Application of an iterative Golub-Kahan algorithm to structural mechanics problems with multi-point constraints

Carola Kruse,Nicolas Tardieu,Ulrich Rüde,Vincent Darrigrand,Mario Arioli

doi:10.1186/s40323-020-00181-2

Abstract

Kinematic relationships between degrees of freedom, also named multi-point constraints, are frequently used in structural mechanics. In this paper, the Craig variant of the Golub-Kahan bidiagonalization algorithm is used as an iterative method to solve the arising linear system with a saddle point structure. The condition number of the preconditioned operator is shown to be close to unity and independent of the mesh size. This property is proved theoretically and illustrated on 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. The Golub-Kahan algorithm converges in only a small number of steps for all considered test problems and discretization sizes. Furthermore, it is robust in practical cases that are otherwise considered to be difficult for iterative solvers.

Highlights

In structural mechanics, it is very common to impose kinematic relationships between degrees of freedom (DOF) in a finite element model
One can consider the prestressed concrete structure of a reactor containment building that can be modeled as a problem in elasticity for which one, two- and three-dimensional finite elements are coupled by multi-point constraints (MPC)
We present a systematic convergence study for three sets of linear elastic test problems augmented with MPC and increasing complexity for which commonly used iterative solvers show poor performances

Summary

Introduction

It is very common to impose kinematic relationships between degrees of freedom (DOF) in a finite element model (see [1, section 35.2.2] or [2, section 2.6]). Rigid body conditions of a stiff part of a mechanical system or cyclic periodicity conditions on a mesh representing only a section of a periodic structure are typical examples of this approach. Such conditions can be used to glue non-conforming meshes or meshes containing different types of finite elements. We could link a thin structure modeled by shell finite elements to a massive 3D structure modeled with continuum finite elements These kinematic relationships are often called multi-point constraints (MPC) in standard finite element software and can be linear or nonlinear [2, section 3.4]. One can consider the prestressed concrete structure of a reactor containment building that can be modeled as a problem in elasticity for which one-, two- and three-dimensional finite elements (representing metallic cables, the inner shell, and the concrete block) are coupled by MPC

Findings

Discussion

Conclusion