Real ferromagnetic materials exhibit complex hysteresis effects, which require memory-dependent constitutive mapping between the magnetic field H and the flux density B for solving low-frequency Maxwell’s equations. While the thermodynamically formulated generalized Prandtl–Ishlinskii [1] stop-type vector hysteresis operator effectively captures hysteresis nonlinearities, it requires a local Newton iterative procedure to update each hysteron (or local hysteresis operator), resulting in considerable computational effort, particularly in finite element (FE) simulations. For isotropic materials, a recent simplification [2] replaces the nonlinear mapping on the stop operator with an identity, such that the hysteresis operators are directly weighted through their outputs. This enables a fully analytical return-point mapping for the local operator update, thereby eliminating per-hysteron iterative updates and achieving significant runtime reductions with comparable accuracy. This work aims to extend the simplification to anisotropic materials, where the admissible domain of the dissipative element in each hysteron becomes an ellipsoid rather than a sphere, and a closed-form update is not straightforward. The core research direction is therefore to exploit the eigendecomposition of the anisotropy matrix (similar to [3]), reducing the local hysteron update to a scalar Newton solve, and to integrate the resulting constitutive mapping into an FE solver for validation.

The specific tasks are:

* Literature review on the generalized and simplified isotropic Prandtl-Ishlinskii stop-type model formulations within the thermodynamic framework

* Formulate operator simplification for the anisotropic case and derive a fast update of the local hysteresis operators

* Identify necessary parameters for the anisotropic simplified model using measured B–H data for reference hysteresis material

* Integrate the anisotropic simplified constitutive mapping into an existing FE solver and assess the accuracy and runtime performance against the generalized anisotropic model

References:

[1] X. Xiao, F. Müller, M. M. Nell and K. Hameyer, “Modeling anisotropic magnetic hysteresis properties with vector stop model by using finite element method,” COMPEL, vol. 41, no. 2, pp. 752–763, 2022.

[2] A. Shankar, K. Kuhnen, I. Kulchytska-Ruchka and S. Schöps, „Simplification of the Isotropic Generalized Stop-Type Prandtl-Ishlinskii Vector Hysteresis Operator Using Analytical Return-Point Mapping,“ in IEEE Trans. on Magn., doi: 10.1109/TMAG.2026.3731649, 2026.

[3] D. Versino and K. Bennett, „Generalized radial-return mapping algorithm for anisotropic von Mises plasticity framed in material eigenspace: Generalized radial-return mapping algorithm,“ Int. J. Numer. Methods Eng., vol. 116, doi: 10.1002/nme.5921, 2018.

Conducting numerical simulations to evaluate performance characteristics is an integral part of the electric machine design workflow. Traditionally, these simulations are conducted using the Finite Element Method (FEM), which operates on the mesh — a discrete representation of the underlying design geometry that consists of piecewise-polynomial elements. Curved geometries, e.g. the circular airgap between stator and rotor in an electric machine, are usually represented by Non-uniform rational B-Splines (NURBS) and can not be represented exactly, but only approximated by these finite elements. The NURBS-enhanced Finite Element Method (NEFEM) is a recently proposed extension, in which select elements are modified to allow exact representation of NURBS-curved geometries. The goal of this thesis project is to investigate the applicability of NEFEM to electric machine simulations using an experimental implementation based on the Finite Element solver GetDP. Tasks will include geometric modelling and simulation setup for electric machines and interpretation of results.

Integral equation methods are widely used in computational electromagnetics to model open-domain problems such as electrostatic, magnetostatic, or magnetoquasistatic fields. After discretization with basis functions, the resulting linear systems are typically dense, leading to quadratic memory consumption and computational cost.

Hierarchical compression techniques exploit the fact that interactions between well-separated regions of the domain are often of low numerical rank. By approximating these interactions with low-rank representations, both memory usage and computational complexity can be significantly reduced. One such approach is recursive skeletonization, implemented in the MATLAB library FLAM (Fast Linear Algebra in MATLAB).

The goal of this thesis is to investigate the use of FLAM to compress system matrices arising from integral equation formulations. A simple electromagnetic model problem will be discretized and solved both with and without compression. The student will analyze the influence of hierarchical compression on solution accuracy, memory consumption, and computation time. If time permits, the impact of higher-order basis functions on compression efficiency may also be explored.

Systems with port-Hamiltonian (pH) structure have several beneficial properties, such as power balance and passivity. Moreover, the interconnection of pH systems results in a system that preserves the pH structure. Recently, an energy-based pH framework for field/circuit coupled systems has been proposed. The framework enables the pH structure to be proven for systems containing both field models and circuits, for example in the three-phase transformers. The goal of this project is to show the effectiveness of the proposed framework for systems containing foil conductors. This necessitates implementing a structure-preserving interconnection for the foil conductor model.

