Sofie E. Leon Glaucio H. Paulino 1 e-mail: paulino@illinois.edu Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, 205 North Matthews Avenue, Urbana, IL 61801 Anderson Pereira Ivan F. M. Menezes Group of Technology in Computer Graphics, Tecgraf, Pontifical Catholic University of Rio de Janeiro, Rio de Janeiro, Rio de Janeiro, 22451, Brazil Eduardo N. Lages Center for Technology, Federal University of Alagoas, Maceio, Alagoas, 57072, Brazil A Unified Library of Nonlinear Solution Schemes Nonlinear problems are prevalent in structural and continuum mechanics, and there is high demand for computational tools to solve these problems. Despite efforts to develop efficient and effective algorithms, one single algorithm may not be capable of solving any and all nonlinear problems. A brief review of recent nonlinear solution techniques is first presented. Emphasis, however, is placed on the review of load, displacement, arc length, work, generalized displacement, and orthogonal residual control algorithms, which are unified into a single framework. Each of these solution schemes differs in the use of a constraint equation for the incremental-iterative procedure. The governing finite element equations and constraint equation for each solution scheme are combined into a single matrix equation, which characterizes the unified approach. This conceptual model leads naturally to an effective object-oriented implementation. Within the unified framework, the strengths and weaknesses of the various solution schemes are examined through numerical examples. [DOI: 10.1115/1.4006992] 1 Introduction Nonlinear problems are prevalent in structural and continuum mechanics; however, one single nonlinear solution method may not be capable of solving any general nonlinear problem. Depend- ing on the problem and the severity of the nonlinearities, modifi- cations to solution algorithms are necessary to recover the entire equilibrium path. In an early work, Bergan et al. [1] stated: “a computer program for nonlinear analysis should possess sev- eral alternative algorithms for the solution of the nonlinear sys- tem. These procedures should also allow for the possibility of an extensive control over the solution process by parameters that are input to the program. Such a scheme would lead to increased flexi- bility, and the experienced user has the possibility of obtaining improved reliability and efficiency for the solution of a particular problem.” This philosophy is the main motivating factor for the present work, which provides a modern theoretical and computational framework for the insightful statement by Bergan et al. [1]. Many authors have developed families of nonlinear solution schemes, which can be adjusted by the user depending on the problem. Mondkar and Powell [2] developed a library of algo- rithms based on the Newton-Raphson method. Seven solution schemes were formulated from 11 control parameters (stiffness update type and frequency, convergence tolerance, etc.) and tested on several nonlinear structural systems. Clarke and Hancock [3] used the concept of load increment from the standard or modified Newton-Raphson method to unify several nonlinear solution schemes through a single load factor. The specific incremental- iterative procedure depends on the chosen constraint equation, which is used to calculate the unifying load factor. The constraint equations are based on iterations at constant load, displacement, work, arc length, or minimum residual. Yang and Sheih [4] and Yang and Kuo [5] presented a similar library of nonlinear solvers unified through a single load parameter, and included the general- ized displacement control method. More recently Rezaiee-Pajand et al. [6] unified five nonlinear solution schemes through a single general constraint equation. The schemes were identified by five different constraints, including minimizing error by means of its length, area, or perimeter, and then the strengths and weaknesses of each algorithm were evaluated. The library of nonlinear solution schemes explored in this review is similar to its predecessors in that several solution schemes, defined by a constraint equation, are unified into a single space by means of a load parameter. The methods include load control, displacement control, work control, arc length control, generalized displacement control, and the orthogonal residual pro- cedure, which until now has not been incorporated into a collec- tion of unified schemes. The unified schemes are formulated and implemented such that (i) additional nonlinear solution schemes are readily incorporated and (ii) integration into a finite element analysis code is straightforward. The approach taken is aimed at widespread dissemination of the work and follows an educational philosophy. Moreover, a website with all of the components devel- oped, including a tutorial, is provided to the interested reader. 2 1.1 On Nonlinear Systems. Nonlinear behavior can arise from either material or geometric nonlinearity. In the former, the constitutive relation describing the material is itself nonlinear and the structural response associated with physical phenomena such as plasticity or strain-softening must be captured. In the latter, nonlinearity is due to changes in geometry, arising from large strains and/or rotations, which enter the formulation from a non- linear strain-displacement relationship, and may occur even if the constitutive relation is linear [7]. Furthermore, in geometric non- linear problems, the applied loads will either have an effect on the deformed configuration, or the configuration will have an effect on the load (e.g., follower loads [8]). Nonlinear problems arising from either geometric or material nonlinearity feature critical points along the solution path. Critical 1 Corresponding author. Manuscript received November 26, 2011; final manuscript received April 2, 2012; published online August 17, 2012. Assoc. Editor: J. N. Reddy. 2 See www.ghpaulino.com/NLS_tutorial.html. Applied Mechanics Reviews JULY 2011, Vol. 64 / 040803-1 Copyright V C 2011 by ASME Downloaded 31 Oct 2012 to 130.126.242.107. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm