Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-26T04:06:59.987Z Has data issue: false hasContentIssue false
  • English
  • Français

Particular Solution of Polyharmonic Spline Associated with Reissner Plate Problems

Published online by Cambridge University Press:  07 December 2011

C. C. Tsai ,
M. E. Quadir ,
H. H. Hwung  and
T. W. Hsu
C. C. Tsai
Affiliation:
Department of Environmental Engineering, National Kaohsiung Marine University, Kaohsiung, Taiwan 70101, R.O.C.
M. E. Quadir
Affiliation:
Department of Hydraulic and Ocean Engineering, National Cheng Kung University, Tainan, Taiwan 70101, R.O.C.
H. H. Hwung
Affiliation:
Department of Hydraulic and Ocean Engineering, National Cheng Kung University, Tainan, Taiwan 70101, R.O.C.
T. W. Hsu*
Affiliation:
Department of Hydraulic and Ocean Engineering, National Cheng Kung University, Tainan, Taiwan 70101, R.O.C.
*
****Professor, corresponding author
Get access

Abstract

In this paper, analytical particular solutions of the augmented polyharmonic spline (APS) associated with Reissner plate model are explicitly derived in order to apply the dual reciprocity method. In the derivations of the particular solutions, a coupled system of three second-ordered partial differential equations (PDEs), which governs problems of Reissner plates, is initially transformed into a single six-ordered PDE by the Hörmander operator decomposition technique. Then the particular solutions of the coupled system can be found by using the particular solution of the six-ordered PDE derived in the first author's previous study. These formulas are further implemented for solving problems of Reissner plates under arbitrary loadings. In the solution procedure, an arbitrary loading measured at some scattered points is first interpolated by the APS and a corresponding particular solution can then be approximated by using the prescribed formulas. After that the complementary homogeneous problem is formally solved by the method of fundamental solutions (MFS). Numerical experiments are carried out to validate these particular solutions.

Keywords

Augmented polyharmonic splineReissner plateHörmander operator decomposition techniqueMethod of fundamental solutionsDual reciprocity method
Type
Articles
Information
Journal of Mechanics , Volume 27 , Issue 4 , December 2011 , pp. 493 - 501
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2011

