Yoshifumi Okamoto, Hiroshi Masuda, Yutaro Kanda, Reona Hoshino and Shinji Wakao
The purpose of this paper is the improvement of topology optimization. The scope of the paper is focused on the speedup of optimization.
Abstract
Purpose
The purpose of this paper is the improvement of topology optimization. The scope of the paper is focused on the speedup of optimization.
Design/methodology/approach
To achieve the speedup, the method of moving asymptotes (MMA) with constrained condition of level set function is applied instead of solving the Hamilton–Jacobi equation.
Findings
The acceleration of convergence of objective function is drastically improved by the implementation of MMA.
Originality/value
Normally, the level set method is solved through the Hamilton–Jacobi equation. However, the possibility of introducing mathematical programming is clear by the constrained condition. Furthermore, the proposed method is suitable for efficiently solving the topology optimization problem in the magnetic field system.
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Reona Hoshino, Yoshifumi Okamoto and Shinji Wakao
Shape optimization using the level-set method is one of the most effective automatic design tools for electromagnetic machines. While level-set method has the advantage of being…
Abstract
Purpose
Shape optimization using the level-set method is one of the most effective automatic design tools for electromagnetic machines. While level-set method has the advantage of being able to suppress unfeasible shape, it has a weakness of being unable to handle complex topology changes such as perforate at material region. With this method, it is only possible to define simple connected topology, and it is difficult to determine the optimal shape which has holes. Therefore, it is important to efficiently expand the searching area in the optimization process with level-set method.
Design/methodology/approach
In this paper, the authors introduce the newly defined hole sensitivity which is based on concept of topological derivatives, and combine it with level-set method to effectively create holes in the search process. Furthermore, they consider a variable bandwidth of gray scale, which indicates the transition width between air and magnetic body and combine it with the hole creation method described above. With these methods, the authors aim to expand the searching area in comparison with the conventional level-set method.
Findings
As a result of applying the proposed methods to a magnetic shielding problem, the multi-layered shielding which effectively reduces the magnetic flux in the target area, is successfully produced.
Originality/value
The proposed methods enable us to effectively create a hole and to expand the searching area in the topology optimization process unlike in the case of conventional level-set method.
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Shimpei Kakita and Yoshifumi Okamoto
The paper aims to improve convergence characteristics of the Newton–Raphson (NR) method applied to time-periodic finite element method using various line searches, as…
Abstract
Purpose
The paper aims to improve convergence characteristics of the Newton–Raphson (NR) method applied to time-periodic finite element method using various line searches, as time-periodic finite element method causes deterioration of convergence characteristic of nonlinear analysis based on NR method. The study also aims to accelerate and improve accuracy of electromagnetic field analysis for improvement of the performance of electrical machine.
Design/methodology/approach
The paper proposes new type line searches that set approximate step size for NR method. The line search evaluated step size using higher-order interpolation of functional derivative. In addition, two criteria for applying these line search were proposed. First method set one scalar value for every NR iteration that is named constant step size. Second method define different step size in each time step of time-periodic finite element method to update solution vector that is named different step size.
Findings
The paper provides efficient line searches to improve convergence characteristics for NR method. Nonlinear magnetic field analysis of two transformer models is demonstrated. The proposed methods achieve the following results: higher-order functional NR is efficient in improving convergence characteristics, and the proposed methods succeeded about twice faster in both models.
Originality/value
The paper fulfills improvement of convergence characteristics of the NR method applied to time-periodic finite element method using proposed line searches and accelerate electromagnetic field analysis.
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Hiroshi Masuda, Yoshifumi Okamoto and Shinji Wakao
The purpose of this paper is to solve efficiently the topology optimization (TO) in time domain problem with magnetic nonlinearity requiring a large-scale finite element mesh. As…
Abstract
Purpose
The purpose of this paper is to solve efficiently the topology optimization (TO) in time domain problem with magnetic nonlinearity requiring a large-scale finite element mesh. As an actual application model, the proposed method is applied to induction heating apparatus.
