Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-23T22:05:29.886Z Has data issue: false hasContentIssue false
  • English
  • Français

The Method of Fundamental Solutions with Dual Reciprocity for thin Plates on Winkler Foundations with Arbitrary Loadings

Published online by Cambridge University Press:  05 May 2011

C. C. Tsai
C. C. Tsai*
Affiliation:
Department of Information Technology, Toko University, Chia-Yi County, Taiwan 61363, R.O.C.
*
* Assistant Professor
Get access

Abstract

This paper describes the combination of the method of fundamental solutions (MFS) and the dual reciprocity method (DRM) as a meshless numerical method to solve problems of thin plates resting on Winkler foundations under arbitrary loadings, where the DRM is based on the augmented polyharmonic splines constructed by splines and monomials. In the solution procedure, the arbitrary distributed loading is first approximated by the augmented polyharmonic splines (APS) and thus the desired particular solution can be represented by the corresponding analytical particular solutions of the APS. Thereafter, the complementary solution is solved formally by the MFS. In the mathematical derivations, the real coefficient operator in the governing equation is decomposed into two complex coefficient operators. In other words, the solutions obtained by the MFS-DRM are first treated in terms of these complex coefficient operators and then converted to real numbers in suitable ways. Furthermore, the boundary conditions of lateral displacement, slope, normal moment, and effective shear force are all given explicitly for the particular solutions of APS as well as the kernels of MFS. Finally, numerical experiments are carried out to validate these analytical formulas.

