Relationship between Cauer ladder network representation and CGlike Krylov subspace method
Takeshi Mifune, Tetsuji Matsuo
Kyoto University, Japan
We prove an equivalence of ladder network representations and CGlike matrix solvers that use the symmetric or skewsymmetric Lanczos process, from a viewpoint of the Krylov subspace methods. First, an equivalent ConcusGolubWidlund (CGW) method, which uses the skewsymmetric Lanczos process, and a Cauer ladder network (CLN) representation is shown through eddycurrent analysis. Second, an equivalent CG method, which is based on the symmetric Lanczos process, and a CLN representation are proven through fullwave analysis. The results and the discussion that follow provide a comprehensive means to understand the mathematical behavior of ladder network representations (e.g., the convergence property). Furthermore, the applicability of ladder network representation to fullwave analyses involving conductive materials is shown based on the symmetric indefinite Lanczos process.
Implementation of Simplified Model Order Reduction Based on POD for Dynamic Simulation of Electric Motors
Kazuya Okamoto^{1}, Hiroki Sakamoto^{1,2}, Hajime Igarashi^{1}
^{1}Hokkaido University, Japan; ^{2}MEIDENSHA CORPORATION, Japan
This paper proposes a fast, dynamic simulation method of electric motors using the simplified model order reduction based on the proper orthogonal decomposition (POD). In the proposed method, the magnetic fields are represented in terms of linear combination of the basis functions constructed from POD considering magnetic saturation. During the dynamic simulation, fast computation of the field quantities such as torque, radial force on the stator and iron loss are performed via interpolation of the weighting coefficients for given d and qaxis currents. The proposed method makes it possible to evaluate mechanical vibration and losses during the dynamic simulation of motors using the phase variables method.
Efficient Mixedprecision Iterative Methods for Highfrequency Electromagnetic Field Analysis
Kazuaki Sekiya^{1}, Masao Ogino^{1}, Lijun Liu^{2}, Koki Masui^{1}
^{1}Nagoya University, Japan; ^{2}Osaka University, Japan
In the high frequency electromagnetic field analysis using the finite element method, it is necessary to solve a complex symmetric linear system. Various iterative methods such as COCG, COCR, and MINRESlike_CS have been developed and utilized to efficiently solve such systems. However, when solving a largescale system with double precision floatingpoint arithmetic, conventional methods suffer from slow convergence and then cannot obtain the solution within a practical time. In this study, we propose iterative methods using mixed precision arithmetic of double precision and pseudo quadruple precision floatingpoint numbers. Especially, whereas all vectors for the iterative method are kept in double precision, the summation in the dot product is calculated by using pseudo quadruple precision. In numerical experiments of highfrequency electromagnetic field analysis, we successfully improved both the number of iterations and computation time.
Research on the Convergence of Iterative Method Using Mixed Precision Calculation Solving Complex Symmetric Linear Equation
Koki Masui^{1}, Masao Ogino^{2}
^{1}Graduate school of informatics Nagoya University, Japan; ^{2}Information Technology Center, Nagoya University, Japan
This paper details with iterative methods with mixedprecision calculation using double and doubledouble (DD) precision for high frequency electromagnetic field problems. We implemented some mixedprecision iterative methods, and conduced numerical experiments. As a result, mixed precision calculation improved the convergence of the iterative method, moreover we got the approximate solution faster than using double precision calculation.
A MemoryEfficient Formulation of PreciseIntegration TimeDomain Method With Riccati Matrix Differential Equations
Xiaojie Zhu, Xikui Ma, Jinghui Shao, Shuli Yin
Xi'an Jiaotong University, China, People's Republic of
A novel numerical method, referred to as Riccati preciseintegration timedomain (RiccatiPITD) method, is proposed to reduce the memory requirement in the conventional PITD method. In the RiccatiPITD method, the field components of the TE/TM wave are ordered in a special matrix so as to construct the Riccati matrix differential equations about the Maxwell’s curl equations. And then the precise integration technique is adopted for the numerical integration of the Riccati matrix differential equations. Theoretical analysis about the memory requirement of the conventional PITD method and the RiccatiPITD method are given. It is found that by solving the electric and magnetic field components in the form of matrices in the RiccatiPITD method rather than vectors in the conventional PITD method, the memory requirement has been greatly reduced. The numerical examples are given to verify the correctness and the memory efficiency of the proposed method. In addition, the Riccati matrix differential equations can also provide an efficient scheme for other electromagnetic field numerical methods with high memory requirement for matrices.
