  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.
