The Helmholtz equation governing an interior domain with shell discontinuities is not efficiently solvable by the traditional boundary element method.
Computer methods in applied mechanics and englneerlng Comput. Methods Appl. Mech. Engrg. 140 (1997) 393-404
ELSEVIER
Solution
of discontinuous interior Helmholtz problems the boundary and shell element method
by
Stephen M. Kirkup Science Research Institute, University of Saljord, Salford M5 4WT, UK
Received 31 January 1996; revised 17 May 19%
Abstract The Helmholtz equation governing an interior domain with shell discontinuities is not efficiently solvable by the traditional boundary element method. In this paper it is shown how the Helmholtz equation can be recast as an integral equation known as the boundary and shell integral equation. The application of collocation to the integral equation gives rise to a method termed the boundary and shell element method.
The associated problem of finding the eigenvalues and eigenfunctions of the Helmholtz equation in a discontinuous domain via the same method is also considered. This leads to a non-linear eigenvalue problem. Such a problem may be solved through polynomial interpolation of the matrix components. In this paper methods for solving the Helmholtz equation and the associated eigenvalue problem are implemented and applied to a test problem.
1. Introduction
The linear BEM PDE.
boundary element method (BEM) is an established computational method for the solution of elliptic partial differential equations (PDEs) (see [14], for example). However, the standard cannot be applied directly to problems with discontinuities in the variables in the domain of the In this paper boundary value problems and eigenvalue problems of the Hehnholtz equation,
V%&) +/+(p)
= 0
)
(1)
in a discontinuous interior domain are considered. The discontinuity is assumed to have the topology of a shell-an open surface in three-dimensional problems, a line in two dimensions. A two-dimensional illustration of the general domain is given in Fig. 1. The traditional BEM is derived from a boundary integral equation (BIE) formulation of the PDE by dividing the boundary into boundary elements and applying an integral equation method (usually collocation) to obtain the solution. For domains in the form of Fig. 1, the traditional BEM can be applied by subdividing the domain into subdomains, as illustrated in Fig. 2. Boundary integral equation reformulations of the PDEs on each subdomain can now be obtained through coupling these equations across common boundaries by enforcing continuity conditions the solution throughout the domain can be obtained. However, the application of this technique does have general disadvantages. One drawback is that further boundaries for each subdomain need to be introduced which will increase the number of elements required and the computational expense. Furthermore, the division of the domain into subdomains needs to be done with care as this in itself could introduce corners and subsequent computational inefficiency. A similar method to the traditional BEM for the solution of a PDE in the infinite domain exterior to a shell discontinuity can be derived through recasting the PDE as an integral equation termed a 00457825/97/$17.00 @ 1997 Elsevier Science S.A. All rights reserved PII SOO45-7825(96)01117-6
394
S.M. KirkuplComput.
Methods Appl. Mech. Engrg. 140 (1997) 393-404
Fig. 1. Illustration Fig. 2. Preparation
of the general for application
domain. of the BEM.
