Sebastian Schöps

Felix Fritzen, Bernhard Haasdonk, David Ryckelynck et al.

A novel algorithmic discussion of the methodological and numerical differences of competing parametric model reduction techniques for nonlinear problems are presented. First, the Galerkin reduced basis (RB) formulation is presented which fails at providing significant gains with respect to the computational efficiency for nonlinear problems. Renown methods for the reduction of the computing time of nonlinear reduced order models are the Hyper-Reduction and the (Discrete) Empirical Interpolation Method (EIM, DEIM). An algorithmic description and a methodological comparison of both methods are provided. The accuracy of the predictions of the hyper-reduced model and the (D)EIM in comparison to the Galerkin RB is investigated. All three approaches are applied to a simple uncertainty quantification of a planar nonlinear thermal conduction problem. The results are compared to computationally intense finite element simulations.

Zeger Bontinck, Jacopo Corno, Herbert De Gersem et al.

In this communication the advantages and drawbacks of the isogeometric analysis (IGA) are reviewed in the context of electromagnetic simulations. IGA extends the set of polynomial basis functions, commonly employed by the classical Finite Element Method (FEM). While identical to FEM with Nédélec's basis functions in the lowest order case, it is based on B-spline and Non-Uniform Rational B-spline basis functions. The main benefit of this is the exact representation of the geometry in the language of computer aided design (CAD) tools. This simplifies the meshing as the computational mesh is implicitly created by the engineer using the CAD tool. The curl- and div-conforming spline function spaces are recapitulated and the available software is discussed. Finally, several non-academic benchmark examples in two and three dimensions are shown which are used in optimization and uncertainty quantification workflows.

Vivek Parekh, Dominik Flore, Sebastian Schöps

Optimization of rotating electrical machines is both time- and computationally expensive. Because of the different parametrization, design optimization is commonly executed separately for each machine technology. In this paper, we present the application of a variational auto-encoder (VAE) to optimize two different machine technologies simultaneously, namely an asynchronous machine and a permanent magnet synchronous machine. After training, we employ a deep neural network and a decoder as meta-models to predict global key performance indicators (KPIs) and generate associated new designs, respectively, through unified latent space in the optimization loop. Numerical results demonstrate concurrent parametric multi-objective technology optimization in the high-dimensional design space. The VAE-based approach is quantitatively compared to a classical deep learning-based direct approach for KPIs prediction.

Stephanie Friedhoff, Jens Hahne, Iryna Kulchytska-Ruchka et al.

We consider the usage of parallel-in-time algorithms of the Parareal and multigrid-reduction-in-time (MGRIT) methodologies for the parallel-in-time solution of the eddy current problem. Via application of these methods to a two-dimensional model problem for a coaxial cable model, we show that a significant speedup can be achieved in comparison to sequential time stepping.

Maximilian Nolte, Riccardo Torchio, Sebastian Schöps et al.

This contribution investigates the connection between isogeometric analysis and integral equation methods for full-wave electromagnetic problems up to the low-frequency limit. The proposed spline-based integral equation method allows for an exact representation of the model geometry described in terms of non-uniform rational B-splines without meshing. This is particularly useful when high accuracy is required or when meshing is cumbersome for instance during optimization of electric components. The augmented electric field integral equation is adopted and the deflation method is applied, so the low-frequency breakdown is avoided. The extension to higher-order basis functions is analyzed and the convergence rate is discussed. Numerical experiments on academic and realistic test cases demonstrate the high accuracy of the proposed approach.

Zeger Bontinck, Oliver Lass, Sebastian Schöps et al.

The simulation of electric rotating machines is both computationally expensive and memory intensive. To overcome these costs, model order reduction techniques can be applied. The focus of this contribution is especially on machines that contain non-symmetric components. These are usually introduced during the mass production process and are modeled by small perturbations in the geometry (e.g., eccentricity) or the material parameters. While model order reduction for symmetric machines is clear and does not need special treatment, the non-symmetric setting adds additional challenges. An adaptive strategy based on proper orthogonal decomposition is developed to overcome these difficulties. Equipped with an a posteriori error estimator the obtained solution is certified. Numerical examples are presented to demonstrate the effectiveness of the proposed method.

Louis Denis, Elias Paakkunainen, Paavo Rasilo et al.

