Thomas Rüberg

Shape optimisation with multiresolution subdivision surfaces and immersed finite elements

Abstract We develop a new optimisation technique that combines multiresolution subdivision surfaces for boundary description with immersed finite elements for the discretisation of the primal and adjoint problems of optimisation. Similar to wavelets, multiresolution surfaces represent the domain boundary using a coarse control mesh and a sequence of detail vectors. Based on the multiresolution decomposition efficient and fast algorithms are available for reconstructing control meshes of varying fineness. During shape optimisation the vertex coordinates of control meshes are updated using the computed shape gradient information. By virtue of the multiresolution editing semantics, updating the coarse control mesh vertex coordinates leads to large-scale geometry changes and, conversely, updating the fine control mesh coordinates leads to small-scale geometry changes. In our computations we start by optimising the coarsest control mesh and refine it each time the cost function reaches a minimum. This approach effectively prevents the appearance of non-physical boundary geometry oscillations and control mesh pathologies, like inverted elements. Independent of the fineness of the control mesh used for optimisation, on the immersed finite element grid the domain boundary is always represented with a relatively fine control mesh of fixed resolution. With the immersed finite element method there is no need to maintain an analysis suitable domain mesh. In some of the presented two and three-dimensional elasticity examples the topology derivative is used for introducing new holes inside the domain. The merging or removing of holes is not considered.

Time Domain BEM: Numerical Aspects of Collocation and Galerkin Formulations

Time domain Boundary Element formulations are very well suited to treat wave propagation phenomena in semi-infinte domains, e.g., to simulate phenomena in earthquake engineering. Beside an analytical integration within each time step there is the formulation based on the Convolution Quadrature Method which utilizes the Laplace domain fundamental solutions. Within this technique not only the extension to inelastic material behavior is easy also the formulation of a symmetric Galerkin procedure can be established because the regularisation has to be performed only for the Laplace domain kernels.

An unstructured immersed finite element method for nonlinear solid mechanics

We present an immersed finite element technique for boundary-value and interface problems from nonlinear solid mechanics. Its key features are the implicit representation of domain boundaries and interfaces, the use of Nitsche’s method for the incorporation of boundary conditions, accurate numerical integration based on marching tetrahedrons and cut-element stabilisation by means of extrapolation. For discretisation structured and unstructured background meshes with Lagrange basis functions are considered. We show numerically and analytically that the introduced cut-element stabilisation technique provides an effective bound on the size of the Nitsche parameters and, in turn, leads to well-conditioned system matrices. In addition, we introduce a novel approach for representing and analysing geometries with sharp features (edges and corners) using an implicit geometry representation. This allows the computation of typical engineering parts composed of solid primitives without the need of boundary-fitted meshes.

Non-conforming Coupled Time Domain Boundary Element Analysis

The time domain Boundary Element Method (BEM) has been found to be well suited for modeling wave propagation phenomena in large or unbounded media. Nevertheless, material discontinuities or local non-linear effects are beyond the scope of classical BEM and require special techniques. Here, a (possibly hybrid) Domain Decomposition Method is proposed in order to circumvent these limitations and to obtain an efficient solution procedure at the same time.

Simulation of electrical machines: a FEM-BEM coupling scheme

Purpose Electrical machines commonly consist of moving and stationary parts. The field simulation of such devices can be demanding if the underlying numerical scheme is solely based on a domain discretization, such as in the case of the finite element method (FEM). This paper aims to present a coupling scheme based on FEM together with boundary element methods (BEMs) that neither hinges on re-meshing techniques nor deals with a special treatment of sliding interfaces. While the numerics are certainly more involved, the reward is obvious: the modeling costs decrease and the application engineer is provided with an easy-to-use, versatile and accurate simulation tool. Design/methodology/approach The authors present the implementation of a FEM-BEM coupling scheme in which the unbounded air region is handled by the BEM, while only the solid parts are discretized by the FEM. The BEM is a convenient tool to tackle unbounded exterior domains, as it is based on the discretization of boundary integral equations (BIEs) that are defined only on the surface of the computational domain. Hence, no meshing is required for the air region. Further, the BIEs fulfill the decay and radiation conditions of the electromagnetic fields such that no additional modeling errors occur. Findings This work presents an implementation of a FEM-BEM coupling scheme for electromagnetic field simulations. The coupling eliminates problems that are inherent to a pure FEM approach. In detail, the benefits of the FEM-BEM scheme are: the decay conditions are fulfilled exactly, no meshing of parts of the exterior air region is necessary and, most importantly, the handling of moving parts is incorporated in an intriguingly simple manner. The FEM-BEM formulation in conjunction with a state-of-the-art preconditioner demonstrates its potency. The numerical tests not only reveal an accurate convergence behavior but also prove the algorithm to be suitable for industrial applications. Originality/value The presented FEM-BEM scheme is a mathematically sound and robust implementation of a theoretical work presented a decade ago. For the application within an industrial context, the original work has been extended by higher-order schemes, periodic boundary conditions and an efficient treatment of moving parts. While not intended to be used under all circumstances, it represents a powerful tool in case that high accuracies together with simple mesh-handling facilities are required.