Meshless Local Radial Point Interpolation Research Articles

An efficient meshfree computational approach to the analyze of thermoelastic waves of functionally graded materials in a two-dimensional space

In this paper, a meshless local radial point interpolation method (RPIM) is proposed for the thermoelastic analysis of functionally graded materials (FGMs) under thermal shocks. The properties of such materials are temperature-dependent varying and change gradually through thickness based on the Mori–Tanaka scheme and the volume fraction power law distribution (Appendix A). The characterization and visualization of the FGMs under a prescribed temperature gradient or heat flux are then detailed. The proposed method does not need any background cell-based integration and is thus very suitable to deal with complex geometries and scattered data. Moreover, the shape functions of this method possess the properties of the Kronecker delta, wherein the essential boundary conditions can be analytically satisfied. The numerical results are fully discussed and validated by the corresponding outcomes of the moving least squares (MLS) method, demonstrating the high capability and efficiency of RPIM proposed here in simulating the dynamic thermoelastic analysis of FGMs.

Meshless local radial point interpolation (MLRPI) to two dimensional wave equation with Neumann’s boundary conditions

In this article, the meshless local radial point interpolation (MLRPI) methods are applied to simulate two dimensional wave equation subject to given appropriate initial and Neumann’s boundary conditions. In MLRPI method, all integrations are carried out locally over small quadrature domains of regular shapes such as square or circle. The radial point interpolation method is proposed to construct shape functions for MLRPI. A weak formulation with a Heaviside step function transforms the set of governing equations into local integral equations on local sub domains where Neumann’s boundary condition is imposed naturally. A two-step time discretization method with the help of Crank-Nicolson technique is employed to approximate the time derivatives. Convergence studies in the numerical example show that MLRPI method possesses excellent rates of convergence.

A MESHLESS DEFORMATION SIMULATION METHOD FOR VIRTUAL SURGERY

An accurate and computation-effective model of soft tissues is very important for surgery simulation. When soft tissues are interacted by a surgical instrument, their external deformation and internal stress and strain distribution are closely related. These stress and strain data can not only be used to describe the soft tissue deformation but also can be used to calculate interaction force rendered to the force feedback device, making the user to have a realistic sense of touch during the surgical procedure. It is difficult, however, to solve partial differential equations (PDE) in conventional soft tissue models, and stress and strain data cannot be easily extracted. This paper presents a meshless local radial point interpolation method for soft tissue modelling and demonstrates its feasibility by numerical tests.

Capillary formation in tumor angiogenesis through meshless weak and strong local radial point interpolation

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.

Meshless local radial point interpolation (MLRPI) for generalized telegraph and heat diffusion equation with non-local boundary conditions

In this paper, the meshless local radial point interpolation (MLRPI) method is formulated to the generalized one-dimensional linear telegraph and heat diffusion equation with non-local boundary conditions. The MLRPI method is categorized under meshless methods in which any background integration cells are not required, so that all integrations are carried out locally over small quadrature domains of regular shapes, such as lines in one dimensions, circles or squares in two dimensions and spheres or cubes in three dimensions. A technique based on the radial point interpolation is adopted to construct shape functions, also called basis functions, using the radial basis functions. These shape functions have delta function property in the frame work of interpolation, therefore they convince us to impose boundary conditions directly. The time derivatives are approximated by the finite difference time- -stepping method. We also apply Simpson’s integration rule to treat the non-local boundary conditions. Convergency and stability of the MLRPI method are clarified by surveying some numerical experiments.

Local radial basis function interpolation method to simulate 2D fractional‐time convection‐diffusion‐reaction equations with error analysis

In this article, a kind of meshless local radial point interpolation (MLRPI) method is proposed to two‐dimensional fractional‐time convection‐diffusion‐reaction equations and satisfactory agreements are archived. This method is based on meshless methods and benefits from collocation ideas but it does not belong to the traditional global meshless collocation methods. In MLRPI method, it does not need any kind of integration locally or globally over small quadrature domains which is essential in the finite element method and those meshless methods based on Galerkin weak form. Also, it is not needed to determine shape parameter which plays important role in collocation method based on the radial basis functions (Kansa's method). Therefore, computational costs of this kind of MLRPI method is less expensive. The stability and convergence of this meshless approach are discussed and theoretically proven. It is proved that the present meshless formulation is very effective for modeling and simulation of fractional differential equations. Furthermore, the numerical studies on sensitivity analysis and convergence analysis show the stability and reliable rates of convergence. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 974–994, 2017

Local radial point interpolation (MLRPI) method for solving time fractional diffusion-wave equation with damping

The purpose of the current investigation is to determine numerical solution of time-fractional diffusion-wave equation with damping for Caputo's fractional derivative of order α ( 1 < α ? 2 ) . A meshless local radial point interpolation (MLRPI) scheme based on Galerkin weak form is analyzed. The reason of choosing MLRPI approach is that it does not require any background integrations 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 unconditional stability and convergence with order O ( ? 6 - 2 α ) are proved, where ? is time stepping. Also, several numerical experiments are illustrated to verify theoretical analysis.

