A study on the method of fundamental solutions using an image concept

In this paper, both analytical and semi-analytical solutions for Green’s functions are obtained by using the image method which can be seen as a special case of method of fundamental solutions (MFS). The image method is employed to solve the Green’s function for the annular, eccentric and half-plane Laplace problems. In addition, an analytical solution is derived for the fixed-free annular case. For the half-plane problem with a circular hole and an eccentric annulus, semi-analytical solutions are both obtained by using the image concept after determining the strengths of two frozen image points and a free constant by matching boundary conditions. It is found that two frozen images terminated at the two focuses in the bipolar coordinates for the problems with two circular boundaries. A boundary value problem of an eccentric annulus without sources is also considered. Error distribution is plotted after comparing with the analytical solution derived by Lebedev et al. using the bipolar coordinates. The optimal locations for the source distribution in the MFS are also examined by using the image concept. It is observed that we should locate singularities on the two focuses to obtain better results in the MFS. Besides, whether the free constant is required or not in the MFS is also studied. The results are compared well with the analytical solutions.     

A variational-type method of fundamental solutions for a Cauchy problem of Laplace’s equation

In this paper, we propose a new regularization method based on a finite-dimensional subspace generated from fundamental solutions for solving a Cauchy problem of Laplace’s equation in a simply-connected bounded domain. Based on a global conditional stability for the Cauchy problem of Laplace’s equation, the convergence analysis is given under a suitable choice for a regularization parameter and an a-priori bound assumption to the solution. Numerical experiments are provided to support the analysis and to show the effectiveness of the proposed method from both accuracy and stability.     

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.     

The method of fundamental solutions with eigenfunction expansion method for nonhomogeneous diffusion equation

In this article we describe a numerical method to solve a nonhomogeneous diffusion equation with arbitrary geometry by combining the method of fundamental solutions (MFS), the method of particular solutions (MPS), and the eigenfunction expansion method (EEM). This forms a meshless numerical scheme of the MFS‐MPS‐EEM model to solve nonhomogeneous diffusion equations with time‐independent source terms and boundary conditions for any time and any shape. Nonhomogeneous diffusion equation with complex domain can be separated into a Poisson equation and a homogeneous diffusion equation using this model. The Poisson equation is solved by the MFS‐MPS model, in which the compactly supported radial basis functions are adopted for the MPS. On the other hand, utilizing the EEM the diffusion equation is first translated to a Helmholtz equation, which is then solved by the MFS together with the technique of the singular value decomposition (SVD). Since the present meshless method does not need mesh generation, nodal connectivity, or numerical integration, the computational effort and memory storage required are minimal as compared with other numerical schemes. Test results for two 2D diffusion problems show good comparability with the analytical solutions. The proposed algorithm is then extended to solve a problem with irregular domain and the results compare very well with solutions of a finite element scheme. Therefore, the present scheme has been proved to be very promising as a meshfree numerical method to solve nonhomogeneous diffusion equations with time‐independent source terms of any time frame, and for any arbitrary geometry. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

A moving pseudo‐boundary method of fundamental solutions for void detection

We propose a new moving pseudo‐boundary method of fundamental solutions (MFS) for the determination of the boundary of a void. This problem can be modeled as an inverse boundary value problem for harmonic functions. The algorithm for imaging the interior of the medium also makes use of radial polar parametrization of the unknown void shape in two dimensions. The center of this radial polar parametrization is considered to be unknown. We also include the contraction and dilation factors to be part of the unknowns in the resulting nonlinear least‐squares problem. This approach addresses the major problem of locating the pseudo‐boundary in the MFS in a natural way, because the inverse problem in question is nonlinear anyway. The feasibility of this new method is illustrated by several numerical examples. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Multiple solutions for a class of biharmonic equations with a nonlinearity concave at the origin

In this paper, we investigate the existence of multiple solutions for a class of biharmonic equations where the nonlinearity involves a concave term at the origin. The solutions are obtained from the versions of mountain pass lemma and linking theorem.     

Localized method of fundamental solutions for large-scale modeling of two-dimensional elasticity problems

The traditional method of fundamental solutions (MFS) based on the “global” boundary discretization leads to dense and non-symmetric coefficient matrices that, although smaller in sizes, require huge computational cost to compute the system of equations using direct solvers. In this study, a localized version of the MFS (LMFS) is proposed for the large-scale modeling of two-dimensional (2D) elasticity problems. In the LMFS, the whole analyzed domain can be divided into small subdomains with a simple geometry. To each of the subdomain, the traditional MFS formulation is applied and the unknown coefficients on the local geometric boundary can be calculated by the moving least square method. The new method yields a sparse and banded matrix system which makes the method very attractive for large-scale simulations. Numerical examples with up to 200,000 unknowns are solved successfully using the developed LMFS code.     

Study on connections of the MFS, Trefftz method, indirect BIEM and invariant MFS in the three-dimensional Laplace problems containing spherical boundaries

Following the success of a study on the method of fundamental solutions using an image concept [13], we extend to solve the three-dimensional Laplace problems containing spherical boundaries by using the three approaches. The case of eccentric sphere for the Laplace problem is considered. The optimal locations for the source distribution to include the foci in the MFS are also examined by using the image concept in the 3D problems. Whether a free constant is required or not in the MFS is also studied. The error distribution is discussed after comparing with the analytical solution derived by using the bispherical coordinates. Besides, the relationship between the Trefftz bases and the singularity in the MFS for the three-dimensional Laplace problems is also addressed. It is found that one source of the MFS contains several interior and exterior Trefftz sets through a degenerate kernel. On the contrary, one single Trefftz base can be superimposed by some lumped sources in the MFS through an indirect BIEM. Based on this finding, the relationship between the fictitious boundary densities of the indirect BIEM and the singularity strength in the MFS can be constructed due to the fact that the MFS is a lumped version of an indirect BIEM.     

