An accelerated incremental algorithm to trace the nonlinear equilibrium path of structures

Abstract

This paper deals with the convergence acceleration of iterative nonlinear methods. An effective iterative algorithm, named the three-point method, is applied to nonlinear analysis of structures. In terms of computational cost, each iteration of the three-point method requires three evaluations of the function. In this study the effective functions have been proposed to accelerate the convergence process. The proposed method has a convergence order of eight, and it is important to note that its implementation does not require the computation of higher order derivatives compared to most other methods of the same order. To trace the equilibrium path beyond the limit point, a normal flow algorithm is implemented into a computer program. The three-point method is applied as an inner step in the normal flow algorithm. The procedure can be used for structures with complex behavior, including: unloading, snap-through, elastic post-buckling and inelastic post-buckling analyses. Several numerical examples are given to illustrate the efficiency and performance of the new method. Results show that the new method is comparable with the well-known existing methods and gives better results in convergence speed.

Improved Homotopy Perturbation Method for Geometrically Nonlinear Analysis of Space Trusses

“…Moreover, fast algorithms were presented to reduce the computational analysis of various structures [1,[16][17][18][19][20][21][22]. An iterative perturbation method to solve a nonlinear system is proposed by Golbabi and Javidi [23]; however, the proposed method was not able to effectively reduce the convergence time despite reducing the number of iterations.…”

Stability of a half-sine shallow arch under sinusoidal and step loads in thermal environment

“…Using the principle of stationary potential energy is another robust analytical approach to investigate the equilibrium and stability of shallow arches (Moon et al, 2007;Pi et al, 2007;Pi et al, 2008;Pi et al, 2010). On the other hand, the non-linear finite element method has been widely applied by researchers to trace the equilibrium path (Chandra et al, 2012;Saffari et al, 2012;Stanciulescu et al, 2012;Zhou et al, 2015b). Identifying the corresponding critical point(s) and finding the relationship between imperfections and load-bearing capacity are the capability of this numerical technique (Eriksson et al, 1999;Moghaddasie and Stanciulescu, 2013b;Rezaiee-Pajand and Moghaddasie, 2014).…”