Classical Method Of Fundamental Solutions Research Articles

A highly accurate indirect boundary integral equation solution for three dimensional elastic scattering problem

In this paper, we propose a numerical formula for three dimensional elastic wave scattering problem. Different from the classical boundary integral equation method, the three dimensional elastic scattering problem is transformed into a coupled boundary value problem including a scalar Helmholtz equation and a vector Helmholtz equation by the decomposition of the displacement. The coupled boundary value problem can be solved by the boundary integral equations based on the Helmholtz equation. A denseness result is given to support this algorithm. A comparative study with the classical method of fundamental solutions getting from the boundary integral equation method is given to check the accuracy with different types of boundary conditions. From the numerical results, the present method can give more accurate results than the classical method of fundamental solutions method.

Fracture mechanics analysis of bimaterial interface cracks using an enriched method of fundamental solutions: Theory and MATLAB code

In this paper we study the application of the method of fundamental solutions (MFS) to interface crack analysis in linear elastic bimaterial fracture mechanics. Such problem presents some modelling difficulties because of the oscillatory behavior of the local asymptotic fields in the neighborhood of the interface crack-tip. The present approach is based on the combination of the classical MFS approximation for linear elasticity problems and a set of enrichment functions that take into account the asymptotic behavior of the near-tip displacement and stress fields. The enriched MFS technique automatically incorporates the oscillatory crack-tip behavior and thus can significantly improve the computational accuracy of the displacements and stresses in the vicinity of the crack-tip, even with a relatively coarse MFS model. A multi-domain MFS technique and the displacement extrapolation method (DEM) are used to compute the complex stress intensity factor (SIF) of the cracked bimaterials. Several benchmark numerical examples are presented to illustrate the accuracy and efficiency of the present method. Results calculated by using the boundary element (BEM) method are also given for the purpose of comparison.

Localized method of fundamental solutions for two-dimensional anisotropic elasticity problems

The purpose of the presented paper is to develop further the Localized Method of Fundamental Solutions (LMFS) for solving two-dimensional anisotropic elasticity problems. The computational domain is divided into overlapping subdomains. In the LMFS, the classical Method of Fundamental Solutions (MFS) is employed in each of these local subdomains to get an expression of the solution for the main node of this subdomain. The expression is structured by a linear combination of the solutions of the other nodes in this subdomain. Displacement or traction boundary conditions are satisfied at the boundary nodes. The solution is calculated from an equation set formed by the boundary conditions for the boundary nodes and expressions in the subdomain for the interior nodes. The presented three numerical examples demonstrate that the novel method is suitable for solving large-scale problems, and especially, the problems with complicated domains.

Stress analysis of elastic bi-materials by using the localized method of fundamental solutions

<abstract> <p>The localized method of fundamental solutions belongs to the family of meshless collocation methods and now has been successfully tried for many kinds of engineering problems. In the method, the whole computational domain is divided into a set of overlapping local subdomains where the classical method of fundamental solutions and the moving least square method are applied. The method produces sparse and banded stiffness matrix which makes it possible to perform large-scale simulations on a desktop computer. In this paper, we document the first attempt to apply the method for the stress analysis of two-dimensional elastic bi-materials. The multi-domain technique is employed to handle the non-homogeneity of the bi-materials. Along the interface of the bi-material, the displacement continuity and traction equilibrium conditions are applied. Several representative numerical examples are presented and discussed to illustrate the accuracy and efficiency of the present approach.</p> </abstract>

