Hardy space infinite elements for time-harmonic guided elastic waves

DI Martin Halla

Dec. 1, 2015, 3:30 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 [5] 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 [1]. 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.