Fast boundary element methods for the simulation of wave phenomena

This thesis is concerned with the efficient implementation of boundary element methods (BEM) for their application in wave problems. BEM present a particularly useful tool, since they reduce the dimension of the problems by one, resulting in much fewer unknowns. However, this comes at the cost of dense system matrices, whose entries require the integration of singular kernel functions over pairs of boundary elements. Because calculating these four-dimensional integrals by cubature rules is expensive, a novel approach based on singularity cancellation and analytical integration is proposed. In this way, the dimension of the integrals is reduced and closed formulae are obtained for the most challenging cases. This allows for the accurate calculation of the matrix entries while requiring less computational work compared with conventional numerical integration. Furthermore, a new algorithm based on hierarchical low-rank approximation is presented, which compresses the dense matrices and improves the complexity of the method. The idea is to collect the matrices corresponding to different time steps in a third-order tensor and to approximate individual sub-blocks by a combination of analytic and algebraic low-rank techniques. By exploiting the low-rank structure in several ways, the method scales almost linearly in the number of spatial degrees of freedom and number of time steps. The superior performance of the new method is demonstrated in numerical examples.Diese Arbeit befasst sich mit der effizienten Implementierung von Randelementmethoden (REM) für ihre Anwendung auf Wellenprobleme. REM stellen ein besonders nützliches Werkzeug dar, da sie die Dimension der Probleme um eins reduzieren, was zu weit weniger Unbekannten führt. Allerdings ist dies mit vollbesetzten Matrizen verbunden, deren Einträge die Integration singulärer Kernfunktionen über Paare von Randelementen erfordern. Da die Berechnung dieser vierdimensionalen Integrale durch Kubaturformeln aufwendig ist, wird ein neuer Ansatz basierend auf Regularisierung und analytischer Integration verfolgt. Auf diese Weise reduziert sich die Dimension der Integrale und es ergeben sich geschlossene Formeln für die schwierigsten Fälle. Dies ermöglicht die genaue Berechnung der Matrixeinträge mit geringerem Rechenaufwand als konventionelle numerische Integration. Außerdem wird ein neuer Algorithmus beruhend auf hierarchischer Niedrigrangapproximation präsentiert, der die Matrizen komprimiert und die Komplexität der Methode verbessert. Die Idee ist, die Matrizen der verschiedenen Zeitpunkte in einem Tensor dritter Ordnung zu sammeln und einzelne Teilblöcke durch eine Kombination von analytischen und algebraischen Niedrigrangverfahren zu approximieren. Durch Ausnutzung der Niedrigrangstruktur skaliert die Methode fast linear mit der Anzahl der räumlichen Freiheitsgrade und der Anzahl der Zeitschritte. Die überlegene Leistung der neuen Methode wird anhand numerischer Beispiele aufgezeigt

An evolving space framework for Oseen equations on a moving domain

This article considers non-stationary incompressible linear fluid equations in a moving domain. We demonstrate the existence and uniqueness of an appropriate weak formulation of the problem by making use of the theory of time-dependent Bochner spaces. It is not possible to directly apply established evolving Hilbert space theory due to the incompressibility constraint. After we have established the well-posedness, we derive and analyse a first order time discretisation of the system

Model Order Reduction

An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This three-volume handbook covers methods as well as applications. This third volume focuses on applications in engineering, biomedical engineering, computational physics and computer science

Snapshot-Based Methods and Algorithms

An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This two-volume handbook covers methods as well as applications. This second volume focuses on applications in engineering, biomedical engineering, computational physics and computer science

Model Order Reduction

An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This two-volume handbook covers methods as well as applications. This second volume focuses on applications in engineering, biomedical engineering, computational physics and computer science

Solvability and regularity for the electrostatic Born-Infeld equation with general charges

In electrostatic Born-Infeld theory, the electric potential u ρ generated by a charge distribution ρ in R m (typically, a Radon measure) minimizes the action ∫ R m ( 1 − √ 1 − ∣ D ψ ∣ 2 ) d x − ⟨ ρ , ψ ⟩ among functions which decay at infinity and satisfy ∣ D ψ ∣≤ 1 . Formally, its Euler-Lagrange equation prescribes ρ as being the Lorentzian mean curvature of the graph of u ρ in Minkowski spacetime L m + 1 . However, because of the lack of regularity of the functional when ∣ D ψ ∣= 1 , whether or not u ρ solves the Euler-Lagrange equation and how regular is u ρ are subtle issues that were investigated only for few classes of ρ . In this paper, we study both problems for general sources ρ , in a bounded domain with a Dirichlet boundary condition and in the entire R m . In particular, we give sufficient conditions to guarantee that u ρ solves Ethe uler-Lagrange equation and enjoys improved W 2 , 2 loc estimates, and we construct examples helping to identify sharp thresholds for the regularity of ρ to ensure the validity of the Euler-Lagrange equation. One of the main difficulties is the possible presence of light segments in the graph of u ρ , which will be discussed in detail

The Factorization Method for Conducting Transmission Conditions

In this work we consider an inverse problem arising from electromagnetic scattering by a medium covered with a very thin and highly conducting layer. Our main objective is to show that the Factorization Method, which is an inverse problem solution algorithm, can be applied to detect the position and shape of such objects from the measurements of the scattered waves at large distances. Such problems originate from applications such as land-mine detection, radar or seismic imaging