Radial basis functions methods for solving Fokker-Planck equation

In this paper two numerical meshless methods for solving the Fokker-Planck equation are considered. Two methods based on radial basis functions to approximate the solution of Fokker-Planck equation by using collocation method are applied. The first is based on the Kansa's approach and the other one is based on the Hermite interpolation. In addition, to conquer the ill-conditioning of the problem for big number of collocation nodes, two time domain Discretizing schemes are applied. Numerical examples are included to demonstrate the reliability and efficiency of these methods. Also root mean square and N_e errors are obtained to show the convergence of the methods. The errors show that the proposed Hermite collocation approach results obtained by the new time-Discretizing scheme are more accurate than the Kansa's approach.     

The radial basis integral equation method for solving the Helmholtz equation

A meshless method for the solution of Helmholtz equation has been developed by using the radial basis integral equation method (RBIEM). The derivation of the integral equation used in the RBIEM is based on the fundamental solution of the Helmholtz equation, therefore domain integrals are not encountered in the method. The method exploits the advantage of placing the source points always in the centre of circular sub-domains in order to avoid singular or near-singular integrals. Three equations for two-dimensional (2D) or four for three-dimensional (3D) potential problems are required at each node. The first equation is the integral equation arising from the application of the Green’s identities and the remaining equations are the derivatives of the first equation with respect to space coordinates. Radial basis function (RBF) interpolation is applied in order to obtain the values of the field variable and partial derivatives at the boundary of the circular sub-domains, providing this way the boundary conditions for solution of the integral equations at the nodes (centres of circles). The accuracy and robustness of the method has been tested on some analytical solutions of the problem. Two different RBFs have been used, namely augmented thin plate spline (ATPS) in 2D and f(R)=⁴Rln(R) augmented by a second order polynomial. The latter has been found to produce more accurate results.     

A local Heaviside weighted meshless method for two-dimensional solids using radial basis functions

 A meshless method is developed for the stress analysis of two-dimensional solids, based on a local weighted residual method with the Heaviside step function as the weighting function over a local subdomain. Trial functions are constructed using radial basis functions (RBF). The present method is a truly meshless method based only on a number of randomly located nodes. No domain integration is needed, no element matrix assembly is required and no special treatment is needed to impose the essential boundary conditions. Effects of the sizes of local subdomain and interpolation domain on the performance of the present method are investigated. The behaviour of shape parameters of multiquadrics (MQ) has been systematically studied. Example problems in elastostatics are presented and compared with closed-form solutions and show that the proposed method is highly accurate and possesses no numerical difficulties. Received: 10 November 2002 / Accepted: 5 March 2003     

A mesh free approach using radial basis functions and parallel domain decomposition for solving three‐dimensional diffusion equations

The analysis of transient heat conduction problems in large, complex computational domains is a problem of interest in many technological applications including electronic cooling, encapsulation using functionally graded composite materials, and cryogenics. In many of these applications, the domains may be multiply connected and contain moving boundaries making it desirable to consider meshless methods of analysis. The method of fundamental solutions along with a parallel domain decomposition method is developed for the solution of three‐dimensional parabolic differential equations. In the current approach, time is discretized using the generalized trapezoidal rule transforming the original parabolic partial differential equation into a sequence of non‐homogeneous modified Helmholtz equations. An approximate particular solution is derived using polyharmonic splines. Interfacial conditions between subdomains are satisfied using a Schwarz Neumann–Neumann iteration scheme. Outside of the first time step where zero initial flux is assumed, the initial estimates for the interfacial flux is given from the converged solution obtained during the previous time step. This significantly reduces the number of iterations required to meet the convergence criterion. The accuracy of the method of fundamental solutions approach is demonstrated through two benchmark problems. The parallel efficiency of the domain decomposition method is evaluated by considering cases with 8, 27, and 64 subdomains. Copyright 2004 © John Wiley & Sons, Ltd.     

A novel algorithm for shape parameter selection in radial basis functions collocation method

Many Radial Basis Functions (RBFs) contain a free shape parameter that plays an important role for the application of Meshless method to the analysis of multilayered composite and sandwich plates. In most papers the authors end up choosing this shape parameter by trial and error or some other ad hoc means. In this paper a novel algorithm for shape parameter selection, based on a convergence analysis, is presented. The effectiveness of this algorithm is assessed by static analyses of laminated composite and sandwich plates.     

