  The goal of this work is to present a fast and viable approach for the
numerical solution of the high-contrast state problems arising in topology
optimization. The optimization process is iterative, and the gradients are
obtained by an adjoint analysis, which requires the numerical solution of large
high-contrast linear elastic problems with features spanning several length
scales. The size of the discretized problems forces the utilization of
iterative linear solvers with solution time dependant on the quality of the
preconditioner. The lack of clear separation between the scales, as well as the
high-contrast, imposes severe challenges on the standard preconditioning
techniques. Thus, here we propose new methods for the high-contrast elasticity
equation with performance independent of the high-contrast and the multi-scale
structure of the elasticity problem. The solvers are based on two-levels domain
decomposition techniques with a carefully constructed coarse level to deal with
the high-contrast and multi-scale nature of the problem. The construction
utilizes spectral equivalence between scalar diffusion and each displacement
block of the elasticity problems and, in contrast to previous solutions
proposed in the literature, is able to select the appropriate dimension of the
coarse space automatically. The new methods inherit the advantages of domain
decomposition techniques, such as easy parallelization and scalability. The
presented numerical experiments demonstrate the excellent performance of the
proposed methods.

Numerical Analysis;Numerical Analysis;Optimization and Control