A new boundary meshfree method with distributed sources

A new boundary meshfree method, to be called the boundary distributed source (BDS) method, is presented in this paper that is truly meshfree and easy to implement. The method is based on the same concept in the well-known method of fundamental solutions (MFS). However, in the BDS method the source points and collocation points coincide and both are placed on the boundary of the problem domain directly, unlike the traditional MFS that requires a fictitious boundary for placing the source points. To remove the singularities of the fundamental solutions, the concentrated point sources can be replaced by distributed sources over areas (for 2D problems) or volumes (for 3D problems) covering the source points. For Dirichlet boundary conditions, all the coefficients (either diagonal or off-diagonal) in the systems of equations can be determined analytically, leading to very simple implementation for this method. Methods to determine the diagonal coefficients for Neumann boundary conditions are discussed. Examples for 2D potential problems are presented to demonstrate the feasibility and accuracy of this new meshfree boundary-node method.     

Calculating transport of water from a conduit to the porous matrix by boundary distributed source method

This study presents the development of a suitable numerical method for porous media flow with free and moving boundary (Stefan) problems arising in systems with wetted and unwetted regions of porous media. A non-singular version of the method of fundamental solutions (MFS), termed the boundary distributed source method (BDS), is applied. Darcy flow and homogenous isotropic porous media is assumed. The solution is represented in terms of the fundamental solution of the Laplace equation in two-dimensional Cartesian coordinates. The desingularisation is achieved through boundary distributed sources of the fundamental solution and indirect calculation of the derivatives of the fundamental solution. Respectively, the artificial boundary, characteristic for the classical, singular MFS is not present. The novel BDS is compared with the MFS and the analytical solutions for several numerical examples with excellent agreement. A sensitivity study of the solution, regarding the discretization and the free parameters is performed. The main contributions of the study are the application of the BDS to free and moving boundary problems and the comparison of BDS with MFS for these types of problems. The developed model can be applied to various geohydrological problems.     

Solving potential problems by a boundary-type meshless method—the boundary point method based on BIE

In this paper, a novel boundary-type meshless method, the boundary point method (BPM), is developed via an approximation procedure based on the idea of Young et al. [Novel meshless method for solving the potential problems with arbitrary domain. J Comput Phys 2005;209:290–321] and the boundary integral equations (BIE) for solving two- and three-dimensional potential problems. In the BPM, the boundary of the solution domain is discretized by unequally spaced boundary nodes, with each node having a territory (the point is usually located at the centre of the territory) where the field variables are defined. The BPM has both the merits of the boundary element method (BEM) and the method of fundamental solution (MFS), both of these methods use fundamental solutions which are the two-point functions determined by the source and the observation points only. In addition to the singular properties, the fundamental solutions have the feature that the greater the distance between the two points, the smaller the values of the fundamental solutions will be. In particular, the greater the distances, the smaller the variations of the fundamental solutions. By making use of this feature, most of the off-diagonal coefficients of the system matrix will be computed by one-point scheme in the BPM, which is similar to the one in the MFS. In the BPM, the ‘moving elements’ are introduced by organizing the relevant adjacent nodes tentatively, so that the source points are placed on the real boundary of the solution domain where the resulting weak singular, singular and hypersingular kernel functions of the diagonal coefficients of the system matrix can be evaluated readily by well-developed techniques that are available in the BEM. Thus difficulties encountered in the MFS are removed because of the coincidence of the two points. When the observation point is close to the source point, the integrals of kernel functions can be evaluated by Gauss quadrature over territories.In this paper, the singular and hypersingular equations in the indirect and direct formulations of the BPM are presented corresponding to the relevant BIE for potential problems, where the indirect formulations can be considered as a special form of the MFS. Numerical examples demonstrate the accuracy of solutions of the proposed BPM for potential problems with mixed boundary conditions where good agreements with exact solutions are observed.     

