Moving Kriging Interpolation Research Articles

An Element-Free Galerkin Method Based on Complex Variable Moving Kriging Interpolation for Potential Problems

The moving Kriging interpolation (MKI) is an accurate approximation method that has the interpolating property. However, it is rarely used in meshless methods because of its low efficiency. In this paper, we proposed an efficient MKI method, the complex variable moving Kriging interpolation (CVMKI) method, for “domain” type meshless method. Further, we proposed the CVMKI-based element-free Galerkin (CVMKIEFG) method for 2D potential problems. CVMKIEFG is an efficient meshless method and can impose the essential boundary conditions directly and easily. We proposed two formulations for CVMKIEFG: the conventional formulation and the cell-based formulation. The latter formulation is proposed for higher efficiency. Three 2D example problems are presented to demonstrate the efficiency and accuracy of CVMKIEFG.

The meshless local Petrov-Galerkin method based on moving Kriging interpolation for solving the time fractional Navier-Stokes equations.

In this paper, we present a numerical scheme used to solve the nonlinear time fractional Navier–Stokes equations in two dimensions. We first employ the meshless local Petrov–Galerkin (MLPG) method based on a local weak formulation to form the system of discretized equations and then we will approximate the time fractional derivative interpreted in the sense of Caputo by a simple quadrature formula. The moving Kriging interpolation which possesses the Kronecker delta property is applied to construct shape functions. This research aims to extend and develop further the applicability of the truly MLPG method to the generalized incompressible Navier–Stokes equations. Two numerical examples are provided to illustrate the accuracy and efficiency of the proposed algorithm. Very good agreement between the numerically and analytically computed solutions can be observed in the verification. The present MLPG method has proved its efficiency and reliability for solving the two-dimensional time fractional Navier–Stokes equations arising in fluid dynamics as well as several other problems in science and engineering.

Two meshless procedures: moving Kriging interpolation and element-free Galerkin for fractional PDEs

In this paper, the element-free Galerkin (EFG) meshless method and moving Kriging collocation meshless technique are applied for finding the numerical solution of a class of two-dimensional (2D) nonlinear time fractional partial differential equations. The Klein–Gordon, sine-Gordon, diffusion wave and Cattaneo equations with Neumann boundary condition are studied. The time fractional derivative has been described in the Caputo’s sense. Firstly, we use a semi-implicit finite difference scheme of convergence order , and then for obtaining a full discrete scheme, the space derivative is discretized with the EFG and moving Kriging collocation techniques. The EFG method uses a weak form of the considered equation that is similar to the finite element method with the difference that in the EFG method, test and trial functions are moving least squares approximation (MLS) shape functions. Also, in the element-free Galekin method, we do not use any triangular, quadrangular, or other types of meshes. The EFG method is a global method while finite element method is a local one. The EFG method is not a truly meshless method and for integration uses a background mesh. We prove the unconditional stability and obtain an error bound for the EFG method using the energy method. Numerical examples are reported which support the theoretical results and the efficiency of the proposed scheme.

A numerical study of the European option by the MLPG method with moving kriging interpolation.

In this paper, the meshless local Petrov–Galerkin (MLPG) method is applied for solving a generalized Black–Scholes equation in financial problems. This equation is a PDE governing the price evolution of a European call or a European put under the Black–Scholes model. The θ-weighted method and MLPG are used for discretizing the governing equation in time variable and option pricing, respectively. We show that the spectral radius of amplification matrix with the discrete operator is less than 1. This ensures that this numerical scheme is stable. Numerical experiments are performed with time varying volatility and the results are compared with the analytical and the numerical results of other methods.

Complex variable moving Kriging interpolation for boundary meshless method

In this paper, we proposed the complex variable moving Kriging interpolation (CVMKI) to approximate functions on two-dimensional (2D) boundaries. The CVMKI is based on complex variable theory and the moving Kriging interpolation (MKI). It requires no curvilinear coordinate, and can construct shape functions possessing Kronecker delta function property and partition of unity property. Further, the complex variable boundary node method (CVBNM) is proposed for potential problems based on CVMKI and boundary integration equation (BIE). CVBNM is an efficient and accurate method that can directly impose the boundary conditions. Three 2D example problems are presented to verify the accuracy and efficiency of CVBNM.

