Acta Mechanica Solida Sinica

, Volume 25, Issue 1, pp 22–36

# Transient Analysis of Composite Plates by a Local Radial Basis Functions-Finite Difference Techique

• C. M. C. Roque
• D. Cunha
• A. J. M. Ferreira
Article

## Abstract

The use of local numerical schemes, such as finite differences produces much better conditioned matrices than global collocation radial basis functions methods. However, finite difference schemes are limited to special grids. For scattered points, a combination of finite differences and radial basis functions would be a possible solution. In this paper, we use a higher-order shear deformation plate theory and a radial basis function — finite difference technique for predicting the transient behavior of thin and thick composite plates. Through numerical experiments on beams and composite plates, the accuracy and efficiency of this collocation technique is demonstrated.

## Key words

meshless beam composite plate

## Preview

Unable to display preview. Download preview PDF.

## References

1. [1]
Reddy, J.N., Mechanics of Laminated Composite Plates and Shells. CRC Press, 2004.Google Scholar
2. [2]
Reddy, J.N., Simple higher-order theory for laminated composite plates. Journal of Applied Mechanics, Transactions ASME, 1984, 51(4): 745–752.
3. [3]
Hardy, R.L., Multiquadric equations of topography and other irregular surfaces. Journal of Geophysical Research, 1971, 176: 1905–1915.
4. [4]
Hardy, R.L., Research results in the application of multiquadric equations to surveying and mapping problems. Surveying and Mapping, 1975, 35(4): 321–332.Google Scholar
5. [5]
Kansa, E.J., Multiquadrics. a scattered data approximation scheme with applications to computational fluid-dynamics. i. surface approximations and partial derivative estimates. Computers & mathematics with applications, 1990, 19(8–9): 127–145.
6. [6]
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 & mathematics with applications, 1990, 19(8–9): 147–161.
7. [7]
Roque, C.M.C., Ferreira, A.J.M. and Jorge, R.M.N., A radial basis function approach for the free vibration analysis of functionally graded plates using a refined theory. Journal of Sound and Vibration, 2007, 300(3–5): 1048–1070.
8. [8]
Roque, C.M.C., Ferreira, A.J.M. and Jorge, R.M.N., Free vibration analysis of composite and sandwich plates by a trigonometric layerwise deformation theory and radial basis functions. Journal of Sandwich Structures and Materials, 2006, 8(6): 497–515.
9. [9]
Roque, C.M.C., Ferreira, A.J.M. and Jorge, R.M.N., Modelling of composite and sandwich plates by a trigonometric layerwise deformation theory and radial basis functions. Composites Part B: Engineering, 2005, 36(8): 559–572.
10. [10]
Ferreira, A.J.M., Roque, C.M.C. and Martins, P.A.L.S., Analysis of composite plates using higher-order shear deformation theory and a finite point formulation based on the multiquadric radial basis function method. Composites Part B: Engineering, 2003, 34(7): 627–636.
11. [11]
Ferreira, A.J.M., Roque, C.M.C. and Jorge, R.M.N., Natural frequencies of FSDT cross-ply composite shells by multiquadrics. Composite Structures, 2007, 77(3): 296–305.
12. [12]
Ferreira, A.J.M., Roque, C.M.C. and Jorge, R.M.N., Modelling cross-ply laminated elastic shells by a higherorder theory and multiquadrics. Computers and Structures, 2006, 84(19–20): 1288–1299.
13. [13]
Ferreira, A.J.M., Roque, C.M.C. and Jorge, R.M.N., Static and free vibration analysis of composite shells by radial basis functions. Engineering Analysis with Boundary Elements, 2006, 30(9): 719–733.
14. [14]
Ferreira, A.J,M., Roque, C.M.C. and Jorge, R.M.N., Analysis of composite plates by trigonometric shear deformation theory and multiquadrics. Computers and Structures, 2005, 83(27): 2225–2237.
15. [15]
Ferreira, A.J.M., Roque, C.M.C. and Jorge, R.M.N., Free vibration analysis of symmetric laminated composite plates by FSDT and radial basis functions. Computer Methods in Applied Mechanics and Engineering, 2005, 194(39–41): 4265–4278.
16. [16]
Ferreira, A.J.M., Roque, C.M.C., Jorge, R.M.N., Fasshauer, G.E. and Batra, R.C., Analysis of functionally graded plates by a robust meshless method. Mechanics of Advanced Materials and Structures, 2007, 14(8): 577–587.
17. [17]
Ferreira, A.J.M., Batra, R.C., Roque, C.M.C., Qian, L.F. and Martins, P.A.L.S., Static analysis of functionally graded plates using third-order shear deformation theory and a meshless method. Composite Structures, 2005, 69(4): 449–457.
18. [18]
Ferreira, A.J.M., Batra, R.C., Roque, C.M.C., Qian, L.F. and Jorge, R.M.N., Natural frequencies of functionally graded plates by a meshless method. Composite Structures, 2006, 75(1–4): 593–600.
19. [19]
Tolstykh, A.I., Lipavskii, M.V. and Shirobokov, D.A., High-accuracy discretization methods for solid mechanics. Archives of Mechanics, 2003, 55(5–6): 531–553.
20. [20]
Shu, C., Ding, H. and Yeo, K.S., Local radial basis funcion-based differential quadrature method and its application to solve two-dimensional incompressible Navier-Stokes equations. Computer Methods in Applied Mechanics and Engineering, 2003, 192(7–8): 941–954.
21. [21]
Cecil, T., Qian, J. and Osher, S., Numerical methods for high dimensional Hamilton-Jacobi equations using radial basis functions. Journal of Computational Physics, 2004, 196(1): 327–347.
22. [22]
Wright, G.B. and Fornberg, B., Scattered node compact finite difference-type formulas generated from radial basis functions. Journal of Computational Physics, 2006, 212(1): 99–123.
23. [23]
Shan, Y.Y., Shu, C. and Qin, N., Multiquadric finite difference (MQ-FD) Methods and its application. Advances in Applied Mathematics and Mechanics, 2009, 1(5): 615–638.
24. [24]
Roque, C.M.C. and Ferreira, A.J.M., Numerical experiments on optimal shape parameters for radial basis functions. Numerical Methods for Partial Differential Equations, 2010, 26(3): 675–689.
25. [25]
Micchelli, C.A., Interpolation of scattered data distance matrices and conditionally positive definite functions. Constructive Approximation, 1986, 2(1): 11–22.
26. [26]
Fornberg, B., Wright, G. and Larsson, E., Some observations regarding interpolants in the limit of flat radial basis functions. Computers & mathematics with applications, 2004, 47: 37–55.

© The Chinese Society of Theoretical and Applied Mechanics and Technology 2012

## Authors and Affiliations

• C. M. C. Roque
• 1
• D. Cunha
• 1
• A. J. M. Ferreira
• 2
1. 1.INEGIFaculdade de Engenharia da Universidade do PortoPortoPortugal
2. 2.Departamento de Engenharia MecânicaFaculdade de Engenharia da Universidade do PortoPortoPortugal