In spite of its effectiveness in structural design, topology optimization is not yet largely widespread in industry, the principal reason is that the topology optimization problem is a large scale optimization problem; it is characterized by a very significant number of design variables, which amplifies the difficulty of its resolution. It is common to introduce 1000 to 10000 design variables to solve a real problem, thus the computation time is typically very high since the problem requires the repeated solution of finite element analysis of the equilibrium equations.

However, during the last two decades, the parallel computers knew a great evolution, in particular in computing power and storage capacity [2]. Domain decomposition methods (called also subdomains methods) are a valuable approach when solving partial differential equation problems on parallel computers. Any domain decomposition method is based on the assumption that the given computational domain is partitioned into sub-domains which may or may not overlap. Next, the original problem can be reformulated upon each sub-domain, yielding a family of subproblems of reduced size, that are coupled one to another through the values of the unknown solution at sub-domain interfaces.

Reviewing the literature, it seems that the application of parallel computing in topology optimization is rare, and devoted only to the discrete case [3], there is no mathematical formulation of the topology optimization problem in the continuum case. Thus, The main objective of the present work is to propose a new mathematical formulation of the minimum compliance problem of an isotropic linear elastic structure based on domain decomposition methods when the design domain is partitioned into two non-overlapping sub-domains, the domain decomposition method for the problem of linear elasticity is then based on a constrained minimization problem for which the objective functional measures the jumps in the solution across the interface between sub-domains, the constraints are the partial differential equations.

We determine the unknown data through an optimization problem in which the discrepancy between an appropriate defined functional of the difference between solutions in sub-domains is minimized.

This article presents a new formulation of the minimum compliance problem based on the domain decomposition methods then we prove the equivalence of the two problems. We propose a numerical method to solve the problem after the finite element approximations and our numerical results prove the efficiency of our approach.

1

O. Sigmund, "On the design of compliant mechanisms using topology optimization", Mech. Struct. Mach., 25, 495-526, 1997. doi:10.1080/08905459708945415

2

B. Radi and J.-F Estrade, "The ARPEGE model: Some strategies and performances", Parallel Computing, 24, 1167-1175, 1998.

3

K. Vemaganti and W.E. Lawrence, "Parallel methods for topology optimization", Elsevier science, 2004.

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