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块预处理方法，用于处理梁-固体相互作用中的混合维度模型及其在工程中的应用。具体而言，本研究针对将几何精确梁嵌入固体连续体的正则化砂浆型方法所产生的线性系统进行了分析。由于所产生的2×2块系统缺乏块对角优势，本文采用了一种近似块分解预处理方法。该方法利用梁子块的稀疏结构构建稀疏近似逆矩阵，该逆矩阵不仅用于显式构造Schur补的近似形式，还在所提出的SIMPLE型预处理方法的预测步骤中充当平滑算子。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所具有的多层特性在实践中实现了可扩展的预处理效果。在数值测试案例中，本文研究了不同算法参数对稀疏近似逆矩阵质量的影响，并在所提出的预处理方法最终应用于土木工程中钢筋混凝土结构分析之前，展示了该方法在多达1000个MPI进程上的弱扩展性能。