Meshless local radial point interpolation to three-dimensional wave equation with Neumann's boundary conditions

In this article, the meshless local radial point interpolation (MLRPI) method is applied to simulate three-dimensional wave equation subject to given appropriate initial and Neumann's boundary conditions. The main drawback of methods in fully 3-D problems is the large computational costs. In the MLRPI method, all integrations are carried out locally over small quadrature domains of regular shapes such as a cube or a sphere. The point interpolation method with the help of radial basis functions is proposed to form shape functions in the frame of MLRPI. The local weak formulation using Heaviside step function converts the set of governing equations into local integral equations on local subdomains where Neumann's boundary condition is imposed naturally. A two-step time discretization technique with the help of the Crank-Nicolson technique is employed to approximate the time derivatives. Convergence studies in the numerical example show that the MLRPI method possesses reliable rates of convergence.

Application of meshless local radial point interpolation (MLRPI) on a one-dimensional inverse heat conduction problem

Abstract In this paper, the meshless local radial point interpolation (MLRPI) method is applied to one-dimensional inverse heat conduction problems. The meshless LRPIM is one of the truly meshless methods since it does not require any background integration cells. In this case, all integrations are carried out locally over small quadrature domains of regular shapes, such as lines in one dimensions, circles or squares in two dimensions and spheres or cubes in three dimensions. A technique is proposed to construct shape functions using radial basis functions. These shape functions which are constructed by point interpolation method using the radial basis functions have delta function property. The time derivatives are approximated by the time-stepping method. Numerical experiments are carried out and compared with implicit Finite difference method.

On the convergence analysis, stability, and implementation of meshless local radial point interpolation on a class of three‐dimensional wave equations

SummaryIn this article, the meshless local radial point interpolation method is applied to analyze three space dimensional wave equations of the form subject to given initial and Dirichlet boundary conditions. The main difficulty of the great number of methods in full 3‐D problems is the large computational costs. In meshless local radial point interpolation method, it does not require any background integration cells, so that all integrations are carried out locally over small quadrature domains of regular shapes such as circles or squares in two dimensions and spheres or cubes in three dimensions. The point interpolation method with the help of radial basis functions is proposed to construct shape functions that have Kronecker delta function property. A weak formulation with the Heaviside step function converts the set of governing equations into local integral equations on local subdomains. A two‐step time discretization method is employed to evaluate the time derivatives. This suggests Crank‐Nicolson technique to be applied on the right hand side of the equation. The convergence analysis and stability of the method are fully discussed. Three illustrative examples are presented, and satisfactory agreements are achieved. It is shown theoretically that the proposed method is unconditionally stable for the second example whereas it is not for the first and third ones. Copyright © 2015 John Wiley &amp; Sons, Ltd.

Local integration of 2-D fractional telegraph equation via local radial point interpolant approximation

In this article, a general type of two-dimensional time-fractional telegraph equation explained by the Caputo derivative sense for (1 < α ≤ 2) is considered and analyzed by a method based on the Galerkin weak form and local radial point interpolant (LRPI) approximation subject to given appropriate initial and Dirichlet boundary conditions. In the proposed method, so-called meshless local radial point interpolation (MLRPI) method, a meshless Galerkin weak form is applied to the interior nodes while the meshless collocation method is used for the nodes on the boundary, so the Dirichlet boundary condition is imposed directly. The point interpolation method is proposed to construct shape functions using the radial basis functions. In the MLRPI method, it does not require any background integration cells so that all integrations are carried out locally over small quadrature domains of regular shapes, such as circles or squares. Two numerical examples are presented and satisfactory agreements are achieved.

Meshless local radial point interpolation (MLRPI) on the telegraph equation with purely integral conditions

In this paper, the meshless local radial point interpolation (MLRPI) method is applied to the one-dimensional telegraph equation with purely integral conditions. In MLRPI, it does not require any background integration cells but it requires all integrations be carried out locally over small quadrature domains of regular shapes, such as lines in one dimension, circles or squares in two dimensions and spheres or cubes in three dimensions. A technique is proposed to construct shape functions using point interpolation method augmented to the radial basis functions. The time derivatives are approximated by the finite difference method. Some numerical experiments for the mentioned problem are carried out as well.