TRBDF2 (Trapezoidal Rule with Backward Differentiation Formula 2) is one of the standard time integration schemes for circuit simulation. The goal of the thesis is to implement the method, while conforming to the interface specified by the scipy.integrate package. Depending on progress, more recent extensions regarding adaptive time stepping from may also be investigated and implemented.

The thesis aims to explore non-overlapping domain decomposition methods with optimized transmission conditions for high-frequency Helmholtz problems. Students should first derive the Helmholtz formulation and test methods on a simpler model problem. Successful approaches transition to Helmholtz for a prototype proof-of-concept solver in C++ or Python using an FEM library. Model problem experiments analyze convergence behavior; Helmholtz experiments focus on proof-of-concept if performed. Optional: modelling of application examples and/or numerical algorithm analysis.

Most finite element methods can be related to some kind of mesh. In computational electromagnetism, the gradient can be associated with vertices and edges of the mesh. We can exploit this knowledge in simulations by identifying and using a spanning tree, i.e., a selection of #vertices−1 edges that connect all vertices. The vertices and edges are implicitly linked to the topology of the mesh. This can impact simulation as well because a tree might not provide enough information, which necessitates adding additional edges to obtain a belted tree. For the tree construction, the well-known Kruskal’s algorithm is a flexible and simple choice. It requires weights that are assigned to each edge and builds a minimal spanning tree, i.e., it tries to minimize the weights of the edges in the tree. This allows us to control the structure of the tree via specifying appropriate weights.

This project investigates the validity of equivalent circuit models for the transient analysis of superconducting magnets. Superconducting magnets exhibit complex magneto-thermal behavior during dynamic events such as quenches and current ramps. The study compares simplified equivalent circuit approaches with detailed multiphysics simulations to evaluate their accuracy and limitations. Key performance metrics include voltage evolution, and thermal response during transients. The results aim to assess whether equivalent circuit models provide a reliable and computationally efficient alternative.

In particular, reduced-order circuit representations will be implemented in Xyce. They are benchmarked against existing high-fidelity finite element models developed within CERN’s Finite Element Quench Simulator (FiQuS). While Xyce enables efficient transient simulations based on lumped parameter networks, FiQuS resolves the coupled electromagnetic and thermal fields in two- or three-dimensional space. By systematically comparing both approaches under representative operating scenarios, the project quantifies the trade-offs between computational cost, modeling complexity, and predictive accuracy.

Please note that this is not about the application of FEM with commercial tools, but about developing algorithms to solve BEM/FEM problems, with a strong focus on mathematics.

You develop and implement solution algorithms and programs to simulate eddy current problems based on the Boundary Element Method (BEM) and FEM in C++ (BETL2 framework) and Python. Furthermore, you extend existing simulation software with new features (e.g. coupling multiple BEM domains, impedance boundary conditions). You validate the functionality of the software in applications for position sensors. Last but not least, you evaluate the use of AI methods for mesh optimization.

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https://jobs.smartrecruiters.com/BoschGroup/744000100010544

Im Rahmen der externen Abschlussarbeit tauchen Sie in die Welt der magnetischen Positionssensoren ein und entwickeln einen innovativen Design-of-Experiments-(DOE)-Ansatz, der auf der leistungsstarken UQpy-Bibliothek basiert. Sie arbeiten an der mathematischen Modellierung wichtiger Kenngrößen in Abhängigkeit von Eingangsparametern. Die Python-basierte Anbindung der entwickelten Methoden an eine elektromagnetische Finite-Elemente-Simulation (FEM) ist eine Ihrer Aufgaben. Zudem haben Sie gegebenenfalls die Möglichkeit, tiefere Einblicke in die Entwicklung von Modellen basierend auf künstlicher Intelligenz zur Modellierung des Sensorverhaltens zu erhalten.

Details

https://jobs.smartrecruiters.com/BoschGroup/744000096573674

When dealing with complex engineering systems, obtaining many field solutions in multi-query scenarios such as design optimization, uncertainty quantification or coupled simulations can become prohibitively expensive. A possible remedy for this problem are so-called surrogate models, which provide a less accurate but computationally cheaper approximation of the field solution. One common type of surrogate model are Gaussian processes (GPs), which often possess advantages in terms of training cost and quality of approximation compared to standard neural network approaches. In previous work, we developed an approach for constructing GP surrogates of parameter dependent stiff ODE solutions using reparameterizations. The aim of this thesis is to extend that approach to the 2D case, illustrating its flexibility and enabling a larger range of applications.

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.