Fabian Meyer

Heiko Gimperlein, Fabian Meyer, Ceyhun Oezdemir et al.

The solution of the wave equation in a polyhedral domain in $\mathbb{R}^3$ admits an asymptotic singular expansion in a neighborhood of the corners and edges. In this article we formulate boundary and screen problems for the wave equation as equivalent boundary integral equations in time domain, study the regularity properties of their solutions and the numerical approximation. Guided by the theory for elliptic equations, graded meshes are shown to recover the optimal approximation rates known for smooth solutions. Numerical experiments illustrate the theory for screen problems. In particular, we discuss the Dirichlet and Neumann problems, as well as the Dirichlet-to-Neumann operator and applications to the sound emission of tires.

Heiko Gimperlein, Fabian Meyer, Ceyhun Oezdemir et al.

This article considers a unilateral contact problem for the wave equation. The problem is reduced to a variational inequality for the Dirichlet-to-Neumann operator for the wave equation on the boundary, which is solved in a saddle point formulation using boundary elements in the time domain. As a model problem, also a variational inequality for the single layer operator is considered. A priori estimates are obtained for Galerkin approximations both to the variational inequality and the mixed formulation in the case of a flat contact area, where the existence of solutions to the continuous problem is known. Numerical experiments demonstrate the performance of the proposed mixed method. They indicate the stability and convergence beyond flat geometries.

Jan Giesselmann, Fabian Meyer, Christian Rohde

In this article we present an a posteriori error estimator for the spatial-stochastic error of a Galerkin-type discretisation of an initial value problem for a random hyperbolic conservation law. For the stochastic discretisation we use the Stochastic Galerkin method and for the spatial-temporal discretisation of the Stochastic Galerkin system a Runge-Kutta Discontinuous Galerkin method. The estimator is obtained using smooth reconstructions of the discrete solution. Combined with the relative entropy stability framework of Dafermos \cite{dafermos2005hyperbolic}, this leads to computable error bounds for the space-stochastic discretisation error. Moreover, it turns out that the error estimator admits a splitting into one part representing the spatial error, and a remaining term, which can be interpreted as the stochastic error. This decomposition allows us to balance the errors arising from spatial and stochastic discretisation. We conclude with some numerical examples confirming the theoretical findings.

Jan Giesselmann, Fabian Meyer, Christian Rohde

In this article we consider one-dimensional random systems of hyperbolic conservation laws. We first establish existence and uniqueness of random entropy admissible solutions for initial value problems of conservation laws which involve random initial data and random flux functions. Based on these results we present an a posteriori error analysis for a numerical approximation of the random entropy admissible solution. For the stochastic discretization, we consider a non-intrusive approach, the Stochastic Collocation method. The spatio-temporal discretization relies on the Runge--Kutta Discontinuous Galerkin method. We derive the a posteriori estimator using continuous reconstructions of the discrete solution. Combined with the relative entropy stability framework this yields computable error bounds for the entire space-stochastic discretization error. The estimator admits a splitting into a stochastic and a deterministic (space-time) part, allowing for a novel residual-based space-stochastic adaptive mesh refinement algorithm. We conclude with various numerical examples investigating the scaling properties of the residuals and illustrating the efficiency of the proposed adaptive algorithm.

Esther Bischoff, Fabian Meyer, Jairo Inga et al.

In order to fully exploit the advantages inherent to cooperating heterogeneous multi-robot teams, sophisticated coordination algorithms are essential. Time-extended multi-robot task allocation approaches assign and schedule a set of tasks to a group of robots such that certain objectives are optimized and operational constraints are met. This is particularly challenging if cooperative tasks, i.e. tasks that require two or more robots to work directly together, are considered. In this paper, we present an easy-to-implement criterion to validate the feasibility, i.e. executability, of solutions to time-extended multi-robot task allocation problems with cross schedule dependencies arising from the consideration of cooperative tasks and precedence constraints. Using the introduced feasibility criterion, we propose a local improvement heuristic based on a neighborhood operator for the problem class under consideration. The initial solution is obtained by a greedy constructive heuristic. Both methods use a generalized cost structure and are therefore able to handle various objective function instances. We evaluate the proposed approach using test scenarios of different problem sizes, all comprising the complexity aspects of the regarded problem. The simulation results illustrate the improvement potential arising from the application of the local improvement heuristic.