An approximate block factorization preconditioner for mixed-dimensional beam-solid interaction

Abstract

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.