Solutions of partial differential equations with random Dirichlet boundary conditions by multiquadric collocation method

Problems described by deterministic partial differential equations with random Dirichlet boundary conditions are considered. Formulation of the solution to such a problem by the global collocation method using multiquadrics is presented. The quality of the solution to a stochastic problem depends on both its expected value and its variance. It is proposed that the shape parameter of multiquadrics should be chosen to optimize both the accuracy and the variance of the solution. Test problems described by the Poisson, the Helmholtz, and the diffusion–convection equations with random Dirichlet boundary conditions are solved by the multiquadric collocation method. It is found that there is a trade-off between solution accuracy and solution variance for each problem.     

A Meshless Numerical Method for Kirchhoff Plates under Arbitrary Loadings

This paper describes the combination of the method of fundamental solutions (MFS) and the dual reciprocity method (DRM) as a meshless numerical method to solve problems of Kirchhoff plates under arbitrary loadings. In the solution procedure, a arbitrary distributed loading is first approximated by either the multiquadrics (MQ) or the augmented polyharmonic splines (APS), which are constructed by splines and monomials. The particular solutions of multiquadrics, splines and monomials are all derived analytically and explicitly. Then, the complementary solutions are solved formally by the MFS. Furthermore, the boundary conditions of lateral displacement, slope, normal moment, and effective shear force are all given explicitly for the particular solutions of multiquadrics, splines and polynomials as well as the kernels of MFS. Finally, numerical experiments are carried out to validate these analytical formulas. In these numerical experiments, homogeneous problems are first considered to find the best location of the MFS sources by the way proposed by Tsai, Lin, Young and Atluri (2006). Then the corresponding nonhomogeneous problems are solved by the DRM based on both the MQ and APS. The numerical results demonstrate that the MQ is in general more accurate than the thin plate spline, or the first order APS, but less accurate than the high order APSs. Overall, this paper derives a meshless numerical method for solving problems of Kirchhoff plates under arbitrary loadings with all kinds of boundary conditions by both the MQ and APS.     

Use of singularity capturing functions in the solution of problems with discontinuous boundary conditions

A method is proposed to improve the accuracy of the numerical solution of elliptic problems with discontinuous boundary conditions using both global and local meshless collocation methods with multiquadrics as basis functions. It is based on the use of special functions which capture the singular behavior near discontinuities in boundary conditions. In the case of global collocation, the method consists in enlarging the functional space spanned by the RBF basis functions, while in the case of local collocation, the method consists in modifying appropriately the problem in order to eliminate the singularities from the formulation. Numerical results for benchmark problems such as a stationary heat equation in a box (harmonic) and Stokes flow in a lid-driven square cavity, show significant improvements in accuracy and in compliance with the continuity equation.     

Automatic particular solutions of arbitrary high-order splines associated with polyharmonic and poly-Helmholtz equations

The explicit analytical particular solutions of the splines and monomials for the polyharmonic and poly-Helmholtz operators and their products were derived in the author's previous study. These solutions enable an implementation for automatically approximating particular solutions of arbitrary high-order splines. The automatic particular solutions obtained by the proposed implementation are extremely accurate. In the case of a polyharmonic equation, these solutions are more accurate than the numerical solutions obtained using the multiquadrics within the limits of the IEEE double precision. In the case of poly-Helmholtz and product equations, this implementation can also result in very accurate solutions despite the fact that an analytical particular solution for the multiquadrics does not exist. After particular solutions are obtained, boundary-type numerical methods, such as the boundary element method, the method of fundamental solutions and the Trefftz method, can be applied to solve the homogeneous differential equations.     

A meshless model for transient heat conduction in functionally graded materials

A meshless numerical model is developed for analyzing transient heat conduction in non-homogeneous functionally graded materials (FGM), which has a continuously functionally graded thermal conductivity parameter. First, the analog equation method is used to transform the original non-homogeneous problem into an equivalent homogeneous one at any given time so that a simpler fundamental solution can be employed to take the place of the one related to the original problem. Next, the approximate particular and homogeneous solutions are constructed using radial basis functions and virtual boundary collocation method, respectively. Finally, by enforcing satisfaction of the governing equation and boundary conditions at collocation points of the original problem, in which the time domain is discretized using the finite difference method, a linear algebraic system is obtained from which the unknown fictitious sources and interpolation coefficients can be determined. Further, the temperature at any point can be easily computed using the results of fictitious sources and interpolation coefficients. The accuracy of the proposed method is assessed through two numerical examples.     