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

REFERENCES

1.Reissner, E., “On the Theory of Bending of Elastic Plates,” Journal of Mathematical Physics, 23, pp. 184–191 (1945).CrossRefGoogle Scholar
2.Reissner, E., “The Effect of Transverse Shear Deformation on the Bending of Elastic Plates,” Journal of Mathematical Physics, 12, pp. 69–76 (1945).Google Scholar
3.Mindlin, R. D., “Influence of Rotary Inertia and Shear on Flexural Motions of Isotropic, Elastic Plates,” Journal of Mathematical Physics, 18, pp. 336–343 (1951).Google Scholar
4.Cheng, A. H.-D. and Cheng, D. T., “Heritage and Early History of the Boundary Element Method,” Engineering Analysis with Boundary Elements, 29, pp. 268–302 (2005).CrossRefGoogle Scholar
5.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
6.Bogomolny, A., “Fundamental Solutions Method for Elliptic Boundary Value Problems,” SIAM Journal on Numerical Analysis, 22, pp. 644–669 (1985).CrossRefGoogle Scholar
7.Trefftz, E., “Ein Gegenstuck zum Ritz'chen Verfahren,” Proceedings of the Second International Congress of Applied Mechanics, Zurich (1926).Google Scholar
8.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).CrossRefGoogle Scholar
9.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 (1999).Google Scholar
10.Golberg, M. A., “The Method of Fundamental Solutions for Poisson's Equation,” Engineering Analysis with Boundary Elements, 16, pp. 205–213 (1995).CrossRefGoogle Scholar
11.Golberg, M. A., Chen, C. S. and Karur, S. R., “Improved Multiquadric Approximation for Partial Differential Equations,” Engineering Analysis with Boundary Elements, 18, pp. 9–17 (1996).CrossRefGoogle Scholar
12.Muleshkov, A. S., Golberg, M. A. and Chen, C. S., “Particular Solutions of Helmholtz-Type Operators Using Higher Order Polyharmonic Splines,” Computational Mechanics, 23, pp. 411–419 (1999).CrossRefGoogle Scholar
13.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
14.Cheng, A. H.-D, Chen, C. S., Golberg, M. A. and Rashed, Y. F., “BEM for Thermoelasticity and Elasticity with Body Force—A Revisit,” Engineering Analysis with Boundary Elements, 25, pp. 377–387 (2001).CrossRefGoogle Scholar
15.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
16.Tsai, C. C., “The Method of Fundamental Solutions with Dual Reciprocity for Thin Plates on Winkler Foundations with Arbitrary Loadings,” Journal of Mechanics, 24, pp. 163–171 (2008).CrossRefGoogle Scholar
17.Tsai, C. C., Cheng, A. H.-D. and Chen, C. S., “Particular Solutions of Splines and Monomials for Polyharmonic and Products of Helmholtz Operators,” Engineering Analysis with Boundary Elements, 33, pp. 514–521 (2009).CrossRefGoogle Scholar
18.Atkinson, K. E., “The Numerical Evaluation of Particular Solutions for Poisson's Equation,” IMA Journal of Numerical Analysis, 5, pp. 319–338 (1985).CrossRefGoogle Scholar
19.Li, X. and Chen, C. S., “A Mesh-Free Method Using Hyperinterpolation and Fast Fourier Transform for Solving Differential Equation,” Engineering Analysis with Boundary Elements, 28, pp. 1253–1260 (2004).CrossRefGoogle Scholar
20.Janssen, H. L. and Lambert, H. L., “Recursive Construction of Particular Solutions to Nonhomogeneous Linear Partial Differential Equations of Elliptic Type,” Journal of Computational and Applied Mathematics, 39, pp. 227–242 (1992).CrossRefGoogle Scholar
21.Cheng, A. H.-D, Lafe, O. and Grilli, S., “Dual Reciprocity BEM Based on Global Interpolation Function Functions,” Engineering Analysis with Boundary Elements, 13, pp. 303–311 (1994).CrossRefGoogle Scholar
22.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
23.Smyrlis, Y. S. and Karageorghis, A., “Some Aspects of the Method of Fundamental Solutions for Certain Biharmonic Problems,” CMES: Computer Modeling in Engineering & Sciences, 4, pp. 535–550 (2003).Google Scholar
24.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
25.Ding, J. and Chen, C.S., “Particular Solutions of Some Elliptic Partial Differential Equations Via Recursive Formulas,” Journal of University of Science and Technology of China, 37, pp. 1–10 (2007).Google Scholar
26.Tsai, C. C., “Particular Solutions of Chebyshev Polynomials for Polyharmonic and Poly-Helmholtz Equations,” CMES: Computer Modeling in Engineering & Sciences, 27, pp. 151–162 (2008).Google Scholar
27.Tsai, C. C., “The Particular Solutions of Chebyshev Polynomials for Reissner Plates Under Arbitrary Loadings,” CMES: Computer Modeling in Engineering & Sciences, 45, pp. 249–271 (2009).Google Scholar
28.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
29.EL-Zafrany, A., Debbih, M. and Fadhil, S., “Boundary Element Analysis of Thick Reissner Plates in Bending,” Engineering Analysis with Boundary Elements, 14, pp. 159–169 (1995).CrossRefGoogle Scholar
30.AL-Hosani, K., Fadhil, S. and EL-Zafrany, A., “Fundamental Solution and Boundary Element Analysis of Thick Plates on Winkler Foundation,” Computers & Structure, 70, pp. 325–336 (1999).CrossRefGoogle Scholar
31.Fadhil, S. and EL-Zafrany, A., “Boundary Element Analysis of Thick Reissner Plates on Two-Parameter Foundation,” International Journal of Solids and Structures, 31, pp. 2901–2917 (1994).CrossRefGoogle Scholar
32.Wen, P. H., “The Fundamental Solution of Mindlin Plates Resting on an Elastic Foundation in the Laplace Domain and its Applications,” International Journal of Solids and Structures, 45, pp. 1032–1050 (2008).CrossRefGoogle Scholar
33.Wen, P. H., Adetoro, M. and Xu, Y., “The Fundamental Solution of Mindlin Plates with Damping in the Laplace Domain and its Applications,” Engineering Analysis with Boundary Elements, 32, pp. 870–882 (2008).CrossRefGoogle Scholar
34.Wen, P. H., Aliabadi, M. H. and Young, A., “Transformation of Domain Integrals to Boundary Integrals in BEM Analysis of Shear Deformable Plate Bending Problems,” Computational Mechanics, 24, pp. 304–309 (2000).CrossRefGoogle Scholar
35.Hörmander, H., Linear Partial Differential Operators, Springer-Verlag, Berlin (1963).CrossRefGoogle Scholar