Compression of boundary integral operators discretized by anisotropic wavelet bases

The present article is devoted to wavelet matrix compression for boundary integral equations when using anisotropic wavelet bases for the discretization. We provide a compression scheme which amounts to only $O(N)$ relevant matrix coefficients in the system matrix without deteriorating the accuracy offered by the underlying Galerkin scheme. Here, $N$ denotes the degrees of freedom in the related trial spaces. By numerical results we validate our theoretical findings

Isogeometric shape optimization of periodic structures in three dimensions

The development of materials with specific structural properties is of huge practical interest, for example, for medical applications or for the development of lightweight structures in aeronautics. In this article, we combine shape optimization and homogenization for the optimal design of the microstructure in scaffolds. Given the current microstructure, we apply the isogeometric boundary element method to compute the effective tensor and to update the microstructure by using the shape gradient in order to match the desired effective tensor. Extensive numerical studies are presented to demonstrate the applicability and feasibility of the approach