Arc-length procedures with BEM in physically nonlinear problems V. Mallardo * , C. Alessandri Department of Architecture, University of Ferrara, via Quartieri 8, 44100 Ferrara, Italy Received 7 July 2003; revised 6 August 2003; accepted 12 November 2003 Abstract Geometrically or physically nonlinear problems are often characterised by the presence of critical points with snapping behaviour in the structural response. These structural or material instabilities usually lead to inefficiency of standard numerical solution techniques. Special numerical procedures are, therefore, required to pass critical points. The authors mean to present an arc-length procedure combined with the Boundary Element Method (BEM). The arc-length methods are intended to enable solution algorithms to pass limit points. Particularly for snap-back behaviour, the arc-length methods are the only procedures, which enable to follow the equilibrium path. The interest is mainly devoted to softening models where snap-backs and snap-throughs usually occur. No BEM applications have been possible so far due to the lack of a procedure enforcing the arc-length constraint. This paper intends to overcome such a difficulty. The analysis beyond the collapse point is mainly justified by two facts: (1) the investigation concerns a structural component and, therefore, it may be desirable to incorporate the load/deflection response of this component within a further analysis of the complete structure; (2) it may be important to know not just the collapse load but whether or not this collapse is of a ductile or brittle form. The procedure can be easily applied both to plasticity and damage, but numerical results will be presented for 2D elastoplasticity. Accurate results will be obtained in the case of both hardening and softening plasticity. The procedure can be used in BEM applications on nonlocal continuum models of the integral type and it can be easily extended to elasto- plasto-dynamics and to buckling in elastoplasticity. q 2004 Elsevier Ltd. All rights reserved. Keywords: Arc-length; Boundary element method; Nonlinear 1. Introduction The aim of this work is to develop a Boundary Elements (BE) solution technique for a class of physically nonlinear problems in elastostatics in which a critical point, with snapping behaviour in the structural response, occurs. These structural or material instabilities usually lead to ineffi- ciency of standard numerical solution techniques. Special numerical procedures are, therefore, required to pass critical points. The attention is mainly devoted to plasticity and damage analyses in which constitutive laws with strain softening are considered. This paper presents a new solution technique which couples the BEM with the arc-length methods in material nonlinear problems. To the authors’ knowledge no appli- cations of this type have been published so far. The difficulty is mainly related to the fact that in the BEM the nonlinear term is collected in the right-hand part and it is not included in the coefficient matrix as in the Finite Element Method (FEM): the problem will be better underlined in Section 4. A very first application of the arc-length technique to the nonlinear BEM is given by Zhang et al. [1]. The analysis of snap-through phenomena in thin shallow shells is performed by the application of the nonlinear field-boundary element technique. The integral equations contain domain integrals even in the linear case due to the curvature of the shell. The coefficient matrix of the equilibrium equations is equivalent to the tangent stiffness matrix in the FEM; therefore, the application of the arc-length technique is straightforward. The BEM is a well established technique suited to deal with a large number of practical applications in engineering. In physically nonlinear problems the boundary element treatment is to be preferred for those applications in which the size of the nonlinear zone is relatively small if compared to the overall size of the finite domain: in these cases the advantage in reducing the number of the unknown quantities by discretising only the boundary and the subregion in which the nonlinearity occurs is tremendous. 0955-7997/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.enganabound.2003.11.002 Engineering Analysis with Boundary Elements 28 (2004) 547–559 www.elsevier.com/locate/enganabound * Corresponding author. Tel.: þ 39-0532-293621; fax: þ 39-0532- 763146. E-mail address: mlv@unife.it (V. Mallardo).