IEEE TRANSACTIONS ON MAGNETICS, VOL. 51, NO. 3, MARCH 2015 7203704 Efficient Parallel 3-D Computation of Electrical Machines With Elmer Janne Keränen 1 , Jenni Pippuri 1 , Mika Malinen 2 , Juha Ruokolainen 2 , Peter Råback 2 , Mikko Lyly 3 , and Kari Tammi 1 1 VTT Technical Research Centre of Finland, Espoo FI-02044, Finland 2 CSC-IT Center for Science, Espoo FI-02101, Finland 3 ABB Oy, Helsinki FI-00381, Finland After its recent improvements described here, open source finite element software Elmer is shown to be a highly efficient option for electrical machine modeling. The parallelization of computational burden is shown to be a necessity. The methods implemented enable applying fully parallel computation to electrical machine models, including rotation and electrical circuits. Computational experiments performed demonstrate that Elmer can effectively utilize several hundreds of computational cores in parallel, making it an attractive alternative when computational speed is of high importance. Index Terms— Computational efficiency, eddy-current problems, electrical machines, parallelization, preconditioning. I. I NTRODUCTION E LECTROMAGNETIC computation is heavily used in the electromechanical industry. Due to tightening effi- ciency and operational requirements, electrical machines are becoming more engineered. This creates a need for more developed design and analysis strategies and emphasizes the role of 3-D computation in the process. However, the present commercial 3-D finite-element method (FEM) tools are not fast enough for research and development, as the simulations may take weeks to months for transient industrial models, with nonlinearities and eddy currents included. The computation times do not reduce automatically by faster processors anymore, as the processor development is targeted at adding more processor cores, not the single-core speed. Hence, we need a method that utilizes parallel computing as efficiently as possible. Our work is based on Elmer [1], [2], an open-source FEM software for multiphysical problems, mainly developed by the authors from CSC—IT Center for Science Ltd. Emphasis in Elmer development is on parallel performance. Elmer’s parallelization is based on domain partitioning and on the message passing approach (the MPI standard); the model geometry is divided into so-called partitions and computations associated with the partitions are performed in parallel with several computational cores. In outline, the linear system assembly, the matrix–vector products of the iterative Krylov methods and preconditioning are performed in parallel for each partition. The key point is the reduction of the MPI data traffic between the cores by reducing the communication between neighboring partitions. We have recently developed Elmer toward electrical machine computation. Now Elmer’s parallel setup, vastly used in other fields, is introduced to electrical machine computation. In this paper, we aim at evaluating Elmer in electrical machine computation, particularly its parallel efficiency, by thorough scaling tests with 3-D electrical motor cases. Manuscript received May 23, 2014; revised August 12, 2014; accepted August 25, 2014. Date of current version April 22, 2015. Corresponding author: J. Keränen (e-mail: janne.sami.keranen@vtt.fi). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2014.2356256 II. COMPUTATION METHODS A. Electromagnetic Computation Magnetic flux density inside the computation domain can be found from the following partial differential equation, along with proper boundary and initial conditions: curl 1 μ curl A =-σ  ∂ ∂ t A + grad V  + J s (1) where A is the magnetic vector potential, V the elec- trical scalar potential, J s the source current, and μ the material permeability. The above—with current continuity equation—is the so-called A–V formulation for eddy-current problems. After solving (1), magnetic flux density and elec- trical field strength can be solved by B = curl A and E =-(∂/∂ t ) A - grad V . The authors have recently implemented edge element A–V formulation into Elmer with mortar finite elements rotation model and connection to the driving circuits. The compu- tational version of the A–V system is based on the stan- dard weak formulation together with the lowest order finite elements [3]. Due to known numerical problems with tree gauge, iterative methods without gauging are used. This is possible, based on [4], as the source field J s is forced to be divergence free [3]. B. Rotation Model To handle the rotations needed in transient electrical machine simulations, mortar FEM was implemented into Elmer [5], [6]. Extra degrees of freedom (DoFs) corresponding to Lagrange multipliers are used to approximate the solution continuity over the stator and rotor interface Ŵ. In our present implementation, the solution continuity is approximated via the weak formulation  Ŵ (n ×[ A] Ŵ × n) · (n × v × n) dS = 0 (2) where [ A] Ŵ denotes the solution jump across Ŵ, n is the surface normal, and v corresponds to a basis function for approximating the associated Lagrange multiplier. Here, v is chosen to be a basis function used also for approximating A. 0018-9464 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. 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