Boundary Elements XXII, C.A. Brebbia & H. Power (Editors)
© 2000 WIT Press, www.witpress.com, ISBN 1-85312-824-4
Some developments of the natural boundary
element method
D. Yu
LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and
System Sciences, Chinese Academy of Sciences,
700080,
Abstract
In this paper some new developments of the natural boundary element
method are presented. Based on the natural boundary reduction, some
new domain decomposition methods, which are very effective for solving
exterior problems, are suggested, and the natural integral operators play a
very important role in the domain decomposition. The natural boundary
element method for 3-d elliptic problems are developed. This method is also
applied to solve 2-d problems of some parabolic and hyperbolic equations.
Based on the natural boundary integral operator and the Poisson integral
formula for the exterior elliptic domain, a coupling method and a domain
decomposition method are developed for solving problems over some elongated exterior domains. Some numerical examples show the feasibility and
efficiency of these methods.
1 Introduction
Many boundary value problems of the elliptic partial differential equations
can be reduced into the boundary integral equations by different ways.
Based on these boundary reductions variour boundary element methods
have been developed.
The natural boundary element method suggested by Feng and Yu [3]
has many distinctive advantages (also see Yu [16,17]). It can be coupled
naturally and directly with finite element method, and is very effective
for solving some problems involving infinite or cracked subdomains (see
Boundary Elements XXII, C.A. Brebbia & H. Power (Editors)
© 2000 WIT Press, www.witpress.com, ISBN 1-85312-824-4
482
Ushijima [8], Yu [12,14,15]).
For solving problems over unbounded domains, some artificial boundary
conditions are usually used (see Engquist and Majda [2]). The natural
integral equation on the artificial boundary, i.e. the Dirichlet-Neumann
mapping, is just the exact artificial boundary condition. For its applications
can see Feng [4], Han and Wu [6], Keller and Givoli [7], Yu [13].
Some new developments of the natural boundary element method in
recent years are presented in following sections.
2 DDM
based on the natural BEM
Comparing with coupling methods, the domain decomposition methods
have their advantages. The natural integral operators, i.e. the DirichletNeumann operators, play a very key role in DDM (see Feng and Yu [5], Yu
[19]).
The natural boundary reduction can be applied directly to DDM. Based
on the natural boundary reduction some overlapping and non-overlapping
DDM are developed. It is very effective for solving some exterior problems
(see Yu [18,20.21]).
Being completely different from the standard DDM based on the finite
element methods, the DDMs based on the natural boundary reduction are
suitable for unbounded domains. They use FEM only in a small interior
subdomain, while the problem in the exterior sub domain is reduced to a
problem on the artificial boundary, and only some numerical integration is
needed.
For Non-overlapping case the selection of relaxation factor and the estimation of the compression ratio are very important in the D-N method.
There are some papers on domain dependence of convergence rate. For example, in Yu [22] a simple and convenient approach to selection of relaxation
factor is given, and an optimal convergence is obtained.
These methods have already been used for solving the exterior boundary
value problems of harmonic, biharmonic, plane elasticity, Stokes, Helmholtz
and other partial differential equations. Numerical examples show the effeciency of these methods (see Yu and Jia [24], Zheng and Yu [25,26]).
3 Natural BEM
for 3-d problems
Consider 3-D harmonic boundary value problem. Similarly to the 2-D
case, there is a natural integral equation, which is the relation between the
Dirichlet boundary value UQ and the Neumann boundary value Un of the
solution u:
Un = /Cuo,
on 9H,
(3.1)
and the relation between the solution u and its Dirichlet boundary value UQ
Boundary Elements XXII, C.A. Brebbia & H. Power (Editors)
© 2000 WIT Press, www.witpress.com, ISBN 1-85312-824-4
#owWaryE/emefz%A%//
483
is given by the Poisson integral formula:
u = P%o,
in 0.
(3.2)
Especially when fi is half space, interior sphere or exterior sphere, the expressoins of JC and P can be obtained (see Wu and Yu [9]).
