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[1] Bendsoe, M.P. & Kikuchi, N., Generating optimal topologies in structural design using
a homogenization method. Computer Methods in Applied Mechanics and Engineering,
[2] Rozvany, G.I.N., Zhou, M. & Birker, T., Generalized shape optimization without
homogenization. Structural Optimization, 4, pp. 250–252, 1992.
[3] Yamada, T., Izui, K., Nishiwaki, S. & Takezawa, A., A topology optimization method
based on the level set method incorporating a fictitious interface energy. Computer
Methods in Applied Mechanics and Engineering, 199, pp. 2876–2891, 2010.
[4] Isakari, H., Kuriyama, K., Harada, S., Yamada, T., Takahashi, T. & Mat-sumoto, T., A
topology optimisation for three-dimensional acoustics with the level set method and
the fast multipole boundary element method. Mechanical Engineering Journal, 1(4),
[5] Jing, G., Matsumoto, T., Takahashi, T., Isakari, H. & Yamada, T., Topology optimization
for 2D heat conduction problems using boundary element method and level set
method. Transactions of JASCOME, 13(19), pp. 5–10, 2013.
[6] Isakari, H., Nakamoto, K., Kitabayashi, T., Takahashi, T. & Matsumoto, T., A multiobjective
topology optimisation for 2D electro-magnetic wave problems with the level
set method and BEM (to appear in). European Journal of Computational Mechanics,
[7] Bebendorf, M., Hierarchical Matrices, Springer Berlin: Heidelberg, 2008.
[8] Chew, W.C., Waves and Fields in Inhomogenous Media, IEEE Press: New york, 1995.
[9] Kreisselmeier, G., Systematic control design by optimizing a vector performance index.
In IFAC Symp. Computer Aided Design of Control Systems, Zurich, Switzerland, 1979.
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Open Access
Research article

Acceleration of a Multi-objective Topology Optimisation in 2D Electro-Magnetic Field Based on the Level-Set Method and the Boundary Element Method by the $\mathscr{H}$-Matrix Method

k. nakamoto,
h. isakari,
t. takahashi,
t. matsumoto
Nagoya University, Japan
International Journal of Computational Methods and Experimental Measurements
|
Volume 5, Issue 5, 2017
|
Pages 686-695
Received: N/A,
Revised: N/A,
Accepted: N/A,
Available online: N/A
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Abstract:

In this study, we develop an efficient topology optimisation method with the H -matrix method and the boundary element method (BEM). In sensitivity analyses of topology optimisation, we need to solve a set of two algebraic equations whose coefficient matrices are common, particularly in many cases. For such cases, by using a direct solver such as LU decomposition to factorise the coefficient matrix, we can reduce the computational time for the sensitivity analysis. A coefficient matrix derived by the BEM is, however, fully populated, which causes high numerical costs for the LU decomposition. In this research, the LU decomposition is accelerated by using the H -matrix method for the sensitivity analyses of topology optimisation problems. We demonstrate the efficiency of the proposed method by a numerical example of a multi-objective optimisation problem for 2D electromagnetic field.

Keywords: Boundary element method, $\mathscr{H}$-matrix method, Level-set method, Topological sensitivity, Topology optimisation

