Efficient solution of large-scale electromagnetic Eigenvalue problems using the211 implicity restarted Arnoldi method

摘要

The authors are interested in determining the electromagnetic fields within211u001eclosed perfectly conducting cavities that may contain dielectric or magnetic 211u001ematerials. The vector Helmholtz equation is the appropriate partial differential 211u001eequation for this problem. It is well known that the electromagnetic fields in a 211u001ecavity can be decomposed into distinct modes that oscillate in time at specific 211u001eresonant frequencies. These modes are referred to as eigenmodes, and the 211u001efrequencies of these modes are referred to as eigenfrequencies. The authors; 211u001epresent application is the analysis of linear accelerator components. These 211u001ecomponents may have a complex geometry; hence numerical methods are require to 211u001ecompute the eigenmodes and the eigenfrequencies of these components. The 211u001eImplicitly Restarted Arnoldi Method (IRAM) is a robust and efficient method for 211u001ethe numerical solution of the generalized eigenproblem Ax = (lambda)Bx, where A 211u001eand B are sparse matrices, x is an eigenvector, and (lambda) is an eigenvalue. 211u001eThe IRAM is an iterative method for computing extremal eigenvalues; it is an 211u001eextension of the classic Lanczos method. The mathematical details of the IRAM are 211u001etoo sophisticated to describe here; instead they refer the reader to (1). A 211u001eFORTRAN subroutine library that implements various versions of the IRAM is freely 211u001eavailable, both in a serial version named ARPACK and parallel version named 211u001ePARPACK. In this paper they discretize the vector Helmholtz equation using 1st 211u001eorder H(curl) conforming edge elements (also known as Nedelec elements). This 211u001ediscretization results in a generalized eigenvalue problem which can be solved 211u001eusing the IRAM. The question of so-called spurious modes is discussed, and it is 211u001eshown that applying a spectral transformation completely eliminates these modes, 211u001ewithout any need for an additional constraint equation. Typically they use the 211u001eIRAM to compute a small set (n < 30) of eigenvalues and eigenmodes for a 211u001every large systems (N > 100,000).