论文标题
通过Kohn-Vogelius型功能检测空腔的相位场方法
A phase-field approach for detecting cavities via a Kohn-Vogelius type functional
论文作者
论文摘要
我们通过边界数据来处理在有界的各向同性介质中腔体形状重建形状重建的几何反问题。从最佳控制的角度来解决问题：目标是在Lipschitz域中最小化Kohn-Vogelius类型的功能，其外围正则化项，从而惩罚了要重建的腔体周围。为了在数值上解决优化问题，我们使用相位场方法，近似于Modica-Mortola弛豫的周长功能，并将腔体建模为具有很小的弹性张量的包含。我们提供了详细的分析，该分析通过一些数值实验，显示了算法的鲁棒性。
We deal with the geometrical inverse problem of the shape reconstruction of cavities in a bounded linear isotropic medium by means of boundary data. The problem is addressed from the point of view of optimal control: the goal is to minimize in the class of Lipschitz domains a Kohn-Vogelius type functional with a perimeter regularization term which penalizes the perimeter of the cavity to be reconstructed. To solve numerically the optimization problem, we use a phase-field approach, approximating the perimeter functional with a Modica-Mortola relaxation and modeling the cavity as an inclusion with a very small elastic tensor. We provide a detailed analysis showing the robustness of the algorithm through some numerical experiments.