We extend the foil winding homogenization method to magnetic field conforming formulations. We first propose a full magnetic field foil winding formulation by analogy with magnetic flux density conforming formulations. We then introduce the magnetic scalar potential in non-conducting regions to improve the efficiency of the model. This leads to a significant reduction in the number of degrees of freedom, particularly in 3-D applications. The proposed models are verified on two frequency-domain benchmark problems: a 2-D axisymmetric problem and a 3-D problem. They reproduce results obtained with magnetic flux density conforming formulations and with resolved conductor models that explicitly discretize all turns. Moreover, the models are applied in the transient simulation of a high-temperature superconducting coil. In all investigated configurations, the proposed models provide reliable results while considerably reducing the size of the numerical problem to be solved.

Michael Wiesheu, Sebastian Schöps, Idoia Cortes Garcia

This paper compares three different multirate splitting approaches for the application on field-circuit coupled magnetoquasistatic simulations. For these methods, again three different variants for exchanging values between the field and circuit are tested, namely voltages, currents and flux correction terms. All scenarios are applied on two different benchmark problems, i.e. a coil inductor and transformer model coupled to different circuits. The convergence behavior of different time steppers (Implicit Euler and Trapezoidal Rule) is determined for all possible settings, and guidelines for practical applications are derived.

Peter Förster, Idoia Cortes Garcia, Wil Schilders et al.

We present a non-intrusive version of the index-aware learning framework introduced in arXiv:2309.00958. Index-aware learning itself is an approach for learning the time and parameter dependent solutions of differential-algebraic equations (DAEs), in particular those of electrical circuits. A key feature of the approach is that it ensures the learned solutions to remain physics-consistent, i.e.\ inherent constraints of the solution, such as e.g.\ Kirchhoff's laws, remain fulfilled. In general, this is achieved by leveraging a decoupling of the DAE into its differential and algebraic parts, while the non-intrusive version of the approach additionally relies on results from arXiv:2604.20475 and arXiv:2107.07755. We illustrate the overall workflow and compare the non-intrusive and intrusive versions using a buck converter as an example.

Timon Seibel, Sebastian Schöps, Kersten Schmidt

In this work, we study the application of a hybrid mixed domain decomposition (HMDD) method for the rotor-stator coupling of a permanent magnet synchronous machine. For this, we derive a variational formulation on the electric machine inspired by hybridized discontinuous Galerkin methods using a mixed magnetostatics problem, an affine material law and boundary conditions respecting the symmetry of the motor. We are then able to locate the resulting finite element method within the HMDD framework presented in arXiv:2604.22543. This enables us naturally to transfer the well-posedness results and error estimates for the HMDD method to the finite element method considered in this work. Lastly, as a proof of concept, we consider an academic example and compare the resulting magnetic flux density and potential lines to their counterparts obtained by a well-established in-house code using iso-geometric analysis.

Robin Willems, Peter Förster, Sebastian Schöps et al.

Cardio-mechanical models can be used to support clinical decision-making. Unfortunately, the substantial computational effort involved in many cardiac models hinders their application in the clinic, despite the fact that they may provide valuable information. In this work, we present a probabilistic reduced-order modeling (ROM) framework to dramatically reduce the computational effort of such models while providing a credibility interval. In the online stage, a fast-to-evaluate generalized one-fiber model is considered. This generalized one-fiber model incorporates correction factors to emulate patient-specific attributes, such as local geometry variations. In the offline stage, Bayesian inference is used to calibrate these correction factors on training data generated using a full-order isogeometric cardiac model (FOM). A Gaussian process is used in the online stage to predict the correction factors for geometries that are not in the training data. The proposed framework is demonstrated using two examples. The first example considers idealized left-ventricle geometries, for which the behavior of the ROM framework can be studied in detail. In the second example, the ROM framework is applied to scan-based geometries, based on which the application of the ROM framework in the clinical setting is discussed. The results for the two examples convey that the ROM framework can provide accurate online predictions, provided that adequate FOM training data is available. The uncertainty bands provided by the ROM framework give insight into the trustworthiness of its results. Large uncertainty bands can be considered as an indicator for the further population of the training data set.

Idoia Cortes Garcia, Peter F. Förster, Lennart Jansen et al.

We derive a topological decoupling of the equations of modified nodal analysis (MNA) to a semi-explicit index one differential-algebraic equation. The decoupling explicitly allows for controlled sources, which play a crucial role in engineering design workflows. Furthermore, the proof is constructive and provides a graph-based algorithmic framework for the computation of the decoupling, enabling its application to a variety of industry problems. These include the generation of consistent initial conditions, model order reduction, (scientific) machine learning, as well as speeding up conventional circuit simulation. In addition, the decoupling preserves the structure of MNA, i.e. the resulting systems remain sparse and key parts remain positive definite. We illustrate the decoupling using multiple examples, including some of the most common subcircuits containing controlled sources. Lastly, we also provide a first software implementation of the decoupling.

