The Triangular Wavelet-based Beam, Slab Structure Analysis And Calculation

Taking wavelet function as interpolation function or trial function in elastic mechanics to analyze structures is a new method in the field of structural analysis and calculation. However, there are still some prominent issues remain to be solved, such as the processing of boundary conditions, the connection of adjacent elements and realization method of adaptive WFEM, etc. In this paper, trigonometric Hermite wavelet, which has both good approximation characteristics of trigonometric function and multi-resolution, local characteristics of wavelet, is introduced into the field of structural analysis. Taking wavelet function as interpolation function, the wavelet finite element formulation of beam, plane frame structure is derived, and the deformation, vibration and stability of beam and frame structure are calculated; realization of adaptive trigonometric WFEM is discussed, and adaptive trigonometric composite element method which combines the trigonometric wavelet function and polynomial shape function of traditional FEM is presented; taking wavelet function as trial function, the uniform formulation of rectangle thin plate (on elastic foundation) problems under different boundary conditions is derived based on two-dimensional tensor product trigonometric wavelet and the principle of minimum potential energy. The main work and conclusions are as follows:1. Interpolating functions which are appropriate for C1 element are obtained by transforming the scaling functions of trigonometric Hermite wavelet reasonably. Then the FEM formulation of the trigonometric wavelet beam element and frame element are derived. Numerical examples show that this new element can process boundary conditions and connect adjacent elements conveniently and fewer degrees of freedom are needed but with high accuracy when adopting it to analyze bending, vibration and buckling of beam structure, especially for natural vibration feature analysis. 2. Instability of column under axial force and plane frame structure is researched using WFEM, and their corresponding trigonometric wavelet finite element formulations are derived, respectively. Numerical examples show that trigonometric WFEM can solve stability problem with sufficient accuracy while only fewer degrees of freedom are needed.3. Realization of adaptive trigonometric WFEM is discussed via trigonometric wavelet hierachical element and multi-resolution trigonometric wavelet element. Adaptive trigonometric composite element method which combines the trigonometric wavelet function and polynomial shape function of traditional FEM is presented, and the effectiveness of the proposed method is verified by case study of the bending, free vibration and buckling analysis of a beam.4. Taking wavelet function as trial function, the uniform formulation of bending, vibration and buckling problems of rectangle thin plate (on elastic foundation) under different boundary conditions is derived based on two-dimensional tensor product trigonometric wavelet and the principle of minimum potential energy. Numerical examples show that this method performs well, especially for natural vibration feature analysis. Furthermore, two approaches, hierachical and multi-resolution approach, are used to improve accuracy of calculation.