Local thin plate spline collocation for free vibration analysis of laminated composite plates

Abstract This paper focuses for the first time on free vibration analysis of laminated composite plates by a meshless local collocation method based on thin plate spline radial basis function. This method approximates the governing equations using the nodes in the support domain of any data center. Natural frequencies of the laminated composite plates with various boundary conditions, side-to-thickness ratios, material properties, and lamination schemes are computed by present method. The choice of shape parameter, effect of dimensionless sizes of the support domain on accuracy, convergence characteristics are studied by several numerical examples. The results are compared with available published results which demonstrate the accuracy and efficiency of present method.

[1]  Benny Y. C. Hon,et al.  Compactly supported radial basis functions for shallow water equations , 2002, Appl. Math. Comput..

[2]  E. Kansa,et al.  Circumventing the ill-conditioning problem with multiquadric radial basis functions: Applications to elliptic partial differential equations , 2000 .

[3]  S. Xiang,et al.  Meshless global radial point collocation method for three-dimensional partial differential equations , 2011 .

[4]  S. Xiang,et al.  Analysis of isotropic, sandwich and laminated plates by a meshless method and various shear deformation theories , 2009 .

[5]  António J.M. Ferreira,et al.  Analysis of Composite Plates Using a Layerwise Theory and Multiquadrics Discretization , 2005 .

[6]  E. Kansa.  Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates , 1990 .

[7]  H. Power,et al.  A comparison analysis between unsymmetric and symmetric radial basis function collocation methods for the numerical solution of partial differential equations , 2002 .

[8]  Gui-Rong Liu,et al.  An Introduction to Meshfree Methods and Their Programming , 2005 .

[9]  H. Kitagawa,et al.  Vibration analysis of fully clamped arbitrarily laminated plate , 2004 .

[10]  Y. C. Hon,et al.  Numerical comparisons of two meshless methods using radial basis functions , 2002 .

[11]  C.M.C. Roque,et al.  Analysis of composite plates using higher-order shear deformation theory and a finite point formulation based on the multiquadric radial basis function method , 2003 .

[12]  Liviu Librescu,et al.  Analysis of symmetric cross-ply laminated elastic plates using a higher-order theory. II - Buckling and free vibration , 1988 .

[13]  J. Reddy.  Mechanics of laminated composite plates and shells : theory and analysis , 1996 .

[14]  E. Kansa.  MULTIQUADRICS--A SCATTERED DATA APPROXIMATION SCHEME WITH APPLICATIONS TO COMPUTATIONAL FLUID-DYNAMICS-- II SOLUTIONS TO PARABOLIC, HYPERBOLIC AND ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS , 1990 .

[15]  R. L. Hardy.  Theory and applications of the multiquadric-biharmonic method : 20 years of discovery 1968-1988 , 1990 .

[16]  E. J. Kansa,et al.  Application of the Multiquadric Method for Numerical Solution of Elliptic Partial Differential Equations , 2022 .

[17]  Y. Hon,et al.  Multiquadric method for the numerical solution of a biphasic mixture model , 1997 .

[18]  Kwok Fai Cheung,et al.  Multiquadric Solution for Shallow Water Equations , 1999 .

[19]  A. Ferreira,et al.  Radial basis functions and higher-order shear deformation theories in the analysis of laminated composite beams and plates , 2004 .

[20]  Gregory E. Fasshauer,et al.  Computation of natural frequencies of shear deformable beams and plates by an RBF-pseudospectral method , 2006 .

[21]  Qibai Huang,et al.  An improved localized radial basis function meshless method for computational aeroacoustics , 2011 .

[22]  A. Ferreira.  FREE VIBRATION ANALYSIS OF TIMOSHENKO BEAMS AND MINDLIN PLATES BY RADIAL BASIS FUNCTIONS , 2005 .

[23]  Gui-Rong Liu,et al.  A stabilized least-squares radial point collocation method (LS-RPCM) for adaptive analysis , 2006 .

[24]  R. Wyatt,et al.  Radial basis function interpolation in the quantum trajectory method: optimization of the multi-quadric shape parameter , 2003 .

[25]  Kang Tai,et al.  Point interpolation collocation method for the solution of partial differential equations , 2006 .

[26]  S. Xiang,et al.  Free vibration analysis of symmetric laminated composite plates by trigonometric shear deformation theory and inverse multiquadric RBF , 2009 .

[27]  Thin plate spline radial basis functions for vibration analysis of clamped laminated composite plates , 2010 .

[28]  S. C. Fan,et al.  Local multiquadric approximation for solving boundary value problems , 2003 .

[29]  Renato Natal Jorge,et al.  Free vibration analysis of symmetric laminated composite plates by FSDT and radial basis functions , 2005 .

[30]  Benny Y. C. Hon,et al.  An efficient numerical scheme for Burgers' equation , 1998, Appl. Math. Comput..

[31]  Kang Tai,et al.  Radial point interpolation collocation method (RPICM) for partial differential equations , 2005 .

[32]  R. Jorge,et al.  Analysis of composite plates by trigonometric shear deformation theory and multiquadrics , 2005 .

[33]  António J.M. Ferreira,et al.  Polyharmonic (thin-plate) splines in the analysis of composite plates , 2004 .

[34]  R. L. Hardy.  Multiquadric equations of topography and other irregular surfaces , 1971 .

[35]  M. Golberg,et al.  Improved multiquadric approximation for partial differential equations , 1996 .

[36]  B. Fornberg,et al.  A numerical study of some radial basis function based solution methods for elliptic PDEs , 2003 .

[37]  Y. V. S. S. Sanyasiraju,et al.  Local radial basis function based gridfree scheme for unsteady incompressible viscous flows , 2008, J. Comput. Phys..