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Relating phase-field and perimeter based structural optimization of variational inequalities

The paper is concerned with the analytical aspects of the structural optimization problems for elastic bodies in the unilateral contact with a given friction. The contact phenomenon is governed by the second order elliptic variational inequality. The aim of the topology optimization problem is to find such material density distribution function in a domain occupied by the body to minimize its normal contact stress. The original optimization problem is regularized in terms of the phase field model using Ginzburg-Landau free energy functional rather than the domain perimeter functional. In this model the width of the interfaces between the domain material phases is dependent on a small parameter. Therefore the goal of this paper is to investigate the relation between the phase field and the perimeter functionals regularized optimization problems as the width interface parameter tends to zero. Using the shape sensitivity approach the first order necessary optimality conditions for these two optimization problems are formulated. As the interface width parameter tends to zero the convergence of the first order necessary optimality conditions for the phase field regularized optimization problem to the first order necessary optimality conditions for the perimeter regularized optimization problem obtained in the framework of shape calculus is shown. Details of numerical implementation as well as numerical examples are provided and discussed.