Postbuckling analysis stabilized by penalty springs and intermediate corrections

Abstract

Whenever a critical point in a non-linear finite element analysis is reached, an implicit Newton procedure is prevented from proceeding until the stiffness matrix is stabilized. Most stabilization procedures result in a damped Newton scheme. This can cause a reduced convergence rate. Based on documented stabilization strategies, an iterative procedure to reduce negative impact on the convergence rate resulting from the damping is to be proposed here. This will be done by a corrector iteration carried out between two successive equilibrium iterative steps. The equilibrium iteration is a Newton-Raphson scheme. Since the stiffness matrix has already been factorized within the preceding standard equilibrium iterative step, the stiffness matrix will be held constant during the corrector iteration, which in turn allows for a computational efficient treatment of the additional iterations. The proposed procedure has been strongly driven by Wright and Gaylord's (ASCE J. Struct. Div., 94, 1143-1163 investigation.

A robust continuation method to pass limit-point instability

“…Similar techniques have been used in Damped Newton Methods accompanied by line search, see for example [39], and in fictitious penalty spring method [26] and in other stabilization methods [3,4]. However, unlike these methods, the particular construction of the proposed continuation method in this paper allows iterations to converge to an unstable configuration without experiencing ill-conditioning as will be shown by several examples in Section 5.…”

“…Although we have considered here only discrete critical points, the form of stabilization used in Eq. (4.1) lends itself easily for handling coincident (or closely spaced) critical points as discussed by several researchers (see for instance [4]). However, evaluating eigenvalues/ vectors can be computationally costly.…”

“…In the vicinity of a limit point, the tangent stiffness matrix of finite element formulation becomes ill-conditioned giving rise to two problems: (1) the underlying algebraic system of equations becomes harder to solve using numerical solvers [3,4], (2) solution jumps to a distant stable configuration making it harder for a numerical method to converge [5]. Numerous techniques, reviewed below, have been proposed to overcome these two problems.…”

“…However, as Müller [3,4] mentioned, these methods suffer from ill-conditioning in the vicinity of critical points in that ''numerical defect of the stiffness matrix is usually not repaired (exception: Wriggers and Simo [25], Felippa [26]). It is commonly assumed that during iteration the critical point is not precisely hit''.…”

“…To address these challenges, Müller [3,4] proposed a stabilized Newton-Raphson method. Stabilization methods are widely used in commercial FEA packages.…”