Meshless Collocation Method Research Articles

Gaussian thermal shock-induced thermoelastic wave propagation in an FG multilayer hybrid nanocomposite cylinder reinforced by GPLs and CNTs

In this paper, a new modified micromechanical model is proposed to obtain the mechanical and thermal properties of a reinforced functionally graded (FG) multilayer hybrid nanocomposite cylinder for thermoelastic wave propagation analysis with energy dissipation using Green–Naghdi theory. The FG multilayer hybrid nanocomposite cylinder is assumed to be under Gaussian thermal shock loading and also it is reinforced by graphene platelets (GPLs) and carbon nanotubes (CNTs). The cylinder is assumed to be made of multi-layers (sub-cylinders), and each layer is reinforced by a uniform distribution of GPLs and CNTs. A modified micromechanical model is proposed to calculate the thermo-mechanical properties with nonlinear grading patterns of the GPLs and CNTs along the radial direction of the cylinder. The nonlinear grading patterns of the GPLs and CNTs distributions along the radial direction of the whole cylinder can be created using a suitable arrangement of the layers (sub-cylinders). An effective meshless collocation method based on the generalized finite difference (GFD) method and the Newmark method are employed to solve the coupled governing equations of the GN-based coupled thermoelasticity analysis. The effects of the key parameters such as the weight fraction of the GPLs and CNTs and volume fraction index on the thermoelastic wave propagations and dynamic behaviors of the field variables are studied in detail. Also, the effects of the reinforcement by GPLs and/or CNTs in a hybrid nanocomposite on the transient behaviors of cylinder in both temperature and displacement fields are compared. The accuracy and stability of the presented meshless method and also the proposed new modified micromechanical model is verified by the published reference data.

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

Solution of the 3D Helmholtz equation using barycentric Lagrange interpolation collocation method

PurposeThis meshless collocation method is applicable not only to the Helmholtz equation with Dirichlet boundary condition but also mixed boundary conditions. It can calculate not only the high wavenumber problems, but also the variable wave number problems.Design/methodology/approachIn this paper, the authors developed a meshless collocation method by using barycentric Lagrange interpolation basis function based on the Chebyshev nodes to deduce the scheme for solving the three-dimensional Helmholtz equation. First, the spatial variables and their partial derivatives are treated by interpolation basis functions, and the collocation method is established for solving second order differential equations. Then the differential matrix is employed to simplify the differential equations which is on a given test node. Finally, numerical experiments show the accuracy and effectiveness of the proposed method.FindingsThe numerical experiments show the advantages of the present method, such as less number of collocation nodes needed, shorter calculation time, higher precision, smaller error and higher efficiency. What is more, the numerical solutions agree well with the exact solutions.Research limitations/implicationsCompared with finite element method, finite difference method and other traditional numerical methods based on grid solution, meshless method can reduce or eliminate the dependence on grid and make the numerical implementation more flexible.Practical implicationsThe Helmholtz equation has a wide application background in many fields, such as physics, mechanics, engineering and so on.Originality/valueThis meshless method is first time applied for solving the 3D Helmholtz equation. What is more the present work not only gives the relationship of interpolation nodes but also the test nodes.

Solving 2D parabolic equations by using time parareal coupling with meshless collocation RBFs methods

The paper is devoted to solving parabolic equations by using time-parareal coupled unsymmetric collocation meshless method. The well-known meshless methods have the attractive fact that it is a true meshless method and spatial dimension independent, which can easily be extended to solve high dimensional problems. As applications, the parabolic equation in electromagnetism are then solved in different domains with smooth boundaries to illustrate the ability of the method to deal with regular or irregular geometries. It shows that the proposed method performs well. Convergence analysis of the coupling technique is processed. Numerical experiments show that the time is 1st order convergence, and the space is at least 1st order convergence. Additionally, time parallelism is superior to non-parallel results.

