Weak imposition of Signorini boundary conditions on the boundary element method

We derive and analyse a boundary element formulation for boundary conditions involving inequalities. In particular, we focus on Signorini contact conditions. The Calder\'on projector is used for the system matrix and boundary conditions are weakly imposed using a particular variational boundary operator designed using techniques from augmented Lagrangian methods. We present a complete numerical a priori error analysis and present some numerical examples to illustrate the theory

Analysis of an implicitly extended Crank-Nicolson scheme for the heat equation on a time-dependent domain

We consider a time-stepping scheme of Crank-Nicolson type for the heat equation on a moving domain in Eulerian coordinates. As the spatial domain varies between subsequent time steps, an extension of the solution from the previous time step is required. Following Lehrenfeld \& Olskanskii [ESAIM: M2AN, 53(2):\,585-614, 2019], we apply an implicit extension based on so-called ghost-penalty terms. For spatial discretisation, a cut finite element method is used. We derive a complete a priori error analysis in space and time, which shows in particular second-order convergence in time under a parabolic CFL condition. Finally, we present numerical results in two and three space dimensions that confirm the analytical estimates, even for much larger time steps

On temporal homogenization in the numerical simulation of atherosclerotic plaque growth

A temporal homogenization approach for the numerical simulation of atherosclerotic plaque growth is extended to fully coupled fluid-structure interaction (FSI) simulations. The numerical results indicate that the two-scale approach yields significantly different results compared to a simple heuristic averaging, where only stationary long-scale FSI problems are solved, confirming the importance of incorporating stress variations on small time-scales. In the homogenization approach, a periodic fine-scale problem, which is periodic with respect to the heart beat, has to be solved for each long-scale time step. Even if no exact initial conditions are available, periodicity can be achieved within only 2-3 heart beats by simple time-stepping

A locally modified second-order finite element method for interface problems

The locally modified finite element method, which is introduced in [Frei, Richter: SINUM 52(2014), p. 2315-2334] is a simple fitted finite element method that is able to resolve weak discontinuities in interface problems. The method is based on a fixed structured coarse mesh, which is then refined into sub-elements to resolve an interior interface. In this work, we extend the locally modified finite element method to second order using an isoparametric approach in the interface elements. Thereby we need to take care that the resulting curved edges do not lead to degenerate sub-elements. We prove optimal a priori error estimates in the $L^2$-norm and in a modified energy norm, as well as a reduced convergence order of ${\cal O}(h^{3/2})$ in the standard $H^1$-norm. Finally, we present numerical examples to substantiate the theoretical findings

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

Dynamic Feedforward Control for an Active Three Phase EMI Filter

Power electronic systems usually produce high amounts of electromagnetic interferences (EMI) due to PWM operation. To comply with international standards on electromagnetic compatibility (EMC), e. g. EN 61800-3 in industrial applications [1], the EMI shall not exceed given limits. To reach these limits often filters need to be integrated into the system. The common solution is to add passive EMI filters consisting of capacitors and inductors [2]. Passive filters are often bulky and heavy, therefore active cancellation techniques for power electronic systems were introduced [3]. Active EMI filter (AEF) consists of an analog (and rarely also digital) circuit to actively suppress the disturbances by injecting an anti-noise signal. As discussed in [4], well-known closed loop feedforward or feedback structures for AEF have their limitation specially because of the limited amplifiers open-loop gain, delays due to the finite signal propagation speed in the circuit and the measurement accuracy. For this reason, active EMI cancellation by injecting synthesized signals was introduced. This paper introduces a method for generating a synthesized signal for a three-phase grid converter