Robust Block Preconditioning for Embedded Beam–Solid Coupling

4 Juni 2026

Our most recent preprint by Max Firmbach, Ivo Steinbrecher, Alexander Popp, and Matthias Mayr introduces advanced preconditioning techniques for mixed-dimensional structural mechanics problems and evaluates their performance in large-scale simulations.

This work introduces modified augmented Lagrangian block preconditioning for the mixed-dimensional coupling of three-dimensional solids and embedded one-dimensional Kirchhoff–Love beams. By combining a augmented Lagrangian formulation of the coupled saddle-point system and thus exact constraint enforcement with efficient block-triangular preconditioning and tailored Schur complement approximations, the approach delivers robust preconditioner performance across a wide range of model and discretization parameters. Numerical studies demonstrate near mesh-independent convergence and favorable parallel scalability, making the method well suited for large-scale structural mechanics simulations.

Firmbach, M., Steinbrecher, I., Popp, A., Mayr, M. (2026): Modified augmented Lagrangian preconditioning for mixed-dimensional beam-solid coupling. Preprint, submitted for publication. arXiv