Global interpolating meshless shape function based on generalized moving least-square for structural dynamic analysis

A global interpolating meshless shape function based on the generalized moving least-square （GMLS） is formulated by the transformation technique. Both the shape function and its derivatives meet the Kronecker delta function property. With the interpolating GMLS （IGMLS） shape function, an improved element-free Galerkin （EFG） method is proposed for the structural dynamic analysis. Compared with the conven- tional EFG method, the obvious advantage of the proposed method is that the essential boundary conditions including both displacements and derivatives can be imposed by the straightforward way. Meanwhile, it can greatly improve the ill-condition feature of the standard GMLS approximation, and provide good accuracy at low cost. The dynamic analyses of the Euler beam and Kirchhoff plate are performed to demonstrate the feasi- bility and effectiveness of the improved method. The comparison between the numerical results of the conventional method and the improved method shows that the proposed method has better stability, higher accuracy, and less time consumption.

An improved interpolating element-free Galerkin method with a nonsingular weight function for two-dimensional potential problems

In this paper, an improved interpolating moving least-square （IIMLS） method is presented. The shape function of the IIMLS method satisfies the property of the Kronecker 5 function. The weight function used in the IIMLS method is nonsingular. Then the IIMLS method can overcome the difficulties caused by the singularity of the weight function in the IMLS method. The number of unknown coefficients in the trial function of the IIMLS method is less than that of the moving least-square （MLS） approximation. Then by combining the IIMLS method with the Galerkin weak form of the potential problem, the improved interpolating element-free Galerkin （IIEFG） method for two-dimensional potential problems is presented. Compared with the conventional element-free Galerkin （EFG） method, the IIEFG method can directly use the essential boundary conditions. Then the IIEFG method has higher accuracy. For demonstration, three numerical examples are solved using the IIEFG method.

Interpolating Isogeometric Boundary Node Method and Isogeometric Boundary Element Method Based on Parameter Space

In this paper,general interpolating isogeometric boundary node method(IIBNM)and isogeometric boundary element method(IBEM)based on parameter space are proposed for 2D elasticity problems.In both methods,the integral cells and elements are defined in parameter space,which can reproduce the geometry exactly at all the stages.In IIBNM,the improved interpolating moving leastsquare method(IIMLS)is applied for field approximation and the shape functions have the delta function property.The Lagrangian basis functions are used for field approximation in IBEM.Thus,the boundary conditions can be imposed directly in both methods.The shape functions are defined in 1D parameter space and no curve length needs to be computed.Besides,most methods for the treatment of the singular integrals in the boundary element method can be applied in IIBNM and IBEM directly.Numerical examples have demonstrated the accuracy of the proposed methods.

A moving Kriging interpolation-based boundary node method for two-dimensional potential problems

In this paper, a meshfree boundary integral equation （BIE） method, called the moving Kriging interpolation- based boundary node method （MKIBNM）, is developed for solving two-dimensional potential problems. This study combines the DIE method with the moving Kriging interpolation to present a boundary-type meshfree method, and the corresponding formulae of the MKIBNM are derived. In the present method, the moving Kriging interpolation is applied instead of the traditional moving least-square approximation to overcome Kronecker＇s delta property, then the boundary conditions can be imposed directly and easily. To verify the accuracy and stability of the present formulation, three selected numerical examples are presented to demonstrate the efficiency of MKIBNM numerically.

A meshless method based on moving Kriging interpolation for a two-dimensional time-fractional diffusion equation

Fractional diffusion equations have been the focus of modeling problems in hydrology, biology, viscoelasticity, physics, engineering, and other areas of applications. In this paper, a meshfree method based on the moving Kriging inter- polation is developed for a two-dimensional time-fractional diffusion equation. The shape function and its derivatives are obtained by the moving Kriging interpolation technique. For possessing the Kronecker delta property, this technique is very efficient in imposing the essential boundary conditions. The governing time-fractional diffusion equations are transformed into a standard weak formulation by the Galerkin method. It is then discretized into a meshfree system of time-dependent equations, which are solved by the standard central difference method. Numerical examples illustrating the applicability and effectiveness of the proposed method are presented and discussed in detail.

