TY - JOUR
T1 - Improvement of topology optimization method based on level set function in magnetic field problem
AU - Okamoto, Yoshifumi
AU - Masuda, Hiroshi
AU - Kanda, Yutaro
AU - Hoshino, Reona
AU - Wakao, Shinji
N1 - Funding Information:
In this paper, to overcome the slower convergence of the objective function in the conventional LS method, an MMA is applied using a constrained LS function. The performance of the proposed method is verified using 2D magnetic field problems such as a magnetic shielding model and an IPM motor. Consequently, the convergence acceleration is achieved, and the proposed method releases the conventional LS method from the difficulty of regularizing the LS function at every optimization step. Although the ability of TO in the conventional LS method is weakened, a topological search is carried out at an early stage of the optimization process by setting the initial topology to completely grayscale. After that, shape optimization is dominant over the topological search in the proposed method. This work was supported by the JSPS (Japan Society for the Promotion of Science) Grant-in-Aid for Scientific Research (C) Grant Number 16K06240. Figure 1. Relationship between LS and the Heaviside function Figure 2. Approximation results for Taylor series expansion and MMA Figure 3. Procedure for the topology optimization method supported by MMA Figure 4. Optimization model Figure 5. Optimized topologies of magnetic shielding derived from conventional LS method without magnetic nonlinearity Figure 6. Optimized topologies of magnetic shielding derived from MMA with material-density without magnetic nonlinearity Figure 7. Optimized topologies of magnetic shielding without magnetic nonlinearity derived from MMA with Heaviside without magnetic nonlinearity Figure 8. Optimized topologies of magnetic shielding with magnetic nonlinearity Figure 9. Convergence characteristics of objective function on magnetic shielding Figure 10. Optimized topologies of IPM motor Figure 11. Torque characteristics of IPM motor Table I. Specifications of optimization parameters Model P h [mm] n S 0 [mm 2 ] ε [sub] ε opt Shielding 2.0 2.0 3,952 3.0 × 10 3 10 −10 10 −4 IPM motor 2.0 10.0 3,604 1.8 × 10 3 10 −8 10 −6 Table II. Optimization results of magnetic shielding Magnetic property Optimization method k opt f 0 (ϕ (k opt ) ) S 0 [mm 2 ] elapsed time [s] Linear Conv. LS method 453 8.07 × 10 −6 3.0 × 10 3 171 MMA with material-density 90 5.08 × 10 −6 3.0 × 10 3 48 MMA with Heaviside 246 6.46 × 10 −6 3.0 × 10 3 68 Nonlinear Conv. LS method 1,551 5.67 × 10 −4 3.0 × 10 3 982 MMA with material-density 192 4.18 × 10 −4 3.0 × 10 3 165 MMA with Heaviside 441 4.08 × 10 −4 3.0 × 10 3 257 Table III. Optimization results of IPM motor Optimization method k opt f 0 (ϕ (k opt ) ) S 0 [mm 2 ] Elapsed time [h] MMA with material-density 136 1.73 × 10 −3 1.8 × 10 3 5.2 MMA with Heaviside 322 1.64 × 10 −3 1.8 × 10 3 11.8
PY - 2018/3/5
Y1 - 2018/3/5
N2 - 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.
AB - 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.
KW - Adjoint variable method
KW - Finite element analysis
KW - Finite element method
KW - Level set function
KW - Material-density
KW - Method of moving asymptotes
KW - Numerical methods
KW - Optimization techniques
KW - Topology optimization
UR - http://www.scopus.com/inward/record.url?scp=85065649711&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85065649711&partnerID=8YFLogxK
U2 - 10.1108/COMPEL-12-2016-0528
DO - 10.1108/COMPEL-12-2016-0528
M3 - Article
AN - SCOPUS:85065649711
SN - 0332-1649
VL - 37
SP - 630
EP - 644
JO - COMPEL - The International Journal for Computation and Mathematics in Electrical and Electronic Engineering
JF - COMPEL - The International Journal for Computation and Mathematics in Electrical and Electronic Engineering
IS - 2
ER -