Parallel solution of contact shape optimization problems with Coulomb friction based on domain decomposition

摘要

We shall first briefly review the FETI based domain decomposition methodology adapted to the solution of multibody contact problems in 3D with Coulomb friction. These problems play a role of the state problem in contact shape optimization problems with Coulomb friction. We use a modification of FETI that we call Total FETI, which imposes not only the compatibility of a solution across the subdomain interfaces, but also the prescribed displacements. For solving a state problem we use the method of successive approximations. Each iterative step of the method requires us to solve the contact problem with Tresca friction. The discretized problem with Coulomb friction has a unique solution for small coefficients of friction. The uniqueness of the equilibria for fixed controls enables us to apply the so-called implicit programming approach. Its main idea consists in minimization of a nonsmooth composite function generated by the objective and the control-state mapping. The implicit programming approach combined with the differential calculus of Clarke was used for a discretized problem of 2D shape optimization. There is no possibility to extend the same approach to the 3D case. The main problem is the nonpoly hedral character of the second-order cone, arising in the 3D model. To get subgradient information needed in the used numerical method we use the differential calculus of Mordukhovich. Application of the Total FETI method to the solution of the state problem and sensitivity analysis allows massively parallel solution of these problems and stable identification of rigid body modes which are a priori known. The effectiveness of our approach is demonstrated by numerical experiments.