Moving Kriging Shape Functions Research Articles

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 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.

A moving Kriging meshfree approach for free vibration and buckling analyses of porous metal foam plates

In this paper, the free vibration and mechanical buckling analyses of the metal foam plates with porosities via a moving Kriging meshfree method based on the higher-order shear deformation theory are studied. The displacement fields are approximated by the moving Kriging shape functions, which satisfy Kronecker’s delta property. Thanks to that property, the essential boundary conditions can be directly imposed the same with finite element method. In this study, the porosities are distributed by uniform, symmetric, and asymmetric distributions along the thickness direction. The natural frequency and critical buckling load of the metal foam plate are determined by solving the explicit governing equations, which are derived through the variational formulation. The influence of the porosity distribution, porous coefficient, width-to-thickness ratio and boundary condition on the natural frequency and critical buckling load of the porous metal foam plate is investigated and discussed in detail. Numerical examples indicate that the present approach is stable and well accurate predictions for investigating vibration and buckling behaviors of the porous metal foam plate. Moreover, an increase of the porous coefficient makes a decline of the natural frequencies and critical buckling loads of the plate.

Probabilistic-based nonlinear stability analysis of randomly reinforced microshells under combined axial-lateral load using meshfree strain gradient formulations

In the present exploration, an effective numerical strategy is developed to examine for the first time different microstructural-dependent features in the nonlinear stability behavior of nanocomposite microshells containing randomly dispersed nanofillers under combinations of axial and lateral compressive loads. Correspondingly, the modified strain gradient theory of elasticity incorporating dilatation, deviatoric stretch, and rotation gradient tensors is applied to the third-order shear flexible shell framework to accommodate the size dependency. The efficacious material properties are captured via Tsai homogenization approach together with a Monte-Carlo simulation based upon a probabilistic-based homogenization scheme. By employing the constructed moving Kriging meshfree-based numerical strategy, the essential boundary conditions are directly enforced at nodes and the Kronecker delta is satisfied comprehensively using correct type of moving Kriging shape function. For the combined loading condition having domination of axial compression, it is seen that by taking a lateral compressive load into account, the region of snap-through experience becomes wider and the associated minimum load switches to a more induced deflection, while a smaller shortening. Furthermore, for particular values of volume fraction and specific area of graphene nanofillers, combining of axial or lateral compressive load causes that the roles of microscale-dependent gradient tensors in the value of critical stability loads of combined compressed microsized shells becomes a bit less.

A couple-stress-based moving Kriging meshfree shell model for axial postbuckling analysis of random checkerboard composite cylindrical microshells

The current investigation intends to devise a couple stress continuum-based moving Kriging meshfree shell model for axial nonlinear buckling and postbuckling responses of random checkerboard composite cylindrical microshells. Accordingly, the graphene nanoplatelet reinforcements are supposed to be randomly dispersed based upon a checkerboard scheme within the resin matrix, the effective material properties of which are estimated based upon a probabilistic method. The modified couple stress continuum elasticity is implemented into the higher-order shear deformation shell theory incorporating geometrical nonlinearity. The established microstructural-dependent shell model is then numerically analyzed via the moving Kriging meshfree technique with the ability to take the essential boundary conditions into account accurately by employing proper moving Kriging shape function. By tracing the postbuckling paths, the snap-through phenomenon is observed. It is found that for a higher graphene nanoplatelet volume fraction, the significance of the stiffening character associated with the rotation gradient tensor increases. Furthermore, for a specific value of graphene nanoplatelet volume fraction, by reducing the length to width aspect ratio of graphene nanoplatelet reinforcements, the role of couple stress size dependency in the critical buckling load and critical shortening of an axially compressed cylindrical microshell becomes more important.

Microstructural Size Dependency in Nonlinear Lateral Stability of Random Reinforced Microshells via Meshfree-Based Applied Mathematical Modeling

The prime objective of this research work is to develop an efficient small scale-dependent computational framework incorporating microstructural tensors of dilatation gradient, rotation gradient, and deviatoric stretch gradient to analyze nonlinear lateral stability of cylindrical microshells. The numerical strategy is established based upon a mixed formation of the third-order shear deformable shell model and modified strain gradient continuum mechanics. The graphene nanoplatelet reinforcements are assumed to be randomly dispersed in a checkerboard scheme within the resin matrix. Accordingly, to extract the effective material properties, the Monte Carlo simulation together with a probabilistic technique are employed. The numerical solution for the microstructural-dependent nonlinear problem is carried out via the moving Kriging meshfree method having the capability to accommodate accurately the essential boundary conditions using proper moving Kriging shape function. It is represented that the role of the stiffening characters related to the effect of microstructural dilatation gradient, rotation gradient, and deviatoric stretch reduces continuously by going to deeper territory of the load-deflection stability path. Moreover, it is indicated that among various microstructural gradient tensors, the stiffening character of the rotation gradient is higher than deviatoric stretch gradient, and the stiffening character of the latter is more considerable than the dilatation gradient tensor.

