耗散介质中声波形状和拓扑optimization的相场方法
A phase-field approach to shape and topology optimization of acoustic waves in dissipative media
摘要：
本文研究了耗散介质中声透镜系统的最佳形状和拓扑结构问题。声传播由一般的半线性强阻尼波动方程控制。我们引入了一个相场制定这个问题，通过扩散之间的界面的透镜和周围的流体。由此产生的配方被证明是适定的，我们证明了相应的最优化问题有一个极小。通过分析约化目标泛函的性质和伴随问题的适定性，我们严格地得到了该问题的一阶最优性条件。此外，我们考虑了约化目标泛函的$Γ$ -极限，从而建立了扩散界面问题和边界正则化的尖锐界面形状最优化问题之间的关系。
英文摘要：
We investigate the problem of finding the optimal shape and topology of a system of acoustic lenses in a dissipative medium. The sound propagation is governed by a general semilinear strongly damped wave equation. We introduce a phase-field formulation of this problem through diffuse interfaces between the lenses and the surrounding fluid. The resulting formulation is shown to be well-posed and we prove that the corresponding optimization problem has a minimizer. By analyzing properties of the reduced objective functional and well-posedness of the adjoint problem, we rigorously derive first-order optimality conditions for this problem. Additionally, we consider the $Γ$-limit of the reduced objective functional and in this way establish a relation between the diffuse interface problem and a perimeter-regularized sharp interface shape optimization problem.
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