Analysis of meshless local and spectral meshless radial point interpolation (MLRPI and SMRPI) on 3-D nonlinear wave equations

In this paper, the meshless local radial point interpolation (MLRPI) method is applied to simulate three space dimensional nonlinear wave equation of the form utt+αut+βu=uxx+uyy+uzz+δg(u)ut+f(x,y,z,t), 0<x,y,z<1,t>0 subject to given appropriate initial and Dirichlet boundary conditions and then a new spectral meshless radial point interpolation (SMRPI) method is proposed to solve the mentioned problem. The main drawback of methods in fully 3-D problems is the large computational costs. In the MLRPI method, all integrations are carried out locally over small quadrature domains of regular shapes such as spheres or cubes. In innovative SMRPI method, it is not needed to 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 converts easily the governing equations (even high order) into linear system of equations in SMRPI. A two-step time discretization method is employed to approximate the time derivatives. To treat the nonlinearity part, a kind of predictor–corrector scheme combined with one-step time discretization and Crank–Nicolson technique is adopted. A comparison study of the efficiency and accuracy of the MLRPI and SMRPI method is given by applying on mentioned problem. Convergence studies in the numerical examples show that the SMRPI method possesses excellent rates of convergence.

Analysis of meshless local radial point interpolation (MLRPI) on a nonlinear partial integro-differential equation arising in population dynamics

Abstract In this paper the meshless local radial point interpolation (MLRPI) method is applied to simulate a nonlinear partial integro-differential equation arising in population dynamics. This PDE is a competition model in which similar individuals are competing for the same resources. It is a kind of reaction–diffusion equation with integral term corresponding to nonlocal consumption of resources. In MLRPI method, it does not require any background integration cells so that all integrations are carried out locally over small quadrature domains of regular shapes, such as circles or squares in two dimensions and spheres or cubes in three dimensions. The point interpolation method is proposed to construct shape functions using the radial basis functions. A one-step time discretization method is employed to approximate the time derivative. To treat the nonlinearity, a simple predictor–corrector scheme is performed. Also the integral term, which is a kind of convolution, is treated by the cubic spline interpolation. The numerical studies on sensitivity analysis and convergence analysis show that our approach is stable. Finally, two numerical examples are presented showing the behavior of the solution and the efficiency of the proposed method.

Shear Locking Analysis of Plate Bending by Using Meshless Local Radial Point Interpolation Method

The shape function of the meshless local radial point interpolation method is constructed by using the radial basis functions and possesses Kronecker delta function properties. Therefore, the essential boundary conditions can be easily imposed. Causation of shear locking occur in plate bending is analyzed. Bending problems for plate with two sides simply supported, the other two sides clamped boundary conditions, are analyzed by the meshless local radial point interpolation method. The shear locking is easier avoided in the meshless method than in the finite element method, and the measure of avoiding the shear locking are presented.

Numerical simulation of two-dimensional sine-Gordon solitons via a local weak meshless technique based on the radial point interpolation method (RPIM)

In this paper the meshless local radial point interpolation method (LRPIM) is adopted to simulate the two-dimensional nonlinear sine-Gordon (S-G) equation. The meshless LRPIM is one of the “truly meshless” methods since it does not require any background integration cells. In this case, all integrations are carried out locally over small quadrature domains of regular shapes, such as circles or squares in two dimensions and spheres or cubes in three dimensions. A technique is proposed to construct shape functions using radial basis functions. These shape functions which are constructed by point interpolation method using the radial basis functions have delta function property. The time derivatives are approximated by the time-stepping method. In order to eliminate the nonlinearity, a simple predictor–corrector scheme is performed. Numerical results are obtained for various cases involving line and ring solitons. Also the conservation of energy in undamped sine-Gordon equation is investigated.

Elastic dynamic analysis of moderately thick plate using meshless LRPIM

A meshless local radial point interpolation method (LRPIM) for solving elastic dynamic problems of moderately thick plates is presented in this paper. The discretized system equation of the plate is obtained using a locally weighted residual method. It uses a radial basis function (RBF) coupled with a polynomial basis function as a trial function, and uses the quartic spline function as a test function of the weighted residual method. The shape function has the properties of the Kronecker delta function, and no additional treatment is done to impose essential boundary conditions. The Newmark method for solving the dynamic problem is adopted in computation. Effects of sizes of the quadrature sub-domain and influence domain on the dynamic properties are investigated. The numerical results show that the presented method can give quite accurate results for the elastic dynamic problem of the moderately thick plate.

Application of the local radial point interpolation method to solve eddy-current problems

This paper introduces the meshless local radial point interpolation method to model eddy-current problems for the first time. The formulation is described and results are compared with those from the ordinary finite-element simulation and the analytical solution