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：我们提出一种面向固体力学有限元分析的机器学习策略，其中使用数据驱动代理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方法（FEM）解析的"外部"粗尺度域和"内部"细尺度域，然后为内部域对外部域的影响建立机器学习模型。实质上，对于固体力学问题，我们的机器学习代理模型实现了内部域自由度的静态凝聚。这是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对外部域贡献的力的映射来实现的。我们考虑两种映射：一种是无约束条件下直接从位移到力的直接映射；另一种是通过学习对称半正定（SPSD）刚度矩阵实现从位移到力的映射。我们通过简化算例证明，学习SPSD刚度矩阵能够确保粗尺度问题适定且存在唯一解。我们针对从立方体有限变形到含接触的紧固件-衬套几何有限变形等多个算例开展了数值实验。结果表明，强制施加SPSD刚度矩阵约束对于实现精确的FEM-ML耦合仿真至关重要，且所得方法能够准确表征样本外加载构型，相较于标准有限元仿真获得显著加速。