shell integral equation. A numerical method can then be derived in a similar way to the BEM (see, for
example, [5,6]). In Kirkup [7,9] it is shown how the Laplace equation in a domain such as that of Fig. 1 can be reformulated as an integral equation termed a boundary and shell integral equation (BSIE) and how such problems may be solved through collocation. Boundary element methods have traditionally fallen into two distinct classes, direct BEMs and indirect BEMs, based on direct and indirect integral equation formulations. In this paper direct and indirect boundary and shell integral equation formulations are given for the interior two-dimensional Helmholtz equation. The BSIEs are a hybrid of the corresponding direct or indirect boundary integral equation with the shell integral equation for the Helmholtz equation. Hence, new integral equation based methods for the solution of the discontinuous interior Helmholtz equation are introduced in this paper. The same formulations are also suitable for three-dimensional interior Helmholtz problems, with the appropriate selection of Green’s function. An illustration of the domain is given in Fig. 1. It consists of a region D with boundary S and with shell discontinuities lY In order to specify the problem fully, conditions for points on the boundary and on the shell must be stated, these are termed the boundary condition and the shell condition. In [6], the shell integral equations are derived by first assuming that the shells have finite thickness and hence the standard boundary integral equation formulation is valid. The shell thickness is then allowed to approach zero. A similar limiting process can be used to derive the boundary and shell integral equation by assuming S to be fixed and taking the limit as the thickness of the shells approach zero. The integral equation formulations of the interior Helmholtz problems, both for the boundary value problem and the eigenvalue problem, are stated in Section 2. In order to derive a particular method, the boundary and shell are divided into uniform elements and the functions defined on the boundary and shell are approximated by a constant on each element. The integral equation method, termed the boundary and shell element method (BSEM), is then derived through collocation. The application of collocation to the integral equations is described in Section 3 and the resulting formulation of the direct and indirect boundary and shell element methods is described in Section 4. The methods are applied to the test problem where the domain is the unit square and a discontinuity lies between (i, i) and (i, 1). Results from the application of direct and indirect BSEM are given and compared with results from the application of the traditional boundary element method. The solution of the Helmholtz eigenvalue problem in a discontinuous domain is considered in Section 6. That is the computation of the non-trivial solutions of (1) with a homogeneous boundary condition. The method introduced in [8] for solving the Helmholtz eigenvalue problem via the boundary element method is adapted for the BSEM. In Section 7, the direct and indirect methods are demonstrated through their application to the test problem. 2. Integral equation formulation In this section the direct and indirect boundary and shell integral equation formulations
of the interior
S.M. Kirkup/Comput.
Methods Appl. Mech. Engrg. 140 (1997) 393-404
395
Helmholtz equation are given. The boundary and shell integral equations may be regarded as hybrids of their respective standard boundary integral equation formulations as given, for example in [l] and shell integral equation formulations given in [5,6]. 2.1. Notation Let the function u@) for p E S be defined as follows:
where np is the unit outward normal be the upper surface and let r_ be discontinuous at the shell, however q_(p), u+(p) and iy_(p) (p E I’) be
v-(p) = hi /
2
to S at p. Each shell is assumed to have two sides or surfaces, let r+ the lower surface. The potential cp and its derivatives are generally they take limiting values on r+ and r_. Let the functions q+(p), defined as follows:
(p - &np) . P
The geometrical function c(p) (jr E S u I’) is defined to be the angle subtended by the interior region at p for points on S and the angle subtended the region at r+ for points on l-‘, each angle then divided by 2~. It is helpful to introduce the functions Sk), Y(P), Q(p) and V(p) for p E r which are defined as follows:
@(PI= c(P>ro+(P) + Cl- dP>>cp-(PICPE n V(P) = C(p>i~+o?> - (1 - ccp))u-(p) (p E r>. 2.2. Boundary and shell conditions The boundary condition has the form 4r)Voo7) + NP>+>
= Y(P)
(P E 9,
where a(p), P(P) and Y(P) are functions of p on S. The shell conditions following general form at&W
+ bW0)
= f(p)
A~)@~)+J~~)V~)=~~)
07 E r),
(PEG),
where a(p), b(p), ,f(p), A(p), B(p) and F(p) are functions of p on r.
are assumed to have the
S.M. KirkupIComput.
396
Methods Appl. Mech. Engrg. 140 (1997) 393-404
2.3. Integral operator notation The Laplace integral operators
G/&,4)
L k, Mk, ML and Nk are defined as follows:
/-&)d%
07 E EuSur),
where 17 c S u r, n, and rzp are unit outward normal to 17 when Il c S or the unit normal to I7+ when I7 c r at q, p and p(q) is a bounded function defined for q E II. Gk@, q) is the free-space Green’s function for the Helmholtz equation, in two dimensions,
Gk(& 4) = ; @‘)(kr)
where r = p - q and r = Ir I. The function #’ zero.
is the spherical Hankel function of the first kind of order
2.4. Direct integral equation formulation The equations that make up the boundary and shell integral equation formulation of the Helmholtz equation are given in this subsection. For points on the boundary the following equation holds {Mk(P)s(P) +c(P)‘Po(p) =
{Lkvhb>
+ {Mk~)r(P)
-
b’ 6 s>.