The localized RBFs collocation methods for solving high dimensional PDEs

In this paper we present a localized meshless method using radial basis functions (RBFs) for solving up to six dimensional problems. To improve the difficulty of selecting a shape parameter of RBF-MQ, a normalized scheme is introduced. We also make a comparison between the global and local RBF methods in terms of stability and accuracy. To demonstrate the applicability of the localized RBF method for high dimensional problems, two numerical examples with Dirichlet boundary conditions are given.     

The Eulerian–Lagrangian method of fundamental solutions for two-dimensional unsteady Burgers’ equations

The Eulerian–Lagrangian method of fundamental solutions is proposed to solve the two-dimensional unsteady Burgers’ equations. Through the Eulerian–Lagrangian technique, the quasi-linear Burgers’ equations can be converted to the characteristic diffusion equations. The method of fundamental solutions is then adopted to solve the diffusion equation through the diffusion fundamental solution; in the meantime the convective term in the Burgers’ equations is retrieved by the back-tracking scheme along the characteristics. The proposed numerical scheme is free from mesh generation and numerical integration and is a truly meshless method. Two-dimensional Burgers’ equations of one and two unknown variables with and without considering the disturbance of noisy data are analyzed. The numerical results are compared very well with the analytical solutions as well as the results by other numerical schemes. By observing these comparisons, the proposed meshless numerical scheme is convinced to be an accurate, stable and simple method for the solutions of the Burgers’ equations with irregular domain even using very coarse collocating points.     

The method of particular solutions for solving axisymmetric polyharmonic and poly-Helmholtz equations

In this paper we derive analytical particular solutions for the axisymmetric polyharmonic and poly-Helmholtz partial differential operators using Chebyshev polynomials as basis functions. We further extend the proposed approach to the particular solutions of the product of Helmholtz-type operators. By using this formulation, we can approximate the particular solution when the forcing term of the differential equation is approximated by a truncated series of Chebyshev polynomials. These formulas were further implemented to solve inhomogeneous partial differential equations (PDEs) in which the homogeneous solutions were obtained by the method of fundamental solutions (MFS). Several numerical experiments were carried out to validate our newly derived particular solutions. Due to the exponential convergence of Chebyshev interpolation and the MFS, our numerical results are extremely accurate.     

The time-marching method of fundamental solutions for wave equations

In this paper, a meshless numerical algorithm is developed for the solution of multi-dimensional wave equations with complicated domains. The proposed numerical method, which is truly meshless and quadrature-free, is based on the Houbolt finite difference (FD) scheme, the method of the particular solutions (MPS) and the method of fundamental solutions (MFS). The wave equation is transformed into a Poisson-type equation with a time-dependent loading after the time domain is discretized by the Houbolt FD scheme. The Houbolt method is used to avoid the difficult problem of dealing with time evolution and the initial conditions to form the linear algebraic system. The MPS and MFS are then coupled to analyze the governing Poisson equation at each time step. In this paper we consider six numerical examples, namely, the problem of two-dimensional membrane vibrations, the wave propagation in a two-dimensional irregular domain, the wave propagation in an L-shaped geometry and wave vibration problems in the three-dimensional irregular domain, etc. Numerical validations of the robustness and the accuracy of the proposed method have proven that the meshless numerical model is a highly accurate and efficient tool for solving multi-dimensional wave equations with irregular geometries and even with non-smooth boundaries.     

A Kansa-type MFS scheme for two-dimensional time fractional diffusion equations

In this paper, an efficient Kansa-type method of fundamental solutions (MFS-K) is extended to the solution of two-dimensional time fractional sub-diffusion equations. To solve initial boundary value problems for these equations, the time dependence is removed by time differencing, which converts the original problems into a sequence of boundary value problems for inhomogeneous Helmholtz-type equations. The solution of this type of elliptic boundary value problems can be approximated by fundamental solutions of the Helmholtz operator with different test frequencies. Numerical results are presented for several examples with regular and irregular geometries. The numerical verification shows that the proposed numerical scheme is accurate and computationally efficient for solving two-dimensional fractional sub-diffusion equations.     

