Keywords: contact problem, finite element method, boundary element method, domain decomposition, parallel programming.

We proposed two efficient algorithms for the solution such problems. The first one is a variant of finite element tearing and interconnecting (FETI) domain decomposition method, which is based on finite element approximation of the state problem. The second one is based on the boundary element approximation and it is known as boundary element tearing and interconnecting (BETI). Their 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 that is solved iteratively. 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 algorithms enjoy parallel scalability. Observing that the equality constraints may be used to define so called "natural coarse grid", it was possible to prove also its numerical scalability, i.e. asymptotically linear complexity.

The main difference between these two approaches is in the discretization techniques. While for the FETI method a discretization mesh for the whole bodies in the state contact problems must be built, for the BETI method the mesh is only required on the boundaries. This plays very important role in the case of contact shape optimization, because the finite element mesh has to be rebuilt many times during the optimization process and is shown to be computationally very expensive part.

In this paper, we exploit the parallel implementation of the above mentioned algorithms for contact problems to the minimization of the compliance of the system of elastic bodies subject to the volume constraint and some additional constraints. These algorithms are used for efficient solution of the state contact problem as well as efficient tools for sensitivity analysis [1]. The efficiency of the FETI and BETI methods is demonstrated on a model Hertz problem.

1

V. Vondrák, T. Kozubek, A. Markopoulos, Z. Dostál, "Parallel solution of contact shape optimization problems based on Total FETI domain decomposition method", Structural and Multidisciplinary Optimization, 42(6), 955-964, 2010. doi:10.1007/s00158-010-0537-3

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