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.

中文翻译：

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使用伴随方法的亥姆霍兹共振器的形状优化

本文提出了一种航空声学中形状优化的方法，并将其应用于亥姆霍兹共振器。目的是在考虑到激励水平的情况下，通过优化谐振器的颈部形状来实现所需的声阻抗。优化问题由合适的目标函数表述，其中Navier–Stokes方程在Lagrangian函数中充当偏微分方程（PDE）约束。通过利用对相关流动物理学的理解，尽管优化目标（即声阻抗）是在频域中定义的量，但是可以在时域中制定目标函数。通过基于梯度的优化解决了该优化问题。目标函数的形状梯度是通过伴随方法确定的，这需要及时解决两组PDE：所谓的前向和后向问题。正向问题由Navier–Stokes方程表示，并在正向时间方向上得到解决。在当前研究中，得出了必须在负时间方向上求解的，用于向后问题的方程组。根据向前和向后问题的解决方案，计算出当前优化步骤的形状导数。然后，迭代优化步骤使阻抗达到目标值。是根据当前研究得出的。根据向前和向后问题的解决方案，计算出当前优化步骤的形状导数。然后，迭代优化步骤使阻抗达到目标值。是根据当前研究得出的。根据向前和向后问题的解决方案，计算出当前优化步骤的形状导数。然后，迭代优化步骤使阻抗达到目标值。