A phase-field approach for detecting cavities via a Kohn–Vogelius type functional. (1st September 2022)

• Record Type: Journal Article
• Authors: Aspri, Andrea
• Is Part Of: Inverse problems. Volume 38:Number 9(2022)
• Publication Date: 2022-09-01
• Subjects: Kohn–Vogelius functional -- cavity -- phase-field -- linear elasticity -- primal dual active set method; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/ac82e4
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: 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.