Keywords: contact shape optimization, parallel solvers, domain decomposition.

The contact shape optimization problem is one of the computationally most challenging problems. The reason is that not only the cost function is a nonlinear implicit function of the design variables, but that its evaluation requires also a solution of the highly nonlinear variational inequality which describes the equilibrium of a system of elastic bodies in mutual contact. Since the cost function must be evaluated many times in the solution process, it is obvious that the solution of the contact problem is a key ingredient of any effective algorithm for the solution of contact shape optimization problems.

The approach that we propose is based on the finite element tearing and interconnecting (FETI) domain decomposition method, which was originally proposed by Farhat and Roux [1] for parallel solving of the linear problems described by elliptic partial differential equations. Its key ingredient is decomposition of the spatial domain into non-overlapping subdomains that are "glued" by Lagrange multipliers, so that, after eliminating the primal variables, the original problem is reduced to a small, relatively well conditioned, typically equality constrained quadratic programming problem. The time that is necessary for both the elimination and iterations can be reduced nearly proportionally to the number of the processors, so that the algorithm enjoys parallel scalability. Observing that the equality constraints may be used to define so called "natural coarse grid", Farhat, Mandel and Roux [2] modified the basic FETI algorithm so that they were able to prove its numerical scalability, i.e. asymptotically linear complexity.

If the FETI procedure is applied to an elliptic variational inequality, the resulting quadratic programming problem has not only the equality constraints, but also the non-negativity constraints. Even though the latter is a considerable complication as compared with linear problems, it seems that the FETI procedure should be even more powerful for the solution of variational inequalities than for linear problems. The reason is that the FETI procedure not only reduces the original problem to a smaller and better conditioned one, but it also replaces at no cost all the inequalities by the bound constraints.

In this paper, we exploit the parallel implementation of our scalable algorithms for contact problem to the minimization of the the compliance of the system elastic bodies subject to the volume constraint and some additional constraints. Particularly, we applied our total FETI (TFETI, also all floating) method introduced independently in the thesis by Of and by Dostál et al. [3]. The efficiency of the implemented parallel algorithms will be demonstrated with respect to the compliance minimization of the Hertz problem.

1

C. Farhat, F.-X. Roux, "A method of finite element tearing and interconnecting and its parallel solution algorithm", Int. J. Numer. Methods Engng., 32, 1205-1227, 1991. doi:10.1002/nme.1620320604

2

C. Farhat, J. Mandel, F.-X. Roux, "Optimal convergence properties of the FETI domain decomposition method", Comp. Meth. Appl. Mech. Eng., 115, 365-385, 1994. doi:10.1016/0045-7825(94)90068-X

3

Z. Dostál, D. Horák, R. Kucera, "Total FETI - an easier implementable variant of the FETI method for numerical solution of elliptic PDE", Commun. Numer. Methods Eng., 22, 1155-1162, 2006. doi:10.1002/cnm.881

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