{Lkv)&)
This equation relates cp(p) and u(p) for points p on the boundary the following equations: @i(p) = -{M/&‘)s(p)
+
v(P) = -{N/&h(P)
+ M$M.P)
{Lkv)Sb)
+ {Mk~)r(p)
-
{Lk~)r@>
+ {N/&r(P)
- W,~M.P)
(2)
S. For points on the shell, we have (P
E r)
,
k’ E r> +
(3) (4)
The value of cp(p) for points in the domain are related to the solutions on S and r through the following equation: 9(P) = -{M/‘(P)&)
+
{Lku)S(P>
+ {Mka)i-(P)
-
{Lkv)l-b>
(P E D> .
(5)
2.5. Indirect integral equation formulation The equations that make up the indirect boundary and shell integral equation formulation of the Helmholtz equation are given in this subsection. For points on the boundary the following equations hold: ‘P(P) = di’>
tLkah(P)
= W,%M4
+ {Mka)r(P> + c(P>d~>
-
+ {Nk~)d.P>
(6)
0, E s>,
{Lkv)I-b> -
W,%-W
(P E s>,
(7)
where (+ is generally known as a source density function. These equations relate cp(p) and v(p) for points p on the boundary S. For points on the shell, we have the following equations: Q(P) =
{Lka)S(P)
+ iMk8)dP)
-
{L/P)&‘>
(P E r) 7
(8)
SM. Kirkup/Comput.
v(P) = {@#Sk) + {Nka),(P>
397
Methods Appl. Mech. Engrg. 140 (1997) 393-404
-
{Mb+-(P)
-
{Lkv)&)
(P E r>
(9) The value of cp(p) for points in the exterior domain are related to the solutions on S and r through the following equation &)
=
{Lk+(P>
+ {“k%-@)
(P E O>
.
.
(10)
2.6. Formulation of the Eigenvalue problem The eigenvalue problem is that of finding the values of k for which the Helmholtz a homogeneous boundary condition of the form a(P)+Q) 4&G)
+ ,@(P)u(P) = 0 + bW44
= 0
A@)@@)+W)Vk) has non-trivial solutions.
3. Application
equation (1) with
(P E S), (P E r),
=O
(PE 0,
of collocation
In this section it is shown how collocation is applied to derive the discrete form of the integral equations. The boundary and shell are divided into uniform elements. The boundary S is divided into ns elements ASi, AS:,, . . . , AS,,, , the shell r is divided into IZ~elements Ar,, A&, . . . , AC,,. and the boundary functions and shell functions are approximated by a constant on each element. Let p1,p2,. . . ,pn, and 41,42,..* 7 qn, be the collocation point with pi E ASi for i = 1,2,. . . , rzs and qi E AT;: for i = 1,2,. . . , noand each lying at the centre of the respective element; c(pi) = i (i = 1,2,. . . ,ns) and C(qi) = k (i = 1,2 )... ,Izr). 3.1. Notation
It is helpful to introduce the following notation. collocation points as follows: s
= kfJP*),
6r = [a).,
dP2L
Q2)>
. ..7 dPn,)1*
Define the vectors of the function values at the
>
*-*> Qzn,>1* 7
@ _iV are defined similarly. vectors TV,,9, Cys, & 3, Yr, _i-y Let the matrices Lk,SS, Lk,Sr, Lk,rs and Lk,rr be defined as follows: [Lk,SS]ij
= {Lke}AS,(pi)
[Lk,Sr]ij
=
[Lk,TS]ij
= {Lke}A$(qi)
(i =
1,2,
-.,nr),
[Lk,lTlij
= (Lkelhq((4i)
(i =
172,
“‘7
{&e}Aq(&)
G =
l,&...,d(j
=
1,2T-e7Q)
(i = l, 27-? nS),(i = l7 2y
‘-)
,
nr>
(i = 1,27...~ns)
)
T
nr), (i = 1,2, -, nr)
where e is the unit function. Notation for the other integral operators a similar way.
