Geometric Gradient Flows for Interface Optimization in Transmission and Contact Problems

Source abstract

ABSTRACT This paper investigates the evolution of interfaces in two‐phase systems governed by elliptic transmission problems and linear elasticity. In the first part, we address a shape optimization problem in which an internal interface is identified by minimizing the Kohn–Vogelius functional, comparing solutions arising from different boundary conditions. The interface evolves according to a geometric gradient flow that combines the shape derivative of the functional with curvature‐based regularization. In the second part, we extend the framework to elastic contact problems with Signorini‐type boundary conditions. Using material derivatives, we derive the shape derivative of the compliance functional and develop a regularized gradient flow governing the interface motion. The application of a gradient‐flow‐based strategy to contact problems in elasticity provides a new and efficient approach to this significant shape and topology optimization setting. Numerical experiments for both transmission and elasticity models demonstrate the stability, efficiency, and accuracy of the proposed method in recovering optimal interface geometries under a variety of conditions.