,
the
method
of
matched
asymptotic expansions discussed
by
Datta.
19
The
three
BEM
used
to
resolve this well-documented
problem
are:
1. A time
domain
approach
that
solves
the
problem
as if it
were
a three-dimensional one. In
particular,
the
plane
strain
problem
is
represented
as a circular cylinder of infinite length
and
a numerical integration
is
performed
along
the
axis of
the
cylinder so as
to
collect
the
contributions
to
a
representative
point
from points lying along
the
same
generator.
2.
An
approach
in which
the
integral
equation
formulation is applied
to
the
Laplace transform
of
the
governing
equations
of
motion
for
the
two-dimensional case.
In
this case, a numerical
inverse
transformation
is
required
to bring
the
transformed solution
back
to
the original time
domain.
3.
The
last
approach
is similar
to
case 2, except
that
the
integral transform in question is
the
Fourier
transform
instead
of
the
Laplace transform.
All
of
the
aforementioned
cases
share
some
common
traits. First,
the
direct
BEM
is
employed,
where
the
unknown
boundary
tractions
and
displacements are directly related
to
the known ones.
One
could use
an
indirect
BEM,
as was
done
in Niwa et a!.
11
For
more
details
on
direct
and
indirect
BEM,
reference should
be
made
to
Banerjee
and
Butterfield?
0
Second, all
approaches
utilize a spatial discretization scheme for which
the
boundary
tractions
and
displacements
are
assumed
to
have constant values
over
a given
boundary
segment.
Finally, considerable effort is
directed
towards ensuring
that
all
three
approaches
share
as
many
common
features
as possible, in
order
to
render
the
comparison
study
as fair as possible.
In
the
next few sections,
the
integral
equation
formulation
of
the
general elastodynarnic
problem
along
with
the
numerical
implementation
is discussed. Following that,
the
results
drawn from applying
the
three
aforementioned
BEM
approaches
to
the
same
problem,
which
is shown in
Figure
1,
are
discussed in detail.
STATEMENT
OF
THE
PROBLEM
AND
DEFINITIONS
Let
V
denote
the
area
occupied by a given
body
and
S
denote
its bounding surface. A
co-ordinate
system
(x,
t) is employed,
where
x
denotes
the
cartesian spatial co-ordinates x
1
and
x
2
and
tis
the
time. Associated with
each
surface
point
is
an
outward
pointing unit normal
vector n.

y
Figure
1.
Circular
cavity in
an
infinite
medium
under
the
action
of
an
incoming
pressure
wave
Under
the
assumption
of
small
displacement
theory
and
linear
elastic
isotropic
and
homo~eneous
material
behaviour,
the
equations
of
motion
of
the
body
are
(1)
where
u;(x,
t)
is
the
displacement
vector
and[;
is
the
body
force
vector
per
unit
mass.
The
propagation
velocities
of
the
pressure
and
shear
waves
in
the
body
are
given, respectively, as
ci=(A+2M)/p;
c~=IJ.IP
(2)
where
A
and
IJ.
are
the
Lame
constants
and
p is
the
mass
density.
Furthermore,
commas
indicate
space
differentiation,
dots
indicate
time
differentiation
and
repeated
indices imply
the
summation
convention.
Since in
what
follows
attention
is
restricted
to
two-dimensional
cases,
the
indices
i
and
j
assume
the
values
of
1
and
2, unless
mentioned
otherwise.
The
displacement
vector
uJx,
t)
is
assumed
to
be
twice
differentiable
except
at
possible surfaces
of
discontinuity,
as
discussed in
Eringen
and
Suhubi.
3
The
displacements
and
the
tractions
t; (x,
t)
satisfy
the
boundary
conditions
I;=
U;;n; = p;(X,
t)
ui = q;(x,
t)
where
u;;
is
the
stress
tensor
and
S = S
1
+ Su.
The
displacements
and
velocities also satisfy
the
initial
conditions
U;
(X,
0+) =
U;o(x)
U;(X,
0+) = uiO(x)
XE
V+S
XE
V+S
(3)
(4)

In addition,
the
Sommerfeld
radiation
condition
must
be
satisfied if
the
body
in
question
is of infinite extent.
Finally,
the
constitutive
equations
are
of
the
form
cr··=p[(c
2
1-2c2
2
)u
~--+c2
2
(u-
+u··)]
I]
r,,UI]
1,]
],1
(5)
with
8ii
being
the
Kronecker
delta.
The Laplace transform
The
Laplace
transform
of
a function f(x,
t)
is defined as
/(x,
s)
= .P{f(x, 1)} = r f(x,
t)
e
_,
dt
(6)
where s is
the
Laplace
transform
parameter
and
f(x,
t)
must
be
at
least
piecewise
continuous
in time. A
Laplace
transform
of
the
equations
of
motion
(1), which
are
hyperbolic partial
differential
equations,
results in
the
following system
of
elliptic
partial
differential equations:
(
2
2)
- + 2 - +
f-
2 - + . + 0
C1
-c2
Ui,ii
CzUj,ii
j
-s
Uj
UjO
SUjo
=
(7)
which
is
more
amenable
to
numerical solutions.
It
should
be
noted
in passing
that
the
Navier
equations
of
equilibrium
are
also of elliptic
nature,
i.e.
they
are
similar in form
to
equation
(7). Since
the
boundary
conditions
and
the
constitutive
equations
do
not
involve time deriva-
tives,
their
Laplace
transforms
are
very simple
and
become,
respectively,
and
~
=
iJiini
= Pi(x, s)
ui=qi(x,s)
ii"··
= p [(c
2
1-
2c2
2
)u
~--
+ c
2
2(u· ·
+ u ·
·)]
I]
r,,UIJ
1,]
],1
(8)
(9)
Essentially,
the
solution
to
a
transient
elastodynamic
problem
when
integral transforms
are
employed, such as
the
Laplace
or
Fourier
transform, consists
of
a series of solutions
to
a
static-like
problem
for
a
number
of discrete values
of
the
transformed
parameters.
This series
of solutions
must
finally
be
numerically
inverted
back
to
the
original
time
domain.
In
what
follows, a quiescent
state
is
assumed
before
time
t =
o+,
which implies
that
the
initial conditions
are
zero. Also,
the
body
forces
are
taken
equal
to
zero
as well.
These
assumptions
are
made
for
the
sake
of
convenience
and
no
additional conceptual difficulty in
the
solution
procedure
is
encountered
if these assumptions
are
relaxed.
The Fourier transform
The
Fourier
transform
of
a function f(x,
t)
is defined as
ic
x,
w)
= .'F{f(x, t)} = L: t(x,
t)
e -
<w•
dt
(10)
where
w is
the
frequency.
For
the
quiescent
initial conditions
assumed
previously, it
is
elementary
to
see
that
the
Fourier
transform
of
the
governing
equations
of
motion ( 1) will
be
identical
to
equation
(7)
if
the
Laplace
parameter
s
is
set
equal
to
iw in
that
equation.
Therefore,
the
BEM
solution
in the
Fourier
domain
is
identical
to
the
solution
scheme
that
is
to
be
employed for
the
Laplace

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