This paper presents a scalable approximate block factorization 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 × 2 block system, an approximate Block-LU 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 and correction step of the arising Block-LU preconditioner. The Schur complement equation is tackled with 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 Schur equation 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. In addition, the robustness of the proposed method regarding material parameters and geometric properties is shown, before the preconditioner is finally applied for the analysis of steel-reinforced concrete structures in civil engineering.

中文翻译：

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混合维梁-固体相互作用的近似分块分解预处理器


本文提出了一种用于梁-固体相互作用混合维模型的可扩展近似块分解预处理器及其在工程中的应用。特别是，它研究了由规则化迫击炮型方法产生的线性系统，用于将几何精确的梁嵌入到固体连续体中。由于所产生的 2 × 2 块系统缺乏块对角优势，因此使用近似的 Block-LU 预处理器。它利用波束子块的稀疏结构来构造稀疏近似逆，然后该稀疏近似逆不仅用于显式地形成 Schur 补的近似，而且还可以在出现的块的预测和校正步骤中充当平滑器-LU 预处理器。 Schur 补方程是用代数多重网格方法解决的。尽管目前仅通过单级方法来处理梁子块，但计算要求较高的 Schur 方程的多级性质在实践中提供了可扩展的预处理器。在数值测试案例中，研究了不同算法参数对稀疏近似逆的质量的影响，并证明了所提出的预处理器在高达 1000 MPI 等级上的弱缩放行为。此外，在预处理器最终应用于土木工程中的钢筋混凝土结构分析之前，显示了所提出的方法在材料参数和几何特性方面的鲁棒性。