Trusses Nonlinear Problems Solution with Numerical Methods of Cubic Convergence Order - Souza, Luiz Antonio Farani de; Castelani, Emerson Vitor; Shirabayashi, Wesley Vagner Inês; Aliano Filho, Angelo; Machado, Roberto Dalledone (Trends in Computational and Applied Mathematics, 2018)
A large part of the numerical procedures for obtaining the equilibrium path or load-displacement curve of structural problems with nonlinear behavior is based on the Newton-Raphson iterative scheme, to which is coupled the path-following methods. This paper presents new algorithms based on Potra-Pták, Chebyshev and super-Halley methods combined with the Linear Arc-Length path-following method. The main motivation for using these methods is the cubic order convergence. To elucidate the potential of our approach, we present an analysis of space and plane trusses problems with geometric nonlinearity found in the literature. In this direction, we will make use of the Positional Finite Element Method, which considers the nodal coordinates as variables of the nonlinear system instead of displacements. The numerical results of the simulations show the capacity of the computational algorithm developed to obtain the equilibrium path with force and displacement limits points. The implemented iterative methods exhibit better efficiency as the number of time steps and necessary accumulated iterations until convergence and processing time, in comparison with classic methods of Newton-Raphson and Modified Newton-Raphson.