Traditional Method Of Fundamental Solutions Research Articles

A novel hybrid SBM-MFS methodology for acoustic wave propagation problems

In this paper, a novel hybrid meshless approach that combines the singular boundary method (SBM) and the method of fundamental solutions (MFS) to deal with two-dimensional (2D) exterior acoustic wave propagation problems is proposed and studied. The methodology is particularly devised to solve problems with complex boundary geometries containing geometric singularities such as corners and sharp edges. It employs the SBM to model intricate segments of these geometries and the MFS for the smooth ones. The proposed hybrid SBM-MFS method is studied in a 2D context in the framework of three benchmark examples involving acoustic radiation problems of circular-, square- and L-shaped objects in a full-space acoustic medium. In addition, the applicability of the proposed hybrid SBM-MFS methodology to predict the acoustic performance of a T-shaped thin barrier is also investigated. These examples are specifically designed to assess the feasibility, validity and accuracy of the hybrid SBM-MFS approach in comparison with the available analytical solutions and alternative numerical strategies such as the MFS, the SBM and the boundary element method (BEM). Numerical simulations demonstrate that the proposed method can match the level of accuracy of the MFS while keeps the robustness of the SBM when dealing with complex geometries, overcoming one the most important drawbacks of the traditional MFS. Moreover, the proposed hybrid SBM-MFS naturally avoids the non-uniqueness problem arising at the fictitious eigenfrequencies associated with the corresponding interior problems, a feature that neither the SBM nor the BEM inherently possesses.

Solving the Eigenfrequencies Problem of Waveguides by Localized Method of Fundamental Solutions with External Source

The localized method of fundamental solutions (LMFS) is a domain-type, meshless numerical method. Compared with numerical methods that have a high grid dependence, it does not require grid generation and numerical integration, so it can effectively improve computational efficiency and avoid complex integration processes. Moreover, it is formed using the traditional method of fundamental solutions (MFS) and the localization approach. Previous studies have shown that the MFS may produce a dense and ill-conditioned matrix. However, the proposed LMFS can yield a sparse system of linear algebraic equations, so it is more suitable and effective in solving complicated engineering problems. In this article, LMFS was used to solve eigenfrequency problems in electromagnetic waves, which were controlled using two-dimensional Helmholtz equations. Additionally, the resonant frequencies of the eigenproblem were determined by the response amplitudes. In order to determine the eigenfrequencies, LMFS was applied for solving a sequence of inhomogeneous problems by introducing an external source. Waveguides with different shapes were analyzed to prove the stability of the present LMFS in this paper.

A new approach to solve the anti-plane crack problems by the method of fundamental solutions

This paper expands the method of fundamental solutions (MFS) to solve the anti-plane crack problems based on the traditional fundamental solution by the conformal mapping technique without using the domain decomposition technique and/or the crack Green's function. Based on the complex analysis and the conformal mapping technique, the approximate solution of the complex analytic function is derived for the MFS. The modified boundary conditions are deduced to construct the interpolation equations of the MFS. The conformal mapping technique naturally introduces singularities of the stress fields and the stress intensity factors (SIFs) for the anti-plane crack problems without using the crack Green's function and the domain-decomposition technique. The obtained numerical examples illustrate the validity and accuracy for the anti-plane crack problems. The proposed method is an expansion of the traditional MFS, and can be applied for many anti-plane elastic problems with the simple concept and the easy numerical implementation.

Singularity problems from source functions as source nodes located near boundaries; numerical methods and removal techniques

Consider the Dirichlet problem for Laplace’s/Poisson’s equation in a bounded simply-connected domain S. The source function qln|PQ*¯| is a fundamental solution (FS), and it can be found in many physical problems. The singularity occurs when the boundary value data affected by qln|PQ*¯| as the source node Q* is located near the boundary Γ(=∂S). So far, there is no comprehensive study on this kind of singularity. In this paper, the solution singularity is explored and the reduced convergence rates are derived for the method of particular solutions (MPS) and the method of fundamental solutions (MFS). Classic domains, such as disks, ellipses and polygons, are discussed for analysis and computation. For this new kind of solution singularity, the convergence rates of the MFS and the MPS are very low. The errors caused by numerical integration are critical to the solution accuracy. A new analytic framework for the collocation Trefftz method (CTM) involving numerical integration is established in this paper; this is an advanced development of our previous study [19]. Since the numerical solutions are poor in accuracy, removal techniques are essential in applications. New removal techniques are proposed for a node Q* located near Γ. In this paper, an additional FS as, d0ln|PQ0¯|, is added to the original source nodes in the traditional MFS, and the point charge d0(=q) and the source node Q0 are unknowns to be sought by nonlinear solvers (such as the secant method). When the source node Q* is located inside S but near Γ, both simple domains (such as disks, ellipses and squares) and complicated domains (such as amoeba-like domains) are studied. The validity of the new removal techniques is supported by numerical experiments. The removal techniques in this paper may also be applied to solve source identification problems. A comprehensive study has been completed in this paper for the solitary source function qln|PQ*¯| as the source node Q* is located near Γ.