Evaluation of An Efficient 3D Poisson Solver for Organic FieldEffect Transistors Simulation
Lijun Liu^{1}, Haoyuan Li^{2}, JeanLuc Bredas^{3}
^{1}Osaka University, Japan; ^{2}Georgia Institute of Technology, USA; ^{3}Georgia Institute of Technology, USA
Organic fieldeffect transistors (OFETs) are attracting much attention because of their promising applications as flexible and lowcost devices. To guide the rational development of OFETs, molecularscale device models have been developed, which help understand the device physics and the features of current characteristics. However, in these device models, even though charge transport is a 3dimensional (3D) problem, the electrostatic interactions are in fact evaluated by solving the 2dimensional (2D) Poisson equation, essentially in order to maintain reasonable computational costs. To improve the accuracy in OFET device modeling, we present an efficient and accurate 3D Poisson solver, which benefits from multi and manycore architectures and is applicable to problems with mixed boundary conditions. Through the use of parallel computing, the speed of computation of the electric potential is significantly improved and can outperform earlier methods. For instance, Poisson equation problems with up to 5 million degrees of freedom are solved in less than 1 second on 64processor elements.
Adaptive DegreesofFreedom Finite Element Analysis of 3D Transient Magnetic Problems
Yunpeng Zhang, Huihuan Wu, SiuLau Ho, Weinong Fu, Xinsheng Yang
Department of Electrical Engineering, The Hong Kong Polytechnic University, Hong Kong S.A.R. (China)
In this paper, an adaptive degreesoffreedom finite element method is presented for 3dimensional (3D) transient magnetic problems. For transient problem, the error distribution changes over time, and mesh coarsening is needed to keep the number of degreesoffreedom (DoFs) small. In this novel htype adaptive finite element method, the mesh coarsening is replaced by an implicit elimination of DoFs, which maintains the mesh topology and avoids the subsequent operations after mesh coarsening. The DoFs to be eliminated are constrained by interpolation functions, which are formulated by the master DoFs. To adapt to the 3D field, a constraint with an alterable number of master DoFs and rational coefficients are proposed. The constraints are then integrated into the discretized equation in the element level using the slavemaster technique. Several transient problems are tested to showcase the effectiveness of this method.
A Well Preprocessed Deep Neural Network for Magnetic Field Analysis
Huihuan Wu, Yunpeng Zhang, Weinong Fu, SiuLau Ho
Department of Electrical Engineering, The Hong Kong Polytechnic University, Hong Kong S.A.R. (China)
In this paper, a deep neural network with well preprocessed input layer is proposed for magnetic field analysis. In addition, nonoverlapping Schwarz domain decomposition method is combined with the neural network to further improve the computational effciency. Inspired by the analytical formula of magnetic potential, an excitation distance function is proposed to integrate the input data, such as geometry, excitation, and boundary conditions. This preprocessor roughly reduces the dimension of the input layer by three quarters, while the training process is accelerated by the more integrated input layer. For large scale problems, the computational domain is divided into subdomains, the neural networks of which are trained separately. The neural networks of subdomains are coupled and solved by the nonoverlapping Schwarz method subsequently to predict the results of the test samples.
Machine Learning Approach in Magnetic Field Calculations
Valentin Mateev, Ilana Marinova
Technical University of Sofia, Bulgaria
Here is presented a machine learning approach for 2D steadystate and harmonic magnetic field calculations based on Poisson and Helmholtz equations for Dirichlet boundary problems. The approach is implemented on multilayer convolutional neural network trained over the 1b TEAM. benchmark problem variations dataset. Implementation is suitable for nonhomogeneous magnetic properties domains and distributed excitation sources. Results accuracy is estimated in comparison with Finite Element Method model of the same problem.
Variable Preconditioned Mister R for Linear System Obtained by Edge Element Premise on Parallelization
Akira Matsumoto^{1}, Yoshihisa Fujita^{2}, Taku Itoh^{3}, Kuniyoshi Abe^{4}, Soichiro Ikuno^{1}
^{1}Tokyo University of Technology, Japan; ^{2}Hakodate College; ^{3}Nihon University; ^{4}Gifu Shotoku Gakuen University
The convergence properties of new iterative solver which name is Mister R (MrR) for the linear system obtained from edge element is numerically investigated. Furthermore, the numerical feature of the variable preconditioned MrR premise on parallelization is numerically investigated. In the electromagnetic analysis, finite element method (FEM) with an edge element is often adopted as a discretizing procedure, and a large sparse linear system with rank deficient coefficient matrix is derived by the method. In order to obtain a specific solution of the system, Krylov subspace method is adopted as the solver. In the present study, we select MrR as a candidate solver for the linear system. In addition, the variable preconditioned (VP) strategy is adopted to MrR, VPMrR is developed. In the present study, the Problem 20 in T.E.A.M Workshop is employed for the benchmark, and the problem is discretized by FEM with an edge element. The results of computation show that the residual history of MrR decreases monotonically than that of conjugate gradient (CG) method, and the residual of MrR decreases in the first several iterations. Taking advantage of the above characteristic of MrR, we extended MrR to VPMrR. The iteration count of VPMrR (outerloop: MrR, innerloop: MrR) decreases drastically than that of standard MrR and VPCG (outerloop: CG, innerloop: MrR).
