Adjoint shape optimization of oscillatory rarefied gas flows
A fast-converging and asymptotic-preserving adjoint shape optimization method is proposed for drag reduction of multiscale gas flows in vibrating micro-electro-mechanical systems. The convergence of the Boltzmann kinetic equation is accelerated by macroscopic synthetic equations, whose constitutive relations integrate continuum-limit terms and high-order kinetic corrections to faithfully characterize spatiotemporal rarefaction effects. As such, this method maintains near-continuum limit consistency while retaining high kinetic accuracy in rarefied flow regimes. Fourier stability analysis performed in an infinite domain demonstrates that the present method yields a spectral radius below 0.5, indicating that the numerical deviation from the converged solution is halved per iteration. Numerical simulations are conducted on an oscillating cylinder and a comb-shaped resonator. The results verify the high accuracy of the derived adjoint sensitivities and the excellent drag reduction performance of the proposed method across various Knudsen and Strouhal numbers. Compared with conventional kinetic iteration methods, the present method produces convergent primal and adjoint solutions within dozens of iterations and features asymptotic preserving behavior, permitting spatial cell sizes far larger than the molecular mean free path. This facilitates efficient design of vibrating micro-electro-mechanical systems.