(j&, ML and Nk) is developed
in
3.2. Discrete form of the integral equations
The adoption of the notation above allows us to construct the following linear systems of approxima-
398
S.M. KirkupIComput.
Methods Appl. Mech. Engrg. 140 (1997) 393-404
tions which are the discrete analogues the direct integral equation formulation Mk,.ss + ; fss s I [ C&E-M I&Z:-
=
Lk,SS &!s+ Mk,sr
k,rs C& + Lk,rs Nk,rs &
+@,rs
(11)
&- - Lk,Sr Vr,
!!s +Mk,rr
Lb - Lk,rr 41r,
(12)
3
6r - ML,rr
(13)
+Nk,rr
Kr.
Similarly, the discrete form of the indirect integral equation formulation % M Lk,ss E.s + Mk,sr 3s”
1 Mk,ss + 2 fss
5& M [
Lk,rs 5
JLi- M M:,rs
%
+ Mk,rr
a
+ Nk,sr
fir - @,sr
(15)
El-3
(16)
i%- - Lk,rr Er,
+ Nk,rr &
- ML,rr
(5)-(8) is as follows: (14)
fir - Lk,sr Vr,
1
(2)-(4):
(17)
Kr.
The boundary condition can be written in the following form (18)
D&~+D&=~> where D& = diag(q, the shell condition.
(~2,... . ans) and D& = diag(&, pZ, . .. . a,).
Similar equations can be obtained for
4. The boundary and shell element method To demonstrate the direct and indirect boundary and shell element methods, the test problem with the domain of the unit square and with a discontinuity between ($, $) and (k, 1) is introduced. The boundary conditions are such that q(p) = 1 for 0 < p1 < i and p2 = 1, q(p) = -1 for i < p1 < 1 and p2 = 1 and v(p) = 0 on the remainder of the boundary. The shell condition is such that v@) = V(p) = 0 for all points on the shell. The test problem is illustrated in Fig. 3.
Lp=l
(0=-l
v=o
v=o v=o
v=o
u=o Fig. 3. Illustration of the domain of the test problem.
S.M. Kirkup/Comput.
399
Methods Appl. Mech. Engrg. 140 (1997) 393404
The discrete forms of the integral operators are computed through the use of a subroutine H2LC, described in [lo]. These numerical integrations are computed to sufficient accuracy so the error does not contribute significantly to the overall error in the integral equation methods. 4.1. Direct boundary and shell element method The following linear system of equations follows from approximations 1 Mk,sS + 2 fss
_LSS
OW
-L,rs -M:,rs
Ifr err
and (18),
-Mk,sr
*cs osr
*is
Mk,rs Nk,r,s
(ll)-(13)
%I-
09)
-Mk,rr -Nk,rr
cp Q, &, & are obtained. The discrete form of Eq. On saIlCttion the approximations I&, &, &r, & to s, (5) is then used to compute the solution in the domain D. 4.2. Indirect boundary and shell element method
The following linear system of equations follows from Eqs. (14)-(17),
Iss
Qss
ass
Jss
*;s
Ors _ Ors
-
M:,,,
+ ;Iss
Osr
(20)
Osr Irr Orr
ass
%
0rs 0rs
%r
-Lk,SS
-Lk,rs -M:,rs
On solution the approximations &, 5, &, &, & to s, b, s, of (10) is then used to compute the solution in the domain D.