1. Introduction

2. Formulations

3. Topology Optimization

4. Acceleration of the Sensitivity Analysis

5. Topology Optimisation Example

6. Concluding Remarks

References
[1] Bendsoe, M.P. & Kikuchi, N., Generating optimal topologies in structural design using
a homogenization method. Computer Methods in Applied Mechanics and Engineering,
[2] Rozvany, G.I.N., Zhou, M. & Birker, T., Generalized shape optimization without
homogenization. Structural Optimization, 4, pp. 250–252, 1992.
[3] Yamada, T., Izui, K., Nishiwaki, S. & Takezawa, A., A topology optimization method
based on the level set method incorporating a fictitious interface energy. Computer
Methods in Applied Mechanics and Engineering, 199, pp. 2876–2891, 2010.
[4] Isakari, H., Kuriyama, K., Harada, S., Yamada, T., Takahashi, T. & Mat-sumoto, T., A
topology optimisation for three-dimensional acoustics with the level set method and
the fast multipole boundary element method. Mechanical Engineering Journal, 1(4),
[5] Jing, G., Matsumoto, T., Takahashi, T., Isakari, H. & Yamada, T., Topology optimization
for 2D heat conduction problems using boundary element method and level set
method. Transactions of JASCOME, 13(19), pp. 5–10, 2013.
[6] Isakari, H., Nakamoto, K., Kitabayashi, T., Takahashi, T. & Matsumoto, T., A multiobjective
topology optimisation for 2D electro-magnetic wave problems with the level
set method and BEM (to appear in). European Journal of Computational Mechanics,
[7] Bebendorf, M., Hierarchical Matrices, Springer Berlin: Heidelberg, 2008.
[8] Chew, W.C., Waves and Fields in Inhomogenous Media, IEEE Press: New york, 1995.
[9] Kreisselmeier, G., Systematic control design by optimizing a vector performance index.
In IFAC Symp. Computer Aided Design of Control Systems, Zurich, Switzerland, 1979.
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Nakamoto, K., Isakari, H., Takahashi, T., & Matsumoto, T. (2017). Acceleration of a Multi-objective Topology Optimisation in 2D Electro-Magnetic Field Based on the Level-Set Method and the Boundary Element Method by the $\mathscr{H}$-Matrix Method. Int. J. Comput. Methods Exp. Meas., 5(5), 686-695. https://doi.org/10.2495/CMEM-V5-N5-686-695
K. Nakamoto, H. Isakari, T. Takahashi, and T. Matsumoto, "Acceleration of a Multi-objective Topology Optimisation in 2D Electro-Magnetic Field Based on the Level-Set Method and the Boundary Element Method by the $\mathscr{H}$-Matrix Method," Int. J. Comput. Methods Exp. Meas., vol. 5, no. 5, pp. 686-695, 2017. https://doi.org/10.2495/CMEM-V5-N5-686-695
@research-article{Nakamoto2017AccelerationOA,
title={Acceleration of a Multi-objective Topology Optimisation in 2D Electro-Magnetic Field Based on the Level-Set Method and the Boundary Element Method by the $\mathscr{H}$-Matrix Method},
author={K. Nakamoto and H. Isakari and T. Takahashi and T. Matsumoto},
journal={International Journal of Computational Methods and Experimental Measurements},
year={2017},
page={686-695},
doi={https://doi.org/10.2495/CMEM-V5-N5-686-695}
}
K. Nakamoto, et al. "Acceleration of a Multi-objective Topology Optimisation in 2D Electro-Magnetic Field Based on the Level-Set Method and the Boundary Element Method by the $\mathscr{H}$-Matrix Method." International Journal of Computational Methods and Experimental Measurements, v 5, pp 686-695. doi: https://doi.org/10.2495/CMEM-V5-N5-686-695
K. Nakamoto, H. Isakari, T. Takahashi and T. Matsumoto. "Acceleration of a Multi-objective Topology Optimisation in 2D Electro-Magnetic Field Based on the Level-Set Method and the Boundary Element Method by the $\mathscr{H}$-Matrix Method." International Journal of Computational Methods and Experimental Measurements, 5, (2017): 686-695. doi: https://doi.org/10.2495/CMEM-V5-N5-686-695
Nakamoto K.， Isakari H.， Takahashi T.， et al. Acceleration of a Multi-objective Topology Optimisation in 2D Electro-Magnetic Field Based on the Level-Set Method and the Boundary Element Method by the $\mathscr{H}$-Matrix Method[J]. International Journal of Computational Methods and Experimental Measurements， 2017， 5(5)： 686-695. https://doi.org/10.2495/CMEM-V5-N5-686-695