An improved interpolating element-free Galerkin method for elasticity

Based on the improved interpolating moving least-squares （ⅡMLS） method and the Galerkin weak form, an improved interpolating element-free Galerkin （ⅡEFG） method is presented for two-dimensional elasticity problems in this paper. Compared with the interpolating moving least-squares （IMLS） method presented by Lancaster, the ⅡMLS method uses the nonsingular weight function. The number of unknown coefficients in the trial function of the ⅡMLS method is less than that of the MLS approximation and the shape function of the ⅡMLS method satisfies the property of Kronecker δ function. Thus in the ⅡEFG method, the essential boundary conditions can be applied directly and easily, then the numerical solutions can be obtained with higher precision than those obtained by the interpolating element-free Galerkin （IEFG） method. For the purposes of demonstration, four numerical examples are solved using the ⅡEFG method.

An improved boundary element-free method (IBEFM) for two-dimensional potential problems

The interpolating moving least-squares （IMLS） method is discussed first in this paper. And the formulae of the IMLS method obtained by Lancaster are revised. Then on the basis of the boundary element-free method （BEFM）, combining the boundary integral equation （BIE） method with the IMLS method, the improved boundary element-free method （IBEFM） for two-dimensional potential problems is presented, and the corresponding formulae of the IBEFM are obtained. In the BEFM, boundary conditions are applied directly, but the shape function in the MLS does not satisfy the property of the Kronecker ~ function. This is a problem of the BEFM, and must be solved theoretically. In the IMLS method, when the shape function satisfies the property of the Kronecker 5 function, then the boundary conditions, in the meshless method based on the IMLS method, can be applied directly. Then the IBEFM, based on the IMLS method, is a direct meshless boundary integral equation method in which the basic unknown quantity is the real solution of the nodal variables, and the boundary conditions can be applied directly and easily, thus it gives a greater computational precision. Some numerical examples are presented to demonstrate the method.

A new complex variable meshless method for transient heat conduction problems

In this paper, based on the improved complex variable moving least-square （ICVMLS） approximation, a new complex variable meshless method （CVMM） for two-dimensional （2D） transient heat conduction problems is presented. The variational method is employed to obtain the discrete equations, and the essential boundary conditions are imposed by the penalty method. As the transient heat conduction problems are related to time, the Crank-Nicolson difference scheme for two-point boundary value problems is selected for the time discretization. Then the corresponding formulae of the CVMM for 2D heat conduction problems are obtained. In order to demonstrate the applicability of the proposed method, numerical examples are given to show the high convergence rate, good accuracy, and high efficiency of the CVMM presented in this paper.

A new complex variable element-free Galerkin method for two-dimensional potential problems

In this paper, based on the element-free Galerkin （EFG） method and the improved complex variable moving least- square （ICVMLS） approximation, a new meshless method, which is the improved complex variable element-free Galerkin （ICVEFG） method for two-dimensional potential problems, is presented. In the method, the integral weak form of control equations is employed, and the Lagrange multiplier is used to apply the essential boundary conditions. Then the corresponding formulas of the ICVEFG method for two-dimensional potential problems are obtained. Compared with the complex variable moving least-square （CVMLS） approximation proposed by Cheng, the functional in the ICVMLS approximation has an explicit physical meaning. Furthermore, the ICVEFG method has greater computational precision and efficiency. Three numerical examples are given to show the validity of the proposed method.

A scaled boundary node method applied to two-dimensional crack problems

A boundary-type meshless method called the scaled boundary node method （SBNM） is developed to directly evaluate mixed mode stress intensity factors （SIFs） without extra post-processing. The SBNM combines the scaled boundary equations with the moving Kriging （MK） interpolation to retain the dimensionality advantage of the former and the meshless attribute of the latter. As a result, the SBNM requires only a set of scattered nodes on the boundary, and the displacement field is approximated by using the MK interpolation technique, which possesses the 5 function property. This makes the developed method efficient and straightforward in imposing the essential boundary conditions, and no special treatment techniques are required. Besides, the SBNM works by weakening the governing differential equations in the circumferential direction and then solving the weakened equations analytically in the radial direction. Therefore, the SBNM permits an accurate representation of the singularities in the radial direction when the scaling center is located at the crack tip. Numerical examples using the SBNM for computing the SIFs are presented. Good agreements with available results in the literature are obtained.

