High-order Multiscale Preconditioner for Elasticity of Arbitrary Structures

We present a two-level preconditioner for solving linear systems arising from the discretization of the elliptic, linear-elastic deformation equation, in displacement unknowns, over domains that have arbitrary geometric and topological complexity and heterogeneity in material properties (including fractures). The preconditioner is an algebraic translation of the high-order pore-level multiscale method (hPLMM) proposed recently by the authors, wherein a domain is decomposed into non-overlapping subdomains, and local basis functions are numerically computed over the subdomains to construct a high-quality coarse space (or prolongation matrix). The term "high-order" stands in contrast to the recent low-order PLMM preconditioner, where BCs of local basis problems assume rigidity of all interfaces shared between subdomains. In hPLMM, interfaces are allowed to deform, through the use of suitable mortar spaces, thereby capturing local bending/twisting moments under challenging loading conditions. Benchmarked across a wide range of complex (porous) structures and material heterogeneities, we find hPLMM exhibits superior performance in Krylov solvers than PLMM, as well as state-of-the-art Schwarz and multigrid preconditioners. Applications include risk analysis of subsurface CO2/H2 storage and optimizing porous materials for batteries, prosthetics, and aircraft.