High-order methods applied to electrical machine modeling

High-order methods have been subject of research over the last years to replace the time-consuming meshing operation of Finite Elements Method (FEM) by a structured grid, which is exploiting the tensor product. The problem formulation in these methods is generally the same, i.e., weak form implemented through the Bubnov-Galerkin method. First and second order polynomials functions used in FEM are replaced by high arbitrary-order polynomial functions because of their overall excellent accuracy and fast computation time. The Spectral Element Method (SEM) and Isogeometric Analysis (IGA) among others, are exploiting high order basis with established mathematical framework [1], [2], and available numerical tools [3]. In this paper, the solution for elliptic Laplace equation formulated with SEM and IGA are applied to 2D magnetostatic problems, including both linear and nonlinear materials. The obtained magnetic field distributions and post-processed parameters such as flux linkage, forces, and inductances are validated with FEM. A very low discrepancy is achieved which demonstrates the applicability of the proposed high-order methods, and enables integrated design-through-analysis of electrical machines. In this paper, SEM and IGA are applied to the analysis of two electrical machine benchmarks, in which a nonlinear iron characteristic is considered. Each of these methods uses different basis functions, quadrature rules, and space discretization, although both are based on the same Galerkin method. Modeling Solutions obtained from FEM are known to be very dependent on the quality of the triangular mesh [4]. Moreover, in FEM a curved geometry is approximated by linear elements which influences the accuracy, or comes at the cost of a high number of mesh elements. SEM divides the geometry into elements or patches, as exemplified in Fig. 1. Each patch is mapped to a unique square parent element, where calculations and matrix assembly are conducted. Legendre polynomials are used as basis functions. Lagrangian interpolation subsequently allows the computation of the solution on the Lobatto-Gauss-Legendre roots [1], and obtains the functional coefficients on the grid. IGA basis-functions are formed by the tensor-product of B-splines or NURBS (non-uniform rational B-splines), which is the industry-standard geometrical description used in computer aided design (CAD). The same basis functions allow to represent complex geometrical shapes [2], compute and visualize the solution. The physical domain is mapped to a rectangular computational domain, on which the basis functions and their gradients are known and where the calculations are conducted through numerical Gaussian quadratures. In both proposed methods, the geometry is discretized into 2D conforming patches where continuity is strongly imposed, forcing each basis function on the interface to match one-to-one. The formulation suited for 2D magnetostatic electrical machine modeling is further extended to include nonlinear material properties, such as soft-magnetic iron. The spatial distribution of the remanent magnetization and the magnetic incremental permeability are updated iteratively, according to the considered BH-curve, interpolated by means of a spline. The developed high-order methods allow for modeling curved topologies such as slots in a simpler manner than generally considered in analytical methods [5], in the same time, ensuring both flexibility and accuracy.