Scattered data fitting of Hermite type by a weighted meshless method

In many practical problems, it is often desirable to interpolate not only the function values but also the values of derivatives up to certain order, as in the Hermite interpolation. The Hermite interpolation method by radial basis functions is used widely for solving scattered Hermite data approximation problems. However, sometimes it makes more sense to approximate the solution by a least squares fit. This is particularly true when the data are contaminated with noise. In this paper, a weighted meshless method is presented to solve least squares problems with noise. The weighted meshless method by Gaussian radial basis functions is proposed to fit scattered Hermite data with noise in certain local regions of the problem’s domain. Existence and uniqueness of the solution is proved. This approach has one parameter which can adjust the accuracy according to the size of the noise. Another advantage of the weighted meshless method is that it can be used for problems in high dimensions with nonregular domains. The numerical experiments show that our weighted meshless method has better performance than the traditional least squares method in the case of noisy Hermite data.     

A symmetric integrated radial basis function method for solving differential equations

In this article, integrated radial basis functions (IRBFs) are used for Hermite interpolation in the solution of differential equations, resulting in a new meshless symmetric RBF method. Both global and local approximation‐based schemes are derived. For the latter, the focus is on the construction of compact approximation stencils, where a sparse system matrix and a high‐order accuracy can be achieved together. Cartesian‐grid‐based stencils are possible for problems defined on nonrectangular domains. Furthermore, the effects of the RBF width on the solution accuracy for a given grid size are fully explored with a reasonable computational cost. The proposed schemes are numerically verified in some elliptic boundary‐value problems governed by the Poisson and convection‐diffusion equations. High levels of the solution accuracy are obtained using relatively coarse discretisations.     

On the analysis of a kind of nonlinear Sobolev equation through locally applied pseudo‐spectral meshfree radial point interpolation

In this study, we develop an approximate formulation for two‐dimensional nonlinear Sobolev problems by focusing on pseudospectral meshless radial point interpolation (PSMRPI) which is a kind of locally applied radial basis function interpolation and truthfully a meshless approach. In the PSMRPI method, the nodal points do not need to be regularly distributed and can even be quite arbitrary. It is easy to have high order derivatives of unknowns in terms of the values at nodal points by constructing operational matrices. The convergence and stability of the technique in some sense are studied via some examples to show the validity and trustworthiness of the PSMRPI technique.     

A greedy algorithm for partition of unity collocation method in pricing American options

A greedy algorithm in combination with radial basis functions partition of unity collocation (GRBF‐PUC) scheme is used as a locally meshless method for American option pricing. The radial basis function partition of unity method (RBF‐PUM) is a localization technique. Because of having interpolation matrices with large condition numbers, global approximants and some local ones suffer from instability. To overcome this, a greedy algorithm is added to RBF‐PUM. The greedy algorithm furnishes a subset of best nodes among the points X. Such nodes are then used as points of trial in a locally supported RBF approximant for each partition. Using of greedy selected points leads to decreasing the condition number of interpolation matrices and reducing the burdensome in pricing American options.     

Analytical formulation of meshless local integral equation method

In this paper, the exact forms of integrals in the meshless local boundary integral equation method are derived and implemented for elastostatic problems. A weak form for a set of governing equations with a unit test function or polynomial test functions is transformed into local integral equations. Each node has its own support domain and is surrounded by a local integral domain with different shapes of boundaries. The meshless approximation based on the radial basis function (RBF) is employed for the implementation of displacements. A completed set of closed forms of the local boundary integrals are obtained. As there are no numerical integrations to be carried out the computational time is significantly reduced. Three examples are presented to demonstrate the application of this approach in solid mechanics.     

Nonlinear transient heat conduction analysis of functionally graded materials in the presence of heat sources using an improved meshless radial point interpolation method

An improved meshless radial point interpolation method, for the analysis of nonlinear transient heat conduction problems is proposed. This method is implemented for the heat conduction analysis of functionally graded materials (FGMs) with non-homogenous and/or temperature dependent heat sources. The conventional meshless RPIM is an appropriate numerical technique for the analysis of engineering problems. One advantage of this method is that it is based on the global weak formulation, and also the associated shape functions possess the Kronecker delta function property. However, in the original form, the evaluation of the global domain integrals requires the use of a background mesh. The proposed method benefits from a meshless integration technique, which has the capability of evaluating domain integrals with a better accuracy and speed in comparison with the conventional integration methods, and therefore a truly meshless technique is attained. This integration technique is especially designed for the fast and accurate evaluation of several domain integrals, with different integrands, over a single domain. Some 2D and 3D examples are provided to assess the efficiency of the proposed method.     