Generalized finite differences using fundamental solutions

It is well known that solutions for linear partial differential equations may be given in terms of fundamental solutions. The fundamental solutions solve the homogeneous equation exactly and are obtained from the solution of the inhomogeneous equation where the inhomogeneous term is described by a Dirac delta distribution. Fundamental solutions are the building blocks of the boundary element method and of the method of fundamental solutions and are traditionally used to build boundary‐only global approximations in the domain of interest. In this work the same characteristic of the fundamental solutions, that of solving the homogeneous equation exactly, is used but not to build a global approximation. On the contrary, local approximations are built in such a manner that it is possible to construct finite difference operators that are free from any form of structured grid. Copyright © 2009 John Wiley & Sons, Ltd.     

Methods of fundamental solutions for time‐dependent heat conduction problems

A time‐dependent heat conduction problem can be solved by the method of fundamental solutions using the fundamental solution to the modified Helmholtz equation or the fundamental solution to the heat equation. This paper presents solutions using both formulations in terms of initial and boundary conditions. Such formulations enable calculation of errors and variance, which indicates sensitivities of solutions to uncertainties in initial and boundary conditions. Both errors and variance of solutions to three test problems by the two methods of fundamental solutions are used to compare performances of the methods. Copyright © 2005 John Wiley & Sons, Ltd.     

The method of fundamental solutions for steady-state groundwater flow problems

The method of fundamental solutions is a meshless method. Only boundary collocation points are needed during the whole solution process. It has the merits of mathematical simplicity, ease of programming, high solution accuracy, and others. In this paper, the method of fundamental solutions is applied to simulate 2D steady-state groundwater flow problems. The principle of superposition is used during the whole solution process. Numerical results are compared with the multiquadrics method and the mixed finite element method as well as analytical solutions. It is shown that the method of fundamental solutions is promising in dealing with steady groundwater flow problems.     

Multiple reciprocity hybrid boundary node method for potential problems

Meshless methods have some obvious advantages such as they do not require meshes in the domain and on the boundary, only some nodes are needed in the computation. Furthermore, for the boundary-type meshless methods, the nodes are even not needed in the domain and only distributed on the boundary. Practice shows that boundary-type meshless methods are effective for homogeneous problems. But for inhomogeneous problems, the application of these boundary-type meshless methods has some difficulties and need to be studied further.The hybrid boundary node method (HBNM) is a boundary-only meshless method, which is based on the moving least squares (MLS) approximation and the hybrid displacement variational principle. No cell is required either for the interpolation of solution variables or for numerical integration. It has a drawback of ‘boundary layer effect’, so a new regular hybrid boundary node method (RHBNM) has been proposed to avoid this pitfall, in which the source points of the fundamental solutions are located outside the domain. These two methods, however, can only be used for solving homogeneous problems. Combining the dual reciprocity method (DRM) and the HBNM, the dual reciprocity hybrid boundary node method (DRHBNM) has been proposed for the inhomogeneous terms. The DRHBNM requires a substantial number of internal points to interpolate the particular solution by the radial basis function, where approximation based only on boundary nodes may not guarantee sufficient accuracy.Now a further improvement to the RHBNM, i.e., a combination of the RHBNM and the multiple reciprocity method (MRM), is presented and called the multiple reciprocity hybrid boundary node method (MRHBNM). The solution comprises two parts, i.e., the complementary and particular solutions. The complementary solution is solved by the RHBNM. The particular solution is solved by the MRM, i.e., a sum of high-order homogeneous solutions, which can be approximated by the same-order fundamental solutions. Compared with the DRHBNM, the MRHBNM does not require internal points to obtain the particular solution for inhomogeneous problems. Therefore, the present method is a real boundary-only meshless method, and can be used to deal with inhomogeneous problems conveniently. The validity and efficiency of the present method are demonstrated by a series of numerical examples of inhomogeneous potential problems.     

The method of fundamental solutions for nonlinear elliptic problems

A meshless method was presented, which couples the method of fundamental solutions (MFS) with radial basis functions (RBFs) and the analog equation method (AEM), to solve nonlinear problems. In this method, the AEM is used to convert the nonlinear governing equation into a corresponding linear inhomogeneous equation, so that a simpler fundamental solution can be employed. Then, the RBFs and the MFS are, respectively, used to construct the expressions of particular and homogeneous solution parts of the substitute equation, from which the approximate solution of the original problem and its derivatives involved in the governing equation are represented via the unknown coefficients. After satisfying all equations of the original problem at collocation points, a nonlinear system of equations can be obtained to determine all unknowns. Some numerical tests illustrate the efficiency of the method proposed.     

