ELSEVIER HhS0955-7997(98)00004-6 Engineering Analysis with Boundary Elements 21 (1998) 81-85 © 1998 Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain 0955-7997/981519.00 Research Note Axisymmetric augmented thin plate splines v Bo~,idar Sarler Laboratory for Fluid Dynamics and Thermodynamics, Faculty of Mechanical Engineering, University of Ljubljana, Agker~eva 6, 1000 Ljubljana, Slovenia This paper reviews the previous axisymmetric global interpolation functions used in the context of the dual reciprocity boundary element method and the axisymmetric Laplace operator. It upgrades the previous heuristic attempts with the axisymmetdc form of the augmented thin plate splines. This new approach, based on the theory of radial basis functions, gives more formal mathematical support to this class of problems. The basic equations are accompanied by a set of related expressions that permit straightforward use of the developed global interpolation functions in a broad spectrum of dual reciprocity boundary element methods like discrete approximative procedures. © 1998 Published by Elsevier Science Ltd. All fights reserved Key words: Boundary element method, radial basis functions, augmented thin plate splines, axisymmetry. 1 INTRODUCTION Axisymmetric geometry and field problems occur very frequently in science and engineering. The discrete approx- imative solutions of the different governing equations in such situations are of pronottnced importance. The fusion of the boundary element method and global interpolation emerges in a variety of dual reciprocity boundary element (DRBEM) discrete approximative procedures 1'2 which give reasonable evaluations of the governing equations. Recently, two very comprehensive overviews have been published 3'4 regarding the use of the different global approximation functions in the BEM con- text. However, the mathematical properties of such methods are nowadays far from being ,;ufficiently understood. Because of the unresolved theoretical answers to related existence, uniqueness, convergence and stability issues, many numeri- cal experiments and comparisons have been traditionally made in an ad-hoc manner in the DRBEM literature. The problem of global interpolation outside of the BEM context has been much more closely investigated mathema- tically. 5 Corresponding analyses show that the use of the radial basis class of functions represents a proper choice for multidimensional global interpolation. Most of the related advances focus on the augmented thin plate splines (ATPS) and multiquadrics (lVlQ). The ATPS are known to give the minimized curvature of the interpolation and the MQ could, depending on the choice of the free parameter, converge very rapidly. Karur and Ramachadran 6 first gave 81 DRBEM numerical examples with ATPS and claim a superior solution to the heuristic 'one-plus-r' global approximation functions in two-dimensional planar problems. Very recently, Golberg et al. 7 used MQ in a method of fundamental solutions, a variant of the BEM with global interpolation. They demonstrate up to three orders of magnitude of improvement in accuracy over ATPS and 'one-plus-r' func- tions provided that the free parameter is properly chosen. Surprisingly, not many DRBEM solutions structured with the fundamental solution of the Laplace equation deal with axisymmetric problems. In the pioneering work concerning this DRBEM aspect, Wrobel and Telles s heuristically use the global approximation functions of the form 9~/n [(,Pp- Pnp) 2 -Jr- (Pz ,2~1/2 ( lpp~ = -Pnz) J 1- (1) 4 pnp/ with the notation elaborated in the next chapter. Masse6 and Marcouiller9 found this function inadequate and after several numerical experiments proposed 2 2 9~J/n = Pnp [ (Pp -- Pno)2 + (Pz -- Pnz) ] (Po [ (Po -- Pnp) +-- ,2]1/2 + 3~ (Pz --Pnz) PO) (2) with P0 representing a small positive constant which was set to 0.01. The principal incitement for this paper is the fact that the axisymmetric form of ATPS and MQ have not yet been deduced. The present paper thus focuses on a relatively complex derivation of the axisymmetric ATPS and related