Keywords: radial basis functions, adaptive methods, composite materials, beams, plates.

Radial basis function (RBF) methods are known to be a good alternative method for the numerical solution of partial differential equations (PDEs) [1,2]. The RBFs have been applied to the study of isotropic and composite materials [3,4]. Driscoll and Heryudono [5] developed a residual sub-sampling technique for RBFs. Tests were performed in interpolation, boundary-value, and time-dependent problems and obtained results are encouraging.

We run tests on the analysis of beams and square plates in bending using Driscoll and Heryudono's residual sub-sampling technique. We considered various ways to calculate the shape parameter and discuss its influence on the solution accuracy. Various laminates and thickness-side ratio were considered. Simply-supported and clamped boundary conditions were analysed.

The method used allows for a more natural and automatic selection of the problem grid, where the user must only define the error tolerance and the initial number of points at each node. The domain is initially partitioned in boxes and each one has a center and residual points. The iteration process begins and centers are added to or removed from the grid depending on the evaluated residual.

The static analysis of plates was performed considering a first-order shear deformation theory, with three degrees of freedom, the transverse displacement w, and the rotation theta_x and theta_y of the normal about the y- and x-axes respectively.

The results obtained show an interesting and promising approach to the static analysis of composite laminates. The number of iterations performed and the final number of points in the grid depend on many parameters such as the material properties, the boundary conditions, the side to thickness ratio and the shape parameter.

Both for beams and plates the grid is more dense near the boundary and the error obtained is very small, even starting with a very low number of points.

1

E.J. Kansa, "Multiquadrics . A scattered data approximation scheme with applications to computational fluid-dynamics. I. Surface approximations and partial derivative estimates", Computers & mathematics with applications, 19(8-9):127-145, 1990. doi:10.1016/0898-1221(90)90270-T

2

E.J. Kansa, "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, 19(8-9):147-161, 1990. doi:10.1016/0898-1221(90)90270-T

3

A.J.M. Ferreira, C.M.C. Roque, P.A.L.S. Martins, "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, Elsevier, Vol. 34, 627-636, 2003. doi:10.1016/S1359-8368(03)00083-0

4

A.J.M. Ferreira, C.M.C. Roque, P.A.L.S. Martins, "Radial basis functions and higher-order theories in the analysis of laminated composite beams and plates", Composite Structures, Vol. 66, 287-293, 2004. doi:10.1016/j.compstruct.2004.04.050

5

T.A. Driscoll, A.R.H. Heryudono, "Adaptive Residual Subsampling Methods for Radial Basis Function Interpolation and Collocation Problems", Computers & Mathematics with Applications, 53, 6, 927-939, 2007. doi:10.1016/j.camwa.2006.06.005

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