Bending analysis of simply supported and clamped thin elastic plates by using a modified version of the LMFS

The localized method of fundamental solutions is a recent domain-type meshless collocation method with the fundamental solutions of governing equations as the radial basis functions. This approach forms a sparse system matrix and has a higher efficiency than the traditional method of fundamental solutions. In this paper, a modified version of the localized method of fundamental solutions is developed for bending analysis of simply supported and clamped thin elastic plates. Some auxiliary nodes on the boundary are firstly introduced to provide additional weight coefficients, which contribute to the construction of the determined system of equations and avoid the over-determined equation system of the localized method of fundamental solutions for bending analysis of thin elastic plates. Several numerical experiments with simply supported or clamped boundary conditions are provided, and numerical results are in good agreement with the analytical or COMSOL Multiphysics solutions.

Localized method of fundamental solutions for 2D harmonic elastic wave problems

In this paper, a localized version of the method of fundamental solutions (MFS) named as the localized MFS (LMFS) is applied to two-dimensional (2D) harmonic elastic wave problems. Due to the dense and ill-conditioned coefficient matrices, the traditional MFS is computationally expensive and time-consuming to solve the above-mentioned problems which require a large amount number of nodes to obtain excellent numerical results. In the LMFS, the domain is divided into small subdomains, in which the traditional MFS is applied to represent the physical variables as linear combinations of the fundamental solution. A sparse and banded system of linear equations is yielded in this method, so that it is attractive to solve large-scale problems. A numerical example demonstrates the effectiveness and accuracy of the proposed LMFS.

Inverse Heat Conduction Problem in Two-Dimensional Anisotropic Medium

The study proposes a novel mesh-free scheme for a two-dimensional space and tests its efficiency for the inverse heat conduction problem in an anisotropic medium. It aims at determining unknown data at the inaccessible boundary with the help of an approximate solution constructed as a linear combination of fundamental solutions and heat polynomials. The advantage of using this combination lies in the fact that it yields ‘shape functions’ which are in turn solution of the considered equation. The crucial point to note here is that the proposed method yields highly accurate results and the number of collocation points required by the altered scheme are lesser than the standard method of fundamental solutions (MFS). As a direct consequence of the ill-posed problem, a highly ill-conditioned system is obtained, which is solved using regularization. Further, numerical results are provided for two-dimensional convex and non-convex domains to prove that the efficiency of the new scheme is slightly better than the traditional MFS, especially for noisy data.

Localized MFS for the inverse Cauchy problems of two-dimensional Laplace and biharmonic equations

This paper makes a first attempt to use a new localized method of fundamental solutions (LMFS) to accurately and stably solve the inverse Cauchy problems of two-dimensional Laplace and biharmonic equations in complex geometries. The LMFS firstly divides the whole physical domain into several small overlapping subdomains, and then employs the traditional method of fundamental solutions (MFS) formulation in every local subdomain for calculating the unknown coefficients on the local fictitious boundary. After that, a sparse linear system is formed by using the governing equation for interior nodes and the nodes on under-specified boundary, and by using the given boundary conditions for the nodes on over-specified boundary. Finally, the numerical solutions of the inverse problems can be obtained by solving the resultant sparse system. Compared with the traditional MFS with the “global” boundary discretization, the LMFS requires less computational cost, which may make the LMFS suitable for solving large-scale problems. Numerical experiments demonstrate the validity and accuracy of the proposed LMFS for the inverse Cauchy problems of two-dimensional Laplace and biharmonic equations with noisy boundary data.

Localized method of fundamental solutions for interior Helmholtz problems with high wave number

This paper introduces a localized version of the method of fundamental solutions (MFS) named as the localized MFS (LMFS) for two-dimensional (2D) interior Helmholtz problems with high wave number. Due to its full interpolation matrix, the traditional MFS is low-efficiency to solve the above-mentioned problem that requires a large number of boundary nodes for obtaining availably numerical results. For the LMFS, the computational domain is first divided into some overlap subdomains based on the distributed nodes. In each subdomain, physical variables are then represented as linear combinations of the fundamental solution of the governing equation as same as in the traditional MFS. A sparse and banded system matrix is finally formed for the LMFS by satisfying Helmholtz equation and boundary conditions, and thus the developed method is inherently efficient for large-scale problems. Three numerical examples are provided to verify the accuracy and the stability of the LMFS.