An improved Moving Kriging-based meshfree method for static, dynamic and buckling analyses of functionally graded isotropic and sandwich plates

A meshfree method with a modified distribution function of Moving Kriging (MK) interpolation is investigated. This method is then combined with a high order shear deformation theory (HSDT) for static, dynamic and buckling analyses of functionally graded material (FGM) isotropic and sandwich plates. A meshfree method uses the normalized form of MK interpolation under a new quartic polynomial correlation to build the basis shape functions in high order approximations. The Galerkin weak form is used to separate the system equations which is numerically solved by meshfree method. A rotation-free technique extracted from isogeometric analysis is introduced to eliminate the degrees of freedom of slopes. Then, the method retains a highly computational effect with a lower number of degrees of freedom. In addition, the requirement of shear correction factors is ignored and the traction free is at the top and bottom surfaces of FGM plates. Various thickness ratios, boundary conditions and material properties are studied to validate the numerical analyses of the rectangular and circular plates. The numerical results show that the present theory is more stable and well accurate prediction as compared to three-dimensional (3D) elasticity solution and other meshfree methods in the literature.

The use of element free Galerkin method based on moving Kriging and radial point interpolation techniques for solving some types of Turing models

In this paper two numerical procedures are presented for solving a class of Turing system. Firstly, we obtain a time discrete scheme by approximating time derivative via finite difference technique. Then we introduce the moving Kriging interpolation and radial point interpolation and also obtain their shape functions. We use the element free Galerkin method for approximating the spatial derivatives. This method uses a weak form of the considered equation that is similar to the finite element method with the difference that in the classical element free Galerkin method test and trial functions are moving least squares (MLS) approximation shape functions. Since the shape functions of moving least squares (MLS) approximation do not have Kronecker delta property, we cannot implement the essential boundary condition, directly. Thus we employ the shape functions of moving Kriging interpolation and radial point interpolation technique which have the mentioned property. Also, in the element free Galerkin method, we do not use any triangular, quadrangular or other type of meshes. The element free Galerkin method is a global method while finite elements method is a local one. This technique employs a background mesh for integration which makes it different from the truly mesh procedures. The coefficient matrix of the element free Galerkin is symmetric. Also, using numerical algorithms, we can conclude that the eigenvalues of the coefficient matrix are positive. Thus, for solving the obtained linear system of equations from the discretization, we use the conjugant gradient method. To keep away from solving a nonlinear algebraic system of equations and obtaining the acceptable numerical results, we use a predictor–corrector algorithm. Several test problems are solved and numerical simulations are reported which confirm the efficiency of the proposed schemes.

The meshless local Petrov–Galerkin based on moving kriging interpolation for solving fractional Black–Scholes model

In this paper, the fractional Black–Scholes equation in financial problem is solved by using the numerical techniques for the option price of a European call or European put under the Black–Scholes model. The MLPG and implicit finite difference method are used for discretizing the governing equation in option price and time variable, respectively. In MLPG method, the shape function is constructed by a moving kriging approximation. The Dirac delta function is chosen to be the test function. The numerical examples for varieties of variables are also included.

Meshless Method Based on Moving Kriging Interpolation for Solving Simply Supported Thin Plate Problems

Meshless method choosing Heaviside function as a test function for solving simply supported thin plates under various loads as well as on regular and irregular domains is presented in this paper. The shape functions using regular and irregular nodal arrangements as well as the order of polynomial basis choice are constructed by moving Kriging interpolation. Alternatively, two-field-variable local weak forms are used in order to decompose the governing equation, biharmonic equation, into a couple of Poisson equations and then impose straightforward boundary conditions. Selected mechanical engineering thin plate problems are considered to examine the applicability and the accuracy of this method. This robust approach gives significantly accurate numerical results, implementing by maximum relative error and root mean square relative error.

