### Trusses Nonlinear Problems Solution with Numerical Methods of Cubic Convergence Order

#### 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.

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P. F. N. Rodrigues, W. D. Varela, and R. A. Souza, "Análise de estratégias de solução do problema não-linear," Revista Ciência e Tecnologia, vol. 8, no. 2, pp. 36-49, 2008.

D. P. Maximiano, A. R. D. Silva, and R. A. M. Silveira, "Iterative strategies associated with the normal flow technique on the nonlinear analysis of structural arches," Revista Escola de Minas(Impresso), vol. 67, pp. 143-150, 2014.

W. T. Matias, "El control variable de los desplazamientos en el análisis no lineal elástico de estructuras de barras," Revista Internacional de Métodos Numéricos para Cálculo y Diseño en Ingeniería, vol. 18, no. 4, pp. 549-572, 2002.

W. E. Haisler, J. A. Stricklin, and F. J. Stebbins, "Development and evaluation of solution procedures for geometrically nonlinear structural analysis," AIAA Journal, vol. 10, no. 3, pp. 264-272, 1972.

R. A. M. Silveira, G. Rocha, and P. B. Gonçalvez, "Estratégias numéricas para análises geometricamente não lineares," in XV Congresso Brasileiro de Engenharia Mecânica, Águas de Lindóia, pp. 117-120, 1999.

J. L. Batoz and D. G., "Incremental displacement algorithms for nonlinear problems," International Journal for Numerical Methods in Engineering, vol. 14, pp. 1262-1267, 1979.

Y. B. Yang and S. R. Kuo, Theory & Analysis of Nonlinear Framed Structures. Singapore: Prentice-Hall, 1994.

Y. B. Yang and M. S. Shieh, "Solution method for nonlinear problems with multiple critical points," AIAA Journal, vol. 28, no. 12, pp. 2110-2116, 1990.

E. Riks, "The application of newtons method to the problem of elastic stability," Journal of Applied Mechanics, vol. 39, no. 4, pp. 1060-1065, 1972.

E. Riks, "An incremental approach to the solution of snapping and buckling problems," International Journal of Solids and Structures, vol. 15, pp. 529-551, 1979.

E. Ramm, Strategies for tracing the non-linear response near limit-points, nonlinear finite element analysis in structural mechanics. Berlin,GE: Springer-Verlag, 1981.

M. A. Crisfield, Non-Linear Finite Element Analysis of Solids and Structures: Essentials. New York, NY, USA: John Wiley & Sons, Inc., 1991.

V. Candela and A. Marquina, "Recurrence relations for rational cubic methods ii: the chebyshev method," Computing, vol. 45, no. 4, pp. 355-367, 1990.

J. M. Gutierrez and M. A. Hernández, "An acceleration of newtons method: Super-halley method," Applied Mathematics and Computation, vol. 117, no. 2, pp. 223-239, 2001.

S. Amat, S. Busquier, and J. M. Gutiérrez, "Geometric constructions of iterative functions to solve nonlinear equations," Journal of Computational and Applied Mathematics, vol. 157, pp. 197-205, 2003.

J. Ezquerro and M. Hernández, "An optimization of Chebyshev's method,"Journal of Complexity, vol. 25, no. 4, pp. 343-361, 2009.

H. B. Coda and M. Greco, "A simple FEM formulation for large deflection 2d frame analysis based on position description," Computer Methods in Applied Mechanics and Engineering, vol. 193, pp. 3541-3557, 2004.

K. Bathe, Finite Element Procedures. Prentice Hall, 2006.

S. E. Leon, G. H. Paulino, A. Pereira, I. F. M. Menezes, and E. N. Lages, "A unified library of nonlinear solution schemes," Applied Mechanics Reviews, vol. 64, pp. 1-26, 2011.

MATLAB, version 8.6.0 (R2015b). Natick, Massachusetts: The MathWorks Inc., 2015.

G. A. Wempner, "Discrete approximations related to nonlinear theories of solids," International Journal of Solids and Structures, vol. 7, no. 11, pp. 1581-1599, 1971.

F. A. Potra and V. Pták, Nondiscrete induction and iterative processes, vol. 103. Pitman Advanced Publishing Program, 1984.

Y. B. Yang and S. R. Kuo, Theory and analysis of nonlinear framed structures. Prentice Hall PTR, 1994.

T. Steihaug and S. Suleiman, "Rate of convergence of higher order methods,"Applied Numerical Mathematics, vol. 67, pp. 230-242, 2013.

M. Frontini and E. Sormani, "Third-order methods from quadrature formulae for solving systems of nonlinear equations," Applied Mathematics and Computation, vol. 149, no. 3, pp. 771-782, 2004.

J. Ezquerro and M. Hernández, "A modification of the convergence conditions for picards iteration," Computational & Applied Mathematics, vol. 23, no. 1, pp. 55-65, 2004.

J. M. Gutiérrez and M. A. Hernández, "A family of chebyshev-halley type methods in banach spaces," Bulletin of the Australian Mathematical Society, vol. 55, no. 1, pp. 113-130, 1997.

M. R. Pajand and M. T. Hakkak, "Nonlinear analysis of truss structures using dynamic relaxation," IJE Transactions B: Applications, vol. 19, no. 1, pp. 11-22, 2006.

S. Krenk, Non-linear modeling analysis of solids and structures. Cambridge, UK: Cambridge, 2009.

S. Krenk and O. Hededal, "A dual orthogonality procedure for non-linear finite element equations," Computer Methods in Applied Mechanics and Engineering, vol. 123, pp. 95-107, 1995.

E. G. M. Lacerda, D. N. Maciel, and A. C. Scudelari, "Geometrically static analysis of trusses using the arc-length method and the positional formulation of finite element method," in XXXV Iberian Latin-American Congress on Computational Methods in Engineering, 2014.

J. Bonet, A. J. Gil, and R. D. Wood, Worked examples in nonlinear continuum mechanics for finite element analysis. Cambridge University Press, 2012.

K. K. Choong and Y. Hangai, "Review on methods of bifurcation analysis for geometrically nonlinear structures," Bulletin of the International Association for Shell and Spatial Structures, vol. 34, no. 112, pp. 133-149, 1993.

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

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