Moving Kriging Interpolation Research Articles

Development of integrated radial basis function Kriging interpolation for linear and nonlinear parabolic integro-differential equations

In this study, we explore linear and nonlinear parabolic integro-differential equations in one and two dimensions. We employ a semi-implicit scheme to discretize the temporal variable and discretize the spatial variable using an integrated radial basis function based on the moving Kriging interpolation (MKI) method. Unlike the global integrated radial basis function (IRBF) method, our proposed approach is particularly well-suited for addressing large-scale problems. Moreover, issues such as the selection of shape parameters leading to matrices with high condition numbers can be mitigated by devising an algorithm to determine optimal shape parameters, resulting in lower condition numbers. To assess the accuracy of the proposed method, we utilize two criteria, namely, L2 and H1. We compare the computed results obtained from the IRBF-MKI method with those from two grid and finite element methods. To demonstrate the accuracy and efficiency of our approach, we consider irregular areas and scattered Halton points in the two-dimensional case.

Vibration analysis of higher-order nonlocal strain gradient plate via meshfree moving Kriging interpolation method

In this paper, a higher-order nonlocal strain gradient plate model combining nonlocal elasticity theory with strain gradient theory in the form of adding signs is constructed. The governing equation for the higher-order nonlocal strain gradient plate model is derived by Hamilton’s principle. Two higher-order parameters with nonlocal stress and strain gradient are introduced here to explain the size effect and dispersive behaviour. The meshfree moving Kriging interpolation method is utilized to address the frequency of higher-order nonlocal strain gradient plate under complicated boundary conditions. Nonlocal elasticity theory, strain gradient theory in the form of add signs and minus signs on the effect of frequency are compared by degenerating the higher-order nonlocal strain gradient plate model. The other goal of this paper is to compare the present model with higher-order nonlocal strain gradient theory in the form of minus signs on the effect of frequency. It is shown that these two items can describe the size effect well. The present method can also produce lower natural frequencies corresponding to ultrahigh-order mode shapes and applied in a discrete medium.

A Moving Kriging Interpolation Meshless for Bending and Free Vibration Analysis of the Stiffened FGM Plates in Thermal Environment

This paper adopts the Moving Kriging (MK) interpolation meshless method to analyze the static and dynamic behaviors of stiffened functionally graded material (FGM) plate in thermal environment based on the physical neutral surface. The ribbed FGM plate is regarded as a composite structure of a FGM plate and ribs. The displacement transformation relationship between stiffeners and FGM plates is obtained through the displacement compatible conditions and MK interpolation. The meshfree model for ribbed FGM plate is obtained by superimposing the total energy of the FGM plate and the stiffeners based on the first-order shear deformation theory (FSDT) and physical neutral surface. The nonlinear temperature field along thickness direction is introduced into the meshless model of stiffened FGM plate. The equations governing the bending and free vibration of the ribbed FGM plate in thermal environment are obtained according to the principle of Minimum Potential Energy and Hamilton’s Principle. Thereafter, several ribbed FGM plate examples in different temperatures and with different locations of ribs are calculated. The results are compared with those given by the ABAQUS and literature. The results show that the effectiveness and accuracy of the proposed method in analyzing the ribbed FGM plate in thermal environment.

A Meshfree Approach Based on Moving Kriging Interpolation for Numerical Solution of Coupled Reaction-Diffusion Problems

In this paper, a meshfree approach based on moving kriging interpolation is presented for numerical solution of coupled reaction-diffusion problems. The proposed approach is developed based upon local collocation using moving Kriging shape function. It is truly meshless and having the Kronecker delta property for accurate imposition of boundary conditions. In the proposed model, the weight function is used with correlation parameter treated as the model internal length factor. This produces a local moving kriging method with improved accuracy together with an ease to choose the weight function factor. The method can hence be used in an efficient manner without cumbersome effort for choosing its parameter. The meshless approach is presented for the first time for numerical solution of reaction-diffusion systems. Problems of Turing system and pattern formation in several 2D domains are solved in this study. The efficacy and accuracy of the proposed method for the reaction-diffusion systems in different problem domains are presented in comparison to available exact solution and other numerical methods. It is found that the present method is accurate and effective as a computational procedure for solving reaction-diffusion problems.