Reliability Analysis of Composite Laminate Structures by Moving Kriging Interpolation Response Surface Method

A response surface method (RSM) for composite laminate structures is proposed in this paper, which is based on the moving Kriging interpolation. The substitute limit state function for failure criteria is discussed and constructed on series of deterministic finite element analysis. Combined with first order reliability method, reliabilities of composite laminate structures are subsequently obtained. Reliability analysis of a composite laminate plate, as a numerical example, is illustrated by the proposed method. The results demonstrate the practicability of the method.

MLPG method based on moving kriging interpolation for solving convection–diffusion equations with integral condition

A formulation of the meshless local Petrov–Galerkin (MLPG) method based on the moving kriging interpolation (MK) is presented in this paper. The method is used for solving time-dependent convection–diffusion equations in two-dimensional spaces with the Dirichlet, Neumann, and non-local boundary conditions on a square domain. The method is developed based on the moving kriging interpolation method for constructing shape functions which have the Kronecker delta property. In the method, the test function in each sub-domain is chosen as the indicator function. The Crank–Nicolson method is chosen for temporal discretization. Two test problems are presented which demonstrate the easiness and accuracy of this method as shown by the relative error.

A novel meshless local Petrov–Galerkin method for dynamic coupled thermoelasticity analysis under thermal and mechanical shock loading

A meshless local Petrov–Galerkin method (MLPG) based on the moving Kriging interpolation is further developed for two-dimensional linear dynamic coupled thermoelasticity problems under thermal and mechanical shock loading. In this method the moving Kriging interpolation is employed to construct the shape functions, and the Heaviside step function is applied in the local weak-form as the test functions. The equations of motion and transient heat conduction equations of the coupled thermoelasticity interact on each other. So these equations must be solved simultaneously. In this paper the backward difference method is applied to approximate the time derivatives. The main idea of treating the time-dependent problem is based on the reduction procedure of the equations of motion, in which the displacements, velocities and temperature are taken as the field variables. Then a unified explicit expression of the system of differential algebraic equations is obtained. Numerical examples are presented to illustrate the accuracy of this method.

An effective sub-domain smoothed Galerkin method for free and forced vibration analysis

Free and forced vibration analysis is significant in structural mechanics and engineering. A sub-domain smoothed Galerkin method is developed for free and forced vibration analyses of 2D solids. This method integrates the advantages of mesh-free Galerkin method and finite element method (FEM). The arbitrarily shaped sub-domains are predefined in the problems domain represented by mesh-free nodes. In each sub-domain, based on mesh-free Galerkin weak formulation, the local discrete equation can be obtained by using moving kriging interpolation, which is similar to the discrete equation of a high-order element in FEM. The strain smoothing technique can be subsequently applied to the nodal integration of sub-domain by dividing the sub-domain into several smoothing cells. Moreover, condensation of degrees of freedom can also be introduced into discrete equations by transfer equations of inner nodes to equations of boundary nodes based on sub-domains. The global dynamic equations of the present method are obtained based on the scheme of FEM by assembling all local discrete equations of sub-domains and are solved by using the standard implicit Newmark’s time integration scheme. Numerical examples proved that the sub-domain smoothed Galerkin method is a robust technique to simulation the elastic dynamic problems, which has high computational efficiency and good accuracy.

Two-field-variable meshless method based on moving kriging interpolation for solving simply supported thin plates under various loads

Meshless method choosing Heaviside step function as a test function for solving simply supported thin plates under various loads is presented in this paper. The shape functions using regular and irregular nodal distribution as well as order of polynomial basis choice are constructed by moving kriging interpolation. Alternatively, two-field-variable local weak forms are used in order to decompose the governing equation, biharmonic equation, into a couple of Poisson equations and then impose straightforward boundary conditions. Selected numerical examples are considered to examine the applicability, the easiness, and the accuracy of the proposed method. Comparing to an exact solution, this robust method gives significantly accurate numerical results, implementing by maximum relative error and root mean square relative error.

