AN ITERATIVE GENERALIZED GOLUB-KAHAN ALGORITHM FOR PROBLEMS IN STRUCTURAL MECHANICS

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