The Collaborative Research Centre TRR 361/F90 CREATOR (2022-2030) aims at establishing a new paradigm for the simulation-driven design of electric machines. Increasing demands on efficiency, power density and sustainability require the integration of multiphysical effects, advanced materials and complex geometries into the design process. Traditional sequential workflows are no longer sufficient to address these challenges. CREATOR therefore combines expertise from electrical engineering, applied mathematics, fluid dynamics and materials science to establish integrated modelling, simulation and optimisation methodologies in a single large-scale project funded by the German and Austrian national funding agencies. This article provides an overview of the research vision, key achievements from the first funding period (2022-2026) and current developments towards a computational electric machine laboratory.

Solidification, coupled with melt flow, plays a critical role in determining the microstructure and properties of materials in several manufacturing processes. Phase-field models coupled with the Navier-Stokes equations are widely used to model and simulate these dynamics. However, most existing models neglect essential thermodynamic couplings, particularly the capillary (Korteweg) stress in the momentum equation. This stress, which arises from the coupling between the phase field and the melt flow, accounts for thermal capillary effects during non-isothermal solidification. Neglecting it leads to models inconsistent with non-equilibrium thermodynamics and incapable of capturing capillarity-driven melt flow. In this work, we present a thermodynamically consistent, non-isothermal phase-field model for solidification coupled with melt flow, incorporating cross-coupling terms and explicitly including the Korteweg stress in the momentum equation. Model validation is performed for solidification-only cases, followed by simulations of dendritic growth under melt flow. The results show that thermal capillary effects induce flow near the interface, influencing dendrite tip velocity and morphology. Simulations under forced convection further demonstrate asymmetric dendrite growth due to the imposed flow field. Additionally, we numerically demonstrate the influence of viscosity interpolation schemes on enforcing the no-slip boundary condition in phase-field models with melt flow.

The modeling of electric machines and power transformers typically involves systems of nonlinear magnetostatics or magneto-quasistatics, and their efficient and accurate simulation is required for the reliable design, control, and optimization of such devices. We study the numerical solution of the vector potential formulation of nonlinear magnetostatics by means of higher-order finite element methods. Numerical quadrature is used for the efficient handling of the nonlinearities, and domain mappings are employed for the consideration of curved boundaries. The existence of a unique solution is proven on the continuous and discrete level, and a full convergence analysis of the resulting finite element schemes is presented, indicating order-optimal convergence rates under appropriate smoothness assumptions. For the solution of the nonlinear discretized problems, we consider a Newton method with line search for which we establish global linear convergence with convergence rates that are independent of the discretization parameters. We also prove local quadratic convergence in a mesh size and polynomial degree-dependent neighborhood of the solution which becomes effective when high accuracy of the nonlinear solver is demanded. The assumptions required for our analysis cover inhomogeneous, nonlinear, and anisotropic materials, which may arise in typical applications, including the presence of permanent magnets. The theoretical results are illustrated by numerical tests for some typical benchmark problems.

Trace theorems are an indispensable tool for the analysis of kinetic equations. They have been established in wide generality by Cessenat and co-workers for radiative transfer and related applications. A variety of trace estimates have been established for the kinetic Fokker-Planck and Kolmogorov equation, typically requiring smoothness of the underlying domain; see the recent survey by Niebel & Valentini. In this work, we prove a new trace estimate for kinetic Sobolev spaces over the sphere for ρ-convex domains which, in general, may have a non-smooth boundary. Similar to the work of Cessenat, we use characteristics to obtain trace estimates in weighted trace spaces with explicit constants. For completeness, we also present a density result for the corresponding function spaces on Lipschitz domains.

We study the numerical evaluation of an energy-based vector hysteresis model and its incorporation into finite element simulations based on a vector potential formulation. The inherent non-smoothness of the hysteresis model poses challenges for both numerical analysis and implementation. Using tools from convex analysis, we characterize the forward and inverse hysteresis operators in terms of energy densities. This characterization yields well-posedness of the resulting models and leads to robust algorithms for their evaluation based on generalized semi-smooth Newton methods. It furthermore enables a seamless integration into magnetic field simulations, leading to nonlinear and non-smooth optimization problems at every load step. We discuss the finite element discretization, present a semi-smooth Newton method for the iterative solution, and establish global linear convergence with mesh-independent convergence rates. The theoretical results are illustrated by numerical experiments.

We consider a systematic numerical approximation of a viscoelastic phase separation model that describes the demixing of a polymer-solvent mixture. An unconditionally stable discretisation method is proposed based on a finite element approximation in space and a variational time discretization strategy. The proposed method preserves the energy-dissipation structure of the underlying system exactly and allows us to establish a fully discrete nonlinear stability estimate in natural norms based on the concept of relative energy. These estimates are used to derive order optimal error estimates for the method under minimal smoothness assumptions on the problem data, despite the presence of various strong nonlinearities in the equations. The theoretical results and main properties of the method are illustrated by numerical simulations, which also demonstrate the capability to reproduce the relevant physical effects observed in experiments.

