不确定条件下超材料声学隐身衣的最优设计
Optimal design of acoustic metamaterial cloaks under uncertainty
摘要：
在这项工作中，我们考虑的问题的最优设计的声学斗篷下的不确定性和发展的可扩展的近似和最优化方法来解决这个问题。设计变量被视为代表材料特性的无限维空间变化场，而附加的无限维随机场代表材料特性的可变性或制造误差。这种最优设计问题的离散化导致高维设计变量和不确定参数。为了解决这个问题，我们开发了一个计算方法的基础上泰勒近似和近似牛顿法的优化，这是基于一个海森导出的平均值的随机场。我们表明，我们的方法是可扩展的设计变量和不确定参数的尺寸方面，在这个意义上，声波传播的必要数量基本上是独立于这些尺寸，高达一百万个设计变量和半百万个不确定参数的数值实验。我们证明，使用我们的计算方法，一个最优设计的声学斗篷，是强大的材料的不确定性是在一个易于处理的方式实现。针对单方向单频入射波和多方向多频入射波共同作用下的经典圆形障碍物，提出并求解了不确定性条件下的最优设计问题.最后，我们将该方法应用到一个确定性的大规模最优隐身问题与复杂的几何形状，证明近似牛顿方法的海森计算是可行的大型，复杂的问题。
英文摘要：
In this work, we consider the problem of optimal design of an acoustic cloak under uncertainty and develop scalable approximation and optimization methods to solve this problem. The design variable is taken as an infinite-dimensional spatially-varying field that represents the material property, while an additive infinite-dimensional random field represents the variability of the material property or the manufacturing error. Discretization of this optimal design problem results in high-dimensional design variables and uncertain parameters. To solve this problem, we develop a computational approach based on a Taylor approximation and an approximate Newton method for optimization, which is based on a Hessian derived at the mean of the random field. We show our approach is scalable with respect to the dimension of both the design variables and uncertain parameters, in the sense that the necessary number of acoustic wave propagations is essentially independent of these dimensions, for numerical experiments with up to one million design variables and half a million uncertain parameters. We demonstrate that, using our computational approach, an optimal design of the acoustic cloak that is robust to material uncertainty is achieved in a tractable manner. The optimal design under uncertainty problem is posed and solved for the classical circular obstacle surrounded by a ring-shaped cloaking region, subjected to both a single-direction single-frequency incident wave and multiple-direction multiple-frequency incident waves. Finally, we apply the method to a deterministic large-scale optimal cloaking problem with complex geometry, to demonstrate that the approximate Newton method's Hessian computation is viable for large, complex problems.
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