Topology optimization of turbulent fluid flow via the TOBS method and a geometry trimming procedure

Abstract

One of the current challenges for topology optimization methods is the consideration of high Reynolds fluid flow analysis, especially including turbulence models. The issues in current pseudo-density-based methods are threefold. The fluid boundaries are unknown during optimization, the convergence to $\left\{ 0,1\right\}$ designs might be highly dependent on the tuning of the optimization parameters and it is difficult to specify the maximum value of the inverse permeability to avoid the presence of fluid flowing inside the modeled solid medium. This paper proposes a methodology to tackle these three problems. The Topology Optimization of Binary Structures (TOBS) method and a geometry trimming procedure are employed to create the TOBS-GT method. This method uses a binary $\left\{ 0,1\right\}$ design variable, which naturally creates explicit fluid boundaries during optimization and avoids the need for tuning the material model interpolation parameters. The geometry trimming procedure removes the solid regions and create a CAD model with only the fluid analysis domain and smooth walls. Since there is no solid region inside the analysis mesh, the problem of having fluid flowing through a solid region is avoided. The k-$\varepsilon$ and k-$\omega$ turbulence models are chosen to illustrate that the method may be applied to any turbulence model. The equilibrium equations are solved using the finite element method. The total fluid energy dissipation is minimized considering a fluid volume constraint. Numerical results show that the TOBS-GT method is well-fitted for topology optimization of turbulent fluid flow problems.

The sensitivity analysis

The present study adopts the adjoint sensitivity analysis via automatic differentiation to compute the sensitivity values of the fluid energy dissipation function. To verify the accuracy of this computation, the sensitivities of the pipe-bend example from Fig. 9 are compared to the finite difference method. The parameters and properties used are the same as for the pipe-bend example, except that $\kappa _{\max} = 10^{6}$ kg/(m³ s) and Re = 2000. Fig. 25a presents the 100$\times$100 finite element mesh used. In the current computational set up, the density variables are defined at the nodes of the mesh. All densities are defined as $\alpha = 1$. Figure 25b presents the volume-averaged adjoint sensitivity field computed via automatic differentiation, in 1/m$^2$. Figure 25c presents the local fluid energy dissipation $(\mu + \mu _T) (\nabla {\varvec{v}} + (\nabla {\varvec{v}})^T)\cdot (\nabla {\varvec{v}} + (\nabla {\varvec{v}})^T)$, in W/m, clipped at 1 for better visualization. Some points inside low and high dissipation regions are chosen to compare the sensitivities against finite differences. These points, the perturbation, the computed sensitivities via the adjoint (AD) and the finite difference (FD) method and the relative errors are presented in Table 1. The errors showed to be between 0.5$\%$ and 8.5$\%$. Eventually, some of the errors might be caused by some terms that are ignored in the computation of the Jacobian matrix for the adjoint problem, so-called “nojac” terms by the automatic differentiation module used. Otherwise, the tolerance on the residual of the governing equations by the FEA solver can also increase the error. Anyway, in general, the sensitivities showed throughout this work to provide good directions to minimize the objective function, verified by some cross-check analyses.

Table 1 Comparison between the obtained adjoint sensitivities against finite differences

Fig. 25

Sensitivity analysis of the pipe-bend example