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

Abstract

When solving nonlinear algebraic equations that arise from discretization using the finite element method (FEM), it is often observed that the standard Newton-Raphson (N-R) iteration either fails to converge or necessitates a large number of iterations in the vicinity of critical points. This work proposes an additional numerical strategy, known as the Potra-Pták iterative cycle, to improve the efficiency of solving highly nonlinear structural problems. Therefore, the focus here is on making the nonlinear solver more robust and efficient, allowing the analysis of more complex nonlinear structures. In the Potra-Pták iterative cycle, two corrections of the objective function (energy function) are performed. The introduction of a second correction in the iterative cycle makes the Potra-Pták strategy more efficient than the standard or modified N-R iterations. This numerical strategy was implemented in the homemade Computational System for Advanced Structural Analysis (CS-ASA) program. The program is based on the FEM and is capable of performing static and dynamic nonlinear analysis of steel, concrete, and composite structures, and its efficiency is then verified through the analysis of slender frames and arches. The algorithm details for solving the nonlinear structural problem, characterized by the Potra-Pták scheme, are provided.

Keywords:

• Potra-Pták iterative cycle
• Newton-Raphson method
• CS-ASA program
• nonlinear structure problem
• finite element method

Acknowledgements

The authors thank CNPq and CAPES (Brazil Federal Research Agencies), FAPEMIG (Minas Gerais State Research Agency), PROPEC/UFOP, PROPPI/UFOP and UFLA for their support in the development of this research.

Disclosure Statement

No potential conflict of interest was reported by the author(s).

Data Availability Statement

The data that support the findings of this study are available from the corresponding author, RAM Silveira, upon reasonable request.

Additional information

Funding

This work was supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico: [Grant Number 307898/2019-9]; Fundação de Amparo à Pesquisa do Estado de Minas Gerais: [Grant Number TEC-PPM-00221-18].