Determination of a control parameter in a one-dimensional parabolic equation using the method of radial basis functions

In this work, the method of radial basis functions is used for finding the solution of an inverse problem with source control parameter. Because a much wider range of physical phenomena are modelled by nonclassical parabolic initial-boundary value problems, theoretical behavior and numerical approximation of these problems have been active areas of research. The radial basis functions (RBF) method is an efficient mesh free technique for the numerical solution of partial differential equations. The main advantage of numerical methods which use radial basis functions over traditional techniques is the meshless property of these methods. In a meshless method, a set of scattered nodes are used instead of meshing the domain of the problem. The results of numerical experiments are presented and some comparisons are made with several well-known finite difference schemes.     

Treatment of singularities in the method of fundamental solutions for two-dimensional Helmholtz-type equations

We investigate a meshless method for the accurate and non-oscillatory solution of problems associated with two-dimensional Helmholtz-type equations in the presence of boundary singularities. The governing equation and boundary conditions are approximated by the method of fundamental solutions (MFS). It is well known that the existence of boundary singularities affects adversely the accuracy and convergence of standard numerical methods. The solutions to such problems and/or their corresponding derivatives may have unbounded values in the vicinity of the singularity. This difficulty is overcome by subtracting from the original MFS solution the corresponding singular functions, without an appreciable increase in the computational effort and at the same time keeping the same MFS approximation. Four examples for both the Helmholtz and the modified Helmholtz equations are carefully investigated and the numerical results presented show an excellent performance of the approach developed.     

瞬态热传导的奇异边界法及其MATLAB实现

基于动力学问题时间依赖基本解的奇异边界法是一种无网格边界配点法.该方法引入源点强度因子的概念从而避免了基本解的源点奇异性,具有数学简单、编程容易、精度高等优点.将该方法用于瞬态热传导问题的数值模拟,运用MATLAB实现该问题的数值研究,并创建相应的MATLAB工具箱.针对二维和三维瞬态热传导问题,进行了基于反插值技术和经验公式的奇异边界法MATLAB算例实现.针对支撑圆坯低温瞬态温度场的模拟结果表明,瞬态热传导奇异边界法的MATLAB工具箱具有简单、方便、精确可靠的优点.研究成果有助于发展瞬态热传导的奇异边界法,并为瞬态热传导问题的数值分析和仿真提供了一种简单高效的模拟工具.     

Dual reciprocity hybrid radial boundary node method for the analysis of Kirchhoff plates

A meshless method of dual reciprocity hybrid radial boundary node method (DHRBNM) for the analysis of arbitrary Kirchhoff plates is presented, which combines the advantageous properties of meshless method, radial point interpolation method (RPIM) and BEM. The solution in present method comprises two parts, i.e., the complementary solution and the particular solution. The complementary solution is solved by hybrid radial boundary node method (HRBNM), in which a three-field interpolation scheme is employed, and the boundary variables are approximated by RPIM, which is applied instead of moving least square (MLS) and obtains the Kronecker’s delta property where the traditional HBNM does not satisfy. The internal variables are interpolated by two groups of symmetric fundamental solutions. Based on those, a hybrid displacement variational principle for Kirchhoff plates is developed, and a meshless method of HRBNM for solving biharmonic problems is obtained, by which the complementary solution can be solved.     

