An Improved Meshless Method for the Static and Vibration Analysis of Plates

An application of a meshless technique for the static and vibration analysis of plates is presented in this paper. The method is based on the radial basis functions-QR (RBF-QR) algorithm. Static deformations and free vibration analysis of plates are obtained by Gaussian RBF-QR technique, previously proposed by Fornberg and Piret (2007) for interpolation problems. Circular and square plates are analyzed considering free-edge, simply-supported, and fully-clamped boundary conditions. The presented technique can be applied to any geometry and node sets embedded in the 2D unitary disk. The RBF-QR approach is stable for decreasing values of the shape parameter, corresponding to flat RBFs and the conversion to polar coordinates brings an extra improvement in precision. The method produces highly accurate displacements, natural frequencies, and modes of vibrations.

[1]  Herbert Reismann,et al.  Elastic Plates: Theory and Application , 1988 .

[2]  Bengt Fornberg,et al.  A Stable Algorithm for Flat Radial Basis Functions on a Sphere , 2007, SIAM J. Sci. Comput..

[3]  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 .

[4]  Marco Amabili,et al.  Analysis of vibrating circular plates having non-uniform constraints using the modal properties of free-edge plates : Application to bolted plates , 1997 .

[5]  Yang Zhong,et al.  On new symplectic superposition method for exact bending solutions of rectangular cantilever thin plates , 2011 .

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

[7]  Bengt Fornberg,et al.  On choosing a radial basis function and a shape parameter when solving a convective PDE on a sphere , 2008, J. Comput. Phys..

[8]  M. Amabili,et al.  EFFECT OF FINITE FLUID DEPTH ON THE HYDROELASTIC VIBRATIONS OF CIRCULAR AND ANNULAR PLATES , 1996 .

[9]  K. M. Liew,et al.  Differential cubature method for static solutions of arbitrarily shaped thick plates , 1998 .

[10]  Robert L. Taylor,et al.  Linked interpolation for Reissner‐Mindlin plate elements: Part II—A simple triangle , 1993 .

[11]  R. D. Mindlin,et al.  Influence of rotary inertia and shear on flexural motions of isotropic, elastic plates , 1951 .

[12]  Erasmo Carrera,et al.  Analysis of laminated shells by a sinusoidal shear deformation theory and radial basis functions collocation, accounting for through-the-thickness deformations , 2011 .

[13]  Tarun Kant,et al.  Free vibration of composite and sandwich laminates with a higher-order facet shell element , 2004 .

[14]  Erasmo Carrera,et al.  Analysis of laminated doubly-curved shells by a layerwise theory and radial basis functions collocation, accounting for through-the-thickness deformations , 2011 .

[15]  D. J. Dawe,et al.  Rayleigh-Ritz vibration analysis of Mindlin plates , 1980 .

[16]  B. Fornberg,et al.  Radial basis functions: Developments and applications to planetary scale flows , 2011 .

[17]  K. M. Liew,et al.  An eight-node curvilinear differential quadrature formulation for Reissner/Mindlin plates , 1997 .

[18]  R. Batra,et al.  Plane wave solutions and modal analysis in higher order shear and normal deformable plate theories , 2002 .

[19]  J. Reddy.  Mechanics of laminated composite plates : theory and analysis , 1997 .

[20]  Elisabeth Larsson,et al.  Stable Computations with Gaussian Radial Basis Functions , 2011, SIAM J. Sci. Comput..

[21]  Erasmo Carrera,et al.  Radial basis functions-differential quadrature collocation and a unified formulation for bending, vibration and buckling analysis of laminated plates, according to Murakami's Zig-Zag theory , 2012 .

[22]  J. Reddy,et al.  Buckling analysis of isotropic and laminated plates by radial basis functions according to a higher-order shear deformation theory , 2011 .

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

[24]  H. Nguyen-Xuan,et al.  A smoothed finite element method for plate analysis , 2008 .

[25]  K. M. Liew,et al.  Vibration analysis of symmetrically laminated plates based on FSDT using the moving least squares differential quadrature method , 2003 .

[26]  A. U.S.,et al.  Stable Computation of Multiquadric Interpolants for All Values of the Shape Parameter , 2003 .

[27]  J. N. Reddy,et al.  Stability and natural vibration analysis of laminated plates by using a mixed element based on a refined plate theory , 1986 .

[28]  C.M.C. Roque,et al.  Analysis of thick plates by local radial basis functions-finite differences method , 2012 .

[29]  Jun Liu,et al.  Free vibration analysis of arbitrary shaped thick plates by differential cubature method , 2005 .

[30]  Marco Amabili,et al.  Theory and experiments for large-amplitude vibrations of rectangular plates with geometric imperfections , 2006 .