Topology optimization for lift–drag problems incorporated with distributed unstructured mesh adaptation

Abstract

This note introduces the distributed unstructured mesh adaptation into the fluid-related topology optimization which is a first step in that direction. We incorporate three different remeshing techniques (isotropic, anisotropic, or body-fitted adaptive mesh refinement) into the reaction–diffusion equation-based fluid topology optimization method. It requires a fully distributed framework (including scalable domain decomposition, matrix assembly, parallel interpolation, linear solver) that very few general purpose libraries offer. In addition, this note is the first attempt to conduct a comparative study by showcasing two different flow modeling strategies with their advantages and disadvantages. More specifically, the “separate” modeling, relying on the surface-capturing technique, i.e., body-fitted mesh, allows the disjoint reunion of a global mesh that contains several (fluid/solid) subdomains. The no-slip boundary conditions can be applied on the moving fluid–solid interface. The “hybrid” modeling, on the other hand, relying on the monolithic formulation, can be incorporated with iso-/anisotropic meshes. For comparison and for accessing the constructed framework, a lift–drag optimization problem and a classical minimal power dissipation problem are formulated. Various two- and three-dimensional numerical examples are presented to validate the computational efficiency of this framework.

Appendices

Appendix A: “Separate” flow modeling

The workflow of the “separate” flow modeling strategy is illustrated here. First, the fluid domain mesh ${\mathcal {T}}_\text {fluid, it}$ is truncated from the entire domain mesh ${\mathcal {T}}_\text {it}$ using the function trunc built in FreeFEM, as shown in Fig. 8a, b. Next, the forward and adjoint problems are successively solved on ${\mathcal {T}}_\text {fluid,it}$, as shown in Fig. 8c, d. Then, the sensitivity is computed on ${\mathcal {T}}_\text {it}$, as shown in Fig. 8e followed by updating the level-set function $\phi$ shown in Fig. 8f. After that, the mesh is evolved based on the newly updated level-set function and is body fitted to the zero level-set interface, as shown in Fig. 8g. Finally, ${\mathcal {T}}_\text {fluid,it+1}$ is extracted for the next iteration computation, as shown in Fig. 8h, i.

Fig. 8

Schematic of the “separate” modeling strategy

Appendix B: Minimal power dissipation problem

We present a test case for the classical minimal power dissipation problem to fully reflect the topological evolution capability of the proposed method. The optimization mathematical model is formulated as follows:

$$\begin{aligned}&\inf _{H_{\phi } \in {\mathcal {H}}} J(\Omega )=\int _{\partial \Omega _{\text {f}}^{D} \cup \partial \Omega _{\text {f}}^{N}}\left( p+\frac{1}{2}|\varvec{v}|^{2}\right) \left( -\varvec{v} \cdot \varvec{n}_{\text {f}}\right) d \Gamma, \end{aligned}$$

(B.1a)

$$\begin{aligned}&\text{ s.t. } \left\{ \begin{aligned}&G_{1}=\dfrac{\int _{D}{\left( H_{\phi } \right) }d\Omega }{\int _{D}{}d\Omega }-V_{\text {max}} = 0 \\&Eq.\,\,(4) \text { (Navier--Stokes equations)}. \end{aligned}\right. \end{aligned}.$$

(B.1b)

The adjoint equations and the sensitivity analysis can be referred to in Li et al. (2022a). In this test case, $\tau$ is set to $10^{-3}$ and $\Delta t$ is set to 0.1, and they are remained unchanged throughout the optimization. The “hybrid” flow modeling is used incorporated with the anisotropic adaptive mesh. The design model is shown in Fig. 9. The computational domain is the unit cubic $\Omega =[0;1]^3$. The radius of the in- and outlets are $\frac{1}{6}$ and $\frac{1}{12}$, respectively. The flow enters the domain from the inlet with a parabolic velocity profile $\varvec{v}_x = 2\left( 1-\left( 36y^2+36z^2\right) \right)$. The flow exits the domain from the four outlets with a zero normal stress. The rest walls bear a no-slip boundary condition. The Reynolds number is set to $Re=50$ and the maximum allowed volume fraction is set to 30%. Fig. 10 shows the snapshots of the channel layout during the intermediate steps. In this test case, the anisotropic mesh adaptation is used, as shown in Fig. 11.

Fig. 9

Design model for the 3D minimal power dissipation problem

Fig. 10

Topology evolution history of the 3D minimal power dissipation problem

Fig. 11

Cross-section view of the anisotropic mesh for the optimal configuration