Modified method of regularized sources for potential flow

This paper describes the development of the Method of Regularized Sources for potential flow problems. It is based on the modification of the fundamental solution near the source point by replacing the singularity with a blob in form of a steep rational function. This allows to solve the problems in the same way as with Method of Fundamental Solutions, however without an artificial boundary. Method of Regularized Sources gives excellent results for Dirichlet boundary conditions, however it fails for Neumann boundary conditions. To overcome this problem are the source point positions on the segments of the boundary with Neumann boundary positions placed close to the collocation points. This approach somehow represents a blending of the Method of Regularized Sources and the classical Method of Fundamental Solutions. The novel approach is characterized by two free parameters; the blob thickness and the artificial boundary displacement position. A two-dimensional numerical example of potential flow around circle is analyzed in detail regarding these two free parameters. The modified Method of Regularized Sources gives even more accurate results for potential and derivatives than the Method of Fundamental Solutions. The source point can be placed 2–25 times closer to the boundary collocation points than with the classical Method of Fundamental Solutions and thus reduces the problem of the placement of the artificial boundary.

A multi-level technique for the Method of Fundamental Solutions without regularization and desingularization

The traditional Method of Fundamental Solutions is revisited, based on a special strategy of defining the external source points. Unlike the classical Method of Fundamental Solutions, the sources are categorized into groups; the density of the spatial distribution of the sources decreases rapidly far from the boundary. On each group, the original problem is discretized using the same set of boundary collocation points. Such groups of sources are constructed in a fully automated way by the quadtree/octtree algorithm. The discretized problems are solved in the sense of least squares. A simple multi-level method is built up, using the (conjugate) gradient iteration as a smoothing procedure. The resulting method significantly reduces the computational complexity. Moreover, the problem of evaluation singular integrals as well as the problem of severely ill-conditioned matrices are avoided. The method is generalized to 3D axisymmetric potential problems as well.

Domain decomposition methods with fundamental solutions for Helmholtz problems with discontinuous source terms

The direct application of the classical method of fundamental solutions (MFS) is restricted to homogeneous linear partial differential equations (PDEs). The use of fundamental solutions with different frequencies allowed the extension of the MFS to non-homogeneous PDEs, in particular, for Poisson or Helmholtz equations and for elastostatic or elastodynamic problems. This method has been called method of fundamental solutions for domains (MFS-D), but it faces an approximation problem when the non-homogeneous term presents discontinuities, because the fundamental solutions are analytic functions outside the source point set. In this paper we analyze two domain decomposition techniques for overcoming this approximation problem. The problem is set in the context of the modified Helmholtz equation, and we also establish the missing density results that justify both the MFS and the MFS-D approximations. Numerical results are presented comparing a direct and an iterative domain decomposition technique, with simulations in non-trivial domains.

Non-singular method of fundamental solutions for elasticity problems in three-dimensions

In this paper, the Non-singular Method of Fundamental Solutions (NMFS) is extended to three-dimensional (3D) isotropic linear elasticity problems. In order to avoid the singularities in the classical Method of Fundamental Solutions (MFS), are the source points outside the problem domain replaced by normalizing the volume integral of the fundamental solutions over the sphere around the singularity on the physical boundary. The derivatives of the fundamental solutions at the singularity, required in the traction boundary conditions, are calculated from three reference solutions of the linearly varying simple displacement fields. The artificial boundary appearing in MFS is with this operations removed in NMFS. A comparison between NMFS and MFS solutions and analytical solutions for two single and two bi-material elasticity problems is used to assess the feasibility and the accuracy of the newly developed 3D method. Although NMFS results are slightly less accurate than MFS results in all spectra of performed tests, all NMFS results converge to the analytical solution. The lack of artificial boundary is particularly advantageous when using NMFS in multibody problems. The developments describe a first use of NMFS for 3D solid mechanics problems.

On the application of the method of fundamental solutions to boundary value problems with jump discontinuities

Two meshfree methods are proposed for the numerical solution of boundary value problems (BVPs) for the Laplace equation, coupled with boundary conditions with jump discontinuities. In the first case, the BVP is solved in two steps, using a subtraction of singularity approach. Here, the singular subproblem is solved analytically while the classical method of fundamental solutions (MFS) is applied for the solution of the regular subproblem. In the second case, the total BVP is solved using a variant of the MFS where its approximation basis is enriched with a set of harmonic functions with singular traces on the boundary of the domain. The same singularity-capturing functions, motivated by the boundary element method (BEM), are used for the singular part of the solution in the first method and for augmenting the MFS basis in the second method. Comparative numerical results are presented for 2D problems with discontinuous Dirichlet boundary conditions. In particular, the inappropriate oscillatory behavior of the classical MFS solution, due to the Gibbs phenomenon, is shown to vanish.

