The construction of 1D wavelet finite elements for structural analysis

Abstract

Adopting the scaling functions of B-spline wavelet on the interval (BSWI) as trial functions, a new finite element method (FEM) of BSWI is presented. Instead of traditional polynomial interpolation, scaling functions at the certain scale have been adopted to form the shape functions and construct wavelet-based elements. Unlike the process of wavelets added directly in the other wavelet numerical methods, the element displacement field represented by the coefficients of wavelets expansions is transformed from wavelet space to physical space via the corresponding transformation matrix. The transformation matrix is the key to construct wavelet-based elements freely as long as we can ensure its non-singularity. Then, classes of C 0 and C 1 type elements are constructed. And the lifting scheme of BSWI elements is also discussed. The numerical examples indicate that the BSWI elements have higher efficiency and precision than traditional finite element method in solving 1D structural problems especially for geometric nonlinear, variable cross-section and loading cases.

1st‐Order Shear Deformable Beam Formulation Based on Meshless Wavelet Galerkin Method

“…A similar case of sections that are used in this paper is found in [4]. Elements are defined and divided into segments that include nodes in [4].…”

“…Elements are defined and divided into segments that include nodes in [4]. In other words, the elements are composed of many segments assigned with nodes.…”

“…A study that used Daubechies wavelets as the shape functions was performed to solve the Euler beam bending problem [3]. Additionally, a study using B-spline wavelet as the shape functions was conducted [4]. Basu et al did a study comparing the FEM, BEM, Meshless Method, and Wavelet Methods [5,6].…”

A wavelet finite-difference method for numerical simulation of wave propagation in fluid-saturated porous media

“…Mustafa [10] developed wavelet-based numerical methods for solving partial differential equations by using adaptive schemes. Xiang [11] presented a new finite element method adopting the scaling functions of B-spline wavelet on the interval as trial functions. In the solution of differential equations, however, wavelets have not been able to replace other traditional techniques.…”

Vibration Analysis of Frame Structures Using Wavelet Finite Elements

“…To overcome these problems, specialized numerical techniques are required [11], which increase the computational costs but maintain the approximation accuracy and orthogonality properties in the analysis. BSWI family of wavelets constructed by Chui and Quak [12], have also been used in some WFEM formulations [13][14]. This family of wavelets has, in addition to the multiresolution and compact support properties, explicit expressions and therefore there is no need to carry out additional calculations to obtain the integral of the products of the scaling function and/or their derivatives.…”