6.3  Solution of the Non-linear Eigenvalue Problem

The discrete eigenvalue problems are each of the form (6.4). The method employed for solving the general problem (6.4) requires that in an interval [kA, kB] of values of k the matrix Ak is approximated by a matrix polynomial in k
Ak A[0] + k A[1] + ... + km A[m]   for k real.
(6.12)

The non-linear eigenvalue problem (6.4) can be replaced with the following eigenvalue problem:

[A[0] + k A[1] + ... + km A[m]]m = 0.
(6.13)
The solutions of (6.13) are the same as those of the following generalised linear eigenvalue problem:








A[0]
A[1]
A[2]
.
.
A[m-2]
A[m-1]
0
I
0
.
.
0
0
:
:
:
:
:
:
:
0
0
0
.
.
I
0
0
0
0
.
.
0
I
















m
k m
:
km-2 m
km-1 m








(6.14)
               =   k 







0
0
0
.
.
0
-A[m]
I
0
0
.
.
0
0
:
:
:
:
:
:
:
0
0
0
.
.
0
0
0
0
0
.
.
I
0
















m
k m
:
km-2 m
km-1 m








  .

Equation (6.14) is amenable to solution by the QZ algorithm [61], which is available on a number of numerical libraries. Methods for solving problems of the form (6.13) are considered in references [67], [55], [28] and [84].

Since the eigenvalues k* of the underlying Helmholtz problem are all real and we are interested only in positive values, it is sufficient to compute the interpolant (6.12) for the positive real numbers k. However, as a result of numerical error, the computed wavenumber and mode shape will have small imaginary parts.

The generalised eigenvalue problem (6.14) will generally have m ×n solutions. Half of these can be immediately discounted since the eigenvalues occur in pairs [^k], -[^k]. The full set of solutions will contain approximations to the true eigenvalues of the underlying Helmholtz problem. However, many spurious solutions are generally produced as a result of the collocation method and approximation (6.12). These spurious eigenvalues do not have small imaginary parts and hence they can be sorted from the true eigenvalues. Approximations to the true eigenvalues lying outside the range [kA, kB] may also be produced. These approximations will generally be poor and they can be excluded from the results.

Let [^k], [^(m)] be a typical non-spurious solution to (6.14). The eigenvalue [^k] is an approximation to the eigenfrequencies k* of the Helmholtz problem. The approximation to the eigenfunctions in D S can be recovered through the substitution of the approximation [^(m)] for s in equation (6.5) or [^(m)] for f (Dirichlet) or [(f)/( n)] (Neumann) in equations (6.8)-(6.9).


Return to boundary-element-method.com/helmholtz
Return to boundary-element-method.com