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# Geometrically Nonlinear Analysis of Structures Using Various Higher Order Solution Methods: A Comparative Analysis for Large Deformation

Ali Maghami1, Farzad Shahabian1, Seyed Mahmoud Hosseini2,*
1 Civil Engineering Department, Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, Iran.
2 Industrial Engineering Department, Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, Iran.
* Corresponding Author: Seyed Mahmoud Hosseini. Email: .

Computer Modeling in Engineering & Sciences 2019, 121(3), 877-907. https://doi.org/10.32604/cmes.2019.08019

### Abstract

The suitability of six higher order root solvers is examined for solving the nonlinear equilibrium equations in large deformation analysis of structures. The applied methods have a better convergence rate than the quadratic Newton-Raphson method. These six methods do not require higher order derivatives to achieve a higher convergence rate. Six algorithms are developed to use the higher order methods in place of the NewtonRaphson method to solve the nonlinear equilibrium equations in geometrically nonlinear analysis of structures. The higher order methods are applied to both continuum and discrete problems (spherical shell and dome truss). The computational cost and the sensitivity of the higher order solution methods and the Newton-Raphson method with respect to the load increment size are comparatively investigated. The numerical results reveal that the higher order methods require a lower number of iterations that the Newton-Raphson method to converge. It is also shown that these methods are less sensitive to the variation of the load increment size. As it is indicated in numerical results, the average residual reduces in a lower number of iterations by the application of the higher order methods in the nonlinear analysis of structures.

### Keywords

Geometrically nonlinear analysis, higher order methods, predictor-corrector algorithms, convergence rate, sensitivity to the increment size.

Maghami, A., Shahabian, F., Hosseini, S. M. (2019). Geometrically Nonlinear Analysis of Structures Using Various Higher Order Solution Methods: A Comparative Analysis for Large Deformation. CMES-Computer Modeling in Engineering & Sciences, 121(3), 877–907.

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