Modeling and simulation in electromagnetics

April 3-4th 2024, SP2 416-1

Program

Wednesday, April 3, 2024: "Complex material behavior"

Thursday, April 4, 2024: "Discretization methods"

Abstracts

Alessio Cesarano (RICAM)

Shape optimization of rotating electric machines

The talk will focus on an attempt to include time-dependent effects in the opti- mization of electric machines. In particular, the minimization of eddy currents is achieved solving a sequence of 2D magnetostatics problems, with the eddy currents being calculated in a finite difference in time approach.

Herbert Egger (JKU/RICAM)

Duality-based error estimation for nonlinear magnetostatics

We review the primal and dual variational formulations of nonlinear magnetostatics and employ a nonlinear version of the Prager-Synge theorem to establish computable error bounds for the disccretization by standard finite elements. The resulting a-posteriori error estimates are constant free and suitable for adaptive mesh refinement. Numerical results are presented for illustration.

Felix Engertsberger (JKU Linz)

Convergence of iterative solvers for nonlinear magnetostatics

We consider the iterative solution of systems arising in nonlinear magnetostatics. Using the equivalence to an underlying minimization problem, we can establish global linear convergence of a large class of methods, including the damped Newton-method, fixed-point iteration, and the Kacanov iteration, which can all be interpreted as generalized gradient descent methods. Armijo backtracking is considered for an adaptive choice of the stepsize. The general assumptions required for our analysis cover inhomogeneous, nonlinear, and anisotropic materials, as well as permanent magnets. The main results are proven on the continuous level, but they carry over almost verbatim to various approximation schemes, including finite elements and isogeometric analysis, leading to bounds on the iteration numbers, which are independent of the particular discretization. The theoretical results are illustrated by numerical tests for a typical benchmark problem.

Mario Gobrial (TU Graz)

Space-Time FEM for Electromagnetism including Hysteresis

Space-time discretization methods are well suited to handle moving domains in electric machines such as the electric motor. The space-time domain is discretized at once, and in contrast to time stepping methods the movement can be captured by a fixed space-time mesh. For the solution of the magento quasi-static Maxwell equations in the electric motor we formulate space-time finite element methods, considering a two-dimensional spatial domain and the time as the third dimension of the domain. As in the case of a fixed domain we are able to prove an inf-sup stability condition to ensure unique solvability. These space-time finite element methods allow for a parallel iterative computation for the magnetic flux density. The incorporation of hysteresis into the magneto quasi-static Maxwell equation leads to two approaches for the discrete problem. One of them appears to be successful which applied to an electric motor gives BH-curves with hysteresis effects.

Christian Köthe (TU Graz)

Space-time least-squares finite element methods

For the numerical solution of an operator equation $Bu=f$ we consider a least-squares approach. We assume that $B:X\to Y^*$ is an isomorphism and $A:Y\to Y^*$ implies a norm, where $X$ and $Y$ are Hilbert spaces.
Firstly, we assume the differential operator $B$ to be linear. The minimizer of the least-squares functional $\frac{1}{2} {\left\lVert{Bu-f}\right\rVert}_{A^{-1}}$ is then characterized by the gradient equation which involves an elliptic operator $S=B^*A^{-1}B:X\to X^*$. We introduce the adjoint $p=A^{-1}(f-Bu)$ and reformulate the first order optimality system as a saddle point system. Based on a discrete inf-sup condition we discuss related a priori error estimates and use the discrete adjoint $p_h$ to drive an adaptive refinement scheme. Numerical examples will be presented which confirm our theoretical findings. Secondly, we demonstrate how to apply the least-squares approach for the numerical solution of a non linear operator equation $B(u)=f$. We derive the related first order optimality system and discuss its solution via Newton's method. Numerical examples involving the semi-linear heat equation and the quasi-linear Poisson equation will be presented.
Finally, we will conclude with some remarks on future work in this area which needs to be done.

Richard Löscher (TU Graz)

Stable and Adaptive Space-Time Finite Element Methods for the Wave Equation

For common discretizations of the wave equation, e.g., finite element methods in space and explicit time stepping schemes, stability can only be expected when the time step size is chosen sufficiently small with respect to the spatial discretization, as is well-known as CFL-condition. This is an issue, in particular, when aiming to phrase fully adaptive schemes in space and time simultaneously. To overcome the stability constraint, we will consider a well-posed space-time formulation of the wave equation on the continuous level and consider its equivalent residual minimization problem. For a conforming discretization, we are able to characterize the optimal, though at first unrealizable, test space and we will outline two computable/practical realizations relying on space-time discretizations, which cannot be interpreted as a semi-discretiza\-tion in space paired with a time stepping method. Firstly, by discretizing the optimal ansatz to test operator, which results in a saddle point formulation that can be realized on fully unstructured space-time finite element meshes and comes with an inbuilt error estimator. Secondly, by using a modified Hilbert transformation, leading to an unconditionally stable Galerkin-Bubnov formulation on space-time tensor product meshes. The theory for both approaches will be complemented by numerical examples and fast solution strategies will be discussed.

Andreas Schafelner (JKU)

Fast solvers for nonlinear time-periodic problems

In this talk, we review existing and novel methods for the solution of time-periodic quasilinear parabolic problems, where the 2d eddy current problem serves as a model problem. In particular, we will focus on two solution methods: a fixed-point iteration method and a time multigrid method. We present the current state of the theory as well as open challenges and questions.

Olaf Steinbach (TU Graz)

Space-time finite element methods in thermodynamics

In this talk we first review space-time variational formulations for parabolic and hyperbolic initial boundary value problems. This includes formulations in Bochner and anisotropic Sobolev spaces, also using a modified Hilbert transformation, and first order systems. We then apply these results to analyze space-time variational formulations for the hyperbolic-parabolic system of thermodynamics. When eliminating the temperature, we end up with a Schur complement system where the coupling term is non-negative. First numerical results are given. This is ongoing work with M. Reichelt.

Julien Taurines (Aalto University and Tampere University)

Magneto-elasto-plastic coupling: using thermodynamics for anhysteretic and hysteretic modeling of magnetization and magnetostriction.

The magnetic properties of ferromagnetic materials used in transformers or electrical machines cores are directly related to the mechanical loading. Writing the constitutive laws linking the physical variables involved in this strong coupling is still the subject of ongoing research. A convenient way of dealing with the problem is to first treat the coupling on the anhysteretic part and then add a dissipative model. This presentation proposes the formulation of an anhysteretic magneto-elasto-plastic constitutive law based on the derivation of a Gibbs free energy, which is itself a function of cubic invariants. A hysteresis model ensuring positive dissipation is then presented. This will be illustrated by application to a wide range of magnetic materials, from permanent magnets to soft magnetic materials. This work was carried out in collaboration with Boris Kolev, Olivier Hubert and Rodrigue Desmorat (ENS Paris-Saclay) for the anhysteretic part, and Anouar Belahcen, Paavo Rasilo and Floran Martin (Aalto University and Tampere University) for the hysteretic part. It was supported by Research Council of Finland and European Research Council.

Stefan Tyoler (RICAM)

Adaptive Multipatch-IgA for electrical machines

In Isogeometric Analysis tensor product B-splines are employed as the underlying Galerkin subspaces. For certain computational domains (like an electric motor) it is neccessary to handle discontinuities in the material coefficients and in order to resolve these material interfaces we employ a Multi-patch Isogeometric function space. Since IgA is based on tensor product spaces there is no canonical way to adaptively refine meshes in higher dimensions. In this talk we present an approach of patchwise splitting the Multipatch configuration and the treatment of emerging hanging nodes.