An improved pseudospectral meshless radial point interpolation (PSMRPI) method for 3D wave equation with variable coefficients

In this article, recently proposed spectral meshless radial point interpolation (SMRPI) method is applied to the two-dimensional diffusion equation with a mixed group of Dirichlet's and Neumann's and non-classical boundary conditions. The present method is based on meshless methods and benefits from spectral collocation ideas. The point interpolation method with the help of radial basis functions is proposed to construct shape functions which have Kronecker delta function property. Evaluation of high-order derivatives is possible by constructing and using operational matrices. The computational cost of the method is modest due to using strong form equation and collocation approach. A comparison study of the efficiency and accuracy of the present method and other meshless methods is given by applying on mentioned diffusion equation. Stability and convergence of this meshless approach are discussed and theoretically proven. Convergence studies in the numerical examples show that SMRPI method possesses excellent rates of convergence.

A new spectral meshless radial point interpolation (SMRPI) method for the two-dimensional Fredholm integral equations on general domains with error analysis

This paper considered the spectral meshless radial point interpolation (SMRPI) method to unravel for the nonlinear p-Laplacian equation with mixed Dirichlet and Neumann boundary conditions. Through this assessment, which includes meshless methods and collocation techniques based on radial point interpolation, we construct the shape functions, with the Kronecker delta function property, as basis functions in the framework of spectral collocation methods. Studies in this regard require one to evaluate the high-order derivatives without any kind of integration locally over the small quadrature domains. Finally, some examples are given to illustrate the low computing costs and high enough accuracy and efficiency of this method to solve a p-Laplacian equation and it would be of great help to fulfill the implementation related to the element-free Galerkin (EFG) method. Both the SMRPI and the EFG methods have been compared by similar numerical examples to show their application in strongly nonlinear problems.

This article is devoted to showing the efficiency and accuracy of two numerical procedures for the numerical solution of the capillary formation in tumor angiogenesis. A meshless local radial point interpolation (MLRPI) scheme based on Galerkin weak form is analyzed. The reason for choosing MLRPI approach is that it does not require any background integration cells; instead, integrations are implemented over local quadrature domains, which are further simplified for reducing the complication of computation using regular and simple shape. The spectral meshless radial point interpolation (SMRPI) method does not need any integration. The radial point interpolation method is proposed to construct shape functions for MLRPI and basis functions for SMRPI. A weak formulation with a Heaviside step function transforms the set of governing equations into local integral equations on local subdomains in MLRPI, whereas the operational matrices convert easily the governing equations (even high order) into a linear system of equations in SMRPI. The finite difference technique is employed to approximate the time derivatives. A stability analysis technique based on the eigenvalue of the matrices is employed to check the stability condition for the presented meshless methods. Furthermore, the methods are successfully applied to solve the problem with high values of the cell diffusion constant which many of the available methods did not investigate for solving these cases. Because of non-availability of the exact solution, we consider two strategies for checking the stability of time difference scheme and for surveying the convergence of the fully discrete scheme. From numerical results, it is considerable that the present methods provide the similar results. Moreover, less CPU time and less computational complexity are two advantages of the SMRPI method with respect to the MLRPI method.

For the better management of available groundwater, it is essential to understand the flow behavior within the aquifer systems. The groundwater flow behavior in the complex aquifer system can be studied by solving the governing equations using either analytical or numerical methods. As the analytical solutions are available only for simple idealized cases, numerical methods such as finite difference method (FDM) and finite element method (FEM) are generally used. The Meshless method is a recently developed alternative numerical approach for the solution of existing problem. A variety of Meshless methods like point interpolation method, smooth particle hydrodynamics, Galerkin methods, etc. are under the development for the solution of many solid mechanics and fluid mechanics problems. Meshless method eliminates many of the drawbacks of grid-based methods such as difficulties in meshing and re-meshing as in FDM and FEM which require huge efforts in pre-processing and with Meshless method it translates in the lesser computational time and costs. In this paper, Meshless Local Petrov-Galerkin (MLPG) method with strong form of collocation with exponential/Gaussian radial basis function is used for solving the groundwater flow problem. A computer model in MATLAB has been developed in 1D and 2D for the solution of confined aquifer problems. The developed 1D model is verified with available analytical solution for a hypothetical problem and found to be satisfactory. Further, the 2D model is applied to two case studies and results are compared with the FEM based solutions and found to be satisfactory. The present study shows that the MLPG based Meshless method is very effective in the simulation of groundwater flow problems.