A C0-HSDT free vibration of magneto-electro-elastic functionally graded porous plates using a moving Kriging meshfree method

This research examines the natural vibration analysis of magneto-electro-elastic functionally graded porous (MEE-FGP) plates. The study employs a moving Kriging (MK) meshfree method to analyze the MEE-FGP plates composed of piezoelectric and piezomagnetic materials with both even and uneven distributions of porosity. The electric and magnetic potentials in the three-dimension are assumed to follow a combination of cosine and linear variations in the thickness direction to satisfy the Maxwell equations in the two-dimension. The displacement field, electric potential and magnetic potential are approximated by the MK interpolation shape function, which satisfies the Kronecker's delta property. This property allows to impose directly essential boundary conditions as those in the finite element method. The governing equilibrium equations are derived using a C0-type higher-order shear deformation plate theory (C0-HSDT) in conjunction with the virtual work principle and then solved using the MK meshfree method. Various geometries are performed to determine the natural frequency of MEE-FGP plates. The impact of the porosity distribution, porosity volume fraction, volume fraction index, initial external electric voltage, initial magnetic potential and geometry on the natural frequency of the MEE-FGP plates is fully investigated. Numerical results provide an effective approach to analyze and calculate the natural vibration of MEE-FGP plates.

Vibration analysis of a cylindrical shell by using strain gradient theory via a moving Kriging interpolation-based meshfree method

The strain gradient theory is introduced into the Donnell shell model to investigate the vibration characteristics of a thin-walled cylindrical shell under classical and elastic boundary conditions. An energy equation concurrently involving the in-plane shell displacements and out-plane shell deflections is constructed. Meshfree discretization scheme is adopted, in which the moving Kriging interpolation, possessing C2 continuum and the Kronecker delta function property, is used to formulate the shape functions. The accuracy of the proposed method is first validated, and then parametric studies in terms of small-scale parameters, boundary conditions and the length-to-radius and thickness-to-radius ratios on the frequencies and mode shapes are carried out in detail. Natural frequencies evaluated by strain gradient elasticity theory are found to be lower than those of the classical Donnell cylindrical shell model. Some lower natural frequencies corresponding to ultrahigh-order mode shapes appear. This demonstrates that the size effect has a significant influence on the vibration of the thin-walled cylindrical shell. Five types of elastic boundary conditions are listed to illustrate the effect on the natural frequency. Meanwhile, it is concluded that the mesh-free moving Kriging interpolation method has high precision and good stability in dealing with strain gradient Donnell cylindrical shells.

A capable numerical meshless scheme for solving distributed order time-fractional reaction–diffusion equation

Distributed order fractional differential equations are efficient in describing physical phenomena because of the differential order distribution. In this paper, the distributed order time-fractional reaction–diffusion equation is considered in the sense of Caputo fractional derivative. A hybrid method is developed based on the Moving Kriging (MK) interpolation and finite difference method for solving this distributed order equation. First, the distributed integral is discretized by the M-point Gauss Legendre quadrature rule. Then, the L2−1σ method is applied to approximate the solution of the fractional derivative discretization. Also, the unconditionally stability and rate of convergence O(τ2) of the time-discrete technique are illustrated. Furthermore, the MK interpolation is resulted in the space variables discretization. Finally, some examples are presented to indicate the efficiency of this method and endorsement the theoretical results.

Large amplitude vibration and bistable jump of functionally graded graphene-platelet reinforced porous composite plates

Using high-performance lightweight functionally graded graphene-platelet reinforced (FG-GPLR) porous composite plates opens up an avenue for a new generation of structural and material designs, which could be tailored to the need of specific applications. Consider the von Karman plate model with geometrical nonlinearity, the equation of motion can be formulated. The Halpin − Tsai model is employed to predict the elastic properties of FG-GPLR porous composite plates. The moving Kriging interpolation mesh-free computational framework, which is confirmed to have high computing performance in terms of accuracy, convergence and robustness, is used to discretize the nonlinear vibration equation. Then, the linearized updated mode (LUM) method is used to obtain iterative solutions. An intensive work, investigating the influence of boundary constraints, porosity distributions, graphene platelets (GPLs), and vibration amplitudes, is conducted. It is found that a continuous increase in the amplitude parameter ultimately yields an abrupt change of the fundamental frequency, inducing a bistable vibration mode jump phenomenon. The present results can provide useful guidelines for the application design of nano-reinforced composite structures.

Meshless local Petrov-Galerkin method for 2D fractional Fokker-Planck equation involved with the ABC fractional derivative

This paper examines a time fractional version of the 2D Fokker-Planck equation involved with the Atangana-Baleanu-Caputo fractional derivative, under the Dirichlet boundary conditions. A fully discretization approach based on the meshless local Petrov-Galerkin method and ( 3 − β ) -order approximation is proposed for this equation. More precisely, we apply the meshless local Petrov-Galerkin method based on the Moving Kriging interpolation to discretize the space domain, and utilize the ( 3 − β ) -order approximation together with the θ -weighted finite difference method to discretize the temporal domain. By implementing this method, we get a solution for the problem by solving a system of algebraic equations. The validity of the method is investigated by solving four examples.

