In the Galerkin Asymptotic Wave Expansion (GAWE) the finite-element matrix equation is expanded about a specified frequency (expansion point f0) in terms of a power series in f – f0, where the initial value of f0 is chosen adaptively by the solver. The solution at each fi near the expansion point is then approximated by a sum over basis vectors that are determined from the matrix equation expansion. Generally the accuracy of the expansion degrades as |fi – f0| gets larger, causing the process to yield acceptable answers only within a range fmin < f0 < fmax, where the bounds are determined by checking the residual of the original matrix equation. Converged frequencies are archived, and new expansion points are then picked in unconverted regions of the band, with the process continuing until all frequency points are converged. The following discussion assumes all materials are independent of frequency; the terms in the expansion change for frequency-dependent materials.
We start with
where the coefficients in the first three terms are given by
The term in parentheses to the left of the equal sign is the port mode expansion, given by
The term in parentheses to the right of the equal sign is the source expansion, given by
In these equations you have
The combined expression can be written in the following form:
and you can create a vector basis as follows:
You can very efficiently obtain the solution at a particular frequency near the expansion point via projection:
where V is a matrix whose columns are the vj. This approach provides very accurate results for frequencies "near enough" to the expansion point (note that V and the Ai are functions of the expansion frequency, and that only one matrix factorization is required per expansion point). In practice, the solutions obtained from this procedure are inserted into the original matrix equation and their residuals are checked to determine when to generate a new expansion point.