Thin plate spline radial basis function for the free vibration analysis of laminated composite shells

In this paper, the free vibration characteristics of laminated composite cylindrical and spherical shells are analyzed by the first-order shear deformation theory and a meshless global collocation method based on thin plate spline radial basis function. The singularity of thin plate spline radial basis function is eliminated by adding infinitesimal to the zero distance. Several numerical examples are used to show convergence of the present method and choose the proper shape parameter. It is found that the natural frequencies computed by thin plate spline radial basis function with shape parameter m = 4 converge most rapidly. In the comparison study, the present results are in good agreement with the results of Reddy and Liu [8] and Ferreira et al. [21].     

A Cartesian‐grid collocation technique with integrated radial basis functions for mixed boundary value problems

In this paper, high‐order systems are reformulated as first‐order systems, which are then numerically solved by a collocation method. The collocation method is based on Cartesian discretization with 1D‐integrated radial basis function networks (1D‐IRBFN) (Numer. Meth. Partial Differential Equations 2007; 23 :1192–1210). The present method is enhanced by a new boundary interpolation technique based on 1D‐IRBFN, which is introduced to obtain variable approximation at irregular points in irregular domains. The proposed method is well suited to problems with mixed boundary conditions on both regular and irregular domains. The main results obtained are (a) the boundary conditions for the reformulated problem are of Dirichlet type only; (b) the integrated RBFN approximation avoids the well‐known reduction of convergence rate associated with differential formulations; (c) the primary variable (e.g. displacement, temperature) and the dual variable (e.g. stress, temperature gradient) have similar convergence order; (d) the volumetric locking effects associated with incompressible materials in solid mechanics are alleviated. Numerical experiments show that the proposed method achieves very good accuracy and high convergence rates. Copyright © 2009 John Wiley & Sons, Ltd.     

The comparison of three meshless methods using radial basis functions for solving fourth-order partial differential equations

In this paper we apply the newly developed method of particular solutions (MPS) and one-stage method of fundamental solutions (MFS-MPS) for solving fourth-order partial differential equations. We also compare the numerical results of these two methods to the popular Kansa's method. Numerical results in the 2D and the 3D show that the MFS-MPS outperformed the MPS and Kansa's method. However, the MPS and Kansa's method are easier in terms of implementation.     

A numerical solution of nonlinear parabolic-type Volterra partial integro-differential equations using radial basis functions

In this paper, an effective numerical method for solving nonlinear Volterra partial integro-differential equations is proposed. These equations include the partial differentiations of an unknown function and the integral term containing the unknown function which is the “memory” of problem. This method is based on radial basis functions (RBFs) and finite difference method (FDM) which provide the approximate solution. These techniques play the important role to reduce a nonlinear partial integro-differential equation to a linear system of equations. Some illustrative examples are shown to describe the method. Numerical examples confirm the validity and efficiency of the presented method.     

Some observations on unsymmetric radial basis function collocation methods for convection–diffusion problems

In this paper, we re‐investigate the unsymmetric radial basis function (RBF) collocation method for solving convection–diffusion problems with high Péclet number as in Power and Barraco (Computers and Mathematics with Applications 2002; 43: 551). By testing different RBFs and different numbers of nodes, we found that the unsymmetric method can still solve high Péclet number problems reasonably well by using more nodes and domain decomposition techniques. Compared to solving one large problem, the domain decomposition method is shown to be very efficient and can improve the accuracy especially when the Péclet number is not that high. From our tests, it seems that the RBFs r⁴ ln r and r⁸ ln r are not very stable, while r⁶ ln r, r⁵ and r⁷ perform very well. Copyright © 2003 John Wiley & Sons, Ltd.     