A method of fundamental solutions without fictitious boundary

This paper proposes a novel meshless boundary method called the singular boundary method (SBM). This method is mathematically simple, easy-to-program, and truly meshless. Like the method of fundamental solutions (MFS), the SBM employs the singular fundamental solution of the governing equation of interest as the interpolation basis function. However, unlike the MFS, the source and collocation points of the SBM coincide on the physical boundary without the requirement of introducing fictitious boundary. In order to avoid the singularity at the origin, this method proposes an inverse interpolation technique to evaluate the singular diagonal elements of the MFS coefficient matrix. The SBM is successfully tested on a benchmark problems, which shows that the method has a rapid convergence rate and is numerically stable.     

Solution of potential flow problems by the modified method of fundamental solutions: Formulations with the single layer and the double layer fundamental solutions

This paper describes an application of the recently proposed modified method of fundamental solutions (MMFS) to potential flow problems. The solution in two-dimensional Cartesian coordinates is represented in terms of the single layer and the double layer fundamental solutions. Collocation is used for the determination of the expansion coefficients. This novel method does not require a fictitious boundary as the conventional method of fundamental solutions (MFS). The source and the collocation points thus coincide on the physical boundary of the system. The desingularised values, consistent with the fundamental solutions used, are deduced from the direct boundary element method (BEM) integral equations by assuming a linear shape of the boundary between the collocation points. The respective values of the derivatives of the fundamental solution in the coordinate directions, as required in potential flow calculations, are calculated indirectly from the considerations of the constant potential field. The normal on the boundary is calculated by parametrisation of its length and the use of the cubic radial basis functions with the second-order polynomial augmentation. The components of the normal are calculated in an analytical way. A numerical example of potential flow around a two-dimensional circular region is presented. The results with the new MMFS are compared with the results of the classical MFS and the analytical solution. It is shown that the MMFS gives better accuracy for the potential, velocity components (partial derivatives of the potential), and absolute value of the velocity as compared with the classical MFS. The results with the single layer fundamental solution are more accurate than the results with the double layer fundamental solution.     

Comparison of boundary collocation methods for singular and non-singular axisymmetric heat transfer problems

This paper presents a computational study of some boundary collocation solution methods for the Laplace equation in cylindrical coordinates with axisymmetry. The methods compared are (i) the direct boundary element method (BEM), (ii) the method of fundamental solutions (MFS) with fixed sources and (iii) the Trefftz method. Relative accuracy of these methods are compared for two test problems. The first problem is a simple problem of heat transfer through a cylindrical rod which is a standard benchmark problem in this field. The second problem deals with heat transfer in silicon melt for Czochralski (CZ) process which involves a singularity in the boundary conditions at the corner of the crystal–melt interface. All the three methods indicated above are highly successful for the simple (first) problem with MFS and Trefftz being simpler to implement than the BEM. However, the Trefftz method was not effective for the second problem due to the boundary singularity and the MFS showed oscillations near the singularity point. Hence the use of higher order non-conforming elements with accurate Gauss–Kronrod integration schemes in the direct BEM method was investigated. It was found that the boundary singularity does not deteriorate the accuracy of the results if this improved numerical integration procedure is used in the direct BEM. Hence higher order elements with Gauss–Kronrod integration schemes can be used for the solution of many free interface problems encountered in crystal growth.     

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.     

A multipole expansion-based boundary element method for axisymmetric potential problem

The multipole expansion is an approximation technique used to evaluate the potential field due to sources located in the far field. Based on the multipole expansion, we describe a new technique to calculate the far potential field due to ring sources which are encountered in the boundary element method (BEM) formulation of axisymmetric problems. As the sources in the near field are processed by the slower conventional BEM, it is important to maximize the amount of multipole calculations taking advantage of both interior and exterior multipole expansions. Numerical results are presented for an axisymmetric potential test problem with Neumann and Dirichlet boundary conditions. The complexity of the proposed method remains O(N²), which is equal to that of the conventional BEM. However, the proposed technique coupled with an iterative solver speeds up the solution procedure. The technique is significantly advantageous when medium and large numbers of elements are present in the domain.     

