Adaptation of the Newton-Raphson and Potra-Pták methods for the solution of nonlinear systems
DOI:
https://doi.org/10.5433/1679-0375.2021v42n1p63Keywords:
Potra-Pták, Space trusses, Nonlinear analysis, Algorithm, Positional formulationAbstract
In this paper we adapt the Newton-Raphson and Potra-Pták algorithms by combining them with the modified Newton-Raphson method by inserting a condition. Problems of systems of sparse nonlinear equations are solved the algorithms implemented in Matlab® environment. In addition, the methods are adapted and applied to space trusses problems with geometric nonlinear behavior. Structures are discretized by the Finite Element Positional Method, and nonlinear responses are obtained in an incremental and iterative process using the Linear Arc-Length path-following technique. For the studied problems, the proposed algorithms had good computational performance reaching the solution with shorter processing time and fewer iterations until convergence to a given tolerance, when compared to the standard algorithms of the Newton-Raphson and Potra-Pták methods.Downloads
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References
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MUHAMMAD, K.; MAMAT, M.; WAZIRI, M. Y. A Broyden’s-like Method for solving systems of Nonlinear Equations. World Applied Sciences Journal, Pakistan v. 21, p. 168-173, 2013.
PAPADRAKAKIS, M. Post-buckling analysis of spatial structures by vector iteration methods. Computers & Structures, Elmsford, v. 14, n. 5-6, p. 393-402, 1981.
POTRA, F. A.; PTÁK, V. Nondiscrete induction and an inversion-free modification of Newton’s method. Casopis ˇ pro Pˇestování Matematiky, Praha, v. 108, n. 4, p. 333-341, 1983.
REZAIEE-PAJAND, M.; SALEHI-AHMADABAD, M.; GHALISHOOYAN, M. Structural geometrical nonlinear analysis by displacement increment. Asian Journal of Civil Engineering, [s. l.], v. 15, n. 5, p. 633-653, 2014.
RIKS, E. The application of Newton’s method to the problem of elastic stability. Journal of Applied Mechanics, New York, v. 39, n. 4, p. 1060–1065, 1972. RIKS, E. An incremental approach to the solution of snapping and buckling problems. International journal of solids and structures, [s.l.], v. 15, n. 7, p. 529-551, 1979.
SAFFARI, H.; MANSOURI, I. Non-linear analysis of structures using two-point method. International Journal of Non-Linear Mechanics, Elmsford, v. 46, n. 6, p. 834-840, 2011.
SOLEYMANI, F.; SHARMA, R. L. X.; TOHIDI, E. An optimized derivative-free form of the Potra–Pták method. Mathematical and Computer Modelling, Oxford, v. 56, n. 5-6, p. 97-104, 2012.
SOUZA, E. A. Métodos iterativos para problemas não lineares. 2015. Dissertação (Mestrado) - Universidade Federal Fluminense, Volta Redonda, 2015.
SOUZA, L. A. F. Static nonlinear analysis of piles cap based on the Continuum Damage Mechanics. Semina: Ciências Exatas e Tecnológicas, Londrina, v. 36, n. 2, p. 85-94, 2015. DOI: 10.5433/1679-0375.2015v36n2p85
SOUZA, E. A.; ALVAREZ, G. B.; LOBAO, D. C. Comparação Numérica entre Métodos Iterativos para Problemas Não Lineares. Proceeding Series of the Brazilian Society of Computational and Applied Mathematics, São Carlos, v. 5, n. 1, 2017. DOI: https://doi.org/10.5540/03.2017.005.01.0316
SOUZA, L. A. F.; CASTELANI, E. V.; SHIRABAYASHI, W. V. I.; MACHADO, R. D. Métodos iterativos de terceira e quarta ordem associados à técnica de comprimento de arco linear. Ciência & Engenharia, Uberlândia, v. 26, n. 1, p. 39-49, 2017.
SOUZA, L. A. F.; CASTELANI, E. V.; SHIRABAYASHI, W. V. I.; ALIANO FILHO, A.; MACHADO, R. D. Trusses Nonlinear Problems Solution with Numerical Methods of Cubic Convergence Order. TEMA, São Carlos, v. 19, n. 1, p. 161-179, 2018. DOI: https://doi.org/10.5540/tema.2018.019.01.0161
SOUZA, L. A. F.; MARTINS, R. S. V.; XAVIER, J. C.; PORTO, J. H. Application and comparison of numerical methods in the solution of systems of linear equations in space trusses problems. Semina: Ciências Exatas e Tecnológicas, Londrina, v. 39, n. 1, p. 49-60, 2018. DOI: 10.5433/1679-0375.2018v39n1p49
WEERAKOON, S.; FERNANDO, T. G. I. A variant of Newton’s method with accelerated third-order convergence. Applied Mathematics Letters, Elmsford, v. 13, n. 8, p. 87-93, 2000.
