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

## Abstract

Saddle point problems arise in many disciplines and applications, such as constrained optimisation or mixed finite elements . In our application, we look at the critical industrial application of the structural analysis of the containment building of nuclear power plants. These buildings are built of prestressed concrete in order to withstand critical conditions. The example at hand is a problem in linear elasticity, although the detailed variational formulation is unknown. The matrices are generated by the open source software Code_Aster (developed by EDF R&D). The structure of the building is described by the coupling of one-, two- and three-dimensional finite elements (representing the outer shell, metallic wires and the concrete block) by using multi-point constraints (MPCs). The MPCs are enforced by Lagrange multipliers, which leads after discretisation to a linear system with an indefinite matrix of 2x2 block structure with a highly singular leading block. The complexity of the structure of the building is reflected in the size of the stiffness and the constraint matrix which makes solving the linear system a challenging task. Classic iterative solvers do not give satisfactory results, which explains the need for the research on new scalable iterative solvers.

With this as our motivation, we discuss an iterative solver based on a variant of the Golub-Kahan bidiagonalization (GKB) method, which is widely used in solving least-squares problems and in the computation of the singular value decomposition of rectangular matrices . First, we will focus on a simplified test case modelling a rigid structure for which the leading block M is positive-definite. For each iteration of the generalized GKB method, a linear system Mz=b has to be solved. The algorithm needs, for a good parameter choice and a direct solver for the inner system, only few iterations to converge and the number of iterations is independent of the problem size of the finite element discretisation. For larger matrices, the inner solution step requires an iterative solver and we obtain an inner-outer iterative GKB method. We will compare practical choices in terms of compational cost and efficiency for the inner preconditioned iterative solver. Last, we will discuss a problem mimicking the prestressed concrete for which the leading block is singular and show regularization choices for it, such that the generalized GKB is efficient.

 M. Arioli. Generalized Golub-Kahan bidiagonalization and stopping criteria. SIAM J. Matrix Anal. Appl., 34(2):571–592, 2013.
 M. Benzi, G. H. Golub, and J. Liesen. Numerical solution of saddle point problems. Acta Numerica, 14:1—137, 2005.

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