Keywords

Method of fundamental solutionsDual reciprocity methodKirchhoff platesWinkler foundation
Type
Articles
Information
Journal of Mechanics , Volume 24 , Issue 2 , June 2008 , pp. 163 - 171
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Kansa, E. J., “Multiquadrics—A Scattered Data Approximation Scheme with Applications to Computational Fluid Dynamics—II. Solutions to Parabolic, Hyperbolic and Elliptic Partial Differential Equations,” Computers and Mathematics with Applications, 19, pp. 147–161 (1990).CrossRefGoogle Scholar
2.Kupradze, V. D. and Aleksidze, M. A., “The Method of Functional Equations for the Approximate Solution of Certain Boundary Value Problem,” USSR Computational Mathematics and Mathematical Physics, 4, pp. 82–126 (1964).CrossRefGoogle Scholar
3.Young, D. L., Chen, K. H. and Lee, C. W., “Novel Meshless Method for Solving the Potential Problems with Arbitrary Domain,” Journal of Computational Physics, 209 pp. 290–321 (2005).CrossRefGoogle Scholar
4.Young, D. L., Chen, K. H., Chen, J. T. and Kao, J. H., “A Modified Method of Fundamental Solutions with Source on the Boundary for Solving Laplace Equation with Circular and Arbitrary Domains,” CMES: Computer Modeling in Engineering and Sciences, 19, pp. 197–221 (2007).Google Scholar
5.Chen, K. H., Chen, J. T. and Kao, J. H., “Regularized Meshless Method for Solving Acoustic Eigenproblem with Multiply Connected Domain,” Computer Modeling in Engineering Science, 16, pp. 27–39 (2006).Google Scholar
6.Chen, K. H., Kao, J. H., Chen, J. T., Young, D. L. and Lu, M. C., “Regularized Meshless Method for Multiply- Connected—Domain Laplace Problems,” Engineering Analysis with Boundary Elements, 30, pp. 882–896 (2006).CrossRefGoogle Scholar
7.Fairweather, G. and Karageorghis, A., “The Method of Fundamental Solutions for Elliptic Boundary Value Problems,” Advances in Computational Mathematics, 9, pp. 69–95 (1998).CrossRefGoogle Scholar
8.Fairweather, G., Karageorghis, A. and Martin, P. A., “The Method of Fundamental Solutions for Scattering and Radiation Problems,” Engineering Analysis with Boundary Elements, 27, pp. 759–769 (2003).CrossRefGoogle Scholar
9.Li, J., Hon, Y. C. and Chen, C. S., “Numerical Comparisons of Two Meshless Methods Using Radial Basis Function,” Engineering Analysis with Boundary Elements, 26, pp. 205–225 (2002).CrossRefGoogle Scholar
10.Tsai, C. C., Young, D. L. and Fan, C. M., “Method of Fundamental Solutions for Plate Vibrations in Multiply Connected Domains,” Journal of Mechanics, 22, pp. 235–245 (2006).CrossRefGoogle Scholar
11.Chen, J. T., Chen, I. L. and Lee, Y. T., “Eigensolutions of Multiply-Connected Membranes Using Method of Fundamental Solution,” Engineering Analysis with Boundary Elements, 29, pp. 166–174 (2005).CrossRefGoogle Scholar
12.Nardini, D. and Brebbia, C. A., “A New Approach to Free Vibration Analysis Using Boundary Elements,” Brebbia, C. A., Ed., Boundary Element Methods in Engineering, pp. 312–326 (1982).Google Scholar
13.Golberg, M. A., “The Method of Fundamental Solutions for Poisson's Equation,” Engineering Analysis with Boundary Elements, 16, pp. 205–213 (1995).CrossRefGoogle Scholar
14.Golberg, M. A. and Chen, C. S., “The Method of Fundamental Solutions for Potential, Helmholtz and Diffusion Problems,” Golberg, M. A., Ed., Boundary Integral Methods: Numerical and Mathematical Aspects, pp. 103–176(1998).Google Scholar
15.Duchon, J., “Splines Minimizing Rotation Invariant Semi-Norms in Sobolev Spaces,” Schempp, W., and Zeller, K., Eds., Constructive Theory of Functions of Several Variables, pp. 85–100 (1977).CrossRefGoogle Scholar
16.Bogomolny, A., “Fundamental Solutions Method for Elliptic Boundary Value Problems,” SIAM Journal on Numerical Analysis, 22, pp. 644–669 (1985).CrossRefGoogle Scholar
17.Banerjee, P. K., The Boundary Element Methods in Engineering, McGraw-Hill, London (1994).Google Scholar
18.Cheng, A. H.-D., “Particular Solutions of Laplacian, Helmholtz-Type, and Polyharmonic Operators Involving Higher Order Radial Basis Functions,” Engineering Analysis with Boundary Elements, 24, pp. 531–538 (2000).CrossRefGoogle Scholar
19.Golberg, M. A., Muleshkov, A. S., Chen, C. S. and Cheng, A. H.-D., “Polynomial Particular Solutions for Some Partial Differential Operators,” Numerical Methods for Partial Differential Equations, 19, pp. 112–133 (2003).CrossRefGoogle Scholar
20.Reutskiy, S. Y. and Chen, C. S., “Approximation of Multivariate Functions and Evaluation of Particular Solutions Using Chebyshev Polynomial and Trigonometric Basis Functions,” International Journal for Numerical Methods in Engineering, 67, pp. 1811–1829 (2006).CrossRefGoogle Scholar
21.Muleshkov, A. S. and Golberg, M. A., “Particular Solutions of the Multi-Helmholtz-Type Equation,” Engineering Analysis with Boundary Elements, 31, pp. 624–630 (2007)CrossRefGoogle Scholar
22.Costa, J. A. and Brebbia, C. A., “The Boundary Element Method Applied to Plates on Elastic Foundations,” Engineering Analysis with Boundary Elements, 2, pp. 174–182(1995).CrossRefGoogle Scholar
23.Costa, J. A. and Brebbia, C. A., “On the Reduction of Domain Integrals to the Boundary for the BEM Formulation of Plates on Elastic Foundations,” Engineering Analysis with Boundary Elements, 3, pp. 123–126 (1986).CrossRefGoogle Scholar
24.Tsai, C. C., Lin, Y. C., Young, D. L. and Atluri, S. N., “Investigations on the Accuracy and Condition Number for the Method of Fundamental Solutions,” CMES: Computer Modeling in Engineering and Sciences, 16, pp. 103–114(2006).Google Scholar
25.Tikhonov, A. N. and Samarskii, A. A., Equations of Mathematical Physics, The MacMillan Company, New York (1973).Google Scholar