Here /C is a hypersingular integral operator, which is a pseudo-differential
operator of positive order. Numerical computation of the hypersingular
integrals can be implemented by using the spherical harmonic expansion
(see Wu and Yu [9,10]).
Let 0 be the 3-D exterior domain with boundary Eg, take a sphere Si
or two surface S2 and Si, where the inner E2 is a sphere, enclosing EQ as
an or two artificial boundary. The unbounded domain fi is divided into a
bounded subdomain fit and an exterior sphere domain %. For simplicity,
let/ = 0
in%.
Consider 3-D exterior problem:
"
v'
on SQ.
For non-overlapping DDM
(3.3)
we have the D-N method:
in %,
'">
in fii,
^ ^'
(3.5)
, on Si,
and
c\ n ~r
i/i\I ~ a"n)A
\ \n ,
A\n-fl — "n^i
/o^o.uj
f(\
n = 0,1, • • •, where 9^ is a relaxation factor.
The overlapping DDM is the Schwarz alternating method:
. =/,
%^ = #,
%2& = %2&-\
mUi,
on Eg,
Qn]Ci,
(3.7)
and
{ -*1Z
^
"2
A = 0,1,-".
=t
**'
~ "l ' On 1,2,
(3-8)
Boundary Elements XXII, C.A. Brebbia & H. Power (Editors)
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Here in HI FEM is applied, while in % we apply the natural integral
operator on the artificial boundary for D-N method:
or use the Poisson integral formula for Schwarz method. It is not necessary
to solve equation in % . The geometric convergence of Schwarz alternating
method is given in Wu and Yu [11].
For the case of spherical inner and artificial boundaries a fine analysis of
convergence rate is obtained, and the compression ratio of the iteration is
less than
where RQ < R^ < RI, RQ, RI and RI are radiuses of three spheres.
We have following results: If the relaxation factor 0 satisfies
/ti o
then the D-N method converges. Especially, if take
(3.11)
then the compression ratio of iteration is less than
2fli
r max(§\3)
\ HO ' '- KI+HQ
°=
/B, m\ , 2RT"'
/Q 1 r)\
(3.12)
For example, we have (see Yu [23])
Sopt = 1/11,
for RI/RQ = 2, 9 = 5/11.
When RI/RQ > 2, the larger the ratio RI/RQ is, the larger the factor Sopt
is. In fact, we can take RI = 1R$.
We can also develop these methods for other elliptic boundary value
problems. For example, some results for 3-D exterior Helmholtz problem
can be found in Wu and Yu [10].
Numerical Examples show above methods are very effective, and the
geometric convergence rate is independent of the mesh size.
For overlapping DDM, the larger is the overlap offliand %, the smaller
the compression factor, the quicker the convergence of iteration; while for
non-overlapping DDM, the selection of the relaxation factor is important,
when RI > 2Ro, the larger the ratio Ri/Ro is, the larger the compression
factor is.
Boundary Elements XXII, C.A. Brebbia & H. Power (Editors)
© 2000 WIT Press, www.witpress.com, ISBN 1-85312-824-4
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For linear elements, the convergent solution has the accuracy of order
nearly.
4 Natural BEM
for parabolic problems
The natural BEM can also be applied to some initial boundary value problems of parabolic and hyperbolic equations (see Du and Yu [1]).
Consider the 2-d initial boundary value problem:
t - A% = /(z,f),
g% = <,(%,;),
(x,0) = UQ(X),
(%,f)Efr x J,
(%,f) E T x J,
X E fi^,
(4.1)
where 17 E H? , F = dfi, W — ^^
T is a,fixedpositive real number,
J = (o, r].
Let T be the time-step, tk = k - r, u^(x) — u(x, t^), z^(x) — Ut(x,tk). By
the Taylor expansion we have
z& -A%& = /&
(4.2)
^ = ^-1 + ^{(l - 7)^-1 + 7^}
Here 7 E (0, 1] and k = 1, 2, • • - , [T/r], 7 = 1/2 is the best.