Coupling boundary elements to a raytracing procedure

Typical outdoor sound propagation problems are governed by two principal phenomena: (i) diffraction in the vicinity of the noise source due to objects such as buildings or insulation barriers, and (ii) refraction at long distances from the source as a consequence of the effects of wind and temperature. The boundary element method (BEM) is well suited to account for the diffraction phenomena in the near field, while the raytracing method based on geometrical acoustics is more effective to deal with the refraction phenomena. In this paper, a new approach is presented which couples the direct BEM and a raytracing model in order to combine their advantages. Two alternative coupling procedures are developed, one is using a singular indirect BEM and the other is based on the method of fundamental solutions (MFS). The direct boundary element model is applied first for solving the near field and computing the sound pressure along an auxiliary interface which limits the near field extent. Then, a singular indirect BEM or MFS is used to find the intensities of a number of point sources which produce the same sound pressure on the interface to that resulting from the near‐field analysis. Finally, the point sources are the input data for the raytracing model of the far field. A 3D implementation of the proposed method is finally applied to an outdoor sound propagation problem. Copyright © 2007 John Wiley & Sons, Ltd.     

Simple modifications for stabilization of the finite point method

A stabilized version of the finite point method (FPM) is presented. A source of instability due to the evaluation of the base function using a least square procedure is discussed. A suitable mapping is proposed and employed to eliminate the ill‐conditioning effect due to directional arrangement of the points. A step by step algorithm is given for finding the local rotated axes and the dimensions of the cloud using local average spacing and inertia moments of the points distribution. It is shown that the conventional version of FPM may lead to wrong results when the proposed mapping algorithm is not used. It is shown that another source for instability and non‐monotonic convergence rate in collocation methods lies in the treatment of Neumann boundary conditions. Unlike the conventional FPM, in this work the Neumann boundary conditions and the equilibrium equations appear simultaneously in a weight equation similar to that of weighted residual methods. The stabilization procedure may be considered as an interpretation of the finite calculus (FIC) method. The main difference between the two stabilization procedures lies in choosing the characteristic length in FIC and the weight of the boundary residual in the proposed method. The new approach also provides a unique definition for the sign of the stabilization terms. The reasons for using stabilization terms only at the boundaries is discussed and the two methods are compared. Several numerical examples are presented to demonstrate the performance and convergence of the proposed methods. Copyright © 2005 John Wiley & Sons, Ltd.     

Defining an accurate MFS solution for 2.5D acoustic and elastic wave propagation

This paper proposes a simple methodology to assess the accuracy of the method of fundamental solutions (MFS) when applied to 2.5D acoustic and elastic wave propagation. The proposed technique is developed in the frequency domain. It copes with the precision uncertainty difficulty presented by the MFS solution through its dependency on the number and position of virtual sources and collocation points.The methodology relies on the correlation between the errors registered along surfaces, where boundary or continuity conditions are known a priori, with those obtained along the system domain. Circular cylindrical domains are modeled to illustrate the efficiency of the proposed methodology, since in this case analytical solutions are available.A numerical example is used to illustrate the application of the methodology to a more complex case. An elastic column exhibiting an embedded curved crack, with null thickness, is used to illustrate the applicability of the proposed technique. Since, there are no known analytical solutions; the results provided by the traction boundary element method (TBEM) are used as reference solutions.     