Indirect Trefftz method for boundary value problem of Poisson equation

Trefftz method is the boundary-type solution procedure using regular T-complete functions satisfying the governing equation. Until now, it has been mainly applied to numerical analyses of the problems governed with the homogeneous differential equations such as the two- and three-dimensional Laplace problems and the two-dimensional elastic problem without body forces. On the other hand, this paper describes the application of the indirect Trefftz method to the solution of the boundary value problems of the two-dimensional Poisson equation. Since the Poisson equation has an inhomogeneous term, it is generally difficult to determine the T-complete function satisfying the governing equation. In this paper, the inhomogeneous term containing an unknown function is approximated by a polynomial in the Cartesian coordinates to determine the particular solutions related to the inhomogeneous term. Then, the boundary value problem of the Poisson equation is transformed to that of the Laplace equation by using the particular solution. Once the boundary value problem of the Poisson equation is solved according to the ordinary Trefftz formulation, the solution of the boundary value problem of the Poisson equation is estimated from the solution of the Laplace equation and the particular solution. The unknown parameters included in the particular solution are determined by the iterative process. The present scheme is applied to some examples in order to examine the numerical properties.     

Trefftz Methods for Time Dependent Partial Differential Equations

In this paper we present a mesh-free approach to numerically solving a class of second order time dependent partial differential equations which include equations of parabolic, hyperbolic and parabolic-hyperbolic types. For numerical purposes, a variety of transformations is used to convert these equations to standard reaction-diffusion and wave equation forms. To solve initial boundary value problems for these equations, the time dependence is removed by either the Laplace or the Laguerre transform or time differencing, which converts the problem into one of solving a sequence of boundary value problems for inhomogeneous modified Helmholtz equations. These boundary value problems are then solved by a combination of the method of particular solutions and Trefftz methods. To do this, a variety of techniques is proposed for numerically computing a particular solution for the inhomogeneous modified Helmholtz equation. Here, we focus on the Dual Reciprocity Method where the source term is approximated by radial basis functions, polynomial or trigonometric functions. Analytic particular solutions are presented for each of these approximations. The Trefftz method is then used to solve the resulting homogenous equation obtained after the approximate particular solution is subtracted off. Two types of Trefftz bases are considered, F-Trefftz bases based on the fundamental solution of the modified Helmholtz equation, and T-Trefftz bases based on separation of variables solutions. Various techniques for satisfying the boundary conditions are considered, and a discussion is given of techniques for mitigating the ill-conditioning of the resulting linear systems. Finally, some numerical results are presented illustrating the accuracy and efficacy of this methodology.     

Solution of transient direct-chill aluminium billet casting problem with simultaneous material and interphase moving boundaries by a meshless method

This paper uses a recently developed upgrade of the classical meshless Kansa method for solution of the transient heat transport in direct-chill casting of aluminium alloys. The problem is characterised by a moving mushy domain between the solid and the liquid phase and a moving starting bottom block that emerges from the mould during the process. The solution of the thermal field is based on the mixture continuum formulation. The growth of the domain and the movement of the bottom block are described by activation of additional nodes and by the movement of the boundary nodes through the computational domain, respectively. The domain and boundary of interest are divided into overlapping influence areas. On each of them, the fields are represented by the multiquadrics radial basis function collocation on a related sub-set of nodes. Time stepping is performed in an explicit way. The governing equation is solved in its strong form, i.e. no integrations are performed. The polygonisation is not present and the method is practically independent of the problem dimension. Realistic boundary conditions and temperature variation of material properties are included. An axisymmetric transient test case solution is shown at different times and its accuracy is verified by comparison with the reference finite volume method results.     

A boundary-only meshless method for numerical solution of the Eikonal equation

The radial basis function (RBF) collocation methods for the numerical solution of partial differential equation have been popular in recent years because of their advantage. For instance, they are inherently meshless, integration free and highly accurate. In this article we study the RBF solution of Eikonal equation using boundary knot method and analog equation method. The boundary knot method (BKM) is a meshless boundary-type radial basis function collocation technique. In contrast with the method of fundamental solution (MFS), the BKM uses the non-singular general solution instead of the singular fundamental solution to obtain the homogeneous solution. Similar to MFS, the RBF is employed to approximate the particular solution via the dual reciprocity principle. In the current paper, we applied the idea of analog equation method (AEM). According to AEM, the nonlinear governing operator is replaced by an equivalent nonhomogeneous linear one with known fundamental solution and under the same boundary conditions. Finally numerical results and discussions are presented to show the validity and efficiency of the proposed method.     