The radial point interpolation method (RPIM) based on local supported radial basis function (RBF) and the Galerkin weak form has been developed and successfully applied to many engineering problems. Recently, a new meshfree method was proposed based on the universal moving Kriging interpolation. This paper studies the difference between the meshfree shape functions created based on the point interpolation and the Kriging interpolation. It is found that both the two methods yield the same shape function as long as the same radial basis function or semivariogram is adopted for interpolation. Although the two methods lead to the same shape function, the theorem in Kriging formulation may provide an alternative theoretical support for the RPIM. Some common semivariograms used in Kriging may also be incorporated in the RPIM. In addition, in order to satisfy the conformability requirements, a penalty technique is introduced in this paper to form a conforming Kriging, which can pass the standard patch test exactly.

In this paper, we introduce a two-dimensional (2-D) Node-based Meshless Radial Point Interpolation Method (RPIM) for modeling nonmagnetized plasma. The problem domain is firstly discretized by a set of electric field nodes (E-nodes) and a set of magnetic field nodes (H-nodes). E-nodes and H-nodes are defined in a manner similar to the point-matched time-domain finite-element method. With the RPIM scheme, the field variable at the node is interpolated by a scalar expansion function which combines radial basis function with monomial basis function. After some mathematical manipulations, the shape function of the node in the support domain is obtained. The field variables and corresponding spatial derivatives in field equations can be obtained analytically and the central finite-difference is used to approximate the time derivatives. The leapfrog RPIM meshless method formulations are then obtained. TM wave propagation in isotropic nonmagnetized plasma is simulated to validate the accuracy and efficiency of the proposed method.

In this study, a numerical meshless method is used to solve the weak form of the linear elastic equations in solid mechanics. Evaluation and comparison of the numerical meshless methods have been carried out via the radial point interpolation meshless method with multi-quadrics base functions (MQ-RPIM) and meshless local Petrov-Galerkin method (MLPG). Using these two methods, stress intensity factors in an elastic medium containing geometric discontinuities and cracks are estimated based on tensile and bending cyclic loading. The analysis domain has been identified via three-dimensional modeling of the notched and un-notched shafts with an initial surface semi-elliptical crack subjected to tensile or bending cyclic loadings. To enhance the accuracy of calculations, the RPIM meshless method is applied using polynomial and extended-enriched 3D base functions. Shape functions have been developed using standard and optimal parameters and values with Mono-Objective Function in PSO algorithm. In the MLPG meshless method with the extended-enriched functions, discretization is performed via direct and penalty factor methods, to reach more efficient results and meet the boundary conditions. Efficiency comparison of the selected numerical methods with the experimental findings and the numerical analysis of finite elements method indicates that in comparison with the MLPG method, MQ-RPIM enriched meshless method can be utilized with fewer nodes in the analysis domain while reaching the accuracy and convergence with lower stress intensity factors and gentler slope. However, the processing time of the MLPG meshless method is lower than that of the other methods.

This paper presents a numerical model of two-dimensional (2D) contaminant transport through unsaturated porous media using a mesh-free method, called the radial point interpolation method (RPIM), with polynomial reproduction. A 2D form of the advection– dispersion equation, involving a reactive contaminant with linear first-order degradation, is considered in the analysis. In the RPIM, the Galerkin weak form is used to establish the system equation using 2D mesh-free shape functions constructed using thin plate spline radial-basis functions. A Matlab code is developed to obtain the numerical solution. Experimental results are used to validate the proposed RPIM. The applicability and performance of the RPIM are studied using three numerical examples. The results of the RPIM are compared with those obtained from the finite-element method. The RPIM has generated results with no oscillations, and which are insensitive to Péclet constraints.