Topology optimization using the improved element-free Galerkin method for elasticity

The improved element-free Galerkin （IEFG） method of elasticity is used to solve the topology optimization problems. In this method, the improved moving least-squares approximation is used to form the shape function. In a topology opti- mization process, the entire structure volume is considered as the constraint. From the solid isotropic microstructures with penalization, we select relative node density as a design variable. Then we choose the minimization of compliance to be an objective function, and compute its sensitivity with the adjoint method. The IEFG method in this paper can overcome the disadvantages of the singular matrices that sometimes appear in conventional element-free Galerkin （EFG） method. The central processing unit （CPU） time of each example is given to show that the IEFG method is more efficient than the EFG method under the same precision, and the advantage that the IEFG method does not form singular matrices is also shown.

Complex variable element-free Galerkin method for viscoelasticity problems

Based on the complex variable moving least-square （CVMLS） approximation, the complex variable element-free Galerkin （CVEFG） method for two-dimensional viscoelasticity problems under the creep condition is presented in this paper. The Galerkin weak form is employed to obtain the equation system, and the penalty method is used to apply the essential boundary conditions, then the corresponding formulae of the CVEFG method for two-dimensional viscoelasticity problems under the creep condition are obtained. Compared with the element-free Galerkin （EFG） method, with the same node distribution, the CVEFG method has higher precision, and to obtain the similar precision, the CVEFG method has greater computational efficiency. Some numerical examples are given to demonstrate the validity and the efficiency of the method.

Analysis of elastoplasticity problems using an improved complex variable element-free Galerkin method

In this paper, based on the conjugate of the complex basis function, a new complex variable moving least-squares approximation is discussed. Then using the new approximation to obtain the shape function, an improved complex variable element-free Galerkin（ICVEFG） method is presented for two-dimensional（2D） elastoplasticity problems. Compared with the previous complex variable moving least-squares approximation, the new approximation has greater computational precision and efficiency. Using the penalty method to apply the essential boundary conditions, and using the constrained Galerkin weak form of 2D elastoplasticity to obtain the system equations, we obtain the corresponding formulae of the ICVEFG method for 2D elastoplasticity. Three selected numerical examples are presented using the ICVEFG method to show that the ICVEFG method has the advantages such as greater precision and computational efficiency over the conventional meshless methods.

Solving unsteady Schr?dinger equation using the improved element-free Galerkin method

By employing the improved moving least-square （IMLS） approximation, the improved element-free Galerkin （IEFG） method is presented for the unsteady Schrodinger equation. In the IEFG method, the two-dimensional （2D） trial function is approximated by the IMLS approximation, the variation method is used to obtain the discrete equations, and the essential boundary conditions are imposed by the penalty method. Because the number of coefficients in the IMLS approximation is less than in the moving least-square （MLS） approximation, fewer nodes are needed in the entire domain when the IMLS approximation is used than when the MLS approximation is adopted. Then the IEFG method has high computational efficiency and accuracy. Several numerical examples are given to verify the accuracy and efficiency of the IEFG method in this paper.

A meshfree-based local Galerkin method with condensation of degree of freedom for elastic dynamic analysis

Condensation technique of degree of freedom is first proposed to improve the computational efficiency of meshfree method with Galerkin weak form for elastic dy- namic analysis. In the present method, scattered nodes with- out connectivity are divided into several subsets by cells with arbitrary shape. Local discrete equation is established over each cell by using moving Kriging interpolation, in which the nodes that located in the cell are used for approxima- tion. Then local discrete equations can be simplified by con- densation of degree of freedom, which transfers equations of inner nodes to equations of boundary nodes based on cells. The global dynamic system equations are obtained by as- sembling all local discrete equations and are solved by using the standard implicit Newmark＇s time integration scheme. In the scheme of present method, the calculation of each cell is carried out by meshfree method, and local search is imple- mented in interpolation. Numerical examples show that the present method has high computational efficiency and good accuracy in solving elastic dynamic problems.

