Spectral analysis of the finite element matrices approximating 2D linearly elastic structures and multigrid proposals

Topology optimization aims to find the best material layout subject to given constraints. The so-called material distribution methods cast the governing equation as an extended or fictitious domain problem, in which a coefficient field represents the design. When solving the governing equation using the finite element method, a large number of elements are used to discretize the design domain, and an element-wise constant function approximates the coefficient field in the considered design domain. This article presents a spectral analysis of the (large) coefficient matrices associated with the linear systems stemming from the finite element discretization of a linearly elastic problem for an arbitrary coefficient field. Based on the spectral information, we design a multigrid method which turns out to be optimal, in the sense that the (arithmetic) cost for solving the related linear systems, up to a fixed desired accuracy, is proportional to the matrix-vector cost, which is linear in the corresponding matrix size. The method is tested, and the numerical results are very satisfactory in terms of linear cost and number of iterations, which is bounded by a constant independent of the matrix size.