A meshless thermal modelling for functionally graded porous materials under the influence of temperature dependent heat sources

In this study, a meshless thermal modeling with meshless local moving kriging interpolation method is presented for analysis of functionally graded porous (FGP) materials under the influence of temperature dependent heat sources. The method is developed based on local collocation with moving kriging shape function. It is truly meshless and having the Kronecker delta property for accurate imposition of boundary conditions. In the proposed model, the weight function is used with correlation parameter treated as the model internal length factor. This produces a local moving kriging method with improved accuracy together with an ease to choose the weight function factor. The method can hence be used in an efficient manner without cumbersome effort for choosing its parameter. The contribution is highlighted in the present study, which is still unexplored in detail in literature. Different forms of porosity are considered to investigate the effect of porosity to heat transfer performance of FGP materials subjected to temperature dependent heat sources. In addition, the method is applied for thermal analysis of FGP materials considering arbitrary geometries, mixed boundary conditions and varied power-law index. Numerical results show the effectiveness and accuracy of the meshless method for thermal modeling and analysis of FGP materials.

Meshless thermal modeling and analysis of functionally graded materials using a meshless local moving kriging interpolation method

In this study, meshless thermal analysis of functionally graded materials (FGM) is presented by using a meshless local moving kriging interpolation method. The method is developed based on local collocation with moving kriging shape function. It is truly meshless and having the Kronecker delta property for accurate imposition of boundary conditions. To improve its accuracy, the weight function is used with correlation parameter treated as the model internal length factor. This produces a meshless local method with clear meaning of model parameter, improved accuracy and allowing a more straightforward analysis for parametric study. Effects of weight function type, parameter value and number of supporting nodes to the method accuracy and convergence rate are first examined in a detailed parametric study of a benchmark problem. Based on the study, suitable parameter, weight function and effective range for supporting nodes number are elucidated. The proposed method is then applied for complex thermal analysis of FGM in arbitrary geometries, taking into account for both temperature and spatial dependent material properties and varied power-law index, including the presence of porosity as well. Numerical results show the effectiveness and accuracy of the meshless method for thermal modeling and analysis of FGM.

Static and free vibration analysis of stiffened FGM plate on elastic foundation based on physical neutral surface and MK method

This paper analyzes the static and free vibration problems of the stiffened functionally graded material (FGM) plate resting on Pasternak foundation by using the moving Kriging (MK) approximation and the physical neutral surface. The stiffened FGM plates are regarded as composite structures of FGM plates and stiffeners (beams). The displacement transformation relationship between stiffeners and FGM plates is obtained through the displacement compatible conditions and moving kriging interpolation. The total potential energy and kinetic energy of FGM plate resting on Pasternak foundation are derived based on the first-order shear deformation theory (FSDT) and physical neutral surface and by superimposing those energies of the stiffeners and the FGM plate. According to the principle of Minimum Potential Energy and Hamilton's Principle, the meshless governing equations on the static and free vibration of the stiffened FGM plate are obtained, respectively. The convergence and accuracy of the proposed method are demonstrated by calculating several numerical examples. Moreover, the effect of the power-law exponent, boundary conditions and foundation parameter on the static and free vibration behavior of the stiffened FGM plates resting on Pasternak foundation are discussed.

A meshfree model enhanced by NURBS-based Cartesian transformation method for cracks at finite deformation in hyperelastic solids

This paper presents a numerical investigation of single and multiple cracks in hyperelastic solids by using an extended moving Kriging meshfree method. The behavior of crack such as jump of displacement field across crack surface is mathematically captured by enriched functions, which are selected based on asymptotic solution. For meshless numerical integration, the NURBS-enhanced Cartesian transformation method is proposed to evaluate domain integrals of geometries with curved boundaries, without the need for sub-division of problem domain into background cells. It is guaranteed that no discontinuity is located within integration domain, as crack surfaces are viewed as part of the boundaries. Thanks to the Kronecker’s delta property of the moving Kriging interpolation, direct imposition of boundary conditions can thus be made. The correlation parameter, which has certain effects on the accuracy of approximation, is not necessary to be pre-defined in this work. Instead, it is determined according to the characteristic nodal distance in every support domain. The accuracy and efficiency of the present approach are investigated in various examples and computed results are compared with available analytical solutions and/or other numerical methods.

