This paper describes an application of the recently developed sparse scheme of the method of fundamental solutions (MFS) for the simulation of three-dimensional modified Helmholtz problems. The solution to the given problems is approximated by a two-step strategy which consists of evaluating the particular solution and the homogeneous solution. The homogeneous solution is approximated by the traditional MFS. The original dense system of the MFS formulation is condensed into a sparse system based on the exponential decay of the fundamental solutions. Hence, the homogeneous solution can be efficiently obtained. The method of particular solutions with polyharmonic spline radial basis functions and the localized method of approximate particular solutions in combination with the Gaussian radial basis function are employed to approximate the particular solution. Three numerical examples including a near singular problem are presented to show the simplicity and effectiveness of this approach.

This study investigates the applicability of the singular boundary method (SBM), a recent developed meshless boundary collocation method, for the analysis of two-dimensional (2D) thin structural problems. The troublesome nearly-singular kernels, which are crucial in the applications of SBM to thin shapes, are dealt with efficiently by using a non-linear transformation technique. Promising SBM results with only a small number of boundary nodes are obtained for thin structures with the thickness-to-length ratio is as small as 1E-9, which is sufficient for modeling most thin layered coating systems as used in smart materials and micro-electro-mechanical systems. The advantages, disadvantages and potential applications of the proposed method, as compared with the finite element (FEM) and boundary element methods (BEM), are also discussed.

The method of fundamentalsolutions (MFS)is known as aneffective boundary meshless method. However, the formulation of the MFS results in a dense and extremely ill-conditioned matrix. In this paper we investigate the MFS for solving large-scale problems for the nonhomogeneous modified Helmholtz equation. The key idea is to exploit the exponential decay of the fundamental solution of the modified Helmholtz equation, and consider a sparse or diagonal matrix instead of the original dense matrix. Hence, the homogeneous solution can be obtained efficiently and accurately. A standard two-step solution process which consists of evaluating the particular solution and the homogeneous solution is applied. Polyharmonic spline radial basis functions are employed to evaluate the particular solution. Five numerical examples in irregular domains and a large number of boundary collocation points are presented to show the simplicity and effectiveness of our approach for solving large-scale problems.

We propose a new moving pseudo-boundary method of fundamental solutions (MFS) for the determination of the boundary of a three-dimensional void (rigid inclusion or cavity) within a conducting homogeneous host medium from overdetermined Cauchy data on the accessible exterior boundary. The algorithm for imaging the interior of the medium also makes use of radial spherical parametrization of the unknown star-shaped void and its centre in three dimensions. We also include the contraction and dilation factors in selecting the fictitious surfaces where the MFS sources are to be positioned in the set of unknowns in the resulting regularized nonlinear least-squares minimization. The feasibility of this new method is illustrated in several numerical examples.

In this paper, we introduce two Galerkin formulations of the Method of Fundamental Solutions (MFS). In contrast to the collocation formulation of the MFS, the proposed Galerkin formulations involve the evaluation of integrals over the boundary of the domain under consideration. On the other hand, these formulations lead to some desirable properties of the stiffness matrix such as symmetry in certain cases. Several numerical examples are considered by these methods and their various features compared.

In this paper, we investigate the method of fundamental solutions (MFS) for solving exterior Helmholtz problems with high wave-number in axisymmetric domains. Since the coefficient matrix in the linear system resulting from the MFS approximation has a block circulant structure, it can be solved by the matrix decomposition algorithm and fast Fourier transform for the fast computation of large-scale problems and meanwhile saving computer memory space. Several numerical examples are provided to demonstrate its applicability and efficacy in two and three dimensional domains.

An inverse geometric problem for two-dimensional Helmholtz-type equations arising in corrosion detection is considered. This problem involves determining an unknown corroded portion of the boundary of a two-dimensional domain and possibly its surface heat transfer (impedance) Robin coefficient from one or two pairs of boundary Cauchy data (boundary temperature and heat flux), and is solved numerically using the meshless method of fundamental solutions. A nonlinear unconstrained minimisation of the objective function is regularised when noise is added into the input boundary data. The stability of the numerical results is investigated for several test examples, with respect to noise in the input data and various values of the regularisation parameters.

This work integrates the Taylor theorem and the method of fundamental solutions to develop a numerical tool for estimating the added mass coefficient tensor for a solid object of any convex shape moving in potential flow. In potential flow theory, the Taylor theorem calculates the added mass coefficient tensor for a Rankine body with algebraic manipulations of the properties of the internal singularities employed to generate the corresponding flow. To apply this theorem for objects in other shapes, the singularity strength and locations are required information which is facilitated numerically in this work by the method of fundamental solutions (MFS). The developed scheme is tested on a circle, an ellipse, a square, and a rhombus and the numerical results are in good agreement with the corresponding analytical values. A final example of a Cassini oval is also considered to show the potential applications on bio-engineering problems.

In this paper, analytical particular solutions of the augmented polyharmonic spline (APS) associated with Reissner plate model are explicitly derived in order to apply the dual reciprocity method. In the derivations of the particular solutions, a coupled system of three second-ordered partial differential equations (PDEs), which governs problems of Reissner plates, is initially transformed into a single six-ordered PDE by the Hörmander operator decomposition technique. Then the particular solutions of the coupled system can be found by using the particular solution of the six-ordered PDE derived in the first author's previous study. These formulas are further implemented for solving problems of Reissner plates under arbitrary loadings. In the solution procedure, an arbitrary loading measured at some scattered points is first interpolated by the APS and a corresponding particular solution can then be approximated by using the prescribed formulas. After that the complementary homogeneous problem is formally solved by the method of fundamental solutions (MFS). Numerical experiments are carried out to validate these particular solutions.