Radial basis function collocation method for the numerical solution of the two-dimensional transient nonlinear coupled Burgers’ equations

This paper examines the numerical solution of the transient nonlinear coupled Burgers’ equations by a Local Radial Basis Functions Collocation Method (LRBFCM) for large values of Reynolds number (Re) up to 10³. The LRBFCM belongs to a class of truly meshless methods which do not need any underlying mesh but works on a set of uniform or random nodes without any a priori node to node connectivity. The time discretization is performed in an explicit way and collocation with the multiquadric radial basis functions (RBFs) are used to interpolate diffusion-convection variable and its spatial derivatives on decomposed domains. Five nodded domains of influence are used in the local support. Adaptive upwind technique [1] and [54] is used for stabilization of the method for large Re in the case of mixed boundary conditions. Accuracy of the method is assessed as a function of time and space discretizations. The method is tested on two benchmark problems having Dirichlet and mixed boundary conditions. The numerical solution obtained from the LRBFCM for different value of Re is compared with analytical solution as well as other numerical methods [15], [47] and [49]. It is shown that the proposed method is efficient, accurate and stable for flow with reasonably high Reynolds numbers.     

A random integral quadrature method for numerical analysis of the second kind of Volterra integral equations

In this paper, a novel meshless technique termed the random integral quadrature (RIQ) method is developed for the numerical solution of the second kind of the Volterra integral equations. The RIQ method is based on the generalized integral quadrature (GIQ) technique, and associated with the Kriging interpolation function, such that it is regarded as an extension of the GIQ technique. In the GIQ method, the regular computational domain is required, in which the field nodes are scattered along straight lines. In the RIQ method however, the field nodes can be distributed either uniformly or randomly. This is achieved by discretizing the governing integral equation with the GIQ method over a set of virtual nodes that lies along straight lines, and then interpolating the function values at the virtual nodes over all the field nodes which are scattered either randomly or uniformly. In such a way, the governing integral equation is converted approximately into a system of linear algebraic equations, which can be easily solved.     

二维黏弹性力学问题的无网格自然单元法

基于无网格自然单元法,建立了求解二维黏弹性力学问题的一条新途径.基于弹性 黏弹性对应原理和Laplace(拉普拉斯)变换技术,首先将黏弹性问题转换成Laplace域内与弹性力学问题相同的形式,然后推导出基于自然单元法分析黏弹性问题的基本公式.作为一种新兴的无网格数值计算方法,自然单元法的实质是一种基于自然邻近插值的Galerkin(伽辽金)法.相对于其他无网格法,自然单元法的形函数具有插值性和支持域各向异性等特点.算例结果证明了所提分析方法的有效性.     

Local meshless method for PDEs arising from models of wound healing

A meshless collocation procedure is proposed for one- and two-dimensional partial differential equations arising from modeling of wound healing processes (Sherratt and Murray, 1991). Main motivation of this choice is its straightforward application in higher dimensions for both regular and irregular domains on various nodal points distributions. In the case of numerical solution of convection-dominated wound healing PDE models, a stencil based upwind stabilization technique is coupled with the local meshless method to counter instabilities of the computed solution. To assess efficacy, efficiency and accuracy of the proposed method on regular and irregular domains, numerical approximations of different wound healing models are obtained and validated against the exact solution and medically tested healing time duration.     

Application of Cauchy integrals and singular integral equations in scattered data problems

A new approach is suggested for constructing “smooth” surfaces which contain discontinuities in functional values or derivatives at prescribed locations. The approach is based on solving singular integral equations with Cauchy-type kernels. It is applicable in the interpolation or approximation of scattered data. We investigate, in particular, several two-dimensional biharmonic and triharmonic problems which have smooth solutions except on given line segments, across which different types of discontinuities occur. We also discuss some issues concerning the application of the approach in practical problems.     

Flow in a waterfall with large Froude number

An approximate method is developed to solve the full nonlinear equations governing two-dimensional irrotational flow in a free waterfall, falling under the influence of gravity, at high Froude number based on conditions far upstream. Schwarz—Christoffel transformation is used to map the region, in the complex potential-plane, onto the upper half-plane. The Hilbert transformation as well as the perturbation technique, for large Froude number, are used as a basis for the approximate solution of the problem. A complete solution, up to second-order approximation, for the downstream free-surfaces profiles, for different Froude number, is discussed and illustrated. The obtained approximate solutions are compared with those of other authors. Favourable agreement with other results suggests that this method is effective in dealing with flow problems strongly influenced by gravity and high Froude number. The results obtained by this method are sufficiently accurate for practical purposes.     