A LowMemoryRequirement Realization of PreciseIntegration TimeDomain Method Using a Sparse Matrix Technique
Xiaojie Zhu, Xikui Ma, Jinghui Shao, Shuli Yin
Xi'an Jiaotong University, China, People's Republic of
In this digest, a sparse matrix (SM) technique is developed to reduce the computation time and the memory requirement in the precise integration time domain (PITD) method. Two fundamental aspects are there in the proposed technique. As the first aspect, reordering entries of the column vector consisting of all the electric and magnetic field components makes the bandwidth of the coefficient matrix, whose entries revealing the correlation between any individual electric (magnetic) component and its adjacent magnetic (electric) components, decreased a lot. Accordingly, the bandwidth of the exponential matrix is also decreased. The second one is exploiting a sparse storage scheme to store the obtained narrowbandwidth matrices. It is found that the efficiency of the PITD method is markedly enhanced by using the SM technique. In the end, numerical examples are carried out to verify the effectiveness and the excellence of the proposed technique.
An Improved Domain Decomposition with Relaxation Method Used for Calculating Dynamic Characteristic of Electromagnetic Devices
Wenying Yang, Zilan Qiu, Fei Peng, Guofu Zhai
HIT, China, People's Republic of
Finite element method (FEM) is the mainstream tool to analyze the dynamic characteristics of electromagnetic apparatus. Parallel technology makes the computation of FEM more effective. For example, domain decomposition (DD) method is widely used to accelerate the FEM process. In the paper, a new finite element solution technique called nodal domain decomposition with relaxation (NDDR) is used to solve the dynamic characteristics of the electromagnetic device. The method can improve the parallelism to accelerate the dynamic characteristic solving. Moreover, We introduce Robintype transmission condition to NDDR method to raise its solving efficiency.
Performance Evaluation of Parallel Finite Element EddyCurrent Analysis Using Direct Method as Subdomain Solver
Takehito Mizuma^{1}, Amane Takei^{2}
^{1}Interdisciplinary Graduate School of Agriculture and Engineering, University of Miyazaki, Miyazaki, Japan; ^{2}Department of Electrical and Systems Engineering, Faculty of Engineering, University of Miyazaki, Miyazaki, Japan
An eddycurrent analysis (Aϕ method) based on a domain decomposition method is proposed. A conjugate orthogonal conjugate gradient with incomplete Cholesky factorization method is generally applied to solve subdomain problems because the coefficient matrices become singular. However, it is difficult to achieve high accuracy solutions by using iterative methods because the solutions contain truncation errors. Therefore, we propose a method of improving the convergence by applying a direct method based on a singular value decomposition as the subdomain solver. As the result, it is thus confirmed that the improvement of convergence can be realized with the direct method. This paper describes the proposed method and the performance evaluation with some examinations.
The Time Domain Discrete Geometric Approach is a Coupling of Two Explicit DG Schemes with Continuous Fluxes
Bernard Kapidani^{1}, Lorenzo Codecasa^{2}, Ruben Specogna^{3}
^{1}Institute for Analysis and Scientific Computing, TU Wien; ^{2}Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano; ^{3}Dipartimento Politecnico di Ingegneria ed Architettura, University of Udine
The Discrete Geometric Approach (DGA) in the time domain, already introduced by Codecasa et al. in 2008, is recast as a Galerkin Method akin to the Finite Element Method (FEM). In particular for the coupled first order Maxwell equations, it is shown to be a mixed method comprising two explicit Discontinuous Galerkin (DG) FEM schemes formulated on dual meshes, in which each of the two schemes provides a continuous numerical flux choice for its dual mesh scheme. The implemented lowest order version is shown to compare favorably in terms of accuracy and time step size with respect to the classic conforming FEM scheme using lowest order edge elements, when tested on a simple example with known analytic solution.
Toward the Electromagnetic Response of a Moving Ferromagnetic Medium with a PITDBased Methodology
Jinghui Shao, Xikui Ma, Xiaojie Zhu, Jiawei Wang
Xi'an Jiaotong University, China, People's Republic of
The knowledge of the interaction between electromagnetic (EM) wave and moving ferromagnetic medium is expected in many applications such as target recognition and microelectromechanic systems manufacture. In this digest, by using a novel methodology stemming from the precise integration time domain (PITD) method, the response of a moving ferromagnetic medium to the illuminating electromagnetic (EM) wave is numerically modelled with the hysteresis effect taken into account. In the suggested methodology, the EM parameters in each computation cell at an instant are evaluated through a conformal meshing procedure. Because of the inherent nonlinearity and the motion of the ferromagnetic medium, the original coefficient matrix is timevariant and therefore approximated by a piecewise constant one to make the precise integration technique implementable. After the numerical experiments involving the inmotion MnZn composite are performed, the relation between the pattern of modulation effect of the moving ferromagnetic medium on the incoming EM wave and the amplitude and initial phase of excitation is preliminarily established, evidencing the feasibility of the suggested methodology.