Jennifer Dutiné, Markus Clemens, Sebastian Schöps

The spatial discretization of the magnetic vector potential formulation of magnetoquasistatic field problems results in an infinitely stiff differential-algebraic equation system. It is transformed into a finitely stiff ordinary differential equation system by applying a generalized Schur complement. Applying the explicit Euler time integration scheme to this system results in a small maximum stable time step size. Fast computations are required in every time step to yield an acceptable overall simulation time. Several acceleration methods are presented.

Elias Paakkunainen, Louis Denis, Benoît Vanderheyden et al.

Efficient numerical models are required for the design of systems with high temperature superconductor (HTS) coils, as fully resolved finite element simulations of individual coated conductors become computationally prohibitive. This work applies the foil conductor model (FCM) to insulated HTS coils using magnetic field conforming h-(full), h-$ϕ$, and t-$ω$ formulations. The approach replaces individual turns by a homogenized bulk and ensures physically consistent current density distributions in the coils by using additional voltage basis functions in the finite element formulations. The models are verified in 2D axisymmetric and 3D geometries with a pancake coil simulation under AC transport current excitation. All FCM formulations show excellent agreement with reference detailed simulations, with coefficients of determination above 0.99 for instantaneous AC losses. In 3D, the h-$ϕ$ and especially the t-$ω$ formulation substantially reduce the number of degrees of freedom by using the magnetic scalar potential in non-conducting regions. Scalability is demonstrated with a 3D stack of racetrack coils model with a field- and angle-dependent critical current density. For the stack of racetrack coils, while maintaining accurate loss prediction, the t-$ω$ FCM achieves a speedup factor of 22 and reduces degrees of freedom by 78 % with respect to a detailed reference model.

Kersten Schmidt, Timon Seibel, Sebastian Schöps

We present a domain decomposition formulation based on hybridization which is inspired by hybridized discontinuous Galerkin (HDG) methods, that enhance mixed domain decomposition methods by incorporating stabilization terms. Unlike discontinuous Galerkin methods, our analysis of the proposed finite element method is based on a corresponding consistent variational formulation and a perturbed Galerkin method. In the variational formulation the divergence appears not only within subdomains, but also as an $L^2$-surface quantity on the interfaces. Furthermore, the traces of the finite element functions on the interfaces are replaced by $L^2$-distributions. The well-posedness of the perturbed Galerkin method is shown for an appropriate choice of subspaces, in a manner similar to that of the variational formulation. For the finite element method we use Raviart-Thomas elements for the dual variable and piecewise polynomials for the primal and hybrid variables, respectively. We perform an analysis of the discretization error which is explicit in the stabilization parameter $τ$. Numerical experiments for piecewise smooth solutions using finite element spaces of order~$q$ on curved quadrilateral meshes confirm the predicted convergence rate of $q+1$ for small values of $τ$. In the error analysis we observe the discretization error to be uniformly bounded in $τ$. Even for large $τ$ values the observed convergence rates for the primal and for the hybrid variables are $q+1$. For the dual variable the convergence rate depends on the stabilization parameter and the mesh-width, with an asymptotic rate of $q+\tfrac12$.

Vivek Parekh, Dominik Flore, Sebastian Schöps et al.

Magneto-static finite element (FE) simulations make numerical optimization of electrical machines very time-consuming and computationally intensive during the design stage. In this paper, we present the application of a hybrid data-and physics-driven model for numerical optimization of permanent magnet synchronous machines (PMSM). Following the data-driven supervised training, deep neural network (DNN) will act as a meta-model to characterize the electromagnetic behavior of PMSM by predicting intermediate FE measures. These intermediate measures are then post-processed with various physical models to compute the required key performance indicators (KPIs), e.g., torque, shaft power, and material costs. We perform multi-objective optimization with both classical FE and a hybrid approach using a nature-inspired evolutionary algorithm. We show quantitatively that the hybrid approach maintains the quality of Pareto results better or close to conventional FE simulation-based optimization while being computationally very cheap.