Offshore Structural Dynamic Analysis Considering Soil-Structure Interaction by a Coupled BEM and Meshfree Method

An investigation of dynamic offshore structure considering soil-structure interaction by a coupled approach based on boundary element and meshfree methods is presented. The moving Kriging interpolation is used for generating the shape functions because of possessing the Kronecker delta property, which makes easier in imposing the essential boundary conditions. The governing elastodynamic equations are transformed into a standard weak formulation. It is then discretized into a meshfree system of time-dependent equations, which are solved by the standard implicit Newmark time integration scheme. Numerical examples illustrating the applicability and effectiveness of the proposed method are presented and discussed in details. A two dimensional plane strain offshorefoundation model has been used for the time history analysis to compute the stresses and displacements against earthquake and wave loading considering the effect of soil-structure interaction. As a consequence, it is found that the method is very efficient and accurate for dynamic analysis compared with those of other conventional methods.

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 interpolation 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.

A sub‒domain smoothed Galerkin method for solid mechanics problems

SUMMARYA sub‒domain smoothed Galerkin method is proposed to integrate the advantages of mesh‒free Galerkin method and FEM. Arbitrarily shaped sub‒domains are predefined in problems domain with mesh‒free nodes. In each sub‒domain, based on mesh‒free Galerkin weak formulation, the local discrete equation can be obtained by using the moving Kriging interpolation, which is similar to the discretization of the high‒order finite elements. Strain smoothing technique is subsequently applied to the nodal integration of sub‒domain by dividing the sub‒domain into several smoothing cells. Moreover, condensation of DOF can also be introduced into the local discrete equations to improve the computational efficiency. The global governing equations of present method are obtained on the basis of the scheme of FEM by assembling all local discrete equations of the sub‒domains. The mesh‒free properties of Galerkin method are retained in each sub‒domain. Several 2D elastic problems have been solved on the basis of this newly proposed method to validate its computational performance. These numerical examples proved that the newly proposed sub‒domain smoothed Galerkin method is a robust technique to solve solid mechanics problems based on its characteristics of high computational efficiency, good accuracy, and convergence. Copyright © 2014 John Wiley & Sons, Ltd.

AN INTERPOLATING LOCAL PETROV–GALERKIN METHOD FOR POTENTIAL PROBLEMS

In this paper, based on the moving Kriging interpolation (MKI), the meshless interpolating local Petrov–Galerkin (ILPG) method is presented to solve two- and three-dimensional potential problems. In the present method, the shape function constructed by the MKI has the property of the Kronecker δ function. Then in the ILPG method the essential boundary conditions can be implemented directly. The discrete equations are obtained using the local symmetric weak form. The Heaviside step function is used as the test function in each sub-domain to avoid some domain integral in the symmetric weak form, which will greatly improve the effectiveness of the present method. The ILPG method in this paper is a truly meshless method, which does not require a mesh either for obtaining shape function or for numerical integration in the local weak form. Several numerical examples of potential problems show that the ILPG method has higher computational efficiency and convergence rate than the MLPG method.

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 dynamic analysis. In the present method, scattered nodes without 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 approximation. Then local discrete equations can be simplified by condensation 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 assembling 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 implemented in interpolation. Numerical examples show that the present method has high computational efficiency and good accuracy in solving elastic dynamic problems.

A Local Integral Equation Formulation Based on Moving Kriging Interpolation for Solving Coupled Nonlinear Reaction-Diffusion Equations

The meshless local Pretrov-Galerkin method (MLPG) with the test function in view of the Heaviside step function is introduced to solve the system of coupled nonlinear reaction-diffusion equations in two-dimensional spaces subjected to Dirichlet and Neumann boundary conditions on a square domain. Two-field velocities are approximated by moving Kriging (MK) interpolation method for constructing nodal shape function which holds the Kronecker delta property, thereby enhancing the arrangement nodal shape construction accuracy, while the Crank-Nicolson method is chosen for temporal discretization. The nonlinear terms are treated iteratively within each time step. The developed formulation is verified in two numerical examples with investigating the convergence and the accuracy of numerical results. The numerical experiments revealing the solutions by the developed formulation are stable and more precise.