314 research outputs found
Symmetric boundary knot method
The boundary knot method (BKM) is a recent boundary-type radial basis
function (RBF) collocation scheme for general PDEs. Like the method of
fundamental solution (MFS), the RBF is employed to approximate the
inhomogeneous terms via the dual reciprocity principle. Unlike the MFS, the
method uses a nonsingular general solution instead of a singular fundamental
solution to evaluate the homogeneous solution so as to circumvent the
controversial artificial boundary outside the physical domain. The BKM is
meshfree, superconvergent, integration free, very easy to learn and program.
The original BKM, however, loses symmetricity in the presense of mixed
boundary. In this study, by analogy with Hermite RBF interpolation, we
developed a symmetric BKM scheme. The accuracy and efficiency of the symmetric
BKM are also numerically validated in some 2D and 3D Helmholtz and diffusion
reaction problems under complicated geometries
Solving high-order partial differential equations with indirect radial basis function networks
This paper reports a new numerical method based on radial basis function networks (RBFNs) for solving high-order partial differential equations (PDEs). The variables and their derivatives in the governing equations are represented by integrated RBFNs. The use of integration in constructing neural networks allows the straightforward implementation of multiple boundary conditions and the accurate approximation of high-order derivatives. The proposed RBFN method is verified successfully through the solution of thin-plate bending and viscous flow problems which are governed by biharmonic equations. For thermally driven cavity flows, the solutions are obtained up to a high Rayleigh number
A novel algorithm for shape parameter selection in radial basis functions collocation method
Many Radial Basis Functions (RBF) 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 plate
Moving-boundary problems solved by adaptive radial basis functions
The objective of this paper is to present an alternative approach to the conventional level set methods for solving two-dimensional moving-boundary problems known as the passive transport. Moving boundaries are associated with time-dependent problems and the position of the boundaries need to be determined as a function of time and space. The level set method has become an attractive design tool for tracking, modeling and simulating the motion of free boundaries in fluid mechanics, combustion, computer animation and image processing. Recent research on the numerical method has focused on the idea of using a meshless methodology for the numerical solution of partial differential equations. In the present approach, the moving interface is captured by the level set method at all time with the zero contour of a smooth function known as the level set function. A new approach is used to solve a convective transport equation for advancing the level set function in time. This new approach is based on the asymmetric meshless collocation method and the adaptive greedy algorithm for trial subspaces selection. Numerical simulations are performed to verify the accuracy and stability of the new numerical scheme which is then applied to simulate a bubble that is moving, stretching and circulating in an ambient flow to demonstrate the performance of the new meshless approach. (C) 2010 Elsevier Ltd. All rights reserved
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