NURBS-based isogeometric analysis (IGA) has currently been applied as a new numerical method in a considerable range of engineering problems. Due to non-interpolatory characteristic of NURBS basis functions, the properties of Kronecker Delta are not satisfied in IGA, and as a consequence, the imposition of essential boundary condition needs special treatment. The main contribution of this study is to use the well-known Lagrange multiplier method to impose essential boundary conditions for improving the accuracy of the isogeometric solution. Unlike the direct and transformation methods which are based on separation of control points, this method is capable of modeling incomplete Dirichlet boundaries. The solution accuracy and convergence rates of proposed method are compared with direct and transformation methods through various numerical examples.

In this paper, we have proposed a new iterative method for finding the solution of ordinary differential equations of the first order. In this method we have extended the idea of variational iteration method by changing the general Lagrange multiplier which is defined in the context of the variational iteration method. This causes the convergent rate of the method increased compared with the variational iteration method. To prevent consuming large amount of the CPU time and computer memory and to control requires significant amounts of computations, the Taylor expansion of the iterative functions in each iteration are applied. Finally to extend the convergence region of the truncated series, also the Pade approximants are used. Error analysis and convergence of the method are studied. Some examples are given to illustrate the performance and efficiency of the proposed method. For comparison, the results obtained by the our method and the variational iteration method are ‎presented.

The variational iteration method, a powerful tool for solving nonlinear oscillators with initial conditions, has been expanded to accommodate two-point boundary conditions. In this modification, two Lagrange multipliers are introduced, and their identification process mirrors that of the conventional approach. A generalized equation is provided for a category of highly nonlinear mechanical systems, followed by three illustrative examples derived from this generalized equation. These examples serve to demonstrate the effectiveness of the method. The solutions produced by this modified approach not only show remarkable agreement with numerical results but also exhibit superior accuracy when compared to outcomes obtained through other established methods.

We aim to reconstruct a nonlinear force-free magnetic field by minimizing the global departure of an initial field from a force-free and solenoidal state in the presence of helicity to obtain an appropriate representation of the magnetic field compatible with the solar coronal condition. Following the Wheatland et al method we modify their functional to include the magnetic helicity using the Lagrange multiplier technique. We reconstruct the magnetic field by minimizing the new functional in a computational box whose lower side coincides with the artificial magnetogram on the photosphere while the lateral and top sides extends up to the corona and by assuming appropriate boundary conditions for the Lagrange multiplier. The artificial magnetogram is obtained by Low and Lou semi-analytical solutions. A potential field as well as a suitable ansatz is used for the initial input magnetic filed and Lagrange multiplier for the iteration procedure, respectively. The results obtained by different optimization methods are in agreement with those obtained by our approach.

In this paper, we consider the variational homotopy perturbation method (VHPM) to obtain an approximate series solution for the generalized Fisher’s equation which converges to the exact solution in the region of convergence. Comparisons are made among the variational iteration method (VIM), the exact solutions and the proposed method. The results reveal that the proposed method is very effective and simple and can be applied for other nonlinear problems in mathematical.

Kinematic and Kinetic Modeling of a 3PRS mechanism is performed in this paper and its related controller is designed according to Computed Torque Method (CTM). This mechanism is a kind of spatial parallel robot with 6 DOFs, which is controlled using three active prismatic joints and three passive revolute joints and thus the system is categorized as constrained mechanism. Kinematic modeling of the robot is performed using Homogenous transformation matrix and its related Jacobian matrix is exracted. Therefore the position and velocity relation between the joint space and workspace are provided both in its direct and inverse modes. Dynamic equation of the system is also derived using Lagrange equation in the joint space of the robot. Thus both of forward and inverse kinetic of the system are solved for the presented system. Finally, by the aid of the extracted dynamic equation, the robot is controlled using Computed Torque Method (CTM). All of the modeling are verified by simulating the equations in MATLAB and comparing the direct and inverse modes. It is shown that using the extracted modeling and the designed controller for this parallel robot, the desired path can be easily tracked within the workspace of the robot.

In this article complete numerical formulation for nonlinear analysis of planar structures made of hyper elastic materials with application in civil engineering is presented. Structural problems can be solved by finite element method using complete lagrange formulation. The presented formulation is for large strain problems and is applicable for a wide range of hyper elastic nonlinear materials. Generally in nonlinear finite element analysis large deformation and disordered mesh (distorted) is seen. As using disordered elements, isoparametric elements would have low precision. Hyper elastic materials like rubber are one of the most effective materials in engineering application. The high axial strength and large deformation capacity of these materials make them suitable for many applications. Theory of hyper elastic materials is proposed by references [13, 14]. These theories explain hyper elasticity behavior in complicated formulations. Therefore we are interested in studying planar hyper elastic elements with large deformation like rubber, leather etc. Eventually planar nonlinear elements formulation with hyper elastic behavior is presented for structures having large deformations and complete lagrange formulation is used to analyze the structure. This formulation has been developed to analyze quadrilateral finite element models, which comparing to isoparametric 4-node elements is less sensitive to distorted meshing and doesn't have shear locking problem generated by geometrically distorted meshing. In order to deploy 2QACM benefits in nonlinear applications, complete lagrange formulation 3Q4HY is used, numerical examples in geometric nonlinear problems have shown presented formulation's ability to prevent loss of precision in highly distorted meshing for 4-node hyper elastic elements. To show effectiveness of presented elements two numerical examples are presented. Q4HY elements efficiency in nonlinear analysis of planar elements made of hyper elastic materials is significantly obvious. The program is written in matlab environment for nonlinear analysis of hyper elastic problems. Two formulations are proposed and results have been compared with references results. Examples of rubber like problems have shown these formulations ability to analyze large strain structures. Numerical examples express four node element has less sensitivity to distortion in meshing in nonlinear analysis and can be used with good precision when element is diagonal, while calculated responses using 4 node isoparametric element may be inappropriate. 4 noded elements effect on developing a simple, applicable and valid nonlinear geometric analysis is significantly observed. Furthermore, a complete lagrangian formulation with a particular formulation by applying integration techniques to effectively establish stiffness matrix is presented. According to presented examples, the investigated elements have great potential in solving Hyper elastic problems.