Stress Recovery and Error Estimation in Functionally Graded Problems with Isogeometrical Analysis

Document Type : Original Research

Authors
S. emam 1
A. ganjali 2
A. Mirzakhani 2
N. zarif moghaddam basefat 3
1 Ph.D. Student of structural engineering, Department of Civil Engineering, Shahrood Branch, Islamic Azad University, Shahrood, Iran.
2 Assistant Professor, Department of Civil Engineering, Shahrood Branch, Islamic Azad University, Shahrood, Iran.
3 PhD in Structures, Shahid Montazeri Technical and Vocational College, Technical and Vocational University, Mashhad, Iran.
10.22034/23.3.109
Abstract
Today, the use of functionally graded materials is increasing. In these materials, the mechanical properties change as a continuous function throughout the problem domain. Due to these continuous changes, the problems of non-adhesion of materials, delamination and stress concentration at the joint, which can be problematic in composite structures, do not arise. Numerical methods such as the finite element method can be used to analyze functionally graded materials, but due to the limitations of this method, we will face many problems. The most important of these problems are the lack of a suitable element for the analysis of problems that can accommodate changes in the properties of materials, or the inability to accurately model the edges of shapes that have complex geometry, so in this research, the isogeometric method is used in which these weaknesses are eliminated. Also, since the error is an inseparable part of any numerical analysis and the reliability of the results has always been the main concern of the researchers, and in general, there is no exact answer to many problems, finding a solution to estimate the error in the calculations is of special importance. Therefore, in this article, for the first time, the isogeometric method has been developed in the analysis of problems with functionally graded materials with the approach of improving the stress field and estimating the error in it. This error estimator is in the category of error estimation methods based on stress recovery, and the goal is to increase the impact index of the error estimator and more adapt the error distribution method obtained from the proposed error estimator with the exact error estimator in solving problems. In this method, by using superconvergent points, where the order of convergence of the gradient of a function is one order higher than the value expected from the approximation of the shape function related to the approximate solution, a hypothetical surface is made for each stress value. To define this surface, we use the same shape functions used in the isogeometric method to approximate unknown functions. This hypothetical level is created when the coordinates x, y and z of its control points are specified. The x and y coordinates of each control point are used to model the geometric shape. The z component of the control points is calculated by minimizing the distance between this hypothetical level and the stress level obtained from isogeometric solution at the gauss-elements points of each region using the minimum square sum method. From the comparison of the exact error norm and the approximate error norm for sample problems, it can be seen that the proposed error estimation has a suitable efficiency for estimating the error in the analysis of problems with functionally graded materials by isogeometric method, and it can be used as a solution to error estimation and calculate the improved stress field level in solving functionally graded problems by isogeometric method. It is also possible to identify areas of the isogeometric solution domain that have a large error with the help of the proposed error estimator method and achieve local improvement of the network in those areas and increase the accuracy of the isogeometric solution.


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Subjects

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Modares Civil Engineering journal
Volume 23, Issue 3
July and August 2023
Pages 109-122