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

TL;DR

This work addresses the computational challenge of solving large linear systems from mixed-dimensional beam–solid coupling with penalty-regularized mortar embeddings. It introduces a physics-based multi-level block preconditioner that combines an explicit sparse inverse of the beam block with an AMG-treated Schur-complement solver, using SPAI as a smoothing operator in predictor and corrector steps. The approach delivers robust convergence and weak scalability up to 1000 ranks, and demonstrates substantial speedups over direct solvers in engineering-relevant problems such as steel-reinforced concrete walls, while maintaining stability across varying beam radii and stiffness ratios. The methodology enables efficient analysis of large-scale fiber–solid systems and is implemented in Trilinos for public use.

Abstract

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.

Figures (10)

• Figure 1: Spectrum of modeling techniques for fibers embedded into three-dimensional solids Steinbrecher2020a
• Figure 2: Optional (gray) and mandatory (orange) steps of the computation and its flow of information with computed data (in boxes) and user parameters (in circles)
• Figure 3: Geometry and setup for the numerical study of the sparse approximate inverse calculation: a solid cube with edge length $l^{\mathcal{S}} = \qty{1}{\meter}$ is randomly filled with fibers of the same length $l^{\mathcal{B}} = \qty{0.25}{\meter}$, clamped at its bottom and loaded with a distributed external load $q=\qty{1}{\newton/\square\meter}$.
• Figure 4: Partial visualization of the sparsity structure $\mathcal{J}\left(\mathbf{A}\right)$ of matrix $\mathbf{A}$ for test cases I--III
• Figure 5: Intersections of all cubes (IDs 1--9) of the weak scaling studies with planes spanned by basis vectors $\boldsymbol{e}_{\xi}$ and $\boldsymbol{e}_{\eta}$, $\xi,\eta\in\{1,2,3\}, \xi\neq \eta$ in the cartesian frame of reference. Orientation of the cutting planes is sketched in the top left.

Theorems & Definitions (2)

• Definition 3.1: Block diagonal dominance
• Remark 4.1: Challenges for multilevel methods