Fracture mechanics analysis of bimaterial interface cracks using the generalized finite difference method

This paper makes the first attempt to apply the generalized finite difference method (GFDM), a recently developed meshless collocation method, for fracture mechanics analysis of dissimilar elastic materials with interfacial cracks. The main idea of the GFDM is to divide the entire computational domain into a set of overlapping small subdomains in which the local Taylor series expansion and moving-least square approximation are employed for generating the local systems of linear equations. Since the method is meshless and no element connectivity is needed, the burdensome remeshing procedures associated with the finite element method (FEM) is avoided. The multi-domain GFDM technique is used to handle the non-homogeneity of the cracked dissimilar materials. The displacement extrapolation method (DEM), which avoids the direct calculation of the oscillatory near-tip displacement and stress fields, is employed to compute the complex stress intensity factors (SIFs) for cracked composite bimaterials. Several representative numerical examples are presented and discussed to demonstrate that the present method is highly accurate and relatively robust for interface crack analysis of composite bimaterials.

Generalized finite difference method for electroelastic analysis of three-dimensional piezoelectric structures

This short communication makes the first attempt to apply the generalized finite difference method (GFDM), a newly-developed meshless collocation method, for the numerical solutions of three-dimensional (3D) piezoelectric problems. In the present method, the entire computational domain is divided into a set of overlapping subdomains in which the local Taylor series expansion and moving-least square approximation are applied to construct the local systems of linear equations. By satisfying the coupled mechanical and electrical governing equations, a sparse and banded stiffness matrix can be established which makes the method very attractive for large-scale engineering simulations. Preliminary numerical experiments are presented to demonstrate the applicability and accuracy of the present method, where the results obtained are compared with the analytical solutions with very good agreement.

A meshless collocation method based on radial basis functions for free and forced vibration analysis of functionally graded plates using FSDT

The current study presents a meshless collocation method based on radial basis functions (MC-RBF) to analyze the free and forced vibration of functionally graded plates. We used the first-order shear deformation plate theory for the displacements and strains. The governing equations are discretized via the MC-RBF method to obtain the natural frequencies. We suggested a new algorithm (modified LOOCV) to acquire the shape parameters of the radial basis function in the free and forced vibration problems. The discretized equations are converted to reduced equations via the proposed transformation matrix. The run-time of this method is significantly lower than the other methods. In particular, the accuracy of the proposed algorithm is acceptable. The discretized equations are reduced by eigenvectors and are converted to state-space form and solved using the numerical Runge-Kutta method. The various boundary conditions are taken into account, such as their clamped, simply supported, free and mixture, and various power-law indices. The results show that the MC-RBF method using a modified LOOCV algorithm is a convenient method for free and forced vibration of FG plates.

Numerical solution of time-fractional coupled Korteweg–de Vries and Klein–Gordon equations by local meshless method

This article provides numerical simulations of the time-fractional coupled Korteweg–de Vries and Klein–Gordon equations via the local meshless collocation method (LMCM) utilising the radial basis functions. The recommended local meshless technique is utilised for the space derivatives of the models whereas Caputo fractional definition is used for time-fractional derivative. Numerical experiments are performed for one-dimensional coupled Korteweg–de Vries and two-dimensional Klein–Gordon equations. In order to verify the efficiency and accuracy of the proposed meshless method, numerical results are compared with exact and numerical techniques reported in recent literature which reveals that the method is computationally attractive and produces better results.

A novel meshless collocation solver for solving multi-term variable-order time fractional PDEs

This paper presents a novel meshless collocation method to solve multi-term variable-order time fractional partial differential equations (VOTFPDEs). In the proposed method, it employs the Fourier series expansion for spatial discretization, which transforms the original multi-term VOTFPDEs into a sequence of multi-term variable-order time fractional ordinary differential equations (VOTFODEs). Then, these VOTFODEs can be solved using the recent-developed backward substitution method. Several numerical examples verify the accuracy and efficiency of the proposed numerical approach in the solution of multi-term VOTFPDEs.

Local non-singular knot method for large-scale computation of acoustic problems in complicated geometries

This paper presents a local non-singular knot method (LNKM) to accurately solve the large-scale acoustic problems in complicated geometries. The LNKM is a domain-type meshless collocation method, which relies only on scattered nodes. Firstly, a series of subdomains corresponding to every nodes can be searched based on the Euclidean distance between nodes. To each subdomain, a small linear system can be yielded by using the non-singular general solutions of Helmholtz-type equations. Secondly, the unknown variables at every nodes can be explicitly expressed by the function values at their corresponding supporting nodes. Finally, a large sparse system of linear equations is formed and solved to obtain the numerical solutions of physical quantities at every nodes. The proposed LNKM is mathematically simple, numerically accurate, and more applicable to the large-scale computation. Four numerical examples conform its effectiveness and accuracy for the large-scale computation of Helmholtz-type equations in complicated geometries.

