This paper presents modified augmented Lagrangian block preconditioners for the mixed-dimensional coupling of three-dimensional solid bodies with embedded one-dimensional torsion-free Kirchhoff-Love beams using Lagrange multipliers for constraint enforcement. The finite element discretization of this mixed formulation leads to an indefinite saddle-point system. An augmented Lagrangian formulation is employed to regularize the linear system while maintaining exact enforcement of the coupling constraints. Starting from the corresponding ideal augmented Lagrangian block preconditioner, more practical block-triangular variants are derived in which the solid, beam, and Schur complement blocks can be treated independently. In addition, different variants of Schur complement approximations are introduced. Numerical experiments demonstrate robustness with respect to model parameters, near mesh-independent iteration counts, and favorable strong and weak scalability. These results indicate the suitability of the proposed approach for large-scale simulations of mixed-dimensional models in solid and structural mechanics, as demonstrated by an engineering example involving a composite sandwich plate.

翻译：本文提出了一种修正的增广拉格朗日块预处理器，用于三维固体与嵌入其中且无扭转的基尔霍夫-洛夫梁的混合维度耦合问题，其中采用拉格朗日乘子实施约束。该混合公式的有限元离散会导致不定鞍点系统。采用增广拉格朗日公式对线性系统进行正则化处理，同时确保耦合约束的精确实施。从对应的理想增广拉格朗日块预处理器出发，推导出更为实用的块三角变体，其中固体、梁及舒尔补块可独立处理。此外，引入了不同形式的舒尔补近似。数值实验表明，该方法对模型参数具有鲁棒性，迭代次数近乎网格无关，并展现出良好的强扩展性与弱扩展性。这些结果验证了所提方法适用于固体与结构力学中混合维度模型的大规模仿真，并通过涵盖复合材料夹层板的工程实例得以证明。