A New Adjoint Problem for Two-Dimensional Helmholtz Equation to Calculate Topological Derivatives of the Objective Functional Having Tangential Derivative Quantities

A New Adjoint Problem for Two-Dimensional Helmholtz Equation to Calculate Topological Derivatives of the Objective Functional Having Tangential Derivative Quantities

Peijun Tang Toshiro Matsumoto Hiroshi Isakari Toru Takahashi

Nagoya University, Japan

Page: 
74-82
|
DOI: 
https://doi.org/10.2495/CMEM-V9-N1-74-82
Received: 
N/A
|
Revised: 
N/A
|
Accepted: 
N/A
|
Available online: 
N/A
| Citation

© 2021 IIETA. This article is published by IIETA and is licensed under the CC BY 4.0 license (http://creativecommons.org/licenses/by/4.0/).

OPEN ACCESS

Abstract: 

A special topology optimization problem is considered whose objective functional consists of tangential derivative of the potential on the boundary for two-dimensional Helmholtz equation. In order to derive the adjoint problem, the functional of the conventional topology optimizations required a boundary integral of the potential and its flux. For the present objective functional having the tangential derivative, integration by parts is applied to the part having the tangential derivative of the variation of the potential to generate a tractable adjoint problem. The derived adjoint problem is used in the variation of the objective function, and the topological derivative is derived in the conventional expression.

Keywords: 

adjoint problem, boundary element method, tangential derivative of potential, topological derivative, topology optimization

  References

[1] Nakamoto, K., Isakari, H., Takahashi, T. & Matsumoto T., A level-set-based topology optimisation of carpet cloaking devices with the boundary element method. Mechanical Engineering Journal, 4(1), 2016. doi:10.1299/mej.16-00268

[2] Matsushima, K., Isakari, H., Takahshi, T. & Matsumoto, T., An application of topology optimisation to defect identification in two-dimensional elastodynamics with the BEM and H-matrix method. International Journal of Computational Methods and Experimental Measurements, 6(6), 2018, pp. 1033–1042. doi:10.2495/CMEM-V6-N6-1033-1042

[3] Allaire, G., Jouve, F. & Toader, A.M., Structural optimization using sensitivity analysis and a level-set method. Journal of Computational Physics, 194(1), 2004, pp. 363–393. https://doi.org/10.1016/j.jcp.2003.09.032

[4] Wang, M.Y., Wang, X. & Guo, D., A level set method for structural topology optimization. Computer Methods in Applied Mechanics and Engineering, 192, 2003, pp. 227–246. https://doi.org/10.1016/s0045-7825(02)00559-5

[5] Xia, Q. & Wang, M.Y., Topology optimization of thermoelastic structures using level set method. Computational Mechanics, 42, 2008, Article number: 837.

[6] 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, 2010, pp. 2876–2891. https:// doi.org/10.1016/j.cma.2010.05.013

[7] Shu, L., Wang, M.Y., Fang, Z., Ma, Z.-D., Wei, P., Level set based structural topology optimization for minimizing frequency response, Sound and Vibration, 330(24), 2011. pp. 5820–5834. https://doi.org/10.1016/j.jsv.2011.07.026

[8] Jing, G., Isakari, H., Matsumoto, T. & Takahashi, T., Level set-based topology optimization for 2D heat conduction problems using BEM with objective function defined  on design-dependent boundary with heat transfer boundary condition. Engineering Analysis with Boundary Elements, 61, 2015, pp. 61–70. https://doi.org/10.1016/j.enga- nabound.2015.06.012

[9] Yang, R.J. & Chen, C.J., Stress-based topology optimization. Structural Optimization Journal, 12, 1996, pp. 98–105. https://doi.org/10.1007/bf01196941

[10] Picelli, R., Townsend, S., Brampton, C., Norato, J. & Kim, H.A., Stress-based shape and topology optimization with the level set method. Computer Method in Applied Mechanics and Engineering, 329, 2017, pp. 1–29. https://doi.org/10.1016/j.cma.2017.09.001