5. Results for the boundary-value
&, & are obtained. The discrete form
problem
Results from th,e application of the direct and indirect methods to the test problem are given in Tables 1 and 2. The numerical solution is given at the point (0.25,0.5) in the domain. The number of elements used in the experiment are 9 (8 boundary and 1 shell), 18 (16 + 2), 36 (32 + 4), 72 (64 + 8), 144 (128 + 16) and 288 (256 -t 32) uniform elements; the element lengths h are i, $, i, A, _j?iand A, respectively. The test problelm can be solved using the BEM in the way outlined in the introduction and illustrated in Fig. 2. Since the solution is antisymmetric about the centre line then the problem is equivalent to the test problem illustrated in Fig. 4. The direct formulation is now the Green’s formula, the indirect
Table 1 Solution via the direct h
BSEM k=l.O
k = 2.0
k = 4.0 -0.7405
1t
0.5021 + iO.0014
0.7652 + iO.0010
f
0.4423+ iO.0004 0.4157 + iO.OCOl 0.4030+ iO.0000 0.3968+ iO.0000 0.3938+ i0.0000
0.6589+ iO.0004 0.6119+ iO.0001 0.5895 + i0.0000 0.5785+ i0.0000 0.5730+ i0.0000
s & + z
+ iO.1209 -0.5393+ iO.0372 -0.5077+ iO.0095 -0.5013+ iO.0024 -0.5000+ i0.0006 -0.5OOfl + i0.0001
400
S.M. Kirkup/Comput.
Methods Appl.
Table 2 Solution via the indirect BSEM k=l.O h
Mech. Engrg. 140 (1997) 393404
k = 2.0
k = 4.0 -0.8663 + iO.2701
0.4335 + iO.0022
0.7277 + iO.0309 0.6454 + iO.0111
0.4120 + iO.0012
0.6049 + iO.0057
-0.5775 + i0.1223
1T
0.4680 + iO.0055
z !. 8
-0.6509 + iO.2055
1 v
0.4014 + iO.0007
0.5858 + iO.0032
-0.5452 + iO.0743
f-?
0.3961 + iO.0004
0.5764 + iO.0019
-0.5278
+ iO.0459
w
0.3934 + iO.0003
0.5718 + iO.0012
-0.5174
+ iO.0286
‘p=l
u=o v=o cp=o
TJ=o Fig. 4. Equivalent problem to which the BEM is applied.
formulation is the one arising through writing cp as a single-layer potential. The boundary functions are discretised in a similar way to the method in the previous section. The number of elements in the experiments are 6, 12, 24, 48, 96 and 192. The results from the application of the standard direct and indirect BEMs are given in Tables 3 and 4.