Meshless methods are numerical methods that have the advantage of high accuracy without the need of an explicitly described mesh topology. In this class of methods, the Radial Point Interpolation Method (RPIM) is a promising collocation method where the application of radial basis functions yields high interpolation accuracy for even strongly unstructured node distributions. For electromagnetic simulations in particular, this distinguishing characteristic translates into an enhanced capability for conformal and multi-scale modeling. The method also facilitates adaptive discretization refinements, which provides an important tool to decrease memory consumption and computation time. In this paper, a refinement strategy is introduced for RPIM. In the proposed node adaptation algorithm, the accuracy of a solution is increased iteratively based on an initial solution with a coarse discretization. In contrast to the commonly used residual-based adaptivity algorithms, this definition is extended by an error estimator based on the solution gradient. In the studied cases this strategy leads to increased convergence rates compared with the standard algorithm. Numerical examples are provided to illustrate the effectiveness of the algorithm.

Point interpolation collocation method for the solution of partial differential equations

Functionally graded materials (FGMs) are complicated composites created using the concept of continuous variation of material property in one or more predetermined directions. FGMs have been used in the manufacturing of structural parts that are subjected to non-uniform functioning requirements in recent years. In a thermal protection system, for example, FGMs combine the benefits of traditional ceramics, such as corrosion and heat resistance, with those of metal, such as mechanical strength and rigidity. They are common in engineering practice, so static analysis and dynamic analysis for FGM plate structures are necessary. The radial point interpolation method (RPIM) meshless method has been used based on the point interpolation method (PIM) by including the radial basis function (RPF) in the interpolation formulation and has shown good performance in computational engineering. One of the advantages of this method is that it satisfies the Kronecker’s delta function, which overcomes the limitations of critical boundary conditions for the traditional meshless method. Furthermore, RPIM shape function satisfies the high-order continuity constraint, which the low-order finite element methods (FEM) method does not. This paper presents a meshless approach for the static and dynamic analysis of FGM plates whose material properties vary through the thickness. Numerical examples are solved and the results are compared with reference solutions or the results of FEM given by SOMSOL program to confirm the accuracy of the proposed method.

The present paper is devoted to the development of a kind of spectral meshless radial point interpolation (SMRPI) technique in order to obtain a reliable approximate solution for buckling of nano-actuators subject to different nonlinear forces. To end this aim, a general type of the governing equation for nano-actuators, containing integro-differential terms and nonlinear forces is considered. This general type for the nano-actuators is a non-linear fourth-order Fredholm integro-differential boundary value problem. The point interpolation method with the help of radial basis functions is used to construct shape functions which play as basis functions in the frame of SMRPI. In the current work, the thin plate splines (TPSs) are used as radial basis functions. This numerical based technique enables us to overcome all kind of nonlinearities in the mentioned boundary value problem and then to obtain fast convergent solution. Thus, it can facilitate the design of nano-actuators.

This paper presents a finite element-least square point interpolation method (FE-LSPIM) for solving 2D acoustic problem by synthesizing the attractive features of finite element and least-square point interpolation shape functions using the concepts of partition of unity (PU) methods. In the present method, the acoustic domain is discretized using quadrilateral element, and the shape functions of the quadrilateral element are used for PU and the least-square point interpolation method (LSPIM) for local approximation. This enables the proposed method to inherit the compatibility properties of finite element method and the quadratic polynomial completeness properties of meshfree methods, so that the finite element-least-square point interpolation method (FE-LSPIM) will greatly reduce the numerical dispersion error because the numerical dispersion error is essentially caused by the �overly-stiff� nature of the FEM model. Numerical results for benchmark problems show that, the FE-LSPIM achieves more accurate results and higher convergence rates as compared with the corresponding finite elements and Element-free Galerkin method (EFGM), especially for high wave number and irregular meshes

This paper presents the solution of the kinematic wave equation using a meshless radial point interpolation method (RPIM). The partial differential equation is discretized using a Galerkin weighted residual method employing RPIM shape functions. A forward difference scheme is used for temporal discretization, while the direct substitution method is employed to solve the nonlinear system at each time step. The formulation is validated against solutions from conventional numerical techniques and physical observation. In all cases, excellent agreements are achieved and hence the validation of the proposed formulation. Optimum values of the multi-quadrics shape parameters were then determined before the assessment of the performance of the method. Based on the convergence rate, it has been shown that the proposed method performs better than the finite difference method and equivalent to the finite element method. This highlights the potential of RPIM as an alternative method for hydrologic modeling.