We present a machine-learning strategy for finite element analysis of solid mechanics wherein we replace complex portions of a computational domain with a data-driven surrogate. In the proposed strategy, we decompose a computational domain into an "outer" coarse-scale domain that we resolve using a finite element method (FEM) and an "inner" fine-scale domain. We then develop a machine-learned (ML) model for the impact of the inner domain on the outer domain. In essence, for solid mechanics, our machine-learned surrogate performs static condensation of the inner domain degrees of freedom. This is achieved by learning the map from (virtual) displacements on the inner-outer domain interface boundary to forces contributed by the inner domain to the outer domain on the same interface boundary. We consider two such mappings, one that directly maps from displacements to forces without constraints, and one that maps from displacements to forces by virtue of learning a symmetric positive semi-definite (SPSD) stiffness matrix. We demonstrate, in a simplified setting, that learning an SPSD stiffness matrix results in a coarse-scale problem that is well-posed with a unique solution. We present numerical experiments on several exemplars, ranging from finite deformations of a cube to finite deformations with contact of a fastener-bushing geometry. We demonstrate that enforcing an SPSD stiffness matrix is critical for accurate FEM-ML coupled simulations, and that the resulting methods can accurately characterize out-of-sample loading configurations with significant speedups over the standard FEM simulations.

翻译：我们提出一种用于固体力学有限元分析的机器学习策略，其中用数据驱动替代模型替换计算域中的复杂部分。在该策略中，我们将计算域分解为采用有限元法（FEM）求解的"外部"粗尺度域和"内部"细尺度域，然后针对内部域对外部域的影响建立机器学习（ML）模型。本质上，在固体力学中，机器学习替代模型对内部域自由度执行静态凝聚，通过学习从内外域界面边界上的（虚拟）位移到内部域在同一界面上贡献给外部域的力的映射来实现。我们考虑两种映射方法：一种直接从位移映射到无约束力，另一种通过学习对称半正定（SPSD）刚度矩阵实现位移到力的映射。简化算例表明，学习SPSD刚度矩阵可使粗尺度问题具有唯一解的适定性。我们针对从立方体有限变形到带接触的紧固件-衬套几何体有限变形等多个示例开展数值实验，结果表明强制施加SPSD刚度矩阵对FEM-ML耦合模拟的精度至关重要，且该方法能准确表征样本外加载构型，相比标准FEM模拟获得显著加速。