Table 3 Solution via the direct BEM h
k = 1.0
1
k = 2.0
k = 4.0
-
t
0.4825 0.4318 + iO.0004 iO.0003
0.6311 - iO.0055 0.6979 iO.0011
-0.5797 + iO.0311 -0.8005 iO.0100
B
0.4095 + iO.0001
0.5983 - iO.0003
-0.5238 + iO.0014
1 m
0.3997 + i0.0000
0.5828 - iO.0001
-0.5071
- iO.0001
P
0.3951 + iO.WOU
0.5752 - i0.0000
-0.5021
- iO.0002
0.3929 + i0.0000
0.5714 - iO.0000
-0.5006
- io.0001
Table 4 Solution via the indirect BEM h k=l.O
k = 2.0
k = 4.0
1 T
0.4582 + iO.0101
0.6894 + io.0377
-0.7778 + i0.3392
f
0.4243 + iO.0037
0.6237 + iO.0130
-0.6483 + i0.2111
8
0.4070 + iO.0018
0.5943 + iO.0062
-0.5857 + i0.1232
&
0.3988 + iO.0010
0.5805 + iO.0034
-0.5512 + iO.0743
6 1
0.3947 + iO.0006
0.5738 - iO.0020
-0.5314 + iO.0457
0.3927 + iO.0003
0.5706 - io.0012
-0.51%
- io.0284
S. M. Kirkup/Comput.
6. Solution of the eigenvalue
Methods Appl. Mech. Engrg. 140 (1997) 393-404
401
problem
In this section the eigenvalue problem for the same domain as in the previous section is considered. The boundary conditions (in the form of Section 2.6) are such that ~001)= 0 for 0 < p1 < 1 and o(p) = 0 on the remainder of the boundary. The shell condition is such that v(p) = VCp> = 0 for all points on the shell. 6.1. Direct and indirect non-linear Eigenvalue problems
From (19) and (20) it follows that the approximation to the eigenvalues and eigenfunctions boundary and shell are given by finding the solution of the non-linear eigenvalue problem Mk,ss +
;Iss -Lk,SS osr -Mk,sr
%s
D&
Mk,.rs &,rs
-Lk,I-S
Osr h-l-
-@,rs
Orr
Osr -Mk,rr -Nk,rr
for the direct method and
Iss
ass
Oss
Is:; %
0rs
Ors Ors
_ Ors
M:,ss + ;Iss
$ =
(21)
8r 1
Osr Osr
ass
Ds”S
jI &
Osr
-Lk,SS
-
on the
_Lk,i-S
1rr
-“:,rs
Orr
(22)
for the indirect method. The approximation to the eigenfunction in the domain can the be obtained through the substitution of the results from the eigenvectors into the discrete form of Eq. (5) for the direct method and the discrete form of (10) for the indirect method. 6.2. Method of solution
The eigenvalue problems (21) and (22) both have the form A,&== (23) where each component of Ak is a continuously differentiable complex-valued function of k. Non-linear eigenvalue problems of this form are considered in [ll-131. In this paper the eigenvalue problem is solved using the method introduced in Kirkup and Amini [S]. This method involves approximating Ak by a matrix polynomial in an interval [kA, kB] of real values of k, Ak M A~,J $_kAp] + . . . + k”AL,l
for k real.
(24)
The non-linear eigenvalue problem (23) can be replaced with the following eigenvalue problem [Al01+ kAp1 + +. . + k”Ar,&
= Q.
The solutions of (25) are the same as those of the following generalised linear eigenvalue problem
“$3
I0 0
0 0
Al;-21 0 0
I 0
(25)
402
S.M. KirkupIComput.
Methods Appl. Mech. Engrg. 140 (1997) 393404
