Parametric Geometric Metamodel of Nonlinear Magnetostatic Problem Based on POD and RBF Approaches

Abstract

The finite element method (FEM) is widely used to model electromagnetic devices. This method ensures precise results at the expense of a consequent computation time, especially when dealing with parametric problems. Model order reduction methods had proven their efficiency in reducing the computation time for parametric problems. The proper orthogonal decomposition (POD) is a popular model order reduction approach [1]. Based on the solutions of the FE model for different values of parameters (called snapshots), the POD approximates the solution of the FE model in a reduced basis. The parametric FE model is projected onto a reduced basis, decreasing the order of the numerical model to be solved. Another approach consists in constructing a metamodel by interpolation of the parametric FE solution expressed into a reduced basis. A hybrid approach has also been proposed combining the POD with an interpolation-based method such as the radial basis functions (RBF) interpolation method [2].Moreover, when the problem involves geometric parameters, an adaption of the FEM mesh is required. Remeshing the geometry at each modification can deteriorate the precision of the FEM model by introducing significant numerical noise. A solution to preserve the problem's consistency is to deform the initial mesh using the RBF interpolation [3][4]. The method consists of imposing a displacement on a set of nodes and interpolating the remaining nodes' position. Then, a metamodel of a FE solution taking into account geometric and electric parameters, can be defined by using the POD approach and the RBF interpolation.In this work, we propose to build a metamodel of a nonlinear magnetostatic parametric problem involving geometric parameters using the POD-RBF approach. The study case is a single-phase EI inductance, with one electrical parameter and two geometrical parameters. An initial geometry is meshed with 36859 elements; the number of nodes is 18553. The geometric parameters are the thickness of the airgap varying from 0.1 mm to 0.9 mm, the width of the central column of the magnetic core varying from 20 mm to 40 mm. The phase current varying from 0 to 50 A is also a parameter of the model. The mesh deformation is modeled using the RBF interpolation with a multi-quadratic function for two consecutive times to consider the two variating geometric parameters. The results of the original FEM model are taken as a reference.The POD-RBF approach is applied to the parametric FEM model. The method consists of first gathering a set of data, using the method of snapshots, by solving the original FEM model for a set of inputs. The snapshots matrix is decomposed using the singular value decomposition (SVD). The left orthogonal matrix of the SVD decomposition corresponds to the reduced basis, and the right orthogonal matrix is composed of the FE solution projected into the reduced basis for each snapshot parameter. An RBF interpolation is then performed on the right orthogonal matrix to interpolate the solution expressed in the reduced basis for the new inputs set. Finally, an approximation of the FE solution is defined by projecting the reduced solution into the FE basis.The choice of snapshots is made using a greedy algorithm [2], which consists of an iterative process. The metamodel is then improved at each iteration until the required precision is reached.The stop criterion to the greedy algorithm is set to an error between the FE solution and the metamodel solution lower than 0.1%. The evolution of error represented by Frobenius norm as a function of the iterations' number of the greedy algorithm is represented in "Fig. 1".In order to satisfy the selected criterion, 158 snapshots are required. The magnetic flux cartography as a function of phase current and the central column width for an airgap width fixed at 0.1 mm is represented in "Fig.2". The global mean absolute error between the flux cartographies obtained using the metamodel and the original FEM model is 1.7%, while the maximal local error is 4.96%.Then, the metamodel of the parametric FE solution can be used in an optimization process for different applications or to be coupled with other numerical models in order to simulate a system. **