In the paper, a new approach is presented for improving the stress field and estimating the solution Error based on the isogeometrical analysis method. In this approach, by using the superconvergent points for the components of the stress field in each area, an imaginary surface is generated. The surface is described by the same NURBS' basis functions which are employed for approximating the displacement function in the isogeometrical analysis. The performance of the method is demonstrated by comparison of the exact and approximate energy Error norms for a couple of examples that the exact solution is available. It seems that the proposed method can be used as a suitable approach for Error estimation in the isogeometrical analysis method.

In this research, for the first time, the net of control points in the isogeometric analysis has been improved by employing an Error estimator based on a stress Recovery method. First, an Error estimation algorithm based on stress Recovery is used to obtain the energy norm for each element. Then, artificial rods are defined between control points and the estimated values of Errors in the control points located at the vicinity of a typical point is assigned to each rod as a thermal gradient. Now, by analyzing this hypothetical truss problem under temperature changes a new arrangement of control points and consequently the knot vectors can be obtained. Repeating this process in isogeometric analysis will lead to a better distribution of Errors in the domain of the problem and results in an optimal net of control points to calculate the integrals. To evaluate the efficiency of this method, the results of modeling and analysis of two elasticity problem is presented. The obtained results show that this innovative approach has a good performance and can be employed for reducing the analysis Errors.

Since the beginning of modeling physical events by computers, the finite element method has been firmly accepted as one of the most efficient general techniques the numerical solution of a variety of problems encountered in engineering. But no one has provided an answer to accurately determine the discretization Error value in analyzing a structural problem using finite element method and there is almost no accessible tool to select suitable sizes for elements and proper types of solutions and the size of each element is selected based on experts’ judgments.The present paper is an attempt to present a closed-form solution for three-node triangular elements in order to estimate the discretization Error in continuous domains by using the rule of gradient Recovery and h-refinement adaptivity. Computing the discretization Error and diagnosing the suitability of the elements size are possible by the closed-form solution presented.

In this research, the efficiency of Error estimation based on two methods of recovering the equilibrium stress of patches and superconvergent points in guiding the adaptive solution of nonlinear problems by isogeometric method has been investigated. Also, by analyzing elasto-plastic problems based on material properties, and adaptive solution by temperature gradient method, the stress improvement process has been investigated. The method of the adaptive solution of this research is based on the movement of control points and it is used in stress Recovery, taking into account the difference between the exact stress level and the stress level obtained from the isogeometric analysis for each element, as a criterion to determine the amount of Error in it. The element is obtained. For this purpose, the modeling of two problems in the non-linear range, which has an exact solution, has been considered. The results have shown that the total norm difference of exact and approximate Error in both stress Recovery methods used is more than 33% and in the direction of improving the network of control points. Also, the method by equilibrium patches is more effective than the method based on superconvergent points, which can be used as a suitable solution to improve the stress field.

In today's era, it is necessary to analyze functionally graded materials. Since the finite element method in the analysis of these materials has limitations and the Error is an inseparable part of any numerical analysis, therefore, finding a solution to estimate the Error in calculations is important. In the finite element method, modifying or enriching the network to reduce the Error is associated with such things as the overlapping of elements during displacement, the formation of elements with zero areas, and the increase of cost in calculations. In this research, the isogeometric method has been used for the first time in the analysis of problems with functionally graded materials in the adaptive solution using the method of moving control points as adaptive correction based on Error estimation, with the approach of improving the stress field. By comparing the exact and approximate Error norm in the solved examples, the effectivity index is more than 75%, which shows the effectiveness of the proposed Error estimator. In addition, improving the network of control points is effective in reducing the Error rate by more than 60% and can be used to increase the accuracy of the results.