(26) Eq. (26) is amenable to solution by the QZ algorithm [14], which may be invoked, for example, via NAG routine F02GJF [15]. Methods for solving problems of the form (25) are considered in [12,13,16]. On solution, the eigenvalues k* include the approximations to the eigenfrequencies of the Helmholtz problem. Spurious solutions to (26) are also obtained but these can be easily excluded, for further details on this see Kirkup and Amini [8]. The approximation to the eigenfunctions in D can be recovered through the substitution of the relevant sub-vectors of the computed eigenvectys into the discrete form of Eq. (5) for the direct method or (10) for the indirect method. In this paper the method employed for deriving the polynomial approximation (24) involves computing Ak at the m + 1 Chebyshev (a norm) interpolation points for any selected range [kA, ks]. The coefficient matrices Ap],Ap], . . . . Ai,] in (24) are obtained through Newton’s divided differences. The generalised eigenvalue problem (26) is then solved through invoking NAG routine F02GJF. 7. Results for the eigenvalue
problem
In this section results from the application of the method of Section 6 to the test problem are presented. The computed eigenvalues from the application boundary and shell element methods are given for 8+1, Table 5 Computed Eigenvalues via the direct BSEM h
ks - kA
m
1st eigenvalue
2nd eigenvalue
2.0
2 4
1.6007 + iO.0073 1.6134 iO.0080
3.2048 - iO.0138 3.2156 iO.0031
1 i
-
3rd eigenvalue 4.8051 - iO.0118 4.7904 iO.0067
1
1.0
4 2
1.6011 + iO.0025 1.6032 iO.0022
3.2029 3.2046 - iO.0022 iO.0029
4.8053 - iO.0066 4.8069
1
2.0
2
1.5952 + iO.0067
3.2689 - iO.0181
4.7228 - iO.0101
1
1.0 2.0
2 4
1.5825 1.5804 + iO.0057 iO.0013
3.2520 - iO.0035 3.2550 iO.0047
4.7432 - iO.0026 4.7403 iO.0027
7
1.0
4
1.5807 + iO.0011
3.2548 - iO.0044
4.7405 - iO.0026
1
2.0
4 2
1.5737 + iO.0062 1.5892 iO.0047
3.3026 3.2868 - iO.0157 iO.0013
4.7208 - iO.0085 4.7027 iO.0044
f
1.0
4 2
1.5757 1.5740 + iO.0007 iO.0004
3.2835 3.2866 - iO.0001 iO.0010
4.7210 - iO.0002 4.7239 iO.0003
1st eigenvalue
2nd eigenvalue
3rd eigenvalue
Table 6 Computed Eigenvalues via the indirect BSEM h
ks - k,
m
1
-
-
-
I
2.0
4 2
1.6368 - iO.0722 1.6398 iO.0771
3.2031 - iO.1047 3.2100 iO.0908
4.7738 - iO.0961 4.7591 iO.0865
1
1.0
4 2
1.6331 - iO.0851 1.6362 iO.0859
3.2030 3.2010 - iO.0878 iO.0905
4.7741 4.7772 - iO.0863 iO.0852
7 s .! 4
2.0 2.0
2 4 2
3.2967 - iO.0756 3.2835 - iO.0599 3.2802 - iO.0584
4.7293 - iO.0515 4.7455 - iO.0429
1.0
1.6187 - iO.0403 1.6108 - iO.0435 1.6110 - iO.0503
4.7486 - iO.0424
3
1.0
4
1.6085 - iO.0497
3.2833 - iO.0596
4.7458 - iO.0428
t IH
2.0
2
2.0
4
1.6061 - iO.0223 1.5950 - iO.0247
3.3233 - iO.0525 3.3081 - iO.0369
4.7301 - iO.0232
4.7128 - iO.0316
8
1.0
2
1.5958 - iO.0304
3.3048 - iO.0357
4.7332 - iO.0229
g
1.0
4
1.5936 - iO.0301
3.3079 - iO.0366
4.7303 - iO.0231
S.M. Kirkup/Comput.
Methods Appl. Mech. Engrg. 140 (1997) 393-404
403
16+2 and 32+4 boundary + shell elements (h = i, f and $) with intervals [kA, kB] of length 2.0 and 1.0 and interpolating polynomials of second and fourth degree (m=2, 4). The first three computed eigenvalues from the direct method are given in Table 5, the results from the indirect method are given in Table 6.
