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A marker-and-cell method for large-scale flow-based topology optimization on GPU

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Abstract

The focus of this paper is to propose a novel computational approach for the solution of large-scale flow-based topology optimization problems using a graphics processing unit (GPU). A marker-and-cell method is first used to discretize a fluid flow design domain. This is followed by a finite difference method to solve the Stokes equations for steady-state incompressible fluid flow. An adjoint method is then employed to conduct design sensitivity analysis for the optimization. We use a generalized minimal residual method as the base solver for the linear system and develop an efficient geometric multigrid preconditioner on GPU in a matrix-free form. We simplify the treatment of different boundary conditions with improved accuracy based on the theory of discrete exterior calculus. Numerical results utilizing different resolutions are presented and highlight a nearly linear computational time scalability. Consequently, intricate branching flow structures may be automatically and efficiently discovered at high resolutions. Our approach is capable of solving indefinite problems (i.e., one forward solution of the Stokes equations) with over 7 million elements in three dimensions (3D) and over 16 million elements in two dimensions (2D) within two minutes using a single desktop computer. Furthermore, all numerical experiments reported in this paper are performed on a single NVIDIA Quadro RTX 8000 graphics card. We subsequently compare the optimized flow structures obtained using the newly proposed method with those obtained by commercial finite element software in an established optimization loop and find the optimized structures from both methods to be in good agreement. To highlight the advantage of GPU acceleration, a quantitative run-time comparison study with the commercial finite element software is performed. Our implementation is shown to solve fluid flow problems with orders of magnitude higher resolution using only a fraction of the computational time.

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Acknowledgements

This project is supported in part by Toyota Research Institute of North America (TRINA), NSF MRI-1919647, and NSF HCC-2106733.

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Authors and Affiliations

  1. Department of Computer Science, Dartmouth College, Hanover, NH, 03755, USA

    Jinyuan Liu, Zangyueyang Xian & Bo Zhu

  2. Electronics Research Department, Toyota Research Institute of North America, Ann Arbor, MI, 48105, USA

    Yuqing Zhou & Ercan M. Dede

  3. Applied Mathematics Research Domain, Toyota Central R & D Labs., Inc., Tokyo, 112-0004, Japan

    Tsuyoshi Nomura

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  1. Jinyuan Liu
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Correspondence to Ercan M. Dede.

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The source code is available on Github with open access at https://github.com/jyl-pages/Stokes_TO 

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Appendix 1: Sensitivity

Appendix 1: Sensitivity

In this section, we discuss the derivative of the objective, J, with respect to the fluid design density variable, \(\rho\). The adjoint method is used to compute the sensitivity. First, we simplify the notation by defining \(\mathbf {X}=[\mathbf {u},P]^T\) as a stacked vector comprising all state variables. The linear system in (4) can be written as \(\mathcal {A}\mathbf {X}=\mathbf {b}\). Since the system matrix, \(\mathcal {A}\), depends on \(\rho\), so does the solution vector, \(\mathbf {X}\). The objective is a function of the state and design variables. Let us define the Lagrangian,

$$\begin{aligned} \mathcal {L}(\mathbf {X},\rho ,\lambda )=J(\mathbf {X},\rho )+\lambda ^T(\mathcal {A}\mathbf {X}-\mathbf {b}), \end{aligned}$$
(9)

where \(\lambda\) is the vector of Lagrange multipliers. As \((\mathcal {A}\mathbf {X}-\mathbf {b})\) is zero everywhere by construction, we may choose \(\lambda\) freely such that \(J=\mathcal {L}\), and derive the gradients as follows,

$$\begin{aligned} \frac{dJ}{d\rho }&=\frac{d\mathcal {L}}{d\rho }\nonumber \\&=\frac{\partial J}{\partial \rho }+\frac{\partial J}{\partial \mathbf {X}}\frac{\partial \mathbf {X}}{\partial \rho }+\lambda ^T\left( \frac{\partial \mathcal {A}}{\partial \rho }\mathbf {X}+\mathcal {A}\frac{\partial \mathbf {X}}{\partial \rho }\right) \nonumber \\&=\frac{\partial J}{\partial \rho }+\lambda ^T\frac{\partial \mathcal {A}}{\partial \rho }\mathbf {X}+\left( \frac{\partial J}{\partial \mathbf {X}}+\lambda ^T\mathcal {A}\right) \frac{\partial \mathbf {X}}{\partial \rho }. \end{aligned}$$
(10)

If we choose \(\lambda\) so that \(\frac{\partial J}{\partial \mathbf {X}}+\lambda ^T\mathcal {A}=0\), the last term becomes zero, and we can avoid calculating \(\frac{\partial \mathbf {X}}{\partial \rho }\). This condition is the adjoint equation:

$$\begin{aligned} \mathcal {A}^T\lambda =-\left( \frac{\partial J}{\partial \mathbf {X}}\right) ^T. \end{aligned}$$
(11)

Here, \(\mathcal {A}^T=\mathcal {A}\) based on symmetry, and the state variable, \(\mathbf {X}\), is pre-computed at each iteration. Equation (11) is solved the same way as the Stokes equations using our multigrid solver, after which \(\lambda\) is substituted to (10) to get the sensitivity. The objective, J, and matrix, \(\mathcal {A}\), are linear functions of \(\alpha (\rho )\), and their partial derivatives with respect to \(\rho\) can be obtained using the chain rule by differentiating (2). Optionally, the adjoint can also be obtained by using automatic differentiation (Griewank and Walther 2008) for the state equations. Overall, this adjoint formulation is an efficient way to evaluate the sensitivity, especially when the number of design variables is large.

Once the sensitivity is computed, it is used to update the fluid design density variable, \(\rho\), of each grid cell. The updated density value is restricted to a range between 0 and 1.

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Liu, J., Xian, Z., Zhou, Y. et al. A marker-and-cell method for large-scale flow-based topology optimization on GPU. Struct Multidisc Optim 65, 125 (2022). https://doi.org/10.1007/s00158-022-03214-z

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