Transient dynamic crack analysis in piecewise homogeneous, anisotropic and linear elastic composites by a spatial symmetric Galerkin-BEM

Transient elastodynamic analysis of two-dimensional, piecewise homogeneous, anisotropic and linear elastic solids containing interior and interface cracks is presented in this paper. To solve the initial boundary value problem, a spatial symmetric time-domain boundary element method is developed. Stationary cracks subjected to impact loading conditions are considered. Elastodynamic fundamental solutions for homogenous, anisotropic and linear elastic solids are implemented. The piecewise homogeneous, anisotropic and linear elastic solids are modeled by the multi-domain technique. The spatial discretization is performed by a symmetric Galerkin-method, while a collocation method is utilized for the temporal discretization. An explicit time-stepping scheme is obtained for computing the unknown boundary data. Numerical examples are presented and discussed to show the effects of the interface cracks, the material anisotropy, the material combination and the dynamic loading on the dynamic stress intensity factors.     

A meshless local boundary integral equation method for dynamic anti-plane shear crack problem in functionally graded materials

This paper presents a meshless local boundary integral equation method (LBIEM) for dynamic analysis of an anti-plane crack in functionally graded materials (FGMs). Local boundary integral equations (LBIEs) are formulated in the Laplace-transform domain. The static fundamental solution for homogeneous elastic solids is used to derive the local boundary–domain integral equations, which are applied to small sub-domains covering the analyzed domain. For the sub-domains a circular shape is chosen, and their centers, the nodal points, correspond to the collocation points. The local boundary–domain integral equations are solved numerically in the Laplace-transform domain by a meshless method based on the moving least–squares (MLS) scheme. Time-domain solutions are obtained by using the Stehfest's inversion algorithm. Numerical examples are given to show the accuracy of the proposed meshless LBIEM.     

二对边法向支承矩形边界平面问题新解法

新解法将平面问题分为广义静定和广义超静定二类,前者可以直接求解,后者要用叠加法求解。广义静定问题解由角点集中力解、体力分量解和计算边值条件解组成,这三种解要满足不同的变形协调条件,必须分别计算。角点集中力解、体力分量解与双调和方程无关,也不必满足边界条件;它们在边界处的应力和位移值反向作用在相应边界上为虚拟边值条件,实际边值条件和虚拟边值条件之和为计算边值条件。计算边值条件解即为经典的双调和方程解,由通解和特解组成。求解角点集中力解采用了隔离体平衡法。该文详细推导了二对边法向支承矩形边界平面问题所有解的表达式以及求解方法,并附有算例。     

Particular solutions of 3D Helmholtz-type equations using compactly supported radial basis functions

In this paper we show how to obtain analytic particular solutions for inhomogeneous Helmholtz-type equations in 3D when the source terms are compactly supported radial basis functions (CS-RBFs) (J. Approx. Theory, 93 (1998) 258). Using these particular solutions we demonstrate the solvability of boundary value problems for the inhomogeneous Helmholtz-type equation by approximating the source term by CS-RBFs and solving the resulting homogeneous equation by the method of fundamental solutions. The proposed technique is a truly mesh-free method and is especially attractive for large-scale industrial problems. A numerical example is given which illustrates the efficiency of the proposed method.     

非均匀弹簧界面模型下柱形夹杂物对弹性波的散射

利用波函数展开法研究了非均匀弹簧界面模型(弹簧系数沿周向非均匀分布)下单个柱形夹杂物对弹性波的散射问题。在弹簧界面模型中,当弹簧系数沿周向分布均匀时,可利用波函数的正交性简化边界条件;当弹簧系数沿周向分布不均匀时,不能利用波函数的正交性简化边界条件。该文研究了这一问题,通过沿周向的离散化将弹性波散射的边界条件归结为一个超定线性代数方程组。针对Ge-Al纤维增强复合材料数值计算了散射截面和远场散射幅。特别地,通过适当选取弹簧常数的周向分布处理了含裂纹界面的弹性波散射问题,数值计算了裂纹张开位移和错开位移。     

The method of fundamental solutions for inhomogeneous elliptic problems

We investigate the use of the Method of Fundamental Solutions (MFS) for solving inhomogeneous harmonic and biharmonic problems. These are transformed to homogeneous problems by subtracting a particular solution of the governing equation. This particular solution is taken to be a Newton potential and the resulting homogeneous problem is solved using the MFS. The numerical calculations indicate that accurate results can be obtained with relatively few degrees of freedom. Two methods for the special case where the inhomogeneous term is harmonic are also examined.