Trusses Nonlinear Problems Solution with Numerical Methods of Cubic Convergence Order

Authors

  • Luiz Antonio Farani de Souza Federal Technological University of Paraná
  • Emerson Vitor Castelani State University of Maringá
  • Wesley Vagner Inês Shirabayashi State University of Maringá
  • Angelo Aliano Filho Federal Technological University of Paraná
  • Roberto Dalledone Machado Federal University of Paraná

DOI:

https://doi.org/10.5540/tema.2018.019.01.161

Keywords:

Arc-Length, Positional Finite Element, Chebyshev, Potra-Pták, Geometric Nonlinearity

Abstract

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.

Author Biographies

Luiz Antonio Farani de Souza, Federal Technological University of Paraná

Civil Engineering Course

Emerson Vitor Castelani, State University of Maringá

Postgraduate Program in Mathematics

Wesley Vagner Inês Shirabayashi, State University of Maringá

Postgraduate Program in Mathematics

Angelo Aliano Filho, Federal Technological University of Paraná

Mathematics Department

Roberto Dalledone Machado, Federal University of Paraná

Postgraduate Program in Numerical Methods in Engineering

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Published

2018-05-05

How to Cite

Souza, L. A. F. de, Castelani, E. V., Shirabayashi, W. V. I., Aliano Filho, A., & Machado, R. D. (2018). Trusses Nonlinear Problems Solution with Numerical Methods of Cubic Convergence Order. Trends in Computational and Applied Mathematics, 19(1), 161. https://doi.org/10.5540/tema.2018.019.01.161

Issue

Vol. 19 No. 1 (2018)

Section

Original Article

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