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Evaluation of Multi-Order Derivatives by Local Radial Basis Function Differential Quadrature Method

Published online by Cambridge University Press:  16 October 2012

L. H. Shen ,
K. H. Tseng  and
D. L. Young
L. H. Shen
Affiliation:
Department of Civil Engineering, National Taiwan University, Taipei, Taiwan 10617, R.O.C.
K. H. Tseng
Affiliation:
Department of Civil Engineering, National Taiwan University, Taipei, Taiwan 10617, R.O.C.
D. L. Young*
Affiliation:
Department of Civil Engineering, National Taiwan University, Taipei, Taiwan 10617, R.O.C.
*
*Corresponding author ([email protected])
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Abstract

It is difficult to obtain the derivative values from most mesh dependent numerical procedures in general. This study proposes an efficient computational tool to accurately evaluate the multi-order derivatives by the radial basis functions and local differential quadrature (LRBF-DQ) algorithm. Most of the traditional derivative calculations can be only adopted to evaluate the differential values with the regular meshes. Moreover, the traditional numerical schemes are very restricted by the order of the shape functions. The present technique can be applied to both the structured and unstructured meshes as well as meshless numerical algorithms – such as RBFs and LDQ method. In addition, the proposed model can be also used to estimate multi-order or mixed partial derivative values because its test function using RBFs is a multi-order differentiable function. All of the evaluation of derivative results will be compared with the exact solutions and other numerical techniques. Consequently, this study provides an effective algorithm of post process to accurately calculate the multi-order derivative values for any numerical schemes.

Keywords

Multi-order derivativesRadial basis functionsLocal differential quadrature
Type
Articles
Information
Journal of Mechanics , Volume 29 , Issue 1 , March 2013 , pp. 67 - 78
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2012

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References

REFERENCES

1. Spalding, D. B., “A Novel Finite Difference Formulation for Differential Expressions Involving Both First and Second Derivatives,” International Journal for Numerical Methods in Engineering, 4, pp. 551–559, (1972).CrossRefGoogle Scholar
2. Fornberg, B., “Numerical Differentiation of Analytic Functions,” ACM Transactions on Mathematical Software, 7, pp. 512–526 (1981).Google Scholar
3. Squire, W. and Trapp, G., “Using Complex Variables to Estimate Derivatives of Real Functions,” SIAM Review (1998).Google Scholar
4. Slazer, H. E., “Divided Differences for Functions of Two Variables for Irregularly Spaced Argument,” Numerische Mathematik, 6, pp. 68–77 (1964).CrossRefGoogle Scholar
5. Ciarlet, P. G. and Raviat, P. A., “General Lagrange and Hermite Interpolation in Rn with Applications to Finite Element Methods,” Archive for Rational Mechanics and Analysis, 46, pp. 177–199 (1972).Google Scholar
6. Gasca, M. and Maeztu, J. I., “On Lagrange and Hermite Interpolation in Rk,” Numerische Mathematik, 39, pp. 1–14 (1982).Google Scholar
7. Nikiforov, A. F. and Sulov, S. K., “Classical Orthogonal Polynomials of a Discrete Variable on Nonuniform Lattices,” Letters in Mathematical Physics, 11, pp. 27–34 (1985).Google Scholar
8. Martins, J. R. R. A., Sturdza, P. and Alonso, J. J., “The Complex-Step Derivative Approximation,” ACM Transactions on Mathematical Software, 29, pp. 245–262 (2003).Google Scholar
9. Lyness, J. N. and Moler, C. B., “Numerical Differentiation of Analytic Functions,” SIAM Journal of Numerical Analysis, 4, pp. 202–210 (1967).Google Scholar
10. Abate, J. and Dubner, H., “A New Method for Generating Power Series Expansions of Functions,” SIAM Journal of Numerical Analysis, 5, pp. 102–112 (1968).Google Scholar
11. Nayroles, B., Touzot, G. and Villon, P., “Generalizing the Finite Element Method- Diffuse Approximation and Diffuse Elements,” Computational Mechanics, 10, pp. 307–318 (1992).Google Scholar
12. Wu, Y. L. and Shu, C., “Development of RBF-DQ Method for Derivative Approximation and its Application to Simulate Natural Convection in Concentric Annuli,” Computational Mechanics, 29, pp. 477–485 (2002).Google Scholar
13. Tseng, K. H., Shen, L. H. and Young, D. L., “Evaluating Accurate Differential Derivative by Local Differential Quadrature,” 2009 Computational Fluid Dynamics National Conference, Yilan, Taiwan (2009).Google Scholar
14. Shu, C., H., , Yeo, K. S., “Local Radial Basis Differential Quadrature Method and its Application to Solve Two-Dimensional Incompressible Navier-Strokes Equations,” Computer Methods in Applied Mechanics and Engineering, 192, pp. 941–954 (2003).Google Scholar