A regularization method for the approximate particular solution of nonhomogeneous Cauchy problems of elliptic partial differential equations with variable coefficients

Radial basis functions (RBFs) have proved to be very flexible in representing functions. Based on the idea of the analog equation method and radial basis functions, in this paper, ill-posed Cauchy problems of elliptic partial differential equations (PDEs) with variable coefficients are considered for the first time using the method of approximate particular solutions (MAPS). We show that, using the Tikhonov regularization, the MAPS results an effective and accurate numerical algorithm for elliptic PDEs and irregular solution domains. Comparing the proposed MAPS with Kansa's method, numerical results show that the proposed MAPS is effective, accurate and stable to solve the ill-posed Cauchy problems.     

Application of variational mesh generation approach for selecting centers of radial basis functions collocation method

In this paper, a two-dimensional variational mesh generation method is applied to obtain adaptive centers for radial basis functions (RBFs). At first, a set of uniform centers is distributed in the domain, then mesh generation differential equations are used to move the centers to region with high gradients. An iterative algorithm is introduced to solve steady-state mesh generation differential equations with RBFs. Functions with steep variation in the domains are used to validate the adaptive centers generation method. In addition to the centers adaption process is applied to solve elliptic partial differential equations via RBFs collocation method. Numerical results of Helmholtz differential equation show a clear reduction in the error, when the adaptive centers are used for RBFs.     

Unified fluid–structure interpolation and mesh motion using radial basis functions

A multivariate interpolation scheme, using radial basis functions, is presented, which results in a completely unified formulation for the fluid–structure interpolation and mesh motion problems. The method has several significant advantages. Primarily, all volume mesh, structural mesh, and flow‐solver type dependence is removed, and all operations are performed on totally arbitrary point clouds of any form. Hence, all connectivity and user‐input requirements are removed from the computational fluid dynamics–computational structural dynamics (CFD–CSD) coupling problem, as only point clouds are required to determine the coupling. Also, it may equally well be applied to structured and unstructured grids, or structural and aerodynamic grids that intersect, again because no connectivity information is required. Furthermore, no expensive computations are required during an unsteady simulation, just matrix–vector multiplications, since the required dependence relations are computed only once prior to any simulation and then remain constant. This property means that the method is both perfectly parallel, since only the data relevant to each structured block or unstructured partition are required to move those points, and totally independent from the flow solver. Hence, a completely generic ‘black box’ tool can be developed, which is ideal for use in an optimization approach. Aeroelastic behaviour of the Brite–Euram MDO wing is analysed in terms of both static deflection and dynamic responses, and it is demonstrated that responses are strongly dependent on the exact CFD–CSD interpolation used. Mesh quality is also examined during the motion resulting from a large surface deformation. Global grid quality is shown to be preserved well, with local grid orthogonality also being maintained well, particularly at and near the moving surface, where the original orthogonality is retained. Copyright © 2007 John Wiley & Sons, Ltd.     

Boundary knot method for 2D and 3D Helmholtz and convection–diffusion problems under complicated geometry

The boundary knot method (BKM) of very recent origin is an inherently meshless, integration‐free, boundary‐type, radial basis function collocation technique for the numerical discretization of general partial differential equation systems. Unlike the method of fundamental solutions, the use of non‐singular general solution in the BKM avoids the unnecessary requirement of constructing a controversial artificial boundary outside the physical domain. The purpose of this paper is to extend the BKM to solve 2D Helmholtz and convection–diffusion problems under rather complicated irregular geometry. The method is also first applied to 3D problems. Numerical experiments validate that the BKM can produce highly accurate solutions using a relatively small number of knots. For inhomogeneous cases, some inner knots are found necessary to guarantee accuracy and stability. The stability and convergence of the BKM are numerically illustrated and the completeness issue is also discussed. Copyright © 2003 John Wiley & Sons, Ltd.     

Testing complete and compact radial basis functions for solution of eigenvalue problems using the boundary element method with direct integration

This paper analyses the performance of the main radial basis functions in the formulation of the Boundary Element Method (DIBEM). This is an alternative for solving problems modeled by non-adjoint differential operators, since it transforms domain integrals in boundary integrals using radial basis functions. The solution of eigenvalue problem was chosen to performance evaluation. Natural frequencies are calculated numerically using several radial functions and their accuracy is evaluated by comparison with the available analytical solutions and with the Finite Element Method as well. The standard radial basis functions have presented similar performance to compact radial functions, being even slightly superior.