Based on our previously study, the accuracy of derivatives of interpolating functions are usually very poor near the boundary of domain when Compactly Supported Radial Basis Functions (CSRBFs) are used, so that it could result in significant error in solving partial differential equations with Neumann boundary conditions. To overcome this drawback, the Consistent Compactly Supported Radial Basis Functions (CCSRBFs) are developed, which satisfy the predetermined consistency conditions. Meshless method based on point collocation with CCSRBFs is developed for solving partial differential equations. Numerical studies show that the proposed method improves the accuracy of approximation significantly.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Price excludes VAT (USA)
Tax calculation will be finalised during checkout.
Gingold RA, Monaghan JJ. Smoothed particle hydrodynamics: theory and applications to non-spherical stars.Mon Not Roy Astrou Soc, 1977, 181: 375–389
Nayroles B, Touzot G, Villon P. Generalizing the finite element method: diffuse approximation and diffuse elements.Comput Mech, 1992, 10: 307–318
Belytschko T, Lu YY, Gu L. Element free Galerkin methods.Int J Numer Methods Engrg, 1994, 37: 229–256
Liu WK, Chen Y, Jun S et al. Overview and applications of the reproducing kernel particle methods.Archives of Computational Methods in Engineering, 1996, 3: 3–80
Onate E, Idelsohn S, Zienkiewicz OC, et al. A finite point method in computational mechanics. Applications to convective transport and fluid flow.Int J Numer Meth Engrg, 1996, 39: 3839–3866
Liszka TJ, Duarte CAM, Tworzydlo WW. hp-meshless cloud method.Comput Methods Appl Mech Engrg, 1996, 139: 263–288
Atluri SN, Zhu T. A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics.Comput Mech, 1998, 22: 117–127
Atluri SN, Sladek J, et al. The Local boundary integral equation (LBIE) and it's meshles implementation for linear elasticity.Comput Mech, 2000, 25: 180–198
Zhang X, Song KZ, Lu MW. Meshless methods based on collocation with radial basis function.Comput Mech, 2000, 26(4): 333–343
Chen W. Symmetric boundary knot method.Engng Anal Bound Elem, 2002, 26(6): 489–494
Chen W. Meshfree boundary particle method applied to Helmholtz problems.Engng Anal Bound Elem, 2002, 26(7): 577–581
Zhang X, Liu XH, Song KZ, et al. Least square collocation meshless method.Int J Numer Methods Engrg, 2001, 51: 1089–1100
Wu Z. Compactly supported positive definite radial functions.Adv Comput Math, 1995, 4: 283–292
Wendland H. Piecewise polynomial, positive definite and compactly supported radial basis functions of minimal degree.Adv Comput Math, 1995, 4: 389–396
Buhmann MD. Radial functions on compact support.Proceedings of the Edinburgh Mathematical Society, 1998, 41: 33–46
Belytschko T, Krongauz Y, et al. Meshless methods: an overview and recent developments.Computer Methods in Applied Mechanics and Engineering, 1996, 139: 3–47
Timoshenko SP, Goodier JN. Theory of Elasticity. 3rd edition. New York: McGraw-Hill, 1987
The project supported by the National Natural Science Foundation of China (10172052)
About this article
Cite this article
Kangzu, S., Xiong, Z. & Mingwan, L. Meshless method based on collocation with consistent compactly supported radial basis functions. Acta Mech Sinica 20, 551–557 (2004). https://doi.org/10.1007/BF02484278