The dimension split element-free Galerkin method for three-dimensional potential problems

This paper presents the dimension split element-free Galerkin （DSEFG） method for three-dimensional potential problems, and the corresponding formulae are obtained. The main idea of the DSEFG method is that a three-dimensional potential problem can be transformed into a series of two-dimensional problems. For these two-dimensional problems, the improved moving least-squares （IMLS） approximation is applied to construct the shape function, which uses an orthogonal function system with a weight function as the basis functions. The Galerkin weak form is applied to obtain a discretized system equation, and the penalty method is employed to impose the essential boundary condition. The finite difference method is selected in the splitting direction. For the purposes of demonstration, some selected numerical examples are solved using the DSEFG method. The convergence study and error analysis of the DSEFG method are presented. The numerical examples show that the DSEFG method has greater computational precision and computational efficiency than the IEFG method.

The complex variable meshless local Petrov-Galerkin method of solving two-dimensional potential problems

Based on the complex variable moving least-square（CVMLS） approximation and a local symmetric weak form,the complex variable meshless local Petrov-Galerkin（CVMLPG） method of solving two-dimensional potential problems is presented in this paper.In the present formulation,the trial function of a two-dimensional problem is formed with a one-dimensional basis function.The number of unknown coefficients in the trial function of the CVMLS approximation is less than that in the trial function of the moving least-square（MLS） approximation.The essential boundary conditions are imposed by the penalty method.The main advantage of this approach over the conventional meshless local Petrov-Galerkin（MLPG） method is its computational efficiency.Several numerical examples are presented to illustrate the implementation and performance of the present CVMLPG method.

Analysis of the generalized Camassa and Holm equation with the improved element-free Galerkin method

In this paper, we analyze the generalized Camassa and Holm （CH） equation by the improved element-free Galerkin （IEFG） method. By employing the improved moving least-square （IMLS） approximation, we derive the formulas for the generalized CH equation with the IEFG method. A variational method is used to obtain the discrete equations, and the essential boundary conditions are enforced by the penalty method. Because there are fewer coefficients in the IMLS approximation than in the MLS approximation, and in the IEFG method, fewer nodes are selected in the entire domain than in the conventional EFG method, the IEFG method should result in a higher computing speed. The effectiveness of the IEFG method for the generalized CH equation is investigated by numerical examples in this paper.

New complex variable meshless method for advection-diffusion problems

In this paper,an improved complex variable meshless method（ICVMM） for two-dimensional advection-diffusion problems is developed based on improved complex variable moving least-square（ICVMLS） approximation.The equivalent functional of two-dimensional advection-diffusion problems is formed,the variation method is used to obtain the equation system,and the penalty method is employed to impose the essential boundary conditions.The difference method for twopoint boundary value problems is used to obtain the discrete equations.Then the corresponding formulas of the ICVMM for advection-diffusion problems are presented.Two numerical examples with different node distributions are used to validate and investigate the accuracy and efficiency of the new method in this paper.It is shown that ICVMM is very effective for advection-diffusion problems,and has a good convergent character,accuracy,and computational efficiency.

Copy-Move Forgeries Detection and Localization Using Two Levels of Keypoints Extraction

Copy-move offense is considerably used to conceal or hide several data in the digital image for specific aim, and onto this offense some portion of the genuine image is reduplicated and pasted in the same image. Therefore, Copy-Move forgery is a very significant problem and active research area to check the confirmation of the image. In this paper, a system for Copy Move Forgery detection is proposed. The proposed system is composed of two stages: one is called the detection stages and the second is called the refine detection stage. The detection stage is executed using Speeded-Up Robust Feature (SURF) and Binary Robust Invariant Scalable Keypoints (BRISK) for feature detection and in the refine detection stage, image registration using non-linear transformation is used to enhance detection efficiency. Initially, the genuine image is picked, and then both SURF and BRISK feature extractions are used in parallel to detect the interest keypoints. This gives an appropriate number of interest points and gives the assurance for finding the majority of the manipulated regions. RANSAC is employed to find the superior group of matches to differentiate the manipulated parts. Then, non-linear transformation between the best-matched sets from both extraction features is used as an optimization to get the best-matched set and detect the copied regions. A number of numerical experiments performed using many benchmark datasets such as, the CASIA v2.0, MICC-220, MICC-F600 and MICC-F2000 datasets. With the proposed algorithm, an overall average detection accuracy of 95.33% is obtained for evaluation carried out with the aforementioned databases. Forgery detection achieved True Positive Rate of 97.4% for tampered images with object translation, different degree of rotation and enlargement. Thus, results from different datasets have been set, proving that the proposed algorithm can individuate the altered areas, with high reliability and dealing with multiple cloning.