On the choice of source points in the method of fundamental solutions

The method of fundamental solutions (MFS) may be seen as one of the simplest methods for solving boundary value problems for some linear partial differential equations (PDEs). It is a meshfree method that may present remarkable results with a small computational effort. The meshfree feature is particularly attractive when we need to change the shape of the domain, which occurs, for instance, in shape optimization and inverse problems. The MFS may be viewed as a Trefftz method, where the approximations have the advantage of verifying the linear PDE, and therefore we may bound the inner error from the boundary error, in well-posed problems. A main counterpart for these global numerical methods, that avoid meshes, are the associated linear systems with dense and ill conditioned matrices. In these methods a sort of uncertainty principle occurs—we cannot get both accurate results and good conditioning—one of the two is lost. A specific feature of the MFS is some freedom in choosing the source points. This might lead to excellent results, but it may also lead to poor results, or even to impossible approximations. In this work we will discuss the choice of source points and propose a choice along the discrete normal direction (following [Alves CJS, Antunes PRS. The method of fundamental solutions applied to the calculation of eigenfrequencies and eigenmodes of 2D simply connected shapes. Comput Mater Continua 2005;2(4):251–66]), with a possible local criterion to define the distance to the boundary. We will also address some extensions that connect the asymptotic MFS to other methods by choosing the sources on a circle/sphere far from the boundary. We also present a direct connection between the approximation based on radial basis functions (RBF) and the MFS approximation in a higher dimension. This increase in dimension was somehow already present in a previous work [Alves CJS, Chen CS. A new method of fundamental solutions applied to non-homogeneous elliptic problems. Adv Comput Math 2005;23:125–42], where the frequency was used as the extra dimension. The free parameters in RBF inverse multiquadrics 2D approximation correspond in fact to the source point distance to the boundary plane in a Laplace 3D setting. Some numerical simulations are presented to illustrate theoretical issues.     

A fast multipole accelerated method of fundamental solutions for potential problems

The fast multipole method (FMM) is a very effective way to accelerate the numerical solutions of the methods based on Green's functions or fundamental solutions. Combined with the FMM, the boundary element method (BEM) can now solve large-scale problems with several million unknowns on a desktop computer. The method of fundamental solutions (MFS), also called superposition or source method and based on the fundamental solutions but without using integrals, has been studied for several decades along with the BEM. The MFS is a boundary meshless method in nature and offers more flexibility in modeling of a problem. It also avoids the singularity of the kernel by placing the source at some auxiliary points off the problem domain. However, like the traditional BEM, the conventional MFS also requires O(N²) operations to compute the system of equations and another O(N³) operations to solve the system using direct solvers, with N being the number of unknowns. Combining the FMM and MFS can potentially reduce the operations in formation and solution of the MFS system, as well as the memory requirement, all to O(N). This paper is an attempt in this direction. The FMM formulations for the MFS is presented for 2D potential problem. Issues in implementation of the FMM for the MFS are discussed. Numerical examples with up to 200,000 DOF's are solved successfully on a Pentium IV PC using the developed FMM MFS code. These results clearly demonstrate the efficiency, accuracy and potentials of the fast multipole accelerated MFS.     

The method of fundamental solutions with dual reciprocity for diffusion and diffusion–convection using subdomains

The method of fundamental solutions (MFS), first proposed in the 1960s, has recently reappeared in the literature and solutions of an extraordinary accuracy have been reported using relatively few data points. The method requires no mesh and therefore no integration, and has been recently combined with dual reciprocity method (DRM) for treating inhomogeneous terms. The objective of this paper is the combination of the two methods for treating convective terms which are derivatives of the problem variable. First the formulation of the methods for mixed Neumann–Dirichlet boundary conditions is considered, as both these types of boundary condition are necessary for this type of problem. Next a formulation for the usual Crank–Nickleson and Galerkin time-stepping procedures is obtained for both diffusion and diffusion–convection and the use of the subdomain technique with MFS is considered. Finally results obtained for some test problems are presented including a diffusion convection problem with variable velocity using both a single domain and a division into subregions, the convective terms being modeled using DRM. Results are compared with exact solutions and in some cases with DRBEM examples from the literature.     