Design/methodology/approach
To achieve TO with efficient computation time, a multistage topology is proposed. This method can derive the optimum structure by repeatedly reducing the design domain and regenerating the finite element mesh.
Findings
It was clarified that the structure derived from proposed method can be similar to the structure derived from the conventional method, and that the computation time can be made more efficient by parameter tuning of the frequency and volume constraint value. In addition, as a time domain induction heating apparatus problem of an actual application model, an optimum topology considering magnetic nonlinearity was derived from the proposed method.
Originality/value
Whereas the entire design domain must be filled with small triangles in the conventional TO method, the proposed method requires finer mesh division of only the stepwise-reduced design domain. Therefore, the mesh scale is reduced, and there is a possibility that the computation time for TO can be shortened.
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Makoto Kumagai, Shimpei Kakita and Yoshifumi Okamoto
This paper aims to present the affinity of BiCGStab and BiCGStab2 with successive over-relaxation (SOR) preconditioner supported by Eisenstat’s technique for a linear system…
Abstract
Purpose
This paper aims to present the affinity of BiCGStab and BiCGStab2 with successive over-relaxation (SOR) preconditioner supported by Eisenstat’s technique for a linear system derived from the time-periodic finite element method (TP-FEM). To solve the time domain electromagnetic field problem with long transient state, TP-FEM is very useful from the perspective of rapidly achieving a steady state. Because TP-FEM solves all of the state variables at once, the linear system derived from TP-FEM becomes the large scale and nonsymmetric, whereas the detailed performance of some preconditioned Krylov subspace method is not reported.
Design/methodology/approach
In this paper, BiCGStab and BiCGStab2 are used as the linear solver for a large-sparse nonsymmetric linear system derived from TP-FEM. In addition, incomplete LU (ILU) factorization is applied as a preconditioner to compare SOR supported by Eisenstat’s technique. As examples, the pot-type reactor and three-phase transformer is analyzed.
Findings
In the problem of the pot-type reactor, when SOR preconditioner supported by Eisenstat’s technique is applied to BiCGStab and BiCGStab2, the elapsed time can be reduced dramatically. However, in the problem of the three-phase transformer, the iterative process of the linear solvers with SOR preconditioner is not terminated, whereas the iterative process of linear solvers with ILU preconditioner is terminated. The preconditioner that can be supported by Eisenstat’s technique is not necessarily appropriate for the problem to derive from TP-FEM.
Originality/value
In this paper, the affinity of preconditioned linear solver supported by Eisenstat’s technique for the nonsymmetric linear system derived from TP-FEM is demonstrated.
Details
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Yoshifumi Okamoto, Akihisa Kameari, Koji Fujiwara, Tomonori Tsuburaya and Shuji Sato
– The purpose of this paper is the realization of Fast nonlinear finite element analysis (FEA).
Abstract
Purpose
The purpose of this paper is the realization of Fast nonlinear finite element analysis (FEA).
Design/methodology/approach
Nonlinear magnetic field analysis is achieved by using Newton-Raphson method implemented by relaxed convergence criterion of Krylov subspace method.
Findings
This paper mathematically analyzes the reason why nonlinear convergence can be achieved if the convergence criterion for linearized equation is relaxed.
Research limitations/implications
The proposed method is essential to reduce the elapsed time in nonlinear magnetic field analysis of quasi-stationary field.
Practical implications
The proposed method is able to be extended to not only static field but also time domain FEA strongly coupled with circuit equation.
Social implications
Because the speedup of performance evaluation of electrical machines would be achieved using proposed method, the work efficiency in manufacturing would be accelerated.
Originality/value
It can be seen that the nonlinear convergence can be achieved if the convergence criterion for linearized equation is relaxed. The verification of proposed method is demonstrated using practical nonlinear magnetic field problem.