ANTUNES FILHO, A.; XAVIER, A. C. C. Solução de sistemas lineares esparsos utilizando CUDA: uma comparação de desempenho em sistemas Windows e Linux. Revista de Ciências Exatas e Tecnologia, Valinhos, v. 8, n. 8, p. 181-195, 2013. DOI: https://doi.org/10.17921/1890- 1793.2013v8n8p181-195
BATHE, K. J. Finite element procedures. New Jersey: Prentice-Hall, Inc., 2006. CODA, H. B. Análise não linear geométrica de sólidos e estruturas: uma formulação posicional baseada no MEF. 2003. Tese (Doutorado) - Escola de Engenharia de São Carlos, Universidade de São Paulo, São Paulo, 2003.
CODA, H. B.; GRECO, M. A simple FEM formulation for large deflection 2D frame analysis based on position description. Computer methods in applied mechanics and engineering, Amsterdam, v. 193, n. 33-35, p. 3541-3557, 2004. DOI: https://doi.org/10.1016/j.cma.2004.01.005
CRISFIELD, M. A. Non-linear finite element analysis of solids and structures. New York: John Wiley & Sons, Inc., 1991. v. 1.
DARVISHI, M. T.; SHIN, B. C. High-order NewtonKrylov methods to solve systems of nonlinear equations. Journal of the Korean Society for Industrial and Applied Mathematics, Seoul, v. 15, n. 1, p. 19-30, 2011.
DEHGHANI, H.; MANSOURI, I.; FARZAMPOUR, A.; HU, J. W. Improved homotopy perturbation method for geometrically nonlinear analysis of space trusses. Applied Sciences, Basel, v. 10, n. 8, p. 2987, 2020. DOI: 10.3390/app10082987
FAN, J.; YUAN, Y. A regularized Newton method for monotone nonlinear equations and its application. Optimization Methods and Software, New York, v. 29, n. 1, p. 102-119, 2014. DOI: https://doi.org/10.1080/10556788.2012.746344
FORDE, B. W. R.; STIEMER, S. F. Improved arc length orthogonality methods for nonlinear finite element analysis. Computers & Structures, Elmsford, v. 27, n. 5, p. 625-630, 1987. DOI: https://doi.org/10.1016/0045-7949(87)90078-2
FRONTINI, M.; SORMANI, E. Some variant of Newton’s method with third-order convergence. Applied Mathematics and Computation, New York, v. 140, n. 2-3, p. 419-426, 2003.
GRAU-SÁNCHEZ, M.; NOGUERA, M.; GUTIÉRREZ, J. M. On some computational orders of convergence. Applied Mathematics Letters, Elmsford, v. 23, n. 4, p. 472-478, 2010. DOI: https://doi.org/10.1016/j.aml.2012.04.012
GUTIERREZ, J. M.; HERNÁNDEZ, M. A. A family of Chebyshev-Halley type methods in Banach spaces. Bulletin of the Australian Mathematical Society, Cambridge, v. 55, n. 1, p. 113-130, 1997.
JARRATT, P. Some efficient fourth order multipoint methods for solving equations. BIT Numerical Mathematics, Dordrecht, v. 9, n. 2, p. 119-124, 1969.
KANWAR, V.; KUMAR, S.; BEHL, R. Several new families of Jarratt’s method for solving systems of nonlinear equations. Applications and Applied Mathematics, Texas, v. 8, n. 2, p. 701-716, 2013.
KANWAR, V.; SHARMA, J. R. A new family of Secantlike method with super-linear convergence. Applied Mathematics and Computation, New York, v. 171, n. 1, p. 104-107, 2005.
KIM, Y. L.; CHUN, C. New twelfth-order modifications of Jarratt’s method for solving nonlinear equations. Studies in Nonlinear sciences, Pakistan, v. 1, n. 1, p. 14-18, 2010.
LEON, S. E.; PAULINO, G. H.; PEREIRA, A.; MENEZES, I. F.; LAGES, E. N. A unified library of nonlinear solution schemes. Applied Mechanics Reviews, New York, v. 64, n. 4, 2011.
MAHDAVI, S. H.; RAZAK, H. A.; SHOJAEE, S.; MAHDAVI, M. S. A comparative study on application of Chebyshev and spline methods for geometrically nonlinear analysis of truss structures. International Journal of Mechanical Sciences, Oxford, v. 101, p. 241-251, 2015.
MATLAB. Version 8.6.0 (R2015b). Natick, Massachusetts: The Math Works Inc., 2015.
