Chapter 6
Interior Modal Analysis
In this Chapter it is shown how the boundary element method
can be used to obtain
the resonant wavenumbers and the mode shapes of an enclosed
region;
the computational solution of the interior Helmholtz
eigenvalue problem. The problem is that of finding the
values of the wavenumber k and a non-trivial scalar function
f such that the Helmholtz equation
|
Ñ2 f(p) + k2 f(p) = 0 (p Î D) |
| (6.1) |
is satisfied in an interior domain D with boundary S and subject to a
homogeneous boundary condition of the form
|
a(p) f(p) + b(p) |
¶f(p)
¶np
|
= 0 (p Î S) |
| (6.2) |
where a(p) and b(p) are known complex-valued functions of
p ( Î S) and np is the unit outward normal to the
boundary at p. The non-trivial solutions k = k* and
f(p) = f*(p) (p Î D ÈS) are termed the
characteristic wavenumbers
and eigenfunctions and they are dependent on the
boundary S and the boundary functions
a(p) and b(p). The characteristic wavenumbers are all
real numbers and they correspond to the resonant
wavenumbers of the enclosed region. The eigenfunctions are equivalent
to the mode shapes.
The Helmholtz eigenvalue problem is amenable to
solution via finite element or finite difference methods.
In these cases, the problem reduces to that of solving a generalised
linear eigenvalue problem of the form
where the matrices K and M (termed the stiffness and mass matrices)
in (6.3) are sparse and structured and are
independent of k. Standard computational algorithms are available
for solving generalised linear eigenvalue problems. Indeed special
techniques (such as iterative methods)
are available for solving the general problem (6.3), given the special
structure of the matrices and the fact that only
a fraction of the full set of eigenvalues are generally required
[69].
Hence eigenfrequency analysis of the Helmholtz problem via
the finite element or finite difference method is straightforward.
In cases where it is applicable, it is well known that the boundary
element method has an important advantage over the finite element
and finite difference methods: the partial differential equation
governing the domain is reduced to an integral equation relating
values of f and [(¶f)/( ¶n)]
on the boundary only. Hence the dimension of the problem
is effectively reduced by one. However, the application of the
boundary element method reduces the Helmholtz eigenvalue problem to
that of solving an eigenproblem of the form
where the matrix Ak is generally full,
having no particular structure but with each component being
a continuously differentiable complex-valued function of k.
Because of the main advantage of the boundary element method
over finite element and finite difference methods stated earlier,
the matrix in (6.4) is generally much smaller than the
matrices in (6.3), for any given modal analysis problem and a given
level of required accuracy. The disadvantages of this approach
are that the eigenvalue problem (6.4) is non-linear and the
components of the Ak matrix are defined in terms of integrals
and hence may be costly to evaluate.
The solution of non-linear eigenvalue problems are
considered in references [55], [73]
and [84]. Unfortunately, standard algorithms for solving
non-linear eigenvalue problems are not generally available.
The problem of solving the Helmholtz eigenvalue problem
via boundary element-type methods have been given some consideration by
researchers. For example iterative methods such as the secant method
are applied to the problem of finding the roots of the
equation det (Ak ) = 0
in references [81], [27]
and [1].
However, this is not a satisfactory
method when the matrix Ak is large [84].
A similar method, based on finding the values of k for which the
smallest eigenvalues of Ak is zero is considered in [58].
Unfortunately, these methods are
unwieldy since they do not compute the solutions simultaneously;
they require a starting point to be chosen for each required eigenfrequency.
In reference [8]
a hybrid of the boundary element and finite
element method is introduced. The method seems to have the advantage of the
finite element method in that a linear eigenvalue problem results
and the advantage of the boundary element method in that a solution
on the boundary only is sought in the main computation. The method
is considered further in references [24], [2].
In general, both eigenfrequencies and eigenfunctions of the Helmholtz
problem will be of interest
The method considered in this Chapter was introduced
in Kirkup and Amini [46]. The method involves
approximating each
component of the matrix Ak by a polynomial in k in some
given sub-range of the full wavenumber range. This allows us to
re-write the non-linear eigenvalue problem (6.4) in the form
of a standard generalised eigenvalue problem. Thus all of the
eigenvalues in the sub-range are computed simultaneously.