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This research is devoted to the adaptive solution and control point net improvement of axisymmetric problems in isogeometric analysis using the error estimation based methods for stress recovery. For this purpose, after the calculation of the energy norm, the estimated value of error in the vicinity of each control point is assigned to the neighboring members of a hypothetical truss-like structure as an artificial thermal gradient. By analysis of this network of rods under the temperature variations a new arrangement of control points is obtained. Repeating this process of thermal isogeometric analysis will eventually lead to a better distribution of errors in the domain of the problem and results in an optimal net of control points for the calculation of the integrals. To demonstrate the performance and efficiency of the proposed method, two axisymmetric elasticity problems with available analytical solutions are considered. The obtained results indicate that this innovative approach is effective in reducing errors of axisymmetric problems and can be employed for improving the accuracy in the context of the isogeometric analysis method. Innovated method of this research focuses on adaptive analysis and Network improving of axisymmetric problems in isogeometric analysis using error estimation methods based on stress recovery. For this purpose after calculation the energy norm, estimated value of error in the vicinity of control points is assigned to each rod as the thermal gradient. Thus after analyzing the hypothetical rods network under the temperature changes a new arrangement of control points and knot vectors can be obtained. The use of multi-cycle of this process in isogeometric analysis will lead to a better distribution of errors in the domain and thus achieve optimal network to calculate the integrals. To measure the efficiency of this method and demonstrate the increased carefully in axisymmetric problems, which has the analytical solution, two elasticity problem is evaluated. The results show that innovative network improving method has good efficiency to reduce the error rate and can be used to increase the accuracy of isogeometric analysis results. Innovated method of this research focuses on adaptive analysis and Network improving of axisymmetric problems in isogeometric analysis using error estimation methods based on stress recovery. For this purpose after calculation the energy norm, estimated value of error in the vicinity of control points is assigned to each rod as the thermal gradient. Thus after analyzing the hypothetical rods network under the temperature changes a new arrangement of control points and knot vectors can be obtained. The use of multi-cycle of this process in isogeometric analysis will lead to a better distribution of errors in the domain and thus achieve optimal network to calculate the integrals. To measure the efficiency of this method and demonstrate the increased carefully in axisymmetric problems, which has the analytical solution, two elasticity problem is evaluated. The results show that innovative network improving method has good efficiency to reduce the error rate and can be used to increase the accuracy of isogeometric analysis results.

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This article is devoted to the derivation of formulation and isogeometric solution of nonlinear nearly incompressible elastic problems, known as nearly incompressible hyperelasticity. After problem definition, the governing equations are linearized for employing the Newton-Raphson iteration method. Then, the problem is discretized by using concepts of isogeometric analysis method and its solution algorithm is devised. To demonstrate the performance of the proposed approach, the obtained results are compared with finite elements. Due to large deformations in this kind of problems, the finite element method requires a relatively large number of elements, as well as the need for remeshings in some problems, that results in a large system of equations with a high computational cost. In the isogeometric analysis method, using B-Spline and NURBS (Non-Uniform Rational B-Spline) basis functions provides us with a good flexibility in modeling of geometry without any need for further remeshings. The examples studied in this article indicate that by using the isogeometric approach good quality results are obtained with a smaller system of equations and less computational cost. Also, influence of different volumetric functions for the nearly incompressible materials are investigated.

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Applying and combining h and p refinement techniques in isogeometric method with the possibility of continuity elevation that this method provides, convergence and error of using different kinds of shape functions with different orders and continuities is investigated. It is done in a numerical analysis framework of a practical and well known problem called “Diametral Compression Test”. The advantage of this case is its circular geometry, since IGA provides designers with high potency of the possibility of using minimum elements to make the exact circular geometry. The point load inserts singularity to the problem. The refinement is utilized uniformly as the effective parameters are limited to the kind, order and continuity of shape functions. With different refinement techniques the convergence of approximated solution to the exact solution of linear elasticity is examined. It is concluded that with the singularity that mentioned, the error in IGA is not necessarily reduced with raise in order, more precisely the level of continuity is another important issue to determine error raise. It is also seen that in the presence of point load singularity the rate of error converges to the same value for all degrees of NURBS and lagrangian shape functions with any continuity. At the beginning of refinement process the minimum number of elements is used to make the process clearer to understand. In next steps h and p techniques and their combination is used to refine the model.

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Abstract In this article, the Charged System Search (CSS) algorithm is utilized for structural shape optimization that aims to minimize weight of a plane structure under stress constraints. Also, the Isogeometric Analysis (IA) is employed in order to analyze the structure. In the IA method, Non Uniform Rational B-Spline (NURBS) basis functions are used for approximation and interpolation of the displacement field as well as modelling geometry of the structure. Coordinates of the NURBS control points, that construct the geometry, can be considered as the design variables of the shape optimization problem. In earlier studies in structural shape optimization using the Finite Element (FE) method, boundaries of the structure were made by NURBS and the finite element discretization changed when the boundaries were modified in every iteration of the optimization process. As it mentioned, when the IA method is used the geometry is constructed by NURBS, therefore, contrary to using the FE method, the need for remeshing of the domain is eliminated and the computational cost will be remarkably decreased. In this paper, the IA method is briefly reviewed for analysis of the plane-stress elasticity problems. Also, the CSS formulation is derived based on physics laws for shape optimization problems. A few examples are presented to demonstrate the performance of the method and the results are compared when the Sequential Quadratic Programming (SQP) is used as a mathematical based method for structural shape optimization.

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In this work, axial vibration of nanorod was analyzed based on two phase integro-differential nonlocal elasticity theory using isogeometric method. Two phase integro-differential nonlocal elasticity theory not only shows the nonlocal property in an integrated manner based on kernel weight function, but also combines local and nonlocal linear curvature for a two phase nonlocal elastic material. The new isogeometric approach combines finite element method with computational geometry and can present an accurate geometric model for the problem. Also, using b-spline basis functions with arbitrary continuity order, it can be a better alternative for classical finite element methods. The obtained results indicated that isogeometric approach was superior to finite element method in term of speed and convergence quality. Moreover, in this model, the effects of phase and nonlocal parameters on the natural frequencies of the nanorod were investigated and it was shown that increase of parameters of local phase and nonlocal length scale, respectively, increased and decreased the values of natural frequencies of nanorods. Finally, for two special cases, asymptotic frequencies for a single type of nonlocal rod, two phase integro-differential was obtained and the results were compared with corresponding available differential Eringen results.

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This paper presents an isogeometric analysis approach for static and free vibration analysis of laminated composite plates covered with piezoelectric layers using the Reissner-Mindlin theory. Isogeometric analysis (IGA) aims at simplifying the Computer Aided Design (CAD) and Computer Aided Engineering (CAE) by using the functions describe the geometry (CAD) and the unknown fields (Analysis). The isogeometric approach has here been employed that utilizes the Non-Uniform Rational B-Splines (NURBS) of quadratic, cubic and quartic orders to approximate the variables defining geometry as well as the unknown functions. Using the Reissner-Mindlin first order shear deformation theory requires the C0-continuity of generalized displacements and the NURBS basis functions are well suited for this purpose. The electric potential is assumed to vary linearly through the thickness for each piezoelectric sublayer. To alleviate the shear locking problem, a stabilization technique is employed in the stiffness formulation. Since study of the performance and accuracy of IGA in solving laminated composite plates is one of the main objectives of this article. Several numerical examples are presented and compared with those available in the literature. The obtained results indicate desirable accuracy and efficiency of the proposed approach.

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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.

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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.