Radial Point Interpolation Collocation Method (RPICM) for Partial Differential Equations

K e y w o r d s R P I C M , Hermite-type interpolation, Meshfree, Partial differential equations. *Author to whom all correspondence should be addressed. Current address: Department of Mechanics, Zhejiang University, Hangzhou, P.R. China, 310027. 0898-1221/05/$ see front matter @ 2005 Elsevier Ltd. All rights reserved. Typeset by .AA/p%TEX doi:10.1016/j.camwa.2005.02.019 1426 x. Liu et al. 1. I N T R O D U C T I O N In recent years, there has been great interest to improve meshfree methods based on radial basis functions (RBF) in the field of computational mathematics and mechanics [1-8]. However, the primary disadvantage of the traditional RBF approach is that the coefficient matrices obtained from this discretization scheme are fully populated. It will cause great inconvenience for largescale practical problems. At present, there are mainly two approaches to improve the traditional RBF approach. One is to improve the conditioning of the coefficient matrix and the solution accuracy using some special mathematical techniques. The other is to obtain banded coefficient matrices using compactly support radial basis functions. It has been shown that these two approaches cannot always produce satisfactory results. More effective approaches are therefore needed. Point interpolation method (PIM) was developed by Liu et al. [9], and it has further been studied [10-15]. As the name implies, PIM obtains its approximation by letting the interpolation function pass through the function values at each scattered node within the defined local support domain. In PIM, its shape function possesses the Kronocker delta function property so that the essential boundary conditions, which have been troubling meshfree researchers for recent years, can be easily handled like in the traditional finite-element method (FEM). So far, PIM is based on Galerkin or Petrov-Galerkin weak forms, and numerical integrations are required. Like other Galerkin-based meshfree methods, the inevitable background cell must be used in integration processes. In contrast to Galerkin-based approaches, the collocation method is simple and efficient to solve partial differential equations without the need of numerical integrations. Collocation is known as an efficient and highly accurate numerical solution procedure for partial differential equations. Another attractive feature is that its formulation is very simple. In [16], a local multiquadrics (MQ) formulation, which is similar to the MQ-RPICM, has been presented and applied to solve PDEs. However, the research results in [17] showed that the accuracy obtained by using direct collocation scheme is a bit poor especially on boundary. In addition, the collocation scheme, which has difficulties in dealing with Neumann boundary conditions, is very different from the Galerkin scheme that can deal with Neumann boundary conditions naturally. Liszka et al. [17] proposed a Hermite-type interpolation scheme in Generalized Finite Difference Method (GFDM) to improve the accuracy of collocation-based approach for solving solid problems. Zhang et al. [7] applied Hermite-type interpolation in compactly supported radial basis function method successfully. Liu et al. [14] presented an efficient RPICM based on thin plate spline (TPS) for solving 2-D linear elastic problem with especially attention for dealing with force boundary condition. In this paper, the Hermite-type interpolation is adopted in the point interpolation in order to improve the accuracy. Approximate field functions are carried out not only with the nodal values but also with the normal gradient at the Neumann boundaries by taking the advantage of the point interpolation method based on radial basis functions. In this paper, the radial point interpolation collocation method (RPICM) is presented. The formulation for constructing shape functions based on radial point interpolation and Hermite radial point interpolation is described and formulated in Section 2 and Section 3. The detail collocation schemes are discussed in Section 4. In Section 5, the accuracy and simplicity of this presented approach is shown numerically by a series of test examples, and h-convergence of this method is numerically analysed. We conclude with a summary in Section 6. 2. R A D I A L B A S I S P O I N T I N T E R P O L A T I O N The approximation of a function u(x), using radial basis functions, may be written as a linear combination of n radial basis functions, viz., Radial Point Interpolation Collocation Method 1427 ~(x) ~ ~(x) = ~ a,¢ (]lr r, ll, c,), (1) i = 1 where n is the number of points in the support domain near x, ai are coefficients to be determined and ¢ are the MQ, or inverse-MQ, or Gaussian basis function, or thin plate spline (TPS) function. These well-known radial basis functions are as follows. MQ. ¢ ( l l r r i l l , c ~ ) = ] [ r r~ll 2 +c~ . GAUSSIAN BASIS FUNCTION. e--4(ilr--rill2/r~). THIN PLATE SPLINE. Ilr rill 2M log Oily rill), The shape parameter can be defined as ci = (a~r~) [8]. Where r is the distance between two nodes. In 2-D problems, we have ] ) r r i l l = J(x-x~)2+(Y-Y~) 2. (2) The constant ci is a shape parameter. How to choose the optimal shape parameter is a problem that has received the attention of many researchers [8,16]. So far, there is no mathematical theory developed for determining the optimal value• Detailed guidelines on how to choose these parameters can be found in Liu's recent monograph [8]. Optimal values for these parameters for PIMs based on Galerkin and Petrov-Galerkin weak forms were found via numerical experiments and provided in this book• Here, the form of dimensionless shape parameter ac will be employed and investigated for RPICM. The constant rc is the characteristic length that is related to the nodal space in the local support domain of the collocation point and it is usually the average nodal spacing for all the nodes in this support domain. The coefficients ai in equation (1) can be determined by enforcing that the function interpolations pass through all n nodes within the support domain. The interpolations of a function at the U h point can have the form of ~(xk)=al¢(llrk_rll),cl)+a2¢(]lrk-r~ll,e2)÷...+a,d~(llrk-r,dl,cn), k=1,2 . . . . n. (3) The function interpolation can be expressed in a matrix form as follows: a = [al ~.1 e = c I ) a , [ ¢ ( l l n _ r t U , ~ ) . . . ¢ ( l l n r ~ l l , c ~ ) • o, , I ,= [ ¢ ( l l r ~ . r l l l , c l ) ".... ¢(llr~-rill,ei). L¢ (llr,~ nl l , c~) -.. ¢ (llr~ r, l l ,~) ~ = [ ~ ( X l ) . . . a (xk) . . . a (xn) ] T, T • . . a i " " a n ] • "'.i ¢(]lr~ -'r~H 'c~)] i / (4)