Recovery of the temperature and the heat flux by a novel meshless method from the measured noisy data

In this paper, we give an invariant method of fundamental solutions (MFS) for recovering the temperature and the heat flux. The invariant MFS is to keep a very basic natural property, which is called the invariance property under trivial coordinate changes in the problem description. The optimal regularization parameter is chosen by Morozov discrepancy principle. Then the reason for introducing the regularization is explained clearly by using the potential function. Three kinds of boundary value problems are investigated to show the effectiveness of this method with some examples. In especial, when the classical MFS does not give accurate results for some problems, it is shown that the proposed method is effective and stable. For each example, the numerical convergence, accuracy, and stability with respect to the number of source points, the distance between the pseudo and real boundary, and decreasing the amount of noise added into the input data, respectively, are also analyzed.

Method of fundamental solutions for 3D elasticity with body forces by coupling compactly supported radial basis functions

In this paper, a meshless computational model by integrating the method of fundamental solutions (MFS) and the method of particular solutions fulfilled with compactly supported radial basis functions (CSRBF) is developed for three-dimensional (3D) linear elasticity with the presence of body forces. The corresponding displacement and stress particular solution kernels across the support radius are firstly derived using Galerkin vectors and then are used to modify the boundary conditions. Subsequently, the classical meshless MFS, in which the homogeneous part of the full solutions are approximated using the linear combination of displacement and stress fundamental solutions in 3D linear elasticity, is formulated for solving the homogeneous 3D linear elastic system. Finally, several examples are presented to demonstrate the accuracy and efficiency of the present meshless method and also the effect of sparseness of interpolation matrix in CSRBF interpolation is discussed.

Non-singular Method of Fundamental Solutions for anisotropic elasticity

The purpose of the present paper is to develop a Non-singular Method of Fundamental Solutions (NMFS) for two-dimensional anisotropic linear elasticity problems. The NMFS is based on the classical Method of Fundamental Solutions (MFS) with regularization of the singularities. This is achieved by replacing the concentrated point sources with distributed sources over disks around the singularity, as recently developed for isotropic elasticity problem. In case of the displacement boundary conditions, the values of distributed sources are calculated by a simple numerical procedure, since the closed form solution is not available. In case of traction boundary conditions, the respective desingularized values of the derivatives of the fundamental solution in the coordinate directions, as required in the calculations, are calculated indirectly by considering two reference solutions of the linearly varying simple displacement fields. The feasibility and accuracy of the newly developed method are demonstrated through comparison with MFS solutions and analytical solutions for a spectra of anisotropic plane strain elasticity problems, including bi-material arrangements. NMFS turns out to give similar results as the MFS in all spectra of performed tests. The lack of artificial boundary is particularly advantageous for using NMFS in multi-body problems.

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.

A new method of fundamental solutions applied to nonhomogeneous elliptic problems

The classical method of fundamental solutions (MFS) has only been used to approximate the solution of homogeneous PDE problems. Coupled with other numerical schemes such as domain integration, dual reciprocity method (with polynomial or radial basis functions interpolation), the MFS can be extended to solve the nonhomogeneous problems. This paper presents an extension of the MFS for the direct approximation of Poisson and nonhomogeneous Helmholtz problems. This can be done by using the fundamental solutions of the associated eigenvalue equations as a basis to approximate the nonhomogeneous term. The particular solution of the PDE can then be evaluated. An advantage of this mesh-free method is that the resolution of both homogeneous and nonhomogeneous equations can be combined in a unified way and it can be used for multiscale problems. Numerical simulations are presented and show the quality of the approximations for several test examples.