A meshfree approach using naturally stabilized nodal integration for multilayer FG GPLRC complicated plate structures

In this study, a meshfree approach using a naturally stabilized nodal integration (NSNI) integrated with a higher-order shear deformation theory (HSDT) for free vibration analysis of multilayer functionally graded graphene platelets reinforced composite (FG GPLRC) plates with complicated shapes is presented. Four different patterns of graphene platelets (GPLs) including uniform and functionally graded distributions are exampled. The rule of mixtures and modified Halpin–Tsai model are respectively used to compute the Poisson's ratio, density and Young's modulus. Based on the principle of virtual work, discretize governing equations of FG GPLRC plates are derived. These equations are solved by a moving Kriging (MK) meshfree method to determine natural frequencies of the FG GPLRC plates. The advantage of moving Kriging shape function is content with the Kronecker delta function property, thus essential boundary conditions are simply imposed as in the finite element method. In addition, the present meshfree approach uses the direct nodal integration with an additional augmentation of stability components to reduce the computational cost as compared to the high-order Gauss quadrature scheme. As observed in numerical examples, natural frequencies of FG GPLRC plates are impressed by the geometries, boundary conditions and distributed patterns of GPLs. It is also seen that the present approach is simply, stable and good predictions for FG GPLRC plates.

A size-dependent moving Kriging meshfree model for deformation and free vibration analysis of functionally graded carbon nanotube-reinforced composite nanoplates

In this paper, a size-dependent meshfree model using the higher-order shear deformation plate theory in conjunction with the nonlocal Eringen elasticity theory for bending and free vibration analyses of functionally graded carbon nanotube-reinforced composite (FG CNTRC) nanoplates is presented. Configurations of carbon nanotubes (CNTs) are carried out for the uniform and functionally graded distributions via the plate thickness. Effective material properties are computed by the extended rule of mixture. The differential equation form of nonlocal elasticity theory is utilized to take account of size-dependent effects. Based on the principle of virtual work, discretized governing equations for nanoplates are obtained. Thereafter, the displacement and natural frequency of the FG CNTRC nanoplates are determined by a moving Kriging meshfree method. Essential boundary conditions are directly enforced at nodes the same with the finite element method because the moving Kriging shape function satisfies the Kronecker delta function property. Numerical results prove that the present model is simple, stable and well accurate prediction for nanostructures. Moreover, the stiffness-softening mechanisms are found when using the nonlocal elasticity theory leading to a rise of deflection and a decrease of natural frequency of FG CNTRC nanoplates.

Element-free Galerkin scaled boundary method based on moving Kriging interpolation for steady heat conduction analysis

This paper develops an element-free Galerkin scaled boundary method (EFG-SBM) for solving steady heat conduction problems, in which the circumferential direction is constructed by the moving Kriging (MK) interpolation based on the EFG approach. Because the MK interpolation satisfies the Kronecker delta property, it is more convenient in enforcing the essential boundary conditions than the traditional MLS-based EFG-SBM. As a newly boundary-type meshless method, EFG-SBM possesses advantages of EFG and scaled boundary finite element method (SBFEM). This method inherits the semi-analytical property of SBFEM by introducing the normalized radial coordinate system, in which the governing differential equations are weakened in the circumferential direction and solved analytically in the radial direction. Unlike the traditional SBFEM, the preprocessing and postprocessing processes of EFG-SBM are simplified and the more accuracy can be obtained because of higher continuity of the MK shape functions. Computation of EFG-SBM can be reduced since only the boundary needs to be discretized compared with the EFG approach. This proposed method is verified via four heat conduction examples including problems with thermal crack considering the prescribed heat flux and temperature on the side-face and unbounded domain. The numerical solutions show that EFG-SBM has higher accuracy and better convergence than the traditional SBFEM. An accurate smooth heat flux can be obtained directly without necessity of using the recovery procedure.

Extended meshless moving Kriging method for crack propagation analyzing in orthotropic media

Orthotropic composite material is the particular type of anisotropic materials and their products have been extensively used in a wide range of engineering applications. Study on mechanical behaviors of such materials under working conditions is very essential. In this study, an extended meshfree moving Kriging interpolation method (namely as X- MK) is presented for crack analyzing in 2D orthotropic materials models. The Gaussian function is used for constructing the moving Kriging shape functions. Typical advantages of the MK shape function are the high-order continuity and the satisfaction of the Kronecker’s delta property. To calculate the stress intensity factors (SIFs), interaction integral method is used with orthotropic auxiliary fields. Several numerical tests including static SIFs calculating and crack propagation predicting are performed to verify the accuracy of the present approach. The obtained results are compared with available refered results and they have shown a very good performance of the present method.

