89
Views
0
CrossRef citations to date
0
Altmetric
Scientific Paper

Applying Potra-Pták Iterative Cycle for Solving Highly Nonlinear Structural Problems

(Civ. Eng.) ORCID Icon, (Prof.) ORCID Icon, (Civ. Eng.) ORCID Icon, (Civ. Eng.) ORCID Icon & (Civ. Eng.) ORCID Icon

References

  • Riks E. The application of Newton's methods to the problems elastic stability. J Appl Mech. 1972; 39: 1060–1066. doi:10.1115/1.3422829.
  • Riks E. An incremental approach to the solution of snapping and buckling problems. Int J Solids Struct. 1979; 15: 529–551. doi:10.1016/0020-7683(79)90081-7.
  • Crisfield MA. A fast incremental/iterative solution procedure that handles snap-through. Comput Struct. 1981; 13: 52–62. doi:10.1016/0045-7949(81)90108-5.
  • Crisfield MA. Nonlinear Finite Element Analysis of Solids and Structures. vol. 1. Chichester, USA: John Wiley & Sons Inc, 1991.
  • Crisfield MA. NonLinear Finite Element Analysis of Solids and Structures. vol. 2. Chichester, USA: John Wiley & Sons Inc, 1997.
  • Chan SL. Geometric and material nonlinear analysis of beam-columns and frames using the minimum residual displacement method. Int J Numer Methods Eng. 1988; 26: 2657–2669. doi:10.1002/nme.1620261206.
  • Yang YB, Kuo SB. Theory & Analysis of Nonlinear Framed Structures. Singapore: Prentice Hall, 1994.
  • Krenk S. An orthogonal residual procedure for non-linear finite element equations. Int J Numer Methods Eng. 1995; 38: 823–839. doi:10.1002/nme.1620380508.
  • Mohit M, Sharifi Y, Tavakoli A. Geometrically nonlinear analysis of space trusses using new iterative techniques. Asian J Civil Eng. 2020; 21: 785–795. doi:10.1007/s42107-020-00239-x.
  • Saffari H, Mansouri I. Non-linear analysis of structures using two-point method. Int J Non-Linear Mech. 2011; 46: 834–840. doi:10.1016/j.ijnonlinmec.2011.03.008.
  • Morandini M. A two-level nonlinear beam analysis method. Int J Solids Struct. 2020; 203: 224–235. doi:10.1016/j.ijsolstr.2020.08.003.
  • Gutierrez JM, Hernandéz MA. An acceleration of Newtons method: Super-Halley method. Appl Math Comput. 2001; 117: 223–239. doi:10.1016/S0096-3003(99)00175-7.
  • Candela V, Marquina A. Recurrence relations for rational cubic methods II: the Chebyshev method. Computing. 1990; 45: 355–367. doi:10.1007/BF02238803.
  • Frontini M, Sormani E. Third-order methods from quadrature formulae for solving systems of nonlinear equations. Appl Math Comput. 2004; 149: 771–782. doi:10.1016/S0096-3003(03)00178-4.
  • Chun C. A family of composite fourth-order iterative methods for solving nonlinear equations. Appl Math Comput. 2007; 187: 951–956. doi:10.1016/j.amc.2006.09.009.
  • Potra FA, Pták V. Nondiscrete Induction and Iterative Processes. Research Notes in Mathematics. Boston: Pitman Advanced Pub. Program, 1984.
  • Herceg D, Herceg D. A family of methods for solving nonlinear equations. Appl Math Comput. 2015; 259: 882–895. doi:10.1016/j.amc.2015.03.028.
  • Herceg D, Herceg D. Eighth order family of iterative methods for nonlinear equations and their basins of attraction. J Comput Appl Math. 2018; 343: 458–480. doi:10.1016/j.cam.2018.04.040.
  • Petkovic LD, Petkovic MS. A note on some recent methods for solving nonlinear equations. Appl Math Comput. 2007; 185: 368–374. doi:10.1016/j.amc.2006.06.118.
  • Cordero A, Hueso JL, Martínez E, Torregrosa JR. New modifications of Potra-Pták's method with optimal fourth and eighth orders of convergence. J Comput Appl Math. 2010; 234: 2969–2976. doi:10.1016/j.cam.2010.04.009.
