# 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. SchÃ¶berl.