Fourier transform is applied to remove the time-dependent variable in the diffusion equation. Under non-harmonic initial conditions this gives rise to a non-homogeneous Helmholtz equation, which is solved by the method of fundamental solutions and the method of particular solutions. The particular solution of Helmholtz equation is available as shown in [4, 15]. The approximate solution in frequency domain is then inverted numerically using the inverse Fourier transform algorithm. Complex frequencies are used in order to avoid aliasing phenomena and to allow the computation of the static response. Two numerical examples are given to illustrate the effectiveness of the proposed approach for solving 2-D diffusion equations.

The method of fundamental solutions (MFS) and domain decomposition method (DDM) are employed to solve the water-wave diffraction by a thin porous vertical breakwater of semi-infinite extent. Based on the linearized theory of water waves, the problem can be reduced to a boundary value problem with degenerate boundary. In contrast to other mesh dependent numerical method, the MFS is easier and more efficient to handle degenerate boundary value problems. Various incident wave angles and porous parameters are included to validate the power of the proposed numerical scheme. The present results demonstrate the MFS is sufficiently accurate and feasible to be used to predict water-wave diffraction by a thin permeable breakwater by comparing with literature.

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 thin plates resting on Winkler foundations under arbitrary loadings, where the DRM is based on the augmented polyharmonic splines constructed by splines and monomials. In the solution procedure, the arbitrary distributed loading is first approximated by the augmented polyharmonic splines (APS) and thus the desired particular solution can be represented by the corresponding analytical particular solutions of the APS. Thereafter, the complementary solution is solved formally by the MFS. In the mathematical derivations, the real coefficient operator in the governing equation is decomposed into two complex coefficient operators. In other words, the solutions obtained by the MFS-DRM are first treated in terms of these complex coefficient operators and then converted to real numbers in suitable ways. Furthermore, the boundary conditions of lateral displacement, slope, normal moment, and effective shear force are all given explicitly for the particular solutions of APS as well as the kernels of MFS. Finally, numerical experiments are carried out to validate these analytical formulas.

The main purpose of this paper is to investigate the pressure-stream function formulation to solve 2D and 3D Stokes flows by the meshless numerical scheme of the method of fundamental solutions (MFS). The MFS can be regarded as a truly scattered, grid-free (or meshless) and non-singular numerical scheme. By the proposed algorithm, the stream function is governed by the bi-harmonic equation while the pressure is governed by the Laplace equation. The velocity field is then obtained by the curl of the stream function for 2D flows and curl of the vector stream function for 3D flows. We can simultaneously solve the pressure, velocity, vorticity, stream function and traction forces fields. Furthermore during the present numerical procedure no pressure boundary condition is needed which is a tedious and forbidden task. The developed algorithm is used to test several numerical experiments for the benchmark examples, including (1) the driven circular cavity, (2) the circular cavity with eccentric rotating cylinder, (3) the square cavity with traction boundary conditions and (4) the uniform flow past a sphere. The results compare very well with the solutions obtained by analytical or other numerical methods such as finite element method (FEM). It is found that the meshless MFS will give a simpler and more efficient and accurate solutions to the Stokes flows investigated in this study.

This paper develops the method of fundamental solutions (MFS) to solve eigenfrequencies of plate vibrations of multiply connected domains. The complex-valued MFS combined with the mix potential method are utilized in order to avoid the spurious eigenvalues. The benchmarked problems of annular plates with clamped, simply supported and free boundary conditions are studied analytically as well as numerically. Wherein the results demonstrate that all true eigenvalues are contained and no spurious eigenvalues are included. In the analytical studies, the continuous version of the MFS is utilized to obtain the eigenequation by applying the degenerate kernels and Fourier series. The proposed numerical method is free from singularities, meshes, and numerical integrations and thus can be easily utilized to solve plate vibrations free from spurious eigenvalues in multiply connected domains.

This paper describes the applications of the method of fundamental solutions (MFS) as a mesh-free numerical method for the Stokes' first and second problems which prevail in the semi-infinite domain with constant and oscillatory velocity at the boundary in the fluid-mechanics benchmark problems. The time-dependent fundamental solutions for the semi-infinite problems are used directly to obtain the solution as a linear combination of the unsteady fundamental solution of the diffusion operator. The proposed numerical scheme is free from the conventional Laplace transform or the finite difference scheme to deal with the time derivative term of the governing equation. By properly placing the field points and the source points at a given time level, the solution is advanced in time until steady state solutions are reached. It is not necessary to locate and specify the condition at the infinite domain such as other numerical methods. Since the present method does not need mesh discretization and nodal connectivity, the computational effort and memory storage required are minimal as compared to the domain-oriented numerical schemes. Test results obtained for the Stokes' first and second problems show good comparisons with the analytical solutions. Thus the present numerical scheme has provided a promising mesh-free numerical tool to solve the unsteady semi-infinite problems with the space-time unification for the time-dependent fundamental solution.

The Method of Fundamental Solutions (MFS) is a boundary-typemeshless method for the solution of certain elliptic boundaryvalue problems. In this work, we investigate the properties of thematrices that arise when the MFS is applied to theDirichlet problem for Laplace's equation in a disk. In particular,we study the behaviour of the eigenvalues of these matrices andthe cases in which they vanish. Based on this, we propose amodified efficient numerical algorithm for the solution of theproblem which is applicable even in the cases when the MFS matrixmight be singular. We prove the convergence of the method foranalytic boundary data and perform a stability analysis of the methodwith respect to the distance of the singularities from the originand the number of degrees of freedom. Finally, wetest the algorithm numerically.