×

zbMATH — the first resource for mathematics

On the optimal shape parameters of radial basis functions used for 2-D meshless methods. (English) Zbl 1065.74074
Summary: A radial point interpolation meshless (or radial PIM) method was proposed by the authors [Int. J. Numer. Methods Eng. 54, No. 11, 1623–1648 (2002; Zbl 1098.74741)] to overcome the possible singularity associated with only polynomial basis. The radial PIM used multiquadric (MQ) or Gaussian as basis functions. These two radial basis functions all included shape parameters. Although choice of shape parameters has been a hot topic in approximation theory and some empirical formulae were proposed, it has not been studied yet how these shape parameters affect the accuracy of the radial PIM.
This paper studies the effect of shape parameters on the numerical accuracy of radial PIM. A range of suitable shape parameters is obtained from the analysis of the condition number of system matrix, error of energy and irregularity of node distribution. It is observed that the widely used shape parameters for MQ and reciprocal MQ basis are not even close to their optimums. The optimal shape parameters are found in this paper to be simply \(q= 1.03\) and \(R= 1.42\) for MQ basis and \(c= 0.003-0.03\) for Gaussian basis.

MSC:
74S30 Other numerical methods in solid mechanics (MSC2010)
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74K20 Plates
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] J.G. Wang, G.R. Liu, A point interpolation meshless method based on radial basis functions, International Journal for Numerical Methods in Engineering, in press · Zbl 1098.74741
[2] Belytschko, T.; Krongauz, Y.; Organ, D.; Fleming, M.; Krysl, P., Meshless methods: an overview and recent developments, Computer methods in applied mechanics and engineering, 139, 3-47, (1996) · Zbl 0891.73075
[3] Hardy, R.L., Theory and applications of the multiquadrics–biharmonic method (20 years of discovery 1968-1988), Computers and mathematics with applications, 19, 163-208, (1990) · Zbl 0692.65003
[4] Agnantiaris, J.P.; Polyzos, D.; Beskos, D.E., Some studies on dual reciprocity BEM for elastodynamic analysis, Computational mechanics, 17, 270-277, (1996) · Zbl 0841.73066
[5] Schaback, R., Error estimates and condition numbers for radial basis function interpolation, Advances in computational mathematics, 3, 251-264, (1995) · Zbl 0861.65007
[6] Powell, M.J.D., The uniform convergence of thin plate splines in two dimensions, Numerische Mathematik, 68, 1, 107-128, (1994) · Zbl 0812.41005
[7] Kansa, E.J., A scattered data approximation scheme with application to computational fluid-dynamics–I and II, Computers and mathematics with applications, 19, 127-161, (1990) · Zbl 0692.76003
[8] Fasshauer, G.E., Solving partial differential equations by collocation with radial basis functions, (), 131-138 · Zbl 0938.65140
[9] Wendland, H., Meshless Galerkin method using radial basis functions, Mathematics of computation, 68, 228, 1521-1531, (1999) · Zbl 1020.65084
[10] Golberg, M.A.; Chen, C.S.; Bowman, H., Some recent results and proposals for the use of radial basis functions in the BEM, Engineering analysis with boundary elements, 23, 285-296, (1999) · Zbl 0948.65132
[11] Coleman, C.J., On the use of radial basis functions in the solution of elliptic boundary value problems, Computational mechanics, 17, 418-422, (1996) · Zbl 0851.76056
[12] Zhang, X.; Song, K.Z.; Lu, M.W., Meshless methods based on collocation with radial basis functions, Computational mechanics, 26, 4, 333-343, (2000) · Zbl 0986.74079
[13] Franke, R., Scattered data interpolation: test of some methods, Mathematics of computation, 38, 157, 181-200, (1982) · Zbl 0476.65005
[14] Rippa, S., An algorithm for selecting a good value for the parameter c in radial basis function interpolation, Advances in computational mathematics, 11, 193-210, (1999) · Zbl 0943.65017
[15] Carlson, R.E.; Foley, T.A., The parameter R2 in multiquadric interpolation, Computers and mathematics with applications, 21, 29-42, (1991) · Zbl 0725.65009
[16] Golberg, M.A.; Chen, C.S.; Karur, S.R., Improved multiquadric approximation for partial differential equations, Engineering analysis with boundary elements, 18, 9-17, (1996)
[17] Golberg, M.A.; Chen, C.S.; Bowman, H.; Power, H., Some comments on the use of radial basis functions in the dual reciprocity method, Computational mechanics, 21, 141-148, (1998) · Zbl 0931.65116
[18] Madych, W.R.; Nelson, S.A., Bounds on multivariate polynomials and exponential error-estimates for multiquadric interpolation, Journal of approximation theory, 70, 1, 94-114, (1992) · Zbl 0764.41003
[19] Powell, M.J.D., The theory of radial basis function approximation in 1990, (), 105-203 · Zbl 0787.65005
[20] Powell, M.J.D., A review of algorithms for thin plate spline interpolation in two dimensions, (), 303-322 · Zbl 1273.65019
[21] Light, W.; Wayne, H., Error estimates for approximation by radial basis functions, (), 215-246 · Zbl 0843.41005
[22] Timoshenko, S.P.; Goodier, J.N., Theory of elasticity, third ed., (1970), McGraw-Hill New York · Zbl 0266.73008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.