We consider systems of partial differential equations arising in the modeling and simulation of electric machines. The nonlinear problems, eventually obtained after time discretization, are usually solved by employing a vector potential formulation. In the relevant two-dimensional setting, a discretization can be obtained by H^1 -conforming finite elements. We here consider an alternative formulation based on the magnetic field which leads to a nonlinear saddlepoint problem. After commenting on the unique solvability, we study the numerical approximation by H(curl) -conforming finite elements and present the main convergence results. A particular focus is put on the efficient solution of the linearized systems arising in every step of the nonlinear Newton solver. Via hybridization, the linearized saddlepoint systems can be transformed into linear positive definite problems, which can be solved with similar computational complexity as those arising in the vector or scalar potential formulation. In summary, we can thus claim that the mixed finite element method involving the H-field can be considered a competitive alternative to the standard vector or scalar potential formulations for the solution of problems in nonlinear magneto-quasistatics.

Implicit models for magnetic coenergy have been proposed by Pera et al. to describe the anisotropic nonlinear material behavior of electrical steel sheets. This approach aims at predicting magnetic response for any direction of excitation by interpolating measured of B--H curves in the rolling and transverse directions. In an analogous manner, an implicit model for magnetic energy is proposed. We highlight some mathematical properties of these implicit models and discuss their numerical realization, outline the computation of magnetic material laws via implicit differentiation, and discuss the potential use for finite element analysis in the context of nonlinear magnetostatics.

Estimation of eddy-current losses in conducting non-laminated components of electrical devices requires expensive three-dimensional simulations. Various approximations are therefore used in practice to reduce the computational cost in the early design phase. We review some approaches and discuss their modelling assumptions and resulting approximations. In particular, we identify eddy-current reaction fields as a significant contribution that should be accounted for globally. These reaction fields can be approximately reconstructed from two-dimensional magnetostatic simulations by solving a single linearized time-periodic problem. We further discuss different strategies for solving this post-processing problem. Numerical results demonstrate improved loss prediction compared to standard post-processing at moderate additional cost.

We consider the stabilisation of solutions to the Cahn-Hilliard equation towards a given trajectory by means of a finite-dimensional static output feedback mechanism. Exponential stabilisation of the controlled state around the target trajectory is proven using careful energy estimates and a spectral condition which characterizes the strength of the feedback. The analysis is general enough to allow for pointwise and distributed measurements and actuation. The main results are derived via arguments that carry over to appropriate discretisation schemes which allows us to establish corresponding exponential stabilisation results also on the discrete level. The validity of our results and the importance of some of our assumptions are illustrated by numerical tests.

Ferromagnetic materials exhibit anisotropy, saturation, and hysteresis. We here study the incorporation of an incremental vector hysteresis model representing such complex behavior into nonlinear magnetic field problems both, from a theoretical and a numerical point of view. We show that the hysteresis operators, relating magnetic fields and fluxes at every material point, are strongly monotone and Lipschitz continuous. This allows to ensure well-posedness of the corresponding magnetic field problems and appropriate finite element discretizations thereof. We further show that the hysteresis operators are semi-smooth, derive a candidate for their generalized Jacobians, and establish global linear and local superlinear convergence of a the semi-smooth Newton method with line search applied to the iterative solution of the discretized nonlinear field problems. The results are proven in detail for a hysteresis model involving a single pinning force and the scalar potential formulation of magnetostatics. The extension to multiple pinning forces and the vector potential formulation is possible and briefly outlined. The theoretical results are further illustrated by numerical tests.

The energy-based vector hysteresis model of Francois-Lavet et al. establishes an implicit relation between magnetic fields and fluxes via internal magnetic polarizations which are determined by convex but non-smooth minimization problems. The systematic solution of these problems for every material point is a key ingredient for the efficient implementation of the model into standard magnetic field solvers. We propose to approximate the non-smooth terms via regularization which allows to employ standard Newton methods for the evaluation of the local material models while being in control of the error in this approximation. We further derive the inverse of the regularized hysteresis operator which amounts to a regularized version of the inverse hysteresis model. The magnetic polarizations in this model are again determined by local minimization problems which here are coupled across the different pinning forces. An efficient algorithm for solving the Newton systems is proposed which allows evaluation of the inverse hysteresis operator at the same cost as the forward model. Numerical tests on standard benchmark problems are presented for illustration of our results.