8. Concluding
discussion
In this paper a new numerical method termed the boundary and shell element method has been described and d,emonstrated on the two-dimensional discontinuous interior Helmholtz equation. The method is based on an integral equation that models the boundary of the domain and discontinuities directly. For such problems, the standard boundary element method requires the shell discontinuities to be extended and the domain divided into several regions, requiring a greater number of elements. It follows that the BSEM is generally more efficient than the standard boundary element method for discontinuous problems. Tables 1 and 2 demonstrate both the direct and indirect BSEMs on the test problem illustrated in Fig. 3. These results are verified by the results of Tables 3 and 4 where the direct and indirect BEMs are applied. The results for the sample point in each of the tables are converging to the same solution. The exact solution is necessarily real, the imaginary parts to the solution in Tables 1 to 4 is error arising through the application of the method. All of the results in the Tables show O(h) convergence. The boundary and shell element method has been applied for the eigenvalue problem associated with Fig. 3 with homogeneous boundary conditions. In Tables 5 and 6 the eigenvalues of the discontinuous interior Helmholtz equation are found by applying the method introduced in [8] to the boundary and shell element matrices. The results show convergence as the number of elements is increased and the precision of the interpolant is improved. However, as is found in the continuous case [8], the convergence rate does not follow a simple formula. In this paper i.t has been demonstrated that the Boundary and Shell Element Method is a powerful tool in the solution of interior Helmholtz problems with discontinuities in the domain. Results have been given for a two-dimensional domain. The application of the method to a three-dimensional case-as for the exterior Helmholtz equation in [17]-would also be useful.
Acknowledgment
The Cray Supercomputer at Rutherford Appleton Laboratory test problems (Supercomputing Grant GR/F61639).
was used to obtain the results from the
References [l] M.A. Jaswon and G.T. Symm, Integral Equation Methods in Potential Theory and Elastostatics (Academic Press, 1977). [2] C.A. Brebbia, The Boundary Element Method for Engineers (Pentech Press, Plymouth, 1978). [3] PK. Banerjee and R. Butterfield, Boundary Element Methods in Engineering Science (McGraw-Hill, 1981). (41 G. Chen and J. Zhou, Boundary Element Methods, Computational Mathematics and Applications (Academic Press, 1992). (51 J. Ben Mariem and M. A. Hamdi, A new boundary element method for fluid-structure interaction problems, Int. J. Numer. Methods Engrg. 24 (1987) 1251-1267. [6] A.G.P. Warham, The Helmholtz integral equation for a thin shell NPL report DITC 129/88, National Physical Laboratory, Teddington, Middlesex, 1988. [7] SM. Kirkup, Shell elements: their use in conjunction with boundary elements and in application to Laplace problems, Proceedings of the NUMETA 90 Conference, G.N. Pande and J. Middleton, eds., Vol. 1 (Elsevier, 1990) 222-229. [8] SM. Kirkup and S. Amini, Solution of the Helmholtz eigenvalue problem via the boundary element method, Int. J. Numer. Methods Engrg. 36(2) (1993) 321-330. [9] S.M. Kirkup, The boundary and shell element method, Appl. Math. Model. 18 (1994) 418-422. [lo] S.M. Kirkup, Fortran codes for computing the discrete Helmholtz integral operators: User guide, Report MCS-96-06, Department of Mathematics and Computer Science, University of Salford, Salford, UK, 1996. [ll] A. Ruhe, Algorithms for the non-linear Eigenvalue problem, SIAM J. Numer. Anal. 10 (1973) 674-689.
404
S. M. Kirkup/Comput.
Methods Appl. Mech. Engrg. 140 (1997) 393404
[12] I? Lancaster, A Review of Numerical Methods for Eigenvalue Problems Nonlinear in Parameter, Numerik turd Andwendungen
von Eigenwertaufgaben und Verzweigungsproblemen, E. Bohl, L. Collatz and K. I? Hedeler, eds., ISNM 38 (Basel-Stuttgart, Birkhauser, 1977). [13] R. Wobst, The generalized Eigenvalue problem and acoustic surface wave computations, Comput. 39 (1987) 57-69. [14] C.B. Moler and G.W. Stewart, An algorithm for generalized matrix Eigenvalue problems, SIAM J. Numer. Anal. lO(2) (1973) 241-256. [15] NAG Library, The Numerical Algorithms Group Oxford, UK. [16] I. Gohberg, P. Lancaster and L. Rodman, Matrix Polynomials (Academic Press, 1982). [17] S.M. Kirkup, The computational modelling of acoustic shields by the boundary and shell element method, Comput. Struct. 40(5) (1991) 1177-1183.
