In this paper we give an overview of the Method of Fundamental Solutions (MFS) as a heuristic numerical method. It is truly meshless. Its concept and numerical implementation are simple. It has the flexibility of using various forms of fundamental solutions, singular, hypersingular, or nonsingular, and mixing with general solutions and particular solutions, for different purposes. The collocation matrix, however, is not guaranteed to be invertible. There are other issues. For example, in using the logarithmic fundamental solution, a degenerate scale can exist, causing the nonuniqueness of solution. For metaharmonic operators, such as the Helmholtz equation, the zeros of the fundamental solutions can create spurious resonance frequencies. These and other issues are discussed, and remedies are offered. The traditional error analysis for the MFS shows that when the boundary condition is prescribed by a harmonic function, the convergence is exponential either by increasing the number of terms in the approximation, or by increasing the radius of the fictitious boundary. In practical problems, however, a harmonic boundary condition hardly exists. Numerical experiments show that the sources need to stay close to the boundary, and there is an optimal distance. Based on the maximum principle, a posteriori error can be monitored on the boundary to seek the optimal fictitious boundary location. Other topics discussed include the origin of the MFS, the equivalence between the MFS and the Trefftz collocation method, effective condition number, nonsingular MFS, and solving ill-posed and inverse problems.

An overview of the method of fundamental solutions—Solvability, uniqueness, convergence, and stability

Benefiting from the use of two cover systems, that is, the mathematical cover and the physical cover, the numerical manifold method (NMM) is capable of solving both continuous and discontinuous problems in the same platform. Presently, the NMM is further developed to tackle two-dimensional transient heat conduction problems in the functionally graded materials (FGMs). Firstly, the governing equation, the associated boundary conditions and the initial condition are presented. Then, the fundamentals of the NMM are briefly reviewed. Following, the NMM discrete formulations are derived based on the Galerkin-form weighted residual method and then solved with the backward difference scheme. Finally, for verification, three numerical examples with increasing complexity are tested on uniform mathematical covers composed of square mathematical elements, and our results well demonstrate the advantages of the proposed method in discretization and accuracy; besides, the effects of material gradient on the thermal behavior of FGMs are also examined.

The numerical manifold method for 2D transient heat conduction problems in functionally graded materials

Numerical prediction of effective electro-elastic properties of three-dimensional braided piezoelectric ceramic composites

To reduce the negative effect of different kernel functions on calculating accuracy, the radial basis function (RBF) is introduced into the reproducing kernel particle method (RKPM), and the radial basis reproducing kernel particle method (RRKPM) is proposed, the corresponding governing equations are derived. The RRKPM is more efficient to solve the local problem domain, and can improve the accuracy and stability of the RKPM. Then the RRKPM is applied to the numerical simulation of piezoelectric materials, the corresponding formulae for piezoelectric materials are derived. The numerical results illustrate the proposed method is more stable and accurate than the RKPM.

Radial basis reproducing kernel particle method for piezoelectric materials

In this paper, the radial basis function (RBF) is used to construct the approximating function of the reproducing kernel particle method (RKPM), which can eliminate the negative effect of different kernel functions on the calculating accuracy. The radial basis reproducing kernel particle method (RRKPM) is proposed. Furthermore, the RRKPM is applied to solve the elastic mechanical problems of the functionally graded materials (FGM), and the RRKPM for FGM is established. The corresponding formulae of the RRKPM for FGM are derived. Compared with the traditional RKPM, the RRKPM has higher calculating accuracy and stability. Finally, the numerical examples illustrate that the RRKPM is correct and effective to solve the elastic mechanical problems of the FGM.

Numerical solution of functionally graded materials based on radial basis reproducing kernel particle method

On the basis of the complex variable meshless manifold method (CVMMM), the time domain is discretized using the traditional two-point difference method, and the complex variable moving least-squares (CVMLS) method is used in the spatial domain. According to the equivalent integral weak form of the governing equation, the CVMMM for transient heat conduction problem is established, and the corresponding discreted equations are derived by introducing the penalty function method to the essential boundary conditions. The numerical examples show that compared with the meshless manifold method (MMM), the CVMMM for transient heat conduction problem has the advantages of high precision and less calculation. This proposed method is a new numerical calculation method which has great development potential.

Complex Variable Meshless Manifold Method for Transient Heat Conduction Problems

The reproducing kernel particle method (RKPM) for the elastic mechanical problems of the functionally graded materials (FGM) is proposed in this paper. The corresponding formulae of the RKPM for th...

Numerical Analysis of Functionally Graded Materials Using Reproducing Kernel Particle Method

The radial basis reproducing kernel particle method for geometrically nonlinear problem of functionally graded materials

Based on the reproducing kernel particle method (RKPM) and the radial basis function (RBF), the radial basis reproducing kernel particle method (RRKPM) is presented for solving geometrically nonlin

Gaofeng Wei

Papers

Complex Variable Meshless Manifold Method for Transient Heat Conduction Problems

Numerical solution of functionally graded materials based on radial basis reproducing kernel particle method

Numerical Analysis of Functionally Graded Materials Using Reproducing Kernel Particle Method

The radial basis reproducing kernel particle method for geometrically nonlinear problem of functionally graded materials

The Meshfree Analysis of Geometrically Nonlinear Problem Based on Radial Basis Reproducing Kernel Particle Method