The solution procedure is as follows:
1)
%& =%A-i+ r(l- 7)z*-\
(4.3)
(4.4)
2) Solve the problem:
2&,
+00,
% e r,
|z| -4- oo,
(4.5)
where
3)
z& = JL(%6 _ %&)
7T
(4.6)
The main procedure is to solve the elliptic problem (4.5) for each time-step.
By the theory of the natural boundary reduction we have the natural
integral equation
=Kul
(4.7)
;f',r,d).
(4.8)
and the Poisson integral formula
Boundary Elements XXII, C.A. Brebbia & H. Power (Editors)
© 2000 WIT Press, www.witpress.com, ISBN 1-85312-824-4
Boundwv Elements XXII
486
Let n be the exterior circular domain with radius R. By Fourier expansion
we can obtain the Poisson integral formula
27T
(4.9)
r > R, where JT^(-) is Bessel function of the second kind, and the natural
boundary integral equation
,R-f",e), (4.10)
an
where
Now let 5/j(F) C #^(r)
be thefiniteelement subspace, we have the
approximate variational problem:
Find u£ € Sh(T) such that
(4.12)
where
=<
>=
(4.13)
Take S/»(r) = span{^i(#), • • • ,^M(#)}, where ^, i - 1, • • • , M, are piecewise linear basis functions, we get the following system of algebaric equations:
Q(7* = 6\
(4.14)
where Q =
(4.15)
m = 0, 1, • • • , AT — 1, the series is convergent, and
Q=
Boundary Elements XXII, C.A. Brebbia & H. Power (Editors)
© 2000 WIT Press, www.witpress.com, ISBN 1-85312-824-4
Boundary Elements XXII
487
The stiffness matrix Q is a circulant matrix producted by ao, ai, • • • , ajv_i,
and QI = ajv_i (i — 0, 1, • • • , TV — 1). So we only need calculate [^-] + 1
elements for matrix Q. It is easy to be implemented in the calculation and
storage.
The approximatoin of Poisson integral formula as follows (r > R):
^
f *"<**•) . ^
yV
I JTo(AA) + 1^2,^2
n=i
(4.16)
Some numerical examples show that this method is effectual. Based on
above results, the coupling method and the domain decomposition method
can be applied to parabolic and hyperbolic equations.
5. Natural BEM
for elliptic domains
Sometimes an elliptic artificial boundary is necessary or much better than
the standard circular artificial boundary, especially for problems over the
exterior domain of some narrow region.
We use elliptic coordinates (//,</?):
{x
— /och// cos </?,
r i
y = /osh/ism<p,
(5.1)
where /o is a positive constant. Consider Neumann problem in an exterior
elliptic domain:
where 0^ = {Qu,y?)|/2 > a} with boundary Fa — {(ii,(p)\n = a}. We have
the natural integral equation on r%:
Un = K,UQ,
(5.3)
u — PUQ,
(5.4)
and Poisson integral formula:
where
P/=
00
V Fnel"l^-">+'^, V/eH*(I\,),
n= — oo
oo
(5.5)
(5.6)
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Boundary Elements XXII
The natural integral equation is equivalent to:
Find %o € #&r*
such that
Let {Lj((p}} be piecewise linear basis functions on F& and
N
3= 1
We obtain
(5.9)
6,
where Q = [qjk}NxN,
1
. 4 mr
cos
2n(A; - j)?r
n=l
which is same as that for the circular domain.
After finding i&o(y), by using Poisson integral formula, we have
(5.11)
^ > «•
The coupling method and the domain decomposition methods with elliptic
artificial boundary then can be developed.
These methods can also be applied to some other elliptic problems. For
example, we can use them for solving the exterior boundary value problems
of equation
,.
<«.«>
For the detail can see our forthcoming paper.
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Boundary Elements XXII, C.A. Brebbia & H. Power (Editors)
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