Details
Keywords
Tomonori Tsuburaya, Yoshifumi Okamoto and Shuji Sato
The purpose of this paper is to improve the performance of block-multicolor (BMC) ordering for the parallelized incomplete-Cholesky-preconditioned conjugate gradient (ICCG…
Abstract
Purpose
The purpose of this paper is to improve the performance of block-multicolor (BMC) ordering for the parallelized incomplete-Cholesky-preconditioned conjugate gradient (ICCG) method. Then, the BMC ordering based on level structure arising in reverse Cuthill-McKee RCM ordering is newly proposed. The name of proposed method is abbreviated as “RBMC”. This paper shows the validity of proposed method by comparison with greedy-based multicolor (MC) and conventional BMC on the real symmetric linear system derived from the voltage-driven finite element method in time domain.
Design/methodology/approach
In RBMC, the blocking and coloring is performed level by level. The number of synchronizations in forward and backward substitution is reduced so that all blocks can be colored with two colors. However, the load-balance in forward and backward substitution might deteriorate because the irregular block matrices are distributed around diagonal. To uniformize load-balance in forward and backward substitution, the RBMC combined with the concept of block red-black ordering has been developed.
Findings
The modified RBMC was the most effective for reduction of the elapsed time among four orderings (MC, BMC, RBMC, modified RBMC) owing to improvement of convergence characteristic and load-balance.
Originality/value
The proposed method had two advantages: although the number of unknowns per block must be previously determined in BMC, its parameter is automatically determined in proposed method, the number of synchronization in forward and backward substitution can be reduced.
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Keywords
Yoshifumi Okamoto, Yusuke Tominaga, Shinji Wakao and Shuji Sato
The purpose of this paper is to improve the multistep algorithm using evolutionary algorithm (EA) for the topology optimization of magnetostatic shielding, and the paper reveals…
Abstract
Purpose
The purpose of this paper is to improve the multistep algorithm using evolutionary algorithm (EA) for the topology optimization of magnetostatic shielding, and the paper reveals the effectiveness of methodology by comparison with conventional optimization method. Furthermore, the design target is to obtain the novel shape of magnetostatic shielding.
Design/methodology/approach
The EAs based on random search allow engineers to define general-purpose objects with various constraint conditions; however, many iterations are required in the FEA for the evaluation of the objective function, and it is difficult to realize a practical solution without island and void distribution. Then, the authors proposed the multistep algorithm with design space restriction, and improved the multistep algorithm in order to get better solution than the previous one.
Findings
The variant model of optimized topology derived from improved multistep algorithm is defined to clarify the effectiveness of the optimized topology. The upper curvature of the inner shielding contributed to the reduction of magnetic flux density in the target domain.
Research limitations/implications
Because the converged topology has many pixel element unevenness, the special smoother to remove the unevenness will play an important role for the realization of practical magnetostatic shielding.
Practical implications
The optimized topology will give us useful detailed structure of magnetostatic shielding.
Originality/value
First, while the conventional algorithm could not find the reasonable shape, the improved multistep optimization can capture the reasonable shape. Second, An additional search is attached to the multistep optimization procedure. It is shown that the performance of improved multistep algorithm is better than that of conventional algorithm.
Details
Keywords
Yoshifumi Okamoto, Toshio Imai, Daisuke Miyagi, Norio Takahashi, Kazuhiro Ozaki, Hiroshi Ono and Naoki Uchida
Induction heating is widely used in the industry due to high efficiency, precise control, rapid heating and low pollution properties. To process a semiconductor of high quality…
Abstract
Induction heating is widely used in the industry due to high efficiency, precise control, rapid heating and low pollution properties. To process a semiconductor of high quality, uniform heating is necessary, but it is not easy to heat uniformly by using the conventional induction heating equipment. For this purpose, the zone control induction heating equipment is jointly developed with Mitsui Engineering Shipbuilding Co., Ltd. In this paper, the optimization of current in each coil is carried out using the finite element method and the optimization method in order to obtain a uniform eddy current loss distribution on graphite.