Magnetic Field Simulation With Data-Driven Material Modeling

Abstract

This paper develops a data-driven magnetostatic finite-element (FE) solver which directly exploits measured material data instead of a material curve constructed from it. The distances between the field solution and the measurement points are minimized while enforcing Maxwell's equations. The minimization problem is solved by employing the Lagrange multiplier approach. The procedure wraps the FE method within an outer data-driven iteration. The method is capable of considering anisotropic materials and is adapted to deal with models featuring a combination of exact material knowledge and measured material data. Thereto, three approaches with an increasing level of intrusivity according to the FE formulation are proposed. The numerical results for a quadrupole-magnet model show that data-driven field simulation is feasible and affordable and overcomes the need of modeling the material law.

Data‐driven solvers for strongly nonlinear material response

“…The fields are evaluated at the quadrature points of the FEs. 27 Note that, in this work, we consider lowest-order FEs, accordingly, one point per element. To obtain the discrete counterpart of the distance function (13) we introduce the matrix induced scalar products…”

“…A previous work of the authors proposes three modifications of increasing intrusiveness to the data-driven computing framework, in order to deal with mixed models, that is, with exact and data-based materials coexisting in the computational domain. 27 In this work, we shall stick to the least intrusive approach, where the exact material relation is treated by the data-driven solver itself. Under this approach, the distance function (44) is updated with the known material coefficients ex and ex , and the solution of the minimization problem is given by…”

“…Employing the 2D basis function (46) as test and trial functions in (32) leads to two linear systems that must be solved. 27 Note that the constructed 2D basis function (46) have zero divergence, meaning an additional gauging can be omitted. It should also be pointed out that the structure of Algorithm 1 and its discrete counterpart remain unchanged.…”

“…25 Outside the field of computational mechanics, data-driven magnetostatic FEM solvers were developed by the authors and coworkers, for the case of noise-free data and heterogeneous domains where exact and data-based material models coexist. 27 The data-driven solvers which follow the original formulation, 14 employ a global weighting factor, thus treating all finite elements in the computational domain equally, regardless their material filling. However, a global, homogeneous factor may actually hinder the efficiency and accuracy of the data-driven solver in the case of incompatible material data sets, that is, data sets where a solution in the material set is distant from the states that conform to the governing equations.…”

“…The manifold is constructed either locally, by means of local linear embedding, 26 or globally, using kernel methods 25 . Outside the field of computational mechanics, data‐driven magnetostatic FEM solvers were developed by the authors and coworkers, for the case of noise‐free data and heterogeneous domains where exact and data‐based material models coexist 27 …”