Séminaire

14:00 - Andrew Gibbs

Solving multiple scattering problems with the Hybrid Numerical Asymptotic Boundary Element Method, and effective evaluation of singular oscillatory integrals

For a two-dimensional boundary integral equation (BIE) formulation of the exterior Helmholtz problem, approximation in a standard piecewise polynomial basis requires the number of degrees of freedom to grow (at least) in proportion with the frequency of the problem, to maintain accuracy. For certain scattering obstacles, the Hybrid Numerical Asymptotic (HNA) method may be used, which absorbs oscillatory behaviour of the solution into the approximation space. If such a basis is used, the number of degrees of freedom need not increase with the frequency of the problem.

In this talk I will briefly introduce the BIE formulation of the Helmholtz problem, discuss the HNA method and recent work extending it to multiple scattering configurations. We will observe that the computational bottleneck of the corresponding Galerkin method is the evaluation of singular and oscillatory three-dimensional integrals, which are very difficult to evaluate efficiently at high frequencies. For the remainder of the talk I will discuss two areas of recent work which can simplify these tricky integrals:

1. Oversampled collocation, which offers the advantage over standard collocation of being numerically stable. This enables us to reduce the dimension of integration by one.
2. PathFinder, an open source quadrature toolbox for oscillatory and singular integrals. This toolbox is essentially an automation of numerical steepest descent, which I will explain via a series of examples.

15:30 - Marc Lenoir

Approximations haute fréquence pour l'équation de Helmholtz

On décrit quelques aspects pratiques de la diffraction haute fréquence, illustrés de résultats numériques obtenus par Nicolas Salles et Eric Lunéville à l'aide du code XLife++ ; on s'intéresse en particulier au traitement des perturbations découlant de la présence d'obstacles de petite taille dans le champ. On terminera par une petite digression relative aux géodésiques.