Incremental models for magnetic vector hysteresis have been developed in previous works in accordance with basic principles of thermodynamics. In this article, we derive an equivalent representation of the associated hysteresis operator in terms of a co-energy function which is useful for magnetic field computations based on a scalar potential. Using the convex duality, we further define the corresponding energy functional and the associated inverse hysteresis operator which is required for computations based on the vector potential. The equivalence of the two representations with the energy-based hysteresis models proposed in earlier works is demonstrated and numerical results for some typical test problems are presented obtained by finite element simulation of corresponding scalar and vector potential formulations.

Chemotaxis describes the intricate interplay of cellular motion in response to a chemical signal. In the slab geometry, the motion takes place between two infinite membranes. Like in previous investigations, the asymptotic regime of high tumbling rates is of particular interest. This work provides local existence and uniqueness of solutions to the kinetic equation and shows their convergence towards solutions of a parabolic Keller–Segel model in the asymptotic limit. In addition, convergence rates with respect to the asymptotic parameter are established under additional regularity assumptions on the problem data. Particular difficulties in the analysis are caused by vanishing velocities in the kinetic model as well as the occurrence of boundary terms.

The accurate modelling and simulation of electric devices involving ferromagnetic materials requires the appropriate consideration of magnetic hysteresis. We discuss the systematic incorporation of the energy-based vector hysteresis model of Henrotte et al. into vector potential formulations for the governing magnetic field equations. The field model describing a single step in a load cycle is phrased as a convex minimization problem which allows us to establish existence and uniqueness of solutions and to obtain accurate approximations by finite element discretization. Consistency of the model with the governing field equations is deduced from the first order optimality conditions. In addition, two globally convergent iterative methods are presented for the solution of the underlying minimization problems. The efficiency of the approach is illustrated by numerical tests for a typical benchmark problem.

Fixed-point or Newton-methods are typically employed for the numerical solution of nonlinear systems arising from discretization of nonlinear magnetic field problems. We here discuss an alternative strategy which uses Quasi-Newton updates locally, at every material point, to construct appropriate linearizations of the material behavior during the nonlinear iteration. The resulting scheme shows similar fast convergence as the Newton-method but, like the fixed-point methods, does not require derivative information of the underlying material law. As a consequence, the method can be used for the efficient solution of models with hysteresis which involve nonsmooth material behavior. The implementation of the proposed scheme can be realized in standard finite- element codes in parallel to the fixed-point and the Newton method. A full convergence analysis of all three methods is established proving global mesh-independent convergence. The theoretical results and the performance of the nonlinear iterative schemes are evaluated by computational tests for a typical benchmark problem.

We study a class of models for nonlinear acoustics, including the well-known Westervelt and Kuznetsov equations, as well as a model of Rasmussen that can be seen as a thermodynamically consistent modification of the latter. Using linearization, energy estimates, and fixed-point arguments, we establish the existence and uniqueness of solutions that, for sufficiently small data, are global in time and converge exponentially fast to equilibrium. In contrast to previous work, our analysis is based on a velocity-enthalpy formulation of the problem, whose weak form reveals the underlying port-Hamiltonian structure. Moreover, the weak form of the problem is particularly well-suited for a structure-preserving discretization. This is demonstrated in numerical tests, which also highlight typical characteristics of the models under consideration.

The propagation of charged particles through a scattering medium in the presence of a magnetic field can be described by a Fokker-Planck equation with Lorentz force. This model is studied both, from a theoretical and a numerical point of view. A particular trace estimate is derived for the relevant function spaces to clarify the meaning of boundary values. Existence of a weak solution is then proven by the Rothe method. In the second step of our investigations, a fully practicable discretization scheme is proposed based on implicit time-stepping through the energy levels and a spherical-harmonics finite-element discretization with respect to the remaining variables. A full error analysis of the resulting scheme is given, and numerical results are presented to illustrate the theoretical results and the performance of the proposed method.

We study the numerical solution of a Cahn-Hilliard/Allen-Cahn system with strong coupling through state and gradient dependent non-diagonal mobility matrices. A fully discrete approximation scheme in space and time is proposed which preserves the underlying gradient flow structure and leads to dissipation of the free-energy on the discrete level. Existence and uniqueness of the discrete solution is established and relative energy estimates are used to prove optimal convergence rates in space and time under minimal smoothness assumptions. Numerical tests are presented for illustration of the theoretical results and to demonstrate the viability of the proposed methods.

A novel strategy is proposed for the coupling of field and circuit equations when modeling power devices in the low-frequency regime. The resulting systems of differential-algebraic equations have a particular geometric structure which explicitly encodes the energy storage, dissipation, and transfer mechanisms. This implies a power balance on the continuous level which can be preserved under appropriate discretization in space and time. The models and main results are presented in detail for linear constitutive models, but the extension to nonlinear elements and more general coupling mechanisms is possible. The theoretical findings are demonstrated by numerical results.