A numerical study on the nonlinear behavior of corner supported flat and curved panels

The nonlinear behavior of corner supported plates and curved shell panels is investigated here using the first-order shear deformation theory based on Marguerre’s membrane strains for shallow shells and von Karman’s nonlinearity. The nonlinear differential equations are transformed into a set of nonlinear algebraic equations by using the element-free Galerkin method. The moving kriging shape function with two different types of correlation formulae (Gaussian and quartic spline) is employed here. After studying the effectiveness of the method, a detailed parametric study is conducted to examine the effect of support size on the displacements and bending moments of corner supported rectangular plates. Thereafter, the numerical study is extended to the nonlinear bending and stability behaviors of corner supported shallow cylindrical and spherical shell panels.

Moving Kriging shape function modeling of vector TARMA models for modal identification of linear time-varying structural systems

This work proposes a Moving Kriging (MK) shape function modeling method for modal identification of linear time-varying (LTV) structural systems based on vector time-dependent autoregressive moving average (VTARMA) models. It aims to avoid the functional subspaces selection of the conventional functional series VTARMA (FS-VTARMA) models. Instead of the common basis functions, it constructs the time-varying coefficients on the time nodes with the MK shape functions in a compact support domain. The merit of the MK shape function is to determine its shape parameters upon vector random vibration signals adaptively. Model identification is effectively dealt with through an optimization scheme that decomposes the identification problem into two subproblems: estimating model parameters via two-stage least squares (2SLS) method and estimating shape function parameters via a discrete-continuous-variable hybrid optimization. In addition, the model order selection is achieved by the optimization scheme. This method has been validated by a Monte Carlo study of simulation case and further by an experimental test case, and the performance and potential advantages are illustrated.

Moving Kriging reconstruction for high-order finite volume computation of compressible flows

This paper describes the development of a high-order finite volume method for the solution of compressible viscous flows on unstructured meshes. The novelty of this approach is based on the use of moving Kriging shape functions for the computation of the derivatives in the numerical flux reconstruction step at the cell faces. For each cell, the successive derivatives of the flow variables are deduced from the interpolation function constructed from a compact stencil support for both Gaussian and quartic spline correlation models. A particular attention is paid for the study of the influence of the correlation parameter onto the accuracy of the numerical scheme. The effect of the size of the moving Kriging stencil is also investigated. Robustness and convergence properties are studied for various inviscid and viscous flows. Results reveal that the moving Kriging shape function can be considered as an interesting alternative for the development of high-order methodology for complex geometries.

AN ENHANCEMENT OF FINITE ELEMENT METHOD WITH MOVING KRIGING SHAPE FUNCTIONS

This paper presents an enhancement of the finite element method (FEM) by adopting the moving Kriging (MK) interpolation as a substitute for the traditional hat functions. The MK shape functions can be referred as element-free because their construction is not tied to the element geometry. Kriging interpolation is a geostatistical technique for spatial interpolation. The basic idea of Kriging is that any unknown point can be interpolated from known scatter points in a specific domain. Using the moving Kriging interpolation, shape functions can be generated over any finite set of nodes. This leads to an idea to extend the influence of a node beyond the layer of surrounding elements to enhance the global smoothness of the field variable and its derivatives. The present paper thus proposes a concept of layered domain of influence. Hence, characteristic arrays of an element, such as the element stiffness, have contributions from all visible nodes that include a set of satellite nodes unattached to the element. The validation of the method was confirmed through numerical tests of one and two-dimensional problems. The results show remarkable accuracy and global smoothness. Existing general-purposed FE programs can be easily modified to accommodate this new element concept; thus, the method has a higher chance to be accepted in practice.

FURTHER INVESTIGATION OF ELEMENT-FREE GALERKIN METHOD USING MOVING KRIGING INTERPOLATION

Following its first introduction, this study further scrutinizes the new type of shape functions for Element-free Galerkin Method (EFGM) based on the Moving Kriging (MK) interpolation. Kriging is a geostatistical method of spatial interpolation. Its basic premise is that every unknown point can be interpolated from known scattered points in its specified neighborhood. This property is ideal for EFGM. Previously, a shortcoming of EFGM based on Moving Least Square (MLS) approximation is associated with its limitation to satisfy essential boundary conditions exactly. With MK interpolation functions, EFGM solution can satisfy essential boundary conditions automatically. Numerical tests on one and two-dimensional elasticity problems have confirmed the effectiveness of MK in addressing this specific shortcoming of EFGM. Furthermore, the study also finds the accuracy of EFGM to be greatly enhanced with the use of MK shape functions.