A Highly Efficient Domain Decomposition Method Applied to 3-D Finite-Element Analysis of Electromechanical and Electric Machine Problems

Limitations/capabilities of electric machine technologies and modeling approaches for electric motor design and analysis in plug-in electric vehicle applications

“…FEA in principle can take into account the spatial distribution of stator winding, rotor and stator slots, and nonlinear behavior of the magnetic core [44]. In 2D FEA, even when several cross sections are used, it is difficult to obtain useful 3D flux, endeffects, skewing effect, and eddy-current distributions [119]. When 3D FEA is used, the element count is based on the cube of the linear dimensions.…”

A 3D finite element analysis of large‐scale nonlinear dynamic electromagnetic problems by harmonic balancing and domain decomposition

“…At each subdomain interface, the Dirichlet and Neumann boundary conditions should be satisfied, which can be written as where and are the unit normal vectors of subdomain i and j , respectively. The Neumann boundary condition contains an unknown vector Λ , often referred to as the ‘dual’ unknown, which must be solved to recover the field values inside each subdomain . In order to enforce the continuity condition along corner edges, one set of global corner DOFs, A c , often referred to as the ‘primal’ unknowns , is used within the entire computational domain.…”

“…In equations and , N s denotes hierarchical basis functions and contains two sets of orthogonal basis functions, namely the divergence free and pure‐gradient basis functions, which are used to expand the magnetic vector potential and the electric scalar potential. Readers are referred to for a more detailed presentation of tree‐cotree splitting and hierarchical basis functions. In equation , the unknown λ is given by where the Boolean matrix extracts the dual unknowns in the s th subdomain from the global dual unknowns.…”

“…wheren i andn j are the unit normal vectors of subdomain i and j, respectively. The Neumann boundary condition contains an unknown vector Λ, often referred to as the 'dual' unknown, which must be solved to recover the field values inside each subdomain [14]. In order to enforce the continuity condition along corner edges, one set of global corner DOFs, A c , often referred to as the 'primal' unknowns [14], is used within the entire computational domain.…”

“…Consequently, the FEA system increases dramatically in size and becomes time consuming to solve for a general large-scale threedimensional (3D) problem. To overcome this problem, the domain decomposition method [5][6][7][8][9][10][11][12][13][14][15] is employed to increase the efficiency of large-scale nonlinear HB simulations. Among various domain decomposition methods, the dual-primal finite element tearing and interconnecting (FETI-DP) method [9][10][11][12] has been successfully applied in simulating both linear and nonlinear electromagnetic and electromechanical problems [13][14][15].…”