A wavelet-based stochastic finite element method of thin plate bending

Abstract A wavelet-based stochastic finite element method is presented for the bending analysis of thin plates. The wavelet scaling functions of spline wavelets are selected to construct the displacement interpolation functions of a rectangular thin plate element and the displacement shape functions are expressed by the spline wavelets. A new wavelet-based finite element formulation of thin plate bending is developed by using the virtual work principle. A wavelet-based stochastic finite element method that combines the proposed wavelet-based finite element method with Monte Carlo method is further formulated. With the aid of the wavelet-based stochastic finite element method, the present paper can deal with the problem of thin plate response variability resulting from the spatial variability of the material properties when it is subjected to static loads of uncertain nature. Numerical examples of thin plate bending have demonstrated that the proposed wavelet-based stochastic finite element method can achieve a high numerical accuracy and converges fast.

[1]  E. Vanmarcke.  Probabilistic Modeling of Soil Profiles , 1977 .

[2]  Ziad M. Elias.  Theory and methods of structural analysis , 1986 .

[3]  Haym Benaroya,et al.  Finite Element Methods in Probabilistic Structural Analysis: A Selective Review , 1988 .

[4]  S. T. Buckland,et al.  An Introduction to the Bootstrap. , 1994 .

[5]  R. Melosh.  BASIS FOR DERIVATION OF MATRICES FOR THE DIRECT STIFFNESS METHOD , 1963 .

[6]  Zheng Xiaojing,et al.  Applications of wavelet galerkin fem to bending of beam and plate structures , 1998 .

[7]  Rezak Ayad,et al.  A new hybrid‐mixed variational approach for Reissner–Mindlin plates. The MiSP model , 1998 .

[8]  Stuart C. Althorpe,et al.  Wavelet-distributed approximating functional method for solving the Navier-Stokes equation , 1998 .

[9]  G. Schuëller.  A state-of-the-art report on computational stochastic mechanics , 1997 .

[10]  M. Shinozuka,et al.  Random fields and stochastic finite elements , 1986 .

[11]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[12]  T. Pian.  Derivation of element stiffness matrices by assumed stress distributions , 1964 .

[13]  J. Z. Zhu,et al.  The finite element method , 1977 .

[14]  K. Binder,et al.  A Guide to Monte Carlo Simulations in Statistical Physics , 2000 .

[15]  Andrew J. Kurdila,et al.  A class of finite element methods based on orthonormal, compactly supported wavelets , 1995 .

[16]  The finite element method based on interpolating with wavelet basis function , 2000 .

[17]  Zheng,et al.  Applications of wavelet Galerkin FEM to bending of plate structure , 1999 .

[18]  D. Allman.  A quadrilateral finite element including vertex rotations for plane elasticity analysis , 1988 .

[19]  D. H. Young,et al.  Theory of Structures , 1965 .

[20]  Xiaodong Zhou,et al.  Wavelet solutions for the Dirichlet problem , 1995 .

[21]  Zhengjia He,et al.  A study of the construction and application of a Daubechies wavelet-based beam element , 2003 .

[22]  C. Burrus,et al.  Introduction to Wavelets and Wavelet Transforms: A Primer , 1997 .

[23]  Christos C. Chamis,et al.  Probabilistic structural analysis methods for space propulsion system components , 1987 .

[24]  Wei-Xin Ren,et al.  A multivariable wavelet-based finite element method and its application to thick plates , 2005 .

[25]  R. Ghanem,et al.  Stochastic Finite Element Expansion for Random Media , 1989 .

[26]  Fumio Kikuchi,et al.  An improved 4-node quadrilateral plate bending element of the Reissner-Mindlin type , 1999 .