Vibration analysis of a strain gradient plate model via a mesh-free moving Kriging Interpolation Method

The strain gradient elasticity theory is adopted to capture the small-scale effect of micro/nanostructures to derive the governing equation by using Hamilton's principle. A mesh-free method on the basis of moving Kriging interpolation, of which shape function possesses C2 continuum, is developed to investigate the vibration of the strain gradient plate. Owing to the inherent Kronecker delta function property of the moving Kriging interpolation, essential boundary conditions can be directly implemented. One superiority of the moving Kriging interpolation over the traditional moving least square approximation is that it can ensure the continuity and stability at the third order derivatives, which is confirmed by comparison with analytic solutions. Parametric studies in terms of small-scale parameters, boundary conditions and side length on the frequencies and mode shapes are carried out. Natural frequencies evaluated by strain gradient elasticity theory are found to be lower than those of classical plate model. Some lower natural frequencies corresponding to ultrahigh-order mode shapes appear. This demonstrates that the mesh-free moving Kriging interpolation method has high precision and good stability in the construction of the C2 continuum shape functions.

Element-Free Galerkin Scaled Boundary Method Based on Moving Kriging Interpolation for Steady Heat Conduction Analysis with Temperatures on Side-Faces

Element-Free Galerkin Scaled Boundary Method Based on Moving Kriging Interpolation for Steady Heat Conduction Analysis with Temperatures on Side-Faces

A refined quasi-3D logarithmic shear deformation theory-based effective meshfree method for analysis of functionally graded plates resting on the elastic foundation

In this paper, a new refined quasi-3-dimensional logarithmic shear deformation theory (RQ-3DLSDT) and an advanced moving Kriging interpolation (AMKI) based-meshless method is combined for the first time to study the static bending, free vibration, and compressive buckling analysis of isotropic and sandwich functionally graded plates laid on elastic foundations. The RQ-3DLSDT considering the effect of thickness stretching based only on four-unknown variables to determine the displacement field leading to reduce computational efforts. The weak forms of the equilibrium equation are discretized and numerically solved by the AMKI mesh-free method employed a proposed quadric correlation function for getting stable solutions.

A reliable algorithm to determine the pollution transport within underground reservoirs: implementation of an efficient collocation meshless method based on the moving Kriging interpolation

A reliable algorithm to determine the pollution transport within underground reservoirs: implementation of an efficient collocation meshless method based on the moving Kriging interpolation

Static bending and free vibration analysis of functionally graded porous plates laid on elastic foundation using the meshless method

This paper presents a numerical approach for static bending and free vibration analysis of the functionally graded porous plates (FGPP) resting on the elastic foundation using the refined quasi-3D sinusoidal shear deformation theory (RQSSDT) combined with the Moving Kriging–interpolation meshfree method. The plate theory considers both shear deformation and thickness-stretching effects by the sinusoidal distribution of the in-plane displacements, satisfies the stress-free boundary conditions on the top and bottom surfaces of the plate without shear correction coefficient. The advantage of the plate theory is that the displacement field of plate is approximated by only four variables leading to reduce computational efforts. Comparison studies are performed for the square FGPP with simply supported all edges to verify the accuracy of the present approach. The effect of the aspect ratio, volume fraction exponent, and elastic foundation parameters on the static deflections and natural frequency of FGPP are also investigated and discussed.
 Keywords:
 meshless method; Moving Kriging interpolation; refined quasi-3D theory; porous functionally\break graded plate; Pasternak foundation.

Meshfree analysis of functionally graded plates with a novel four-unknown arctangent exponential shear deformation theory

A novel refined arctangent exponential shear deformation theory (RAESDT) is presented for analysis the mechanical behavior of both isotropic and sandwich FGM plates. Material properties are set to be isotropic at each point and varied across the thickness direction obeying to a power-law distribution of the volume fraction gradation with respect to FGM core or skins of the plate. Unlike high-order shear deformation plate theories based on five or more variables, the displacement field of the novel RAESDT using arctangent exponential variations in planed displacements were approximated by only four unknowns, satisfying naturally tangential stress-free conditions at the plate surfaces and leading to reduce computational efforts. In accordance with RAESDT and enhanced moving kriging interpolation (EMKI)-based meshfree method with a new quadrature correlation function is introduced for the numerical modeling. Numerical validations with different plate configurations, geometries, length to thickness ratios and boundaries conditions are conducted. The obtained results are compared with the corresponding solutions available in the literature showing the accuracy and efficiency of the present approach.

An extended element free Galerkin method based on moving kriging interpolation for second-order elliptic interface problems

The aim of this paper is to introduce an efficient meshless element free Galerkin technique for solving elliptic interface problems. In this work, the second-order elliptic equation with discontinuous coefficients and homogeneous and inhomogeneous jump conditions is considered. Moving kriging interpolation is chosen to construct shape functions in the proposed method. To apply the jump conditions in the weak form of the problem, Nitsche's method is used. Some examples are presented to confirm the effectiveness of the proposed method for interface problems.