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We present a machine-learning strategy for finite element analysis of solidmechanics wherein we replace complex portions of a computational domain with adata-driven surrogate. In the proposed strategy, we decompose a computationaldomain into an "outer" coarse-scale domain that we resolve using a finiteelement method (FEM) and an "inner" fine-scale domain. We then develop amachine-learned (ML) model for the impact of the inner domain on the outerdomain. In essence, for solid mechanics, our machine-learned surrogate performsstatic condensation of the inner domain degrees of freedom. This is achieved bylearning the map from (virtual) displacements on the inner-outer domaininterface boundary to forces contributed by the inner domain to the outerdomain on the same interface boundary. We consider two such mappings, one thatdirectly maps from displacements to forces without constraints, and one thatmaps from displacements to forces by virtue of learning a symmetric positivesemi-definite (SPSD) stiffness matrix. We demonstrate, in a simplified setting,that learning an SPSD stiffness matrix results in a coarse-scale problem thatis well-posed with a unique solution. We present numerical experiments onseveral exemplars, ranging from finite deformations of a cube to finitedeformations with contact of a fastener-bushing geometry. We demonstrate thatenforcing an SPSD stiffness matrix is critical for accurate FEM-ML coupledsimulations, and that the resulting methods can accurately characterizeout-of-sample loading configurations with significant speedups over thestandard FEM simulations.

Embedded symmetric positive semi-definite machine-learned elements for  reduced-order modeling in finite-element simulations with application to  threaded fasteners