Hardy space infinite elements for time-harmonic guided elastic waves
DI Martin HallaDec. 1, 2015, 2:30 p.m. S2 059
We consider the time-harmonic elasticity equation posed in wave-guide structures. Such geometries involve cylinders (e.g. R x (-1, 1) in 2d) or plates (e.g. R2 x (-1, 1)) of infinite volume. Since the domain is unbounded, a standard finite element method doesn't suffice. A common technique is to split the domain into a bounded interior part, which contains all inhomogeneities, and an unbounded "simple" exterior part. For the first a standard finite element discretization can be used, while for the latter we use the Hardy space infinite element method (HSIEM), which was introduced in  for acoustic problems and further analyzed in [6, 2]. The method is based on the Laplace transform of test and trial functions in the "unbounded direction". Incoming/outgoing waves are thereby identified by the position of the poles of their Laplace transforms.
Different to acoustics, elastic materials exhibit frequencies for which waves with different signs of group and phase velocity appear. This curious phenomenon poses numerical difficulties for methods such as the standard HSIEM or perfectly matched layer methods . We explain how a modified HSIEM can recover the physical radiation condition. An important feature of the method is that it doesn't depend on a modal representation. This allows an extension to plates and leads to linear matrix eigenvalue problems for resonance studies. We report a convergence analysis for a one-dimensional model problem and numerical computations in two and three dimensions.
The talk is based on joint work with T. Hohage, L. Nannen and J. Schöberl.