Error Estimation and Stress Improvement in Nonlinear Analysis of Materials by Isogeometric Method

Document Type : Original Research

Authors
A. ganjali 1
A. Mirzakhani 1
A. Shahini 2
1 Faculty member of Islamic Azad University, Shahrood Branch
2 PhD student in Structural Islamic Azad University, Shahrood Branch
10.22034/23.2.6
Abstract
With the growth of science and technology, engineering issues are becoming more complex. As problems become more complex and need to be resolved more quickly and accurately, past analytical methods no longer meet the growing needs of societies. With such an attitude, researchers have always tried to develop numerical methods in addition to developing the basics of science. In this direction, several methods have been developed by researchers. Each of these methods has its own applications and still researchers are trying to grow and develop these methods and invent new methods. The most important of these are the nonlinear isogeometric method which is based on non-uniform rational B-Splines (NURBS). In the nonlinear isogeometric method, while using the properties of the basic functions of spline and NURBS in the exact definition of curves and surfaces, they are also used for interpolation and approximation. Using all the capacity of the structure in load bearing causes nonlinear behavior of the structure which is due to improper performance of the structure geometry, weakness of the structural materials and weakness due to the combination of the two previous states. In this study, nonlinearity due to material weakness has been considered. Also, in solving nonlinear equilibrium equations, an incremental and iterative process of load is used and this increase is done until the total loads defined for each problem are entered. In each increase, the iterative process is adopted until convergence or the maximum number of iterations is achieved. Obviously, all numerical methods are approximate methods. The main source of error in numerical methods is related to the discretization error of the continuous environment and is due to the approximation of the displacement field by the shape functions. This group of errors is also reduced by making the elemental mesh smaller and increasing the degree of shape functions used. Error is an integral part of numerical analysis and has always been a concern for researchers in the reliability of the results. Therefore, in this study, the error estimation based on the stress recovery method based on points where the order of gradient convergence of a function is one time higher than the value expected from the approximation of the shape function related to the approximate solution (superconvergent points) is discussed. Thus, by considering the difference between the recovered stress level and the stress level obtained from nonlinear isogeometric analysis for each element, a criterion has been determined approximately to determine the amount of error in that element. All research relationalizations and linearization of equations have been performed using a numerical algorithm with the help of programming in Fortran software environment and the results of the analysis for validation have been compared with its classical solution. The results show acceptable numerical and distributive similarity; Therefore, it can be said that the analysis performed by the program has good performance for nonlinear analysis of problems. Also, the error estimation method used can be called a simple and engineering solution to estimate the error and improve the stress field obtained from elastoplastic analysis of problems by isogeometric method.

Keywords

Subjects

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Modares Civil Engineering journal
Volume 23, Issue 2
May and June 2023
Pages 91-106