Optimal time delays in a class of reaction-diffusion equations

A class of semilinear parabolic reaction diffusion equations with multiple time delays is considered. These time delays and corresponding weights are to be optimized such that the associated solution of the delay equation is the best approximation of a desired state function. The differentiability of the mapping is proved that associates the solution of the delay equation to the vector of weights and delays. Based on an adjoint calculus, first-order necessary optimality conditions are derived. Numerical test examples show the applicability of the concept of optimizing time delays.Comment: 17 pages, 2 figure

Eulerian finite element methods for interface problems and fluid-structure interactions

In this thesis, we develop an accurate and robust numerical framework for interface problems involving moving interfaces. In particular, we are interested in the simulation of fluid-structure interaction problems in Eulerian coordinates. Our numerical model for fluid-structure interactions (FSI) is based on the monolithic "Fully Eulerian" approach. With this approach we can handle both strongly-coupled problems and large structural displacements up to contact. We introduce modified discretisation schemes of second order for both space and time discretisation. The basic concept of both schemes is to resolve the interface locally within the discretisation. For spatial discretisation, we present a locally modified finite element scheme that is based on a fixed patch mesh and a local resolution of the interface within each patch. It does neither require any remeshing nor the introduction of additional degrees of freedom. For discretisation in time, we use a modified continuous Galerkin scheme. Instead of polynomials in direction of time, we define polynomial functions on space-time trajectories that do not cross the interface. Furthermore, we introduce a pressure stabilisation technique based on "Continuous Interior Penalty" method and a simple stabilisation technique for the structural equation that increases the robustness of the Fully Eulerian approach considerably. We give a detailed convergence analysis for all proposed discretisation and stabilisation schemes and test the methods with numerical examples. In the final part of the thesis, we apply the numerical framework to different FSI applications. First, we validate the approach with the help of established numerical benchmarks. Second, we investigate its capabilities in the context of contact problems and large deformations. We study contact problems of a falling elastic ball with the ground, an inclined plane and some stairs including the subsequent bouncing. For the case that no fluid layer remains between ball and ground, we use a simple contact algorithm. Furthermore, we study plaque growth in blood vessels up to a complete clogging of the vessel. Therefore, we use a monolithic mechano-chemical fluid-structure-interaction model and include the fast pulsating flow dynamics by means of a temporal two-scale scheme. We present detailed numerical studies for all three applications including a numerical convergence analysis in space and time, as well as an investigation of the influence of different material parameters