1 Introduction
The boundary element method (BEM) is an established
computational method for the solution of linear elliptic partial differential
equations (PDEs), see references [1-4], for example. However,
the standard BEM cannot be applied directly to problems
with discontinuities in the variables
in the domain of the PDE.
In this paper boundary
value problems and eigenvalue problems of the Helmholtz equation,
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 figure 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 figure 1, the traditional BEM can be applied
by subdividing the domain into subdomains, as illustrated in figure 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 increases 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 shell integral equation. A numerical method
can then be derived in a similar way to the BEM (see, for example,
Ben Mariem and Hamdi [5] or Warham [6]).
In Kirkup [7], [9] it is shown how the Laplace equation
in a domain such as that of figure 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 figure 1. It consists
of a region D with boundary S and with shell discontinuities
G.
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 Warham [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 ([1/2],[1/2]) and ([1/2],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 Kirkup and Amini
[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.