Bending analysis of simply supported and clamped thin elastic plates by using a modified version of the LMFS

The localized method of fundamental solutions is a recent domain-type meshless collocation method with the fundamental solutions of governing equations as the radial basis functions. This approach forms a sparse system matrix and has a higher efficiency than the traditional method of fundamental solutions. In this paper, a modified version of the localized method of fundamental solutions is developed for bending analysis of simply supported and clamped thin elastic plates. Some auxiliary nodes on the boundary are firstly introduced to provide additional weight coefficients, which contribute to the construction of the determined system of equations and avoid the over-determined equation system of the localized method of fundamental solutions for bending analysis of thin elastic plates. Several numerical experiments with simply supported or clamped boundary conditions are provided, and numerical results are in good agreement with the analytical or COMSOL Multiphysics solutions.

Nonlinear Bending Analysis of Hyperelastic Plates Using FSDT and Meshless Collocation Method Based on Radial Basis Function

This paper investigates the nonlinear bending analysis of a hyperelastic plate via neo-Hookean strain energy function. The first-order shear deformation plate theory (FSDPT) is used for the formulation of the field variables. Also, the nonlinear Lagrangian strains are considered via the right Cauchy–Green tensor. The governing equations and nonlinear boundary conditions are derived using Euler–Lagrange relations. The meshless collocation method based on radial basis function is used to discretize the governing equations of the hyperelastic plate. Square and circular plates are studied to evaluate the accuracy of the meshless collocation method based on thin-plate spline (TPS) and multiquadric (MQ) and logarithmic thin-plate spline (LTPS) radial basis function. Also, the results of the meshless method are compared to those of the finite element method. In some cases, the meshless method is more efficient than the finite element method due to no meshing. The linear and nonlinear natural boundary conditions are directly imposed on the stiffness matrix and are compared to each other. The maximum differences between linear and nonlinear natural boundary conditions are 1.43%. The von-Mises stress using meshless collocation method based on TPS basis function is compared to those of the finite element method.

Stress analysis of elastic bi-materials by using the localized method of fundamental solutions

<abstract> <p>The localized method of fundamental solutions belongs to the family of meshless collocation methods and now has been successfully tried for many kinds of engineering problems. In the method, the whole computational domain is divided into a set of overlapping local subdomains where the classical method of fundamental solutions and the moving least square method are applied. The method produces sparse and banded stiffness matrix which makes it possible to perform large-scale simulations on a desktop computer. In this paper, we document the first attempt to apply the method for the stress analysis of two-dimensional elastic bi-materials. The multi-domain technique is employed to handle the non-homogeneity of the bi-materials. Along the interface of the bi-material, the displacement continuity and traction equilibrium conditions are applied. Several representative numerical examples are presented and discussed to illustrate the accuracy and efficiency of the present approach.</p> </abstract>

A localized meshless collocation method based on semi‐analytical basis functions for bandgap calculation of elastic waves in phononic crystals

AbstractThis paper makes the first attempt to develop an efficient localized meshless collocation method (LMCM) based on the semi‐analytical T‐complete functions to calculate the bandgaps of elastic waves in solid phononic crystals. In comparison with the FEM (COMSOL software), the proposed LMCM can provide the similar accurate results with less computational resources for band structure calculation.

A Meshless Collocation Method with Barycentric Lagrange Interpolation for Solving the Helmholtz Equation

In this paper, Chebyshev interpolation nodes and barycentric Lagrange interpolation basis function are used to deduce the scheme for solving the Helmholtz equation. First of all, the interpolation basis function is applied to treat the spatial variables and their partial derivatives, and the collocation method for solving the second order differential equations is established. Secondly, the differential equations on a given test node. Finally, based on three kinds of test nodes, numerical experiments show that the present scheme can not only calculate the high wave numbers problems, but also calculate the variable wave numbers problems. In addition, the algorithm has the advantages of high calculation accuracy, good numerical stability and less time consuming.