Application of the boundary element method to three-dimensional potential problems in heterogeneous media

The present work discusses a solution procedure for heterogeneous media three-dimensional potential problems, involving nonlinear boundary conditions. The problem is represented mathematically by the Laplace equation and the adopted numerical technique is the boundary element method (BEM), here using velocity correcting fields to simulate the conductivity variation of the domain. The integral equation is discretized using surface elements for the boundary integrals and cells, for the domain integrals. The adopted strategy subdivides the discretized equations in two systems: the principal one involves the calculation of the potential in all boundary nodes and the secondary which determines the correcting field of the directional derivatives of the potential in all points. Comparisons with other numerical and analytical solutions are presented for some examples.     

The Method of Fundamental Solutions Applied to the Calculation of Eigenfrequencies and Eigenmodes of 2D Simply Connected Shapes

In this work we show the application of the Method of Fundamental Solutions(MFS) in the determination of eigenfrequencies and eigenmodes associated to wave scattering problems. This meshless method was already applied to simple geometry domains with Dirichlet boundary conditions (cf. Karageorghis (2001)) and to multiply connected domains (cf. Chen, Chang, Chen, and Chen (2005)). Here we show that a particular choice of point-sourcescan lead to very good results for a fairly general type of domains. Simulations with Neumann boundary conditionare also considered.     

The method of fundamental solutions for three-dimensional elastostatic problems of transversely isotropic solids

This paper describes the method of fundamental solutions (MFS) to solve three-dimensional elastostatic problems of transversely isotropic solids. The desired solution is represented by a series of closed-form fundamental solutions, which are the displacement fields due to concentrated point forces acting on the transversely isotropic material. To obtain the unknown intensities of the fundamental solutions, the source points are properly located outside the computational domain and the boundary conditions are then collocated. Furthermore, the closed-form traction fields corresponding to the previously published point force solutions are reviewed and addressed explicitly in suitable forms for numerical implementations. Three numerical experiments including Dirichlet, Robin, and peanut-shaped-domain problems are carried out to validate the proposed method. It is found that the method performs well for all the three cases. Furthermore, a rescaling method is introduced to improve the accuracy of Robin problem with noisy boundary data. In the spirits of MFS, the present meshless method is free from numerical integrations as well as singularities.     

基本解方法解水下刚性目标三维Helmholtz外散射问题

运用基本解方法求解水下刚性目标三维Helmholtz外散射问题。研究了源点位置分布和数目对基本解方法计算结果的影响,比较了最小二乘配点法和等额配点法的计算精度。结果表明,当源点构成的形状与目标边界的形状差异大时,计算精度差,增加源点的数量可提高计算精度,运用较少的源点也可获得令人满意的精度,从而提高计算效率,但源点不宜距离目标边界过远;最小二乘配点法的计算精度较等额配点法高些。     

The method of fundamental solutions with a minimum orderformulation applied to 3D eddy current problems

The most salient feature of the method of fundamental solutions is that by locating the source points on an auxiliary boundary, the singular kernels encountered in most boundary element methods (BEMs) are avoided. Therefore, good results can be obtained. The A*-ψ formulation is used in the 3D MFS. Furthermore, by eliminating the dependent coefficients from the equations, the unknowns are reduced by one on each node, as compared to the conventional BEM. In so doing, a minimum-order formulation is obtained. Two examples of the application of the approach are given. Results are compared with analytical solutions, as well as with the conventional BEM solutions     

Numerical solution of two backward parabolic problems using method of fundamental solutions

This work is devoted to a numerical algorithm based on the method of fundamental solutions (MFS) for solving two backward parabolic problems with different boundary conditions, one with nonlocal Dirichlet boundary conditions, and second one with Robin type boundary conditions. The initial temperature distribution will be identified from the final temperature distribution, which appear in some applied subjects. The Tikhonov regularization method with the L-curve criterion for choosing the regularization parameter is adopted for solving the resulting matrix equation which is highly ill-conditioned. Two numerical examples are provided to show the high efficiency of the suggested method.