A fast-converging and asymptotic-preserving method for adjoint shape optimization of rarefied gas flows

TL;DR

This work develops a fast-converging, asymptotic-preserving adjoint-based optimization framework for three-dimensional rarefied gas flows by extending the general synthetic iterative scheme (GSIS) to the adjoint kinetic equation. It couples BGK kinetic modeling with accurate boundary sensitivities, free-form deformation (FFD) parameterization, and wall-distance-weighted spring smoothing to enable robust gradient-based drag minimization across all Knudsen numbers. The method achieves converged primal and adjoint solutions with only a few dozen velocity-distribution updates and delivers substantial drag reductions (e.g., $34.5\%$ at Kn=$0.1$ and $61.1\%$ at Kn=$0.01$) within roughly ten optimization iterations. This yields a practical, scalable tool for design in atmospheric re-entry, vacuum systems, and micro-electromechanical devices, where rarefied-gas effects are essential.

Abstract

Adjoint based shape optimization is a powerful technique in fluid-dynamics optimization, capable of identifying an optimal shape within only dozens of design iterations. However, when extended to rarefied gas flows, the computational cost becomes enormous because both the six dimensional primal and adjoint Boltzmann equations must be solved for each candidate shape. Building on the general synthetic iterative scheme (GSIS) for solving the primal Boltzmann model equation, this paper presents a fast converging and asymptotic preserving method for solving the adjoint kinetic equation. The GSIS accelerates the convergence of the adjoint kinetic equation by incorporating solutions of macroscopic synthetic equations, whose constitutive relations include the Newtonian stress law along with higher order terms capturing rarefaction effects. As a result, the method achieves asymptotic preservation (allowing the use of large spatial cell sizes in the continuum limit) while maintaining accuracy in highly rarefied regimes. Numerical tests demonstrate exceptional performance on drag minimization problems for 3D bodies, achieving drag reductions of 34.5% in the transition regime and 61.1% in the slip-flow regime within roughly ten optimization iterations. For each candidate shape, converged solutions of the primal and adjoint Boltzmann equation are obtained with only a few dozen updates of the velocity distribution function, dramatically reducing computational cost compared with conventional methods.

Figures (8)

• Figure 1: General procedure for optimizing rarefied gas flows.
• Figure 2: Schematic for the GSIS algorithm for the adjoint equation. First, the intermediate velocity distribution function is obtained by solving Eq. \ref{['eqn:implicit_kinetic0']}, when $n+1$ is replaced by $n+1/2$. Second, with the macroscopic adjoint variables and high-order term, the adjoint-NS-based equation \ref{['eq:adjoint_macro_discreted']} is solved to the steady state (or with maximum 1000 iterations of $m$), to get the prediction of adjoint macroscopic quantities $\hat{\bm{W}}$ at the $(n+1)$-th step. Finally, the adjoint distribution function is updated as per Eq. \ref{['adjoint_vdf_update']}.
• Figure 3: (Left) Schematic of flow simulation around a sphere with its center at the origin (0, 0, 0). (Right) Cross-section of the hybrid computational mesh on the plane $x=0$, consisting of prismatic layers near the inner spherical wall and tetrahedral elements in the outer region.
• Figure 4: The sensitivity of the wall drag force to the $z$-coordinate of the FFD control points ($Q_x = Q_y = -0.114286$) in the numerical simulation of flow around a sphere at Ma = 2, with Kn=0.1 (left) and 0.01 (right).
• Figure 5: The variation of $S_{\rho_{\infty}}$ with iteration steps for the adjoint equations using the conventional DVM and GSIS methods, with Kn=0.1 (left) and 0.01 (right). The first five steps in the GSIS method involve the initial field evolution using the conventional DVM.