Space–time generalized finite difference nonlinear model for solving unsteady Burgers’ equations

In this study, the space–time (ST) generalized finite difference method (GFDM) was combined with Newton’s method to stably and accurately solve two-dimensional unsteady Burgers’ equations. In the coupled ST approach, the time axis is selected as a spatial axis; thus, the temporal derivative in governing equations is treated as a spatial derivative. In general, the GFDM is an optimal meshless collocation method for solving partial differential equations. Moreover, one can avoid the construction of a mesh for simulation by using the GFDM. The derivatives at each node are described as a linear combination of nearby functional values by using weighting coefficients in the computational domain. Due to the property of the moving least-square approximation in the GFDM, the resultant matrix system can be formed as a sparse matrix so that the GFDM is suitable for solving large-scale problems. In this study, two benchmark examples were used to demonstrate the consistency and accuracy of the proposed ST meshless numerical scheme.

Generalized Finite Difference Method for Plate Bending Analysis of Functionally Graded Materials

In this paper, an easy-to-implement domain-type meshless method—the generalized finite difference method (GFDM)—is applied to simulate the bending behavior of functionally graded (FG) plates. Based on the first-order shear deformation theory (FSDT) and Hamilton’s principle, the governing equations and constrained boundary conditions of functionally graded plates are derived. Based on the multivariate Taylor series and the weighted moving least-squares technique, the partial derivative of the underdetermined displacement at a certain node can be represented by a linear combination of the displacements at its adjacent nodes in the GFDM implementation. A certain node of the local support domain is formed according to the rule of “the shortest distance”. The proposed GFDM provides the sparse resultant matrix, which overcomes the highly ill-conditioned resultant matrix issue encountered in most of the meshless collocation methods. In addition, the studies show that irregular distribution of structural nodes has hardly any impact on the numerical performance of the generalized finite difference method for FG plate bending behavior. The method is a truly meshless approach. The numerical accuracy and efficiency of the GFDM are firstly verified through some benchmark examples, with different shapes and constrained boundary conditions. Then, the effects of material parameters and thickness on FG plate bending behavior are numerically investigated.

Meshless modeling of coupled transdermal pharmacokinetics with analytical validation

An efficient localized collocation meshless computational approach is applied to the study of transdermal drug delivery (TDD). To validate the numerical technique, a one-dimensional analytical solution is first derived for the set of coupled differential equations that govern the pharmacokinetic process in the two-compartment compound diffusion model. These equations consist of one partial differential equation (PDE) and one ordinary differential equation (ODE). These equations are solved in one-dimensional space via superposition and separation of variables to arrive at the field solution for the compound concentration in the tissue. The convergence of the numerical result to the analytical result indicates the viability of the localized collocation meshless method (LCMM) as a valid approach to the solution of TDD problems. The analytical solution is used to benchmark the numerical technique for several cases with differing pharmacokinetic coefficients, demonstrating the robustness and accuracy of the meshless numerical technique. Furthermore, the highly efficient LCCM implementation is extended to address multi-dimensional TDD cases achieving stable and physically consistent results.

Analysing time-fractional exotic options via efficient local meshless method

In this article, we analyse the numerical simulation of the time-fractional Black-Scholes model governing butterfly spread option, digital option and double barrier option. For this purpose, a local meshless collocation method based on multiquadric radial basis function is used. The model equation is discretized in the temporal sense by Liouville-Caputo fractional derivative scheme for 0<β<1, whereas the space derivatives are discretized by the suggested meshless method. Numerical experiments are performed for butterfly spread option, digital option and double barrier option. Efficiency and accuracy of the proposed meshless method are assessed in terms of double mesh procedure since the exact solution of these options are not available. Furthermore, the greeks (delta and gamma) of the options are also computed.

A localized meshless collocation method for bandgap calculation of anti-plane waves in 2D solid phononic crystals

In this paper, a localized meshless collocation method, the generalized finite difference method (GFDM), is first applied to calculate the bandgaps of anti-plane transverse elastic waves in 2D solid phononic crystals with square and triangular lattice. The corresponding theoretical consistency analysis of the GFDM is given. The universal algorithm for the uniform/scattered node generation in the GFDM is presented. In comparison with the traditional plane wave expansion (PWE) method and Pressure Acoustics Module in COMSOL software, the proposed GFDM can provide the similar accurate results with less computational times for calculating the band structures of the simple/complicated shape scatterers in the square/triangular lattice. Three influence factors (Filling fractions (Ff), rotation angles (Ra) and arm widths (Aw) in the unit-cell) of the bandgap properties in 2D phononic crystals are numerically discussed.