Parallel algorithms play an increasingly vital role in both research and industry for accelerating simulations. Domain Decomposition methods (DDMs), such as nonoverlapping Schwarz or Dirichlet-Neumann/Neumann-Neumann methods, are effective approaches for introducing spatial concurrency, thereby facilitating parallel computations. When the problems under examination are not only spatially dependent but also time-dependent, waveform relaxation enhances information exchange between different time steps [2, 3]. This, in turn, enhances additional concurrency in time when combined with Parareal methods.

The objective of this project is to implement selected algorithms from the provided references and evaluate their performance using discretized benchmark problems.

Due to the growing importance of e-mobility, the efficient simulation and optimization of electric energy converters, in particular electric machines, is becoming increasingly important. In the manufacturing process of these electric machines imperfections and small deviations from the nominal design can occur. In the worst case, these imperfections can lead to a significant decrease in quality or even failure of the machine. To avoid this, the machine can be optimized robustly, i.e., considering the deviations in the machine design. Robust optimization aims to find a machine design which is robust in terms of deviations from the nominal design, i.e., to find an optimum in terms of a goal function J which does not deteriorate significantly for small changes in the design parameters p, compare Fig. 2. In this project an electric machine is simulated using isogeometric analysis and the shape of the machine shall be optimized robustly considering uncertainties, using uncertainty quantification methods like the Monte Carlo method.

The determination of optimal current distributions on given surfaces is often the starting point for the electromagnetic design of coil dominated electromagnets utilized, e.g. in particle accelerators and magnetic resonance imaging (MRI). Geometrical and mechanical constraints as well as the particle beam size determine the coil winding surface, on which an optimal current density is to be determined.

In this Bachelor thesis, a stream function approach for the determination of optimal current densities on given surfaces shall be developed and implemented in Bembel, the BEM-based engineering library, see www.bembel.eu. The goal is an automated generation of winding paths, on given surfaces and field quality requirements.

The classical way of expressing the field uniformity in accelerator magnets is by means of the Fourier coefficients of the trigonometric eigenfunctions of the Laplace equation also known as multipoles. When the magnets are curved (Fig. 1), higher-order terms appear and the scaling laws derived for straight magnets are no more applicable. The field reconstruction from field simulations or magnetic measurements should therefore be based on the eigenfunctions of the Laplace equation in the toroidal coordinate system, so called toroidal harmonics.

TU Darmstadt, together with the European Organization for Nuclear Research (CERN) formulated a linear inverse problem to determine the toroidal harmonics from the magnetic flux density.

In this thesis, different algorithms to solve this inverse problem shall be studied and compared on magnetic flux density measurements.

High-temperature superconducting (HTS) materials are a promising technology for high-field magnets in particle accelerators. In particular, no-insulation (NI) coils, i.e., coils wound without turn-to-turn insulation, have gained popularity due to their robustness [1]. Numerical methods such as the finite element (FE) method play a key role in developing HTS-based applications. The objective of this project is to extend CERN’s existing open-source Finite Element Quench Simulation (FiQuS) framework with 2D axisymmetric FE models of NI coils [2]. Since they are more efficient but less general than existing 3D models, they will complement the latter as an important tool in FiQuS to analyze NI coils. Following FiQuS’s ideals, a key aspect will be to hide the complexities CERN of the FE formulation from the users who are typically not FE experts.

CERN started a project last year to build a comprehensive open-source quench simulation tool called FiQuS (Finite Element Quench Simulation). It is based on the finite element (FE) framework ONELAB and written mostly in Python. The idea is to build a flexible tool which allows users to build and simulate complex models from human-readable inputs hiding the complexity of the FE kernel. In order to simulate real-world accelerator magnet circuits, this project aims at coupling FE magnet models from FiQuS with electric circuits [1], which can then be solved by circuit simulators such as Xyce. A key aspect will be to hide the complexities of this coupling from the users who are typically not FE experts.

Solving partial differential equations numerically with boundary element methods requires the application of fast methods to be competitive with other numerical methods. At high-frequencies, the existing implementation of H2-matrices breaks down and needs to be adapted in order to work efficiently.

The idea of the approach is to approximate a spherical wave in far distance hierarchically by plane waves. The implementation is carried out in the C++ library Bembel, see www.bembel.eu.

Electric energy conversion is a key issue on the way to decarbonization. Computational design and optimization of electric motors is a very active research area with the aim to increase the efficiency and power density of electric drives. Yet, optimization in commercial solvers is often performed using time-consuming methods such as surrogates or genetic algorithms, taking days or weeks for one optimization.

This work combines the modeling of the motor using Isogeometric Analysis (IGA), which allows to exactly represent the geometry, with fast gradient based optimization. By using present state-of-the-art numerical modeling techniques together with efficient optimization algorithms, it is possible to reduce the optimization time to several minutes.

The need for higher energy efficiency and decarbonization give rise to a steadily increasing importance of electric drives. Simulations allow the physical limits to be pushed in order to increase the power density and make motors more cost-efficient. This work aims to investigate the influence of mechanical stresses in electric motors on the electromagnetic behavior and find out how stress dependent material properties can be mitigated or exploited. Simulations are performed in an Finite Element (FE) framework using Isogeometric Analysis (IGA), which allows to exactly represent the geometry. This enables an efficient coupling of the geometric, magnetic and mechanical systems.

Deviations in the manufacturing process of electronic components may lead to rejections due to malfunctioning. Uncertain design parameters (i.e. geometrical and material parameters) can be modeled as random variables. Then, the failure probability of a realization can be estimated. A standard approach for estimating failure probabilities is a Monte Carlo analysis. In a Monte Carlo analysis a large number of sample points is generated according to a given probability distribution. The percentage of sample points not fulfilling some predefined performance feature specifications denotes the failure probability. In order to obtain a reliable estimation, a large number of sample points is required. This leads to high computing costs, since for each sample point a PDE must be solved, e.g. with the finite element method (FEM). Current research deals with the reduction of computational effort. Importance sampling is an approach to reduce the number of FEM evaluations by generating sample points in critical regions with a higher probability.

Due to the growing importance of e-mobility, the efficient simulation and optimization of electric energy converters, in particular electric machines, is becoming increasingly important. In the manufacturing process of these electric machines imperfections and small deviations from the nominal design can occur. In the worst case, these imperfections can lead to a significant decrease in quality or even failure of the machine. To avoid this, the machine can be optimized robustly, i.e., considering the deviations in the machine design. Robust optimization aims to find a machine design which is robust in terms of deviations from the nominal design, i.e., to find an optimum in terms of a goal function J which does not deteriorate significantly for small changes in the design parameters p, compare Fig. 2. In this project an electric machine is simulated using isogeometric analysis and the shape of the machine shall be optimized robustly considering uncertainties, using uncertainty quantification methods like the Monte Carlo method.