An iterative generalized Golub-Kahan algorithm for problems in structural mechanics

TL;DR

This paper introduces an iterative generalized Golub-Kahan algorithm tailored for saddle point systems in structural mechanics, demonstrating robust convergence and effectiveness in complex finite element models with constraints.

Contribution

It presents a systematic convergence analysis of a modified Golub-Kahan algorithm for challenging saddle point problems in structural engineering applications.

Findings

Exhibits excellent convergence with weak size dependence.

Robust performance on complex, constrained finite element models.

Effective in practical, difficult cases for iterative solvers.

Abstract

This paper studies the Craig variant of the Golub-Kahan bidiagonalization algorithm as an iterative solver for linear systems with saddle point structure. Such symmetric indefinite systems in 2x2 block form arise in many applications, but standard iterative solvers are often found to perform poorly on them and robust preconditioners may not be available. Specifically, such systems arise in structural mechanics, when a semidefinite finite element stiffness matrix is augmented with linear multi-point constraints via Lagrange multipliers. Engineers often use such multi-point constraints to introduce boundary or coupling conditions into complex finite element models. The article will present a systematic convergence study of the Golub-Kahan algorithm for a sequence of test problems of increasing complexity, including concrete structures enforced with pretension cables and the coupled finite element model of a reactor containment building. When the systems are suitably transformed using augmented Lagrangians on the semidefinite block and when the constraint equations are properly scaled, the Golub-Kahan algorithm is found to exhibit excellent convergence that depends only weakly on the size of the model. The new algorithm is found to be robust in practical cases that are otherwise considered to be difficult for iterative solvers.