Roland Pulch, Sebastian Schöps

We consider a model of an electric circuit, where differential algebraic equations for a circuit part are coupled to partial differential equations for an electromagnetic field part. An uncertainty quantification is performed by changing physical parameters into random variables. A random quantity of interest is expanded into the (generalised) polynomial chaos using orthogonal basis polynomials. We investigate the determination of sparse representations, where just a few basis polynomials are required for a sufficiently accurate approximation. Furthermore, we apply model order reduction with proper orthogonal decomposition to obtain a low-dimensional representation in an alternative basis.

Christine Herter, Sebastian Schöps, Winnifried Wollner

In this paper we consider the free-form optimization of eigenvalues in electromagnetic systems by means of shape-variations with respect to small deformations. The objective is to optimize a particular eigenvalue to a target value. We introduce the mixed variational formulation of the Maxwell eigenvalue problem introduced by Kikuchi (1987) in function spaces of (H(\operatorname{curl}; Ω)) and (H^1(Ω)). To handle this formulation, suitable transformations of these spaces are utilized, e.g., of Piola-type for the space of (H(\operatorname{curl}; Ω)). This allows for a formulation of the problem on a fixed reference domain together with a domain mapping. Local uniqueness of the solution is obtained by a normalization of the the eigenfunctions. This allows us to derive adjoint formulas for the derivatives of the eigenvalues with respect to domain variations. For the solution of the resulting optimization problem, we develop a particular damped inverse BFGS method that allows for an easy line search procedure while retaining positive definiteness of the inverse Hessian approximation. The infinite dimensional problem is discretized by mixed finite elements and a numerical example shows the efficiency of the proposed approach.

Boian Balouchev, Jürgen Dölz, Maximilian Nolte et al.

We discuss the advantages of a spline-based freeform shape optimization approach using the example of a multi-tapered coaxial balun connected to a spiral antenna. The underlying simulation model is given in terms of a recently proposed isogeometric integral equation formulation, which can be interpreted as a high-order generalization of the partial element equivalent circuit method. We demonstrate a significant improvement in the optimized design, i.e., a reduction in the magnitude of the scattering parameter over a wide frequency range.

Arvinth Shankar, Klaus Kuhnen, Iryna Kulchytska-Ruchka et al.

While the thermodynamically formulated generalized Prandtl-Ishlinskii stop-type operator effectively captures hysteresis nonlinearities, it requires a local iterative procedure to update each hysteron, resulting in considerable computational effort. In this work, we propose a simplified thermodynamic formulation of the generalized Prandtl-Ishlinskii stop operator. The nonlinear mapping on the stop operator is replaced by an identity, such that the hysteresis operators are directly weighted through their outputs, while the nonlinear anhysteretic response, represented by ramp dead-zone basis functions, is fully preserved. For isotropic cases, this simplification enables a closed-form solution for the local plastic correction, eliminating per-hysteron iterative Newton updates. The resulting constitutive mapping is integrated into a finite element solver, and numerical results show a significant reduction in computation time with accuracy comparable to the generalized model.

Arvinth Shankar, Iryna Kulchytska-Ruchka, Sebastian Schöps

The simulation of electric machines plays a significant role in the design of efficient and competitive products. Faster simulations reduce computational costs, such as CPU hours, and shorten development cycles, thereby enabling faster design iterations and ultimately accelerating time-to-market. In this work, we analyze the dominant computational bottlenecks and demonstrate how targeted acceleration measures can significantly reduce the overall runtime of 2D and 3D finite element simulations of electric machines in an industrial environment.

Theodor Komann, Michael Wiesheu, Stefan Ulbrich et al.

The multiobjective optimization of electric machines always involves a trade-off caused by various competing objectives such as performance and cost. A suitable design is usually determined by comparing variants from the Pareto front, which has been generated by a large number of simulation runs. This paper addresses the efficient generation of the Pareto front using a continuation method based on a homotopy method that exploits second-order derivative information to achieve superlinear convergence, enabling the fast generation of new Pareto-optimal points within only a few iterations. A key contribution is the derivation of formulas to compute the Hessian with respect to geometry parameters and shape, thus enabling direct modifications of the motor geometry in the context of Isogeometric Analysis. We apply our method to nonlinear 2D magnetostatic simulations of a permanent magnet synchronous motor and demonstrate its effectiveness by optimizing the cost, mean torque and torque ripple of the motor. Compared to a first-order optimization method, this approach reduces the number of iterations and function evaluations needed, making the pareto optimization fast and efficient.

Michael Reichelt, Michael Wiesheu, Melina Merkel et al.

