  Volume 3, Issue 1
An Efficient Threshold Dynamics Method for Topology Optimization for Fluids

CSIAM Trans. Appl. Math., 3 (2022), pp. 26-56.

Published online: 2022-03

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• Abstract

We propose an efficient threshold dynamics method for topology optimization for fluids modeled with the Stokes equation. The proposed algorithm is based on minimization of an objective energy function that consists of the dissipation power in the fluid and the perimeter approximated by nonlocal energy, subject to a fluid volume constraint and the incompressibility condition. We show that the minimization problem can be solved with an iterative scheme in which the Stokes equation is approximated by a Brinkman equation. The indicator functions of the fluid-solid regions are then updated according to simple convolutions followed by a thresholding step. We prove mathematically that the iterative algorithm has the total energy decaying property. The proposed algorithm is simple and easy to implement. Extensive numerical experiments in both two and three dimensions show that the proposed iteration algorithm converges in much fewer iterations and is more efficient than many existing methods. In addition, the numerical results show that the algorithm is very robust and insensitive to the initial guess and the parameters in the model.

35K93, 35K05, 65M12, 35Q35, 49Q10, 65M60, 76S05

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@Article{CSIAM-AM-3-26, author = {Chen , HuangxinLeng , HaitaoWang , Dong and Wang , Xiao-Ping}, title = {An Efficient Threshold Dynamics Method for Topology Optimization for Fluids}, journal = {CSIAM Transactions on Applied Mathematics}, year = {2022}, volume = {3}, number = {1}, pages = {26--56}, abstract = {

We propose an efficient threshold dynamics method for topology optimization for fluids modeled with the Stokes equation. The proposed algorithm is based on minimization of an objective energy function that consists of the dissipation power in the fluid and the perimeter approximated by nonlocal energy, subject to a fluid volume constraint and the incompressibility condition. We show that the minimization problem can be solved with an iterative scheme in which the Stokes equation is approximated by a Brinkman equation. The indicator functions of the fluid-solid regions are then updated according to simple convolutions followed by a thresholding step. We prove mathematically that the iterative algorithm has the total energy decaying property. The proposed algorithm is simple and easy to implement. Extensive numerical experiments in both two and three dimensions show that the proposed iteration algorithm converges in much fewer iterations and is more efficient than many existing methods. In addition, the numerical results show that the algorithm is very robust and insensitive to the initial guess and the parameters in the model.

}, issn = {2708-0579}, doi = {https://doi.org/10.4208/csiam-am.SO-2021-0007}, url = {http://global-sci.org/intro/article_detail/csiam-am/20287.html} }
TY - JOUR T1 - An Efficient Threshold Dynamics Method for Topology Optimization for Fluids AU - Chen , Huangxin AU - Leng , Haitao AU - Wang , Dong AU - Wang , Xiao-Ping JO - CSIAM Transactions on Applied Mathematics VL - 1 SP - 26 EP - 56 PY - 2022 DA - 2022/03 SN - 3 DO - http://doi.org/10.4208/csiam-am.SO-2021-0007 UR - https://global-sci.org/intro/article_detail/csiam-am/20287.html KW - Topology optimization, Stokes flow, threshold dynamics method, mixed finite-element method. AB -

We propose an efficient threshold dynamics method for topology optimization for fluids modeled with the Stokes equation. The proposed algorithm is based on minimization of an objective energy function that consists of the dissipation power in the fluid and the perimeter approximated by nonlocal energy, subject to a fluid volume constraint and the incompressibility condition. We show that the minimization problem can be solved with an iterative scheme in which the Stokes equation is approximated by a Brinkman equation. The indicator functions of the fluid-solid regions are then updated according to simple convolutions followed by a thresholding step. We prove mathematically that the iterative algorithm has the total energy decaying property. The proposed algorithm is simple and easy to implement. Extensive numerical experiments in both two and three dimensions show that the proposed iteration algorithm converges in much fewer iterations and is more efficient than many existing methods. In addition, the numerical results show that the algorithm is very robust and insensitive to the initial guess and the parameters in the model.

Huangxin Chen, Haitao Leng, Dong Wang & Xiao-Ping Wang. (2022). An Efficient Threshold Dynamics Method for Topology Optimization for Fluids. CSIAM Transactions on Applied Mathematics. 3 (1). 26-56. doi:10.4208/csiam-am.SO-2021-0007
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