MAXIMIANO, D. P.; SILVA, A. R. D.; SILVEIRA, R. A. M. Iterative strategies associated with the normal flow technique on the nonlinear analysis of structural arches. Rem: Revista Escola de Minas, Ouro Preto, v. 67, n. 2, p. 143-150, 2014.
MOHIT, M.; SHARIFI, Y.; TAVAKOLI, A. Geometrically nonlinear analysis of space trusses using new iterative techniques. Asian Journal of Civil Engineering, [s. l.], v. 21, p. 785–795, 2020.
MUHAMMAD, K.; MAMAT, M.; WAZIRI, M. Y. A Broyden’s-like Method for solving systems of Nonlinear Equations. World Applied Sciences Journal, Pakistan v. 21, p. 168-173, 2013.
PAPADRAKAKIS, M. Post-buckling analysis of spatial structures by vector iteration methods. Computers & Structures, Elmsford, v. 14, n. 5-6, p. 393-402, 1981.
POTRA, F. A.; PTÁK, V. Nondiscrete induction and an inversion-free modification of Newton’s method. Casopis ˇ pro Pˇestování Matematiky, Praha, v. 108, n. 4, p. 333-341, 1983.
REZAIEE-PAJAND, M.; SALEHI-AHMADABAD, M.; GHALISHOOYAN, M. Structural geometrical nonlinear analysis by displacement increment. Asian Journal of Civil Engineering, [s. l.], v. 15, n. 5, p. 633-653, 2014.
RIKS, E. The application of Newton’s method to the problem of elastic stability. Journal of Applied Mechanics, New York, v. 39, n. 4, p. 1060–1065, 1972. RIKS, E. An incremental approach to the solution of snapping and buckling problems. International journal of solids and structures, [s.l.], v. 15, n. 7, p. 529-551, 1979.
SAFFARI, H.; MANSOURI, I. Non-linear analysis of structures using two-point method. International Journal of Non-Linear Mechanics, Elmsford, v. 46, n. 6, p. 834-840, 2011.
SOLEYMANI, F.; SHARMA, R. L. X.; TOHIDI, E. An optimized derivative-free form of the Potra–Pták method. Mathematical and Computer Modelling, Oxford, v. 56, n. 5-6, p. 97-104, 2012.
SOUZA, E. A. Métodos iterativos para problemas não lineares. 2015. Dissertação (Mestrado) - Universidade Federal Fluminense, Volta Redonda, 2015.
SOUZA, L. A. F. Static nonlinear analysis of piles cap based on the Continuum Damage Mechanics. Semina: Ciências Exatas e Tecnológicas, Londrina, v. 36, n. 2, p. 85-94, 2015. DOI: 10.5433/1679-0375.2015v36n2p85
SOUZA, E. A.; ALVAREZ, G. B.; LOBAO, D. C. Comparação Numérica entre Métodos Iterativos para Problemas Não Lineares. Proceeding Series of the Brazilian Society of Computational and Applied Mathematics, São Carlos, v. 5, n. 1, 2017. DOI: https://doi.org/10.5540/03.2017.005.01.0316
SOUZA, L. A. F.; CASTELANI, E. V.; SHIRABAYASHI, W. V. I.; MACHADO, R. D. Métodos iterativos de terceira e quarta ordem associados à técnica de comprimento de arco linear. Ciência & Engenharia, Uberlândia, v. 26, n. 1, p. 39-49, 2017.
SOUZA, L. A. F.; CASTELANI, E. V.; SHIRABAYASHI, W. V. I.; ALIANO FILHO, A.; MACHADO, R. D. Trusses Nonlinear Problems Solution with Numerical Methods of Cubic Convergence Order. TEMA, São Carlos, v. 19, n. 1, p. 161-179, 2018. DOI: https://doi.org/10.5540/tema.2018.019.01.0161
SOUZA, L. A. F.; MARTINS, R. S. V.; XAVIER, J. C.; PORTO, J. H. Application and comparison of numerical methods in the solution of systems of linear equations in space trusses problems. Semina: Ciências Exatas e Tecnológicas, Londrina, v. 39, n. 1, p. 49-60, 2018. DOI: 10.5433/1679-0375.2018v39n1p49
WEERAKOON, S.; FERNANDO, T. G. I. A variant of Newton’s method with accelerated third-order convergence. Applied Mathematics Letters, Elmsford, v. 13, n. 8, p. 87-93, 2000.
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Published
2021-06-02
How to Cite
Souza, L. A. F. de, Castelani, E. V., & Shirabayashi, W. V. I. (2021). Adaptation of the Newton-Raphson and Potra-Pták methods for the solution of nonlinear systems. Semina: Ciências Exatas E Tecnológicas, 42(1), 63–74. https://doi.org/10.5433/1679-0375.2021v42n1p63
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