Space-time methods promise more efficient time-domain simulations, in particular of electrical machines. However, most approaches require the motion to be known in advance so that it can be included in the space-time mesh. To overcome this problem, this paper proposes to use the well-known air-gap element for the rotor-stator coupling of an isogeometric machine model. First, we derive the solution in the air-gap region and then employ it to couple the rotor and stator. This coupling is angle dependent and we show how to efficiently update the coupling matrices to a different angle, avoiding expensive quadrature. Finally, the resulting time-dependent problem is solved in a space-time setting. The spatial discretization using isogeometric analysis is particularly suitable for coupling via the air-gap element, as NURBS can exactly represent the geometry of the air-gap. Furthermore, the model including the air-gap element can be seamlessly transferred to the space-time setting. However, the air-gap element is well known in the literature. The originality of this work is the application to isogeometric analysis and space-time.

Idoia Cortes Garcia, Peter Förster, Lennart Jansen et al.

Electrical circuits are present in a variety of technologies, making their design an important part of computer aided engineering. The growing number of parameters that affect the final design leads to a need for new approaches to quantify their impact. Machine learning may play a key role in this regard, however current approaches often make suboptimal use of existing knowledge about the system at hand. In terms of circuits, their description via modified nodal analysis is well-understood. This particular formulation leads to systems of differential-algebraic equations (DAEs) which bring with them a number of peculiarities, e.g. hidden constraints that the solution needs to fulfill. We use the recently introduced dissection index that can decouple a given system of DAEs into ordinary differential equations, only depending on differential variables, and purely algebraic equations, that describe the relations between differential and algebraic variables. The idea is to then only learn the differential variables and reconstruct the algebraic ones using the relations from the decoupling. This approach guarantees that the algebraic constraints are fulfilled up to the accuracy of the nonlinear system solver, and it may also reduce the learning effort as only the differential variables need to be learned.

Vivek Parekh, Dominik Flore, Sebastian Schöps

In the design phase of an electrical machine, finite element (FE) simulation are commonly used to numerically optimize the performance. The output of the magneto-static FE simulation characterizes the electromagnetic behavior of the electrical machine. It usually includes intermediate measures such as nonlinear iron losses, electromagnetic torque, and flux values at each operating point to compute the key performance indicators (KPIs). We present a data-driven deep learning approach that replaces the computationally heavy FE calculations by a deep neural network (DNN). The DNN is trained by a large volume of stored FE data in a supervised manner. During the learning process, the network response (intermediate measures) is fed as input to a physics-based post-processing to estimate characteristic maps and KPIs. Results indicate that the predictions of intermediate measures and the subsequent computations of KPIs are close to the ground truth for new machine designs. We show that this hybrid approach yields flexibility in the simulation process. Finally, the proposed hybrid approach is quantitatively compared to existing deep neural network-based direct prediction approach of KPIs.

Vivek Parekh, Dominik Flore, Sebastian Schöps

The design of an electrical machine can be quantified and evaluated by Key Performance Indicators (KPIs) such as maximum torque, critical field strength, costs of active parts, sound power, etc. Generally, cross-domain tool-chains are used to optimize all the KPIs from different domains (multi-objective optimization) by varying the given input parameters in the largest possible design space. This optimization process involves magneto-static finite element simulation to obtain these decisive KPIs. It makes the whole process a vehemently time-consuming computational task that counts on the availability of resources with the involvement of high computational cost. In this paper, a data-aided, deep learning-based meta-model is employed to predict the KPIs of an electrical machine quickly and with high accuracy to accelerate the full optimization process and reduce its computational costs. The focus is on analyzing various forms of input data that serve as a geometry representation of the machine. Namely, these are the cross-section image of the electrical machine that allows a very general description of the geometry relating to different topologies and the classical way with scalar parametrization of geometry. The impact of the resolution of the image is studied in detail. The results show a high prediction accuracy and proof that the validity of a deep learning-based meta-model to minimize the optimization time. The results also indicate that the prediction quality of an image-based approach can be made comparable to the classical way based on scalar parameters.

Mona Fuhrländer, Sebastian Schöps

Quantification and minimization of uncertainty is an important task in the design of electromagnetic devices, which comes with high computational effort. We propose a hybrid approach combining the reliability and accuracy of a Monte Carlo analysis with the efficiency of a surrogate model based on Gaussian Process Regression. We present two optimization approaches. An adaptive Newton-MC to reduce the impact of uncertainty and a genetic multi-objective approach to optimize performance and robustness at the same time. For a dielectrical waveguide, used as a benchmark problem, the proposed methods outperform classic approaches.