Data‐driven solvers for strongly nonlinear material response

Abstract

This work presents a data-driven magnetostatic finite-element solver that is specifically well suited to cope with strongly nonlinear material responses. The data-driven computing framework is essentially a multiobjective optimization procedure matching the material operation points as closely as possible to given material data while obeying Maxwell's equations. Here, the framework is extended with heterogeneous (local) weighting factors-one per finite element-equilibrating the goal function locally according to the material behavior. This modification allows the data-driven solver to cope with unbalanced measurement data sets, that is, data sets suffering from unbalanced space filling. This occurs particularly in the case of strongly nonlinear materials, which constitute problematic cases that hinder the efficiency and accuracy of standard data-driven solvers with a homogeneous (global) weighting factor. The local weighting factors are embedded in the distance-minimizing data-driven algorithm used for noiseless data, likewise for the maximum entropy data-driven algorithm used for noisy data. Numerical experiments based on a quadrupole magnet model with a soft magnetic material show that the proposed modification results in major improvements in terms of solution accuracy and solver efficiency. For the case of noiseless data, local weighting factors improve the convergence of the data-driven solver by orders of magnitude. When noisy data are considered, the convergence rate of the data-driven solver is doubled.

A Neural Solver for Variational Problems on CAD Geometries with Application to Electric Machine Simulation

“…One step beyond would be the combination of traditional and neural solvers as to benefit from their individual strengths. Such possibilities can be sought, for example, in the approximation of PDEs with highly nonlinear constitutive (material) laws, which often pose significant problems for standard numerical approximation techniques (Galetzka et al, 2021).…”

Three-Dimensional Data-Driven Magnetostatic Field Computation using Real-World Measurement Data

“…In the following, we briefly introduce the data-driven solver employed in this work. For a detailed exposition, see Galetzka et al (2021).…”

“…Remark 2 A recent work by the authors Galetzka et al (2021) showed that heterogeneous (local) weighting factors μ, respectively ν, improve significantly the accuracy and efficiency of the data-driven solver in the case of unbalanced material data sets. A local weighting factor is adaptively matched to the current operation point of the field state at the quadrature point.…”

“…The switching point from a global (homogeneous) weighting factor to local (heterogeneous) weighting factors was fixed to 4 iterations. Selecting the switching point adaptively is also possible Galetzka et al (2021).…”

“…In this work, we build upon existing data-driven electromagnetic field solvers De Gersem et al (2020); Galetzka et al (2021) and extend their application to the case of computationally demanding three-dimensional simulations, as well as to the case of real-world measurement data. We consider a highly non-trivial example, namely the model of an inductor excited by a DC-current.…”