# Binary Level Set Method for Topology Optimization of Variational Inequalities

Andrzej Myśliński^{*}

The paper deals with topology optimization of systems governed by variational inequalities. Contact phenomenon between the surfaces of the elastic or hyperelastic bodies described by the elliptic or hyperbolic inequality belongs, among others, to the class of such systems. The structural optimization problem for a contact problem is formulated. Shape and/or topological derivatives formulae of the cost functional of this problem are provided using material derivative and asymptotic expansion methods, respectively. These derivatives are employed to formulate necessary optimality condition as well as to determine the descent direction in the numerical algorithm. \newline Recently, the level set based numerical algorithms combined with the shape or topology sensitivity analysis has become the frequently used tool to solve topology optimization problems. The classical level set method, based on Hamilton – Jacobi equation, is a powerful scheme for representing the moving boundary. Under this level set framework, the boundary models can be easily changed without specification of the structure topology during the optimization process. \newline The paper is concerned with the analysis and numerical solution of the topology optimization of system governed by the variational inequalities using the binary rather than classical level set method. In this approach the interface between subdomains is implicitly represented by the discontinuity of the set of basis functions. In the paper the original topology optimization problem is approximated by a two-phase optimization problem. Different phases in a whole design domain are represented by a one level set function only. This function takes values either $+1$ or $-1$ in each subdomain of the computational domain. Each subdomain is characterized by its basis function depending on this level set function. Using this approach the original optimization problem is reformulated as equivalent constrained optimization problem with respect to this level set function. Necessary optimality condition is formulated. The optimization problem is solved numerically using the projected Lagrangian method. Numerical examples are provided and discussed.

Mathematics Subject Classification: 65K10 35J86 49Q10 74P10

Keywords: topology optimization; variational inequalities; level set method; projection Lagrangian method

Minisymposion: Material and Topology Optimization: Theory, Methods and Applications