This paper presents a scalable physics-based block preconditioner for mixed-dimensional models in beam-solid interaction and their application in engineering. In particular, it studies the linear systems arising from a regularized mortar-type approach for embedding geometrically exact beams into solid continua. Due to the lack of block diagonal dominance of the arising 2 x 2 block system, an approximate block factorization preconditioner is used. It exploits the sparsity structure of the beam sub-block to construct a sparse approximate inverse, which is then not only used to explicitly form an approximation of the Schur complement, but also acts as a smoother within the prediction step of the arising SIMPLE-type preconditioner. The correction step utilizes an algebraic multigrid method. Although, for now, the beam sub-block is tackled by a one-level method only, the multi-level nature of the computationally demanding correction step delivers a scalable preconditioner in practice. In numerical test cases, the influence of different algorithmic parameters on the quality of the sparse approximate inverse is studied and the weak scaling behavior of the proposed preconditioner on up to 1000 MPI ranks is demonstrated, before the proposed preconditioner is finally applied for the analysis of steel-reinforced concrete structures in civil engineering.

翻译：本文提出了一种可扩展的基于物理的分块预处理器，用于梁-固体相互作用中的混合维度模型及其工程应用。具体而言，研究了基于正则化mortar方法将几何精确梁嵌入固体连续体时产生的线性系统。针对所生成的2×2分块系统缺乏块对角占优性的特点，采用近似分块分解预处理器。该方法利用梁子块稀疏结构构造稀疏近似逆矩阵，该逆矩阵不仅用于显式构造舒尔补的近似形式，还作为SIMPLE型预处理器预测步骤中的平滑算子。校正步骤采用代数多重网格方法。尽管目前梁子块仅通过单层方法处理，但计算密集的校正步骤的多层特性在实践中提供了可扩展的预处理器。通过数值算例，研究了不同算法参数对稀疏近似逆质量的影响，并展示了所提预处理器在多达1000个MPI进程上的弱扩展性能。最后将该预处理器应用于土木工程中钢筋混凝土结构的分析。