薄板的局部Petrov-Galerkin方法

利用薄板控制微分方程的等效积分对称弱形式和对变量(挠度)采用移动最小二乘近似函数进行插值,研究了薄板弯曲问题的无网格局部Petrov-Galerkin方法．这是一种真正的无网格方法,它不需要任何有限元或边界元网格,不管这种网格是用于能量积分还是进行插值的目的．所有的积分都在规则形状的子域及其边界上进行,并用罚因子法施加本质边界条件．数值例子表明,无网格局部Petrov-Galerkin法不但能够求解二阶微分方程的边值问题,而且求解四阶微分方程的边值问题也很有效,也具有收敛快、稳定性好、对挠度和内力都具有精度高的特点．     

An efficient meshfree point collocation moving least squares method to solve the interface problems with nonhomogeneous jump conditions

We are going to study a simple and effective method for the numerical solution of the closed interface boundary value problem with both discontinuities in the solution and its derivatives. It uses a strong‐form meshfree method based on the moving least squares (MLS) approximation. In this method, for the solution of elliptic equation, the second‐order derivatives of the shape functions are needed in constructing the global stiffness matrix. It is well‐known that the calculation of full derivatives of the MLS approximation, especially in high dimensions, is quite costly. In the current work, we apply the diffuse derivatives using an efficient technique. In this technique, we calculate the higher‐order derivatives using the approximation of lower‐order derivatives, instead of calculating directly derivatives. This technique can improve the accuracy of meshfree point collocation method for interface problems with nonhomogeneous jump conditions and can efficiently estimate diffuse derivatives of second‐ and higher‐orders using only linear basis functions. To introduce the appropriate discontinuous shape functions in the vicinity of interface, we choose the visibility criterion method that modifies the support of weight function in MLS approximation and leads to an efficient computational procedure for the solution of closed interface problems. The proposed method is applied for elliptic and biharmonic interface problems. For the biharmonic equation, we use a mixed scheme, which replaces this equation by a coupled elliptic system. Also the application of the present method to elasticity equation with discontinuities in the coefficients across a closed interface has been provided. Representative numerical examples demonstrate the accuracy and robustness of the proposed methodology for the closed interface problems. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1031–1053, 2015     

A local radial basis function collocation method to solve the variable‐order time fractional diffusion equation in a two‐dimensional irregular domain

The local radial basis function (RBF) method is a promising solver for variable‐order time fractional diffusion equation (TFDE), as it overcomes the computational burden of the traditional global method. Application of the local RBF method is limited to Fickian diffusion, while real‐world diffusion is usually non‐Fickian in multiple dimensions. This article is the first to extend the application of the local RBF method to two‐dimensional, variable‐order, time fractional diffusion equation in complex shaped domains. One of the main advantages of the local RBF method is that only the nodes located in the subdomain, surrounding the local point, need to be considered when calculating the numerical solution at this point. This approach can perform well with large scale problems and can also mitigate otherwise ill‐conditioned problems. The proposed numerical approach is checked against two examples with curved boundaries and known analytical solutions. Shape parameter and subdomain node number are investigated for their influence on the accuracy of the local RBF solution. Furthermore, quantitative analysis, based on root‐mean‐square error, maximum absolute error, and maximum error of the partial derivative indicates that the local RBF method is accurate and effective in approximating the variable‐order TFDE in two‐dimensional irregular domains.     

A meshless method for nonhomogeneous polyharmonic problems using method of fundamental solution coupled with quasi-interpolation technique

This paper proposes a meshless method based on coupling the method of fundamental solutions (MFS) with quasi-interpolation for the solution of nonhomogeneous polyharmonic problems. The original problems are transformed to homogeneous problems by subtracting a particular solution of the governing differential equation. The particular solution is approximated by quasi-interpolation and the corresponding homogeneous problem is solved using the MFS. By applying quasi-interpolation, problems connected with interpolation can be avoided. The error analysis and convergence study of this meshless method are given for solving the boundary value problems of nonhomogeneous harmonic and biharmonic equations. Numerical examples are also presented to show the efficiency of the method.