  • Soleymani F, Sharma R, Li X, Tohidi E. An optimized derivative-free form of the Potra-Pták method. Math Comput Model. 2012; 56: 97–104. doi:10.1016/j.mcm.2011.12.005.
  • Souza LA, Castelani EV, Shirabayashi Machado RD. Trusses nonlinear problems solution with numerical methods of cubic convergence order. Tendências em Matemática Aplicada e Computacional. 2018; 19: 161–179. doi:10.5540/tema.2018.019.01.161.
  • Reis RA. Implementação de Um Código Computacional Destinado à Solução de Sistemas de Equações Lineares e Não Lineares via Métodos Iterativos: Aplicações em Treliças Metálicas. Dissertação de Mestrado, Programa de Pós-Graduação em Engenharia Civil: Ouro Preto – MG/Brasil, 2019.
  • Silva ARD. Sistema Computacional para Análise Avançada Estática e Dinâmica de Estruturas Metálicas. Tese de Doutorado, Programa de Pós-Graduação em Engenharia Civil: Ouro Preto – MG/Brasil, 2009.
  • Lemes ÍJM. Estudo numérico avançado de estruturas de aço, concreto e mistas. Tese de Doutorado, Programa de Pós-Graduação em Engenharia Civil: Ouro Preto – MG/Brasil, 2018.
  • Batoz JL, Dhatt G. Incremental displacement algorithms for nonlinear problems. Int J Numer Methods Eng. 1979; 14: 1262–1267. doi:10.1002/nme.1620140811.
  • Homeier HHH. On Newton-type methods for multiple roots with cubic convergence. J Comput Appl Math. 2009; 231: 249–254. doi:10.1016/j.cam.2009.02.006.
  • Atluri SN, Liu CS, Kuo CL. A modified Newton method for solving non-linear algebraic equations. J Mar Sci Technol. 2009; 17: 238–247. doi:10.51400/2709-6998.1960.
  • Ali F, Aslam W, Rafiq A. Some new iterative techniques for the problems involving nonlinear equations. Int J Comput Methods. 2020; 16: 1–18.
  • Petkovic I, Herceg D. Computers in mathematical research: the study of three-point root-finding methods. Numer Algorithms. 2020; 84: 1179–1198. doi:10.1007/s11075-019-00796-6.
  • Pacoste C, Eriksson A. Beam elements in instability problems. Comput Methods Appl Mech Eng. 1997; 144: 163–197. doi:10.1016/S0045-7825(96)01165-6.
  • Lee S, Manuel FS, Rossow EC. Large deflections and stability of elastic frames. J Eng Mech Div. 1968; 94: 521–547. doi:10.1061/JMCEA3.0000966.
  • Schweizerhof KH, Wriggers P. Consistent linearization for path following methods in nonlinear FE analysis. Comput Methods Appl Mech Eng. 1986; 59: 269–279. doi:10.1016/0045-7825(86)90001-0.
  • Silva JL. Formulações Corrotacionais 2D para Análise Geometricamente Não Linear de Estruturas Reticuladas. Dissertação de Mestrado, Programa de Pós-Graduação em Engenharia Civil: Ouro Preto – MG/Brasil, 2016.
  • Bergan PG. Solution algorithms for nonlinear structural problems. Comput Struct. 1980; 12: 497–509. doi:10.1016/0045-7949(80)90125-X.
  • Galvão AS. Instabilidade Estática e Dinâmica de Pórticos Planos com Ligações Semi rígidas. Tese de Doutorado, Programa de Pós-Graduação em Engenharia Civil: Ouro Preto – MG/Brasil, 2004.
  • Maximiano DP. Uma Técnica Eficiente para Estabilizar a Estratégia do Resíduo Ortogonal na Análise Não Linear de Estruturas. Dissertação de Mestrado, Programa de Pós-Graduação em Engenharia Civil: Ouro Preto – MG/Brasil, 2012.
  • Xu Z, Mirmiran A. Looping behavior of arches using corotational finite element. Comput Struct. 1997; 62: 1059–1071. doi:10.1016/S0045-7949(96)00300-8.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.