Rate of convergence of the method of fundamental solutions and hyperinterpolation for modified Helmholtz equations on the unit ball

Approximate solutions of boundary value problems of homogeneous modified Helmholtz equations on the unit ball are explicitly constructed by the method of fundamental solutions (MFS) with the order of approximation provided. Hyperinterpolation is used to find particular solutions of non-homogeneous equations, and the rate of convergence of solving boundary value problems of non-homogeneous equations is derived. Numerical examples are shown to demonstrate the efficiency of the methods.      

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. AMS subject classification 35J25, 65N38, 65R20, 74J20     

A matrix decomposition MFS algorithm for axisymmetric biharmonic problems

We consider the approximate solution of axisymmetric biharmonic problems using a boundary-type meshless method, the Method of Fundamental Solutions (MFS) with fixed singularities and boundary collocation. For such problems, the coefficient matrix of the linear system defining the approximate solution has a block circulant structure. This structure is exploited to formulate a matrix decomposition method employing fast Fourier transforms for the efficient solution of the system. The results of several numerical examples are presented. AMS subject classification 65N38, 65F30, 65T50, 65Y99     

Numerical calculation of eigensolutions of 3D shapes using the method of fundamental solutions

In this work, we study the application of the Method of Fundamental Solutions (MFS) for the calculation of eigenfrequencies and eigenmodes in two and three‐dimensional domains. We address some mathematical results about properties of the single layer operator related to the eigenfrequencies. Moreover, we propose algorithms for the distribution of the collocation and source points of the MFS in three‐dimensional domains which is an extension of the choices considered by Alves and Antunes (CMC 2(2005), 251–266) for the two‐dimensional case. Also the application of the Plane Waves Method is investigated. Several examples with Dirichlet and Neumann boundary conditions are considered to illustrate the performance of the proposed methods. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1525–1550, 2011     

Lower and upper bounds of condition number for Vandermonde-wise matrices and method of fundamental solutions using pseudo radial-lines

Consider the method of fundamental solutions (MFS) for 2D Laplace's equation in a bounded simply connected domain $S$. In the standard MFS, the source nodes are located on a closed contour outside the domain boundary $\mathrm{\Gamma }(=\partial S)$, which is called pseudo-boundary. For circular, elliptic, and general closed pseudo-boundaries, analysis and computation have been studied extensively. New locations of source nodes are proposed along two pseudo radial-lines outside $\mathrm{\Gamma }$. Numerical results are very encouraging and promising. Since the success of the MFS mainly depends on stability, our efforts are focused on deriving the lower and upper bounds of condition number (Cond). The study finds stability properties of new Vandermonde-wise matrices on nodes $x_i\in [a,b]$ with . The Vandermonde-wise matrix is called in this article if it can be decomposed into the standard Vandermonde matrix. New lower and upper bounds of Cond are first derived for the standard Vandermonde matrix, and then for new algorithms of the MFS using two pseudo radial-lines. Both lower and upper bounds of Cond are intriguing in the stability study for the MFS. Numerical experiments are carried out to verify the stability analysis made. Since the fundamental solutions (as $\{\ln |\overline{{PQ}_i}|\}$) are the basis functions of the MFS, new Vandermonde-wise matrices are found. Since the nodes $x_i\in [a,b]$ with may come from approximations and interpolations by the Laurent polynomials with singular part, the conclusions in this article are important not only to the MFS but also to matrix analysis.     

Efficient MFS algorithms in regular polygonal domains

In this work, we apply the Method of Fundamental Solutions (MFS) to harmonic and biharmonic problems in regular polygonal domains. The matrices resulting from the MFS discretization possess a block circulant structure. This structure is exploited to produce efficient Fast Fourier Transform–based Matrix Decomposition Algorithms for the solution of these problems. The proposed algorithms are tested numerically on several examples.      

The closed-form particular solutions for Laplace and biharmonic operators using a Gaussian function

Particular solutions play a critical role in solving inhomogeneous problems using boundary methods such as boundary element methods or boundary meshless methods. In this short article, we derive the closed-form particular solutions for the Laplace and biharmonic operators using the Gaussian radial basis function. The derived particular solutions are implemented numerically to solve boundary value problems using the method of particular solutions and the localized method of approximate particular solutions. Two examples in 2D and 3D are given to show the effectiveness of the derived particular solutions.     

Numerical analysis of the method of fundamental solution for harmonic problems in annular domains

In this study, we investigate the application of the method of fundamental solutions (MFS) to the Dirichlet problem for Laplace's equation in an annular domain. We examine the properties of the resulting coefficient matrix and its eigenvalues. The convergence of the method is proved for analytic boundary data. An efficient matrix decomposition algorithm using fast Fourier transforms (FFTs) is developed for the computation of the MFS approximation. We also tested the algorithm numerically on several problems confirming the theoretical predictions. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Ground state and nodal solutions for a class of biharmonic equations with singular potentials

In this paper, we are concerned with a class of fourth order elliptic equations of Kirchhoff type with singular potentials in $\mathbb{R}^{N}.$ The existence of ground state and nodal solutions are obtained by using variational methods and properties of Hessian matric. Furthermore, the "energy doubling" property of nodal solutions is also explored.