Analysis of three-dimensional interior acoustic fields by using the localized method of fundamental solutions

• Applying the localized method of fundamental solutions for 3D interior acoustic fields. • The behavior of the present method for oscillating solution is first investigated. • Acoustic fields in a high-speed train and a room with complicated geometries are solved successfully. The traditional method of fundamental solutions has a full interpolation matrix, and thus its solution is computationally expensive, especially for large-scale problems with complicated domains. In this paper, we make a first attempt to apply the localized method of fundamental solutions for analysis of 3D interior acoustic fields. The present method first divides the whole computational domain into some overlapping subdomains, and then expresses physical variables as linear combinations of the fundamental solution in each subdomain. Finally, the method forms a sparse and banded system matrix by satisfying both governing equations at interior nodes and boundary conditions at boundary nodes. We provide four numerical experiments to verify the accuracy and the stability of the method. Comparisons of numerical results and computational time are also made between the present method, the method of fundamental solutions, and the COMSOL software.

Localized method of fundamental solutions for large-scale modelling of three-dimensional anisotropic heat conduction problems – Theory and MATLAB code

The method of fundamental solutions (MFS) belongs to the family of meshless boundary collocation methods and now has been successfully tried for many kinds of engineering problems. The traditional MFS based on the “global” boundary discretization, however, 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. Such an approach will be arduous, time consuming and computationally expensive for analyzing large-scale problems. In the present work, a localized version of the MFS, named as the localized MFS (LMFS), is proposed for large-scale modelling of three-dimensional (3D) anisotropic heat conduction problems. In the LMFS, the computational domain can be divided into small subdomains with a simple geometry such as circle and/or sphere. To each of the subdomains, the MFS formulation is applied and the unknown coefficients on the local simple geometric boundary are approximated by the moving least square (MLS) method. The satisfactions of governing equations at interior points and boundary conditions at boundary nodes lead to a sparse and banded system matrix. Numerical examples with up to 1,000,000 unknowns are solved successfully using the developed LMFS code. The advantages, disadvantages and potential applications of the proposed method, as compared with the traditional MFS and boundary element method (BEM), are discussed. Finally, a fast, reliable and self-contained MATLAB code is provided in the part of Supplementary Materials of the paper.

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.

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.

Generalized method of fundamental solutions (GMFS) for boundary value problems

In order to cope with the instability of the method of fundamental solutions (MFS), which caused by source offset, source location, or a fictitious boundary, a generalized method of fundamental solutions (GMFS) is proposed. The crucial part of the GMFS is using a generalized fundamental solution approximation (GFSA), which adopts a bilinear combination of fundamental solutions to approximate, rather than the linear combination of the MFS. Then the numerical solution of the GMFS is decided by a group of offsets corresponding to an intervention-point diffusion (IPD), instead of the MFS’ offset of a single source. To demonstrate the effectiveness of the proposed approach, five numerical examples are given. The results have shown that the GMFS is more accurate, stable, and has a better convergence rate than the traditional MFS.

Fast Solution of Three-Dimensional Modified Helmholtz Equations by the Method of Fundamental Solutions

AbstractThis 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.

Formulation of the MFS for the two-dimensional Laplace equation with an added constant and constraint

Motivated by the incompleteness of single-layer potential approach for the interior problem with a degenerate-scale domain and the exterior problem with bounded potential at infinity, we revisit the method of fundamental solutions (MFS). Although the MFS is an easy method to implement, it is not complete for solving not only the interior 2D problem in case of a degenerate scale but also the exterior problem with bounded potential at infinity for any scale. Following Fichera׳s idea for the boundary integral equation, we add a free constant and an extra constraint to the traditional MFS. The reason why the free constant and extra constraints are both required is clearly explained by using the degenerate kernel for the closed-form fundamental solution. Since the range of the single-layer integral operator lacks the constant term in the case of a degenerate scale for a two dimensional problem, we add a constant to provide a complete base. Due to the rank deficiency of the influence matrix in the case of a degenerate scale, we can promote the rank by simultaneously introducing a constant term and adding an extra constraint to enrich the MFS. For an exterior problem, the fundamental solution does not contain a constant field in the degenerate kernel expression. To satisfy the bounded potential at infinity, the sum of all source strengths must be zero. The formulation of the enriched MFS can solve not only the degenerate-scale problem for the interior problem but also the exterior problem with bounded potential at infinity. Finally, three examples, a circular domain, an infinite domain with two circular holes and an eccentric annulus were demonstrated to see the validity of the enriched MFS.

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.