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

TL;DR

This paper introduces a fast, asymptotic-preserving adjoint kinetic equation solver for rarefied gas flow shape optimization, significantly reducing computational costs and enabling efficient drag minimization in complex regimes.

Contribution

It presents a novel GSIS-based method for the adjoint Boltzmann equation that accelerates convergence and preserves asymptotic behavior, improving efficiency in shape optimization tasks.

Findings

Achieves 34.5% drag reduction in transition regime

Achieves 61.1% drag reduction in slip-flow regime

Reduces computational cost by requiring only a few dozen updates per shape

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.