Shape optimization of a Helmholtz resonator using an adjoint method

Abstract

This paper proposes a method for shape optimization in aero-acoustics and applies it to a Helmholtz resonator. The objective is to realize a desired acoustic impedance by optimizing the shape of the neck of the resonator, in due consideration of the excitation level. The optimization problem is formulated with a suitable objective functional, where the Navier–Stokes equations act as a partial differential equation (PDE) constraint in a Lagrangian functional. By exploiting the understanding of the relevant flow physics, it is possible to formulate the objective functional in the time domain, although the optimization target, i.e. the acoustic impedance, is a quantity defined in the frequency domain. This optimization problem is solved by a gradient-based optimization. The shape gradient of the objective functional is determined by an adjoint method, which requires solving two sets of PDEs in time: the so-called forward and backward problems. The forward problem is represented by the Navier–Stokes equations and is solved in the positive time direction. The set of equations for the backward problem, which has to be solved in the negative time direction, is derived in the current study. From the solutions of the forward and backward problems, the shape derivative for the current optimization step is calculated. Iterative optimization steps then bring the impedance to the target value.