We propose a novel robust topology optimization for designing acoustic devices that are effective for broadband sound waves. Here, we define the objective function as the weighted sum of the acoustic response to an incident wave consisting of a single frequency and its standard deviation (SD) against the frequency perturbation. By approximating the SD, under the assumption that the incident frequency follows the normal distribution, with the high‐order Taylor expansion of the (conventional) objective function, we deal with significant frequency variations. To calculate such an approximation, we need to calculate the high‐order frequency derivatives of the state variable. Here, we define them by integral representations, which enables us to characterize them even when the state variable is defined in an unbounded domain as is often the case with wave scattering problems. We further show that, based on this definition, the high‐order derivatives can efficiently be computed by a combination of the boundary element method and automatic differentiation. We also present a derivation and calculation of the topological derivative for the newly defined objective function. We install all the proposed techniques into a topology optimization method based on the level‐set method to design wideband acoustic devices. The validity and effectiveness of the proposed topology optimization are confirmed through several numerical examples.

中文翻译：

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强大的拓扑优化，可扩大声学设备的工作带宽

我们提出了一种新颖的鲁棒拓扑优化，用于设计对宽带声波有效的声学设备。在这里，我们将目标函数定义为对入射波的声学响应的加权总和，该入射波由单个频率及其相对于频率扰动的标准偏差（SD）组成。通过假设入射频率服从正态分布，并利用（常规）目标函数的高阶泰勒展开式，对SD进行近似计算，我们可以处理显着的频率变化。要计算这样的近似值，我们需要计算状态变量的高阶频率导数。在这里，我们通过积分表示来定义它们，即使状态变量是在无界域中定义的，这也使我们能够表征它们的特征，这是波散射问题经常发生的情况。我们进一步表明，基于此定义，可以通过边界元素方法和自动微分的组合有效地计算高阶导数。我们还提出了新定义的目标函数的拓扑导数的推导和计算。我们将所有提出的技术安装到基于水平集方法的拓扑优化方法中，以设计宽带声学设备。通过几个数值例子验证了所提出的拓扑优化的有效性和有效性。结合边界元法和自动微分法可以有效地计算出高阶导数。我们还提出了新定义的目标函数的拓扑导数的推导和计算。我们将所有提出的技术安装到基于水平集方法的拓扑优化方法中，以设计宽带声学设备。通过几个数值例子验证了所提出的拓扑优化的有效性和有效性。结合边界元法和自动微分法可以有效地计算出高阶导数。我们还提出了新定义的目标函数的拓扑导数的推导和计算。我们将所有提出的技术安装到基于水平集方法的拓扑优化方法中，以设计宽带声学设备。通过几个数值例子验证了所提出的拓扑优化的有效性和有效性。