Modified augmented Lagrangian preconditioning for mixed-dimensional beam-solid coupling

2026-06-04

AI Summary

This work addresses the challenge of solving indefinite saddle-point systems arising from the coupling of three-dimensional solids with embedded one-dimensional Kirchhoff–Love beams. The authors propose a block triangular preconditioner framework based on the augmented Lagrangian method, which enforces mixed-dimensional coupling constraints exactly and enables decoupled solution of the solid, beam, and constraint subsystems through an efficient approximation of the Schur complement. The resulting preconditioner achieves both high constraint accuracy and computational efficiency. Numerical experiments demonstrate that the iteration count is nearly independent of mesh refinement, robust with respect to material parameters, and exhibits strong and weak scalability. The approach has been successfully applied to engineering simulations such as composite sandwich panels.

Abstract

This paper presents modified augmented Lagrangian block preconditioners for the mixed-dimensional coupling of three-dimensional solid bodies with embedded one-dimensional torsion-free Kirchhoff-Love beams using Lagrange multipliers for constraint enforcement. The finite element discretization of this mixed formulation leads to an indefinite saddle-point system. An augmented Lagrangian formulation is employed to regularize the linear system while maintaining exact enforcement of the coupling constraints. Starting from the corresponding ideal augmented Lagrangian block preconditioner, more practical block-triangular variants are derived in which the solid, beam, and Schur complement blocks can be treated independently. In addition, different variants of Schur complement approximations are introduced. Numerical experiments demonstrate robustness with respect to model parameters, near mesh-independent iteration counts, and favorable strong and weak scalability. These results indicate the suitability of the proposed approach for large-scale simulations of mixed-dimensional models in solid and structural mechanics, as demonstrated by an engineering example involving a composite sandwich plate.

Problem

• mixed-dimensional coupling
• saddle-point system
• augmented Lagrangian
• Kirchhoff-Love beams
• constraint enforcement

Innovation

• augmented Lagrangian preconditioning
• mixed-dimensional coupling
• Kirchhoff-Love beams
• saddle-point systems
• Schur complement approximation