One-dimensional finite elements based on the Daubechies family of wavelets

Abstract

The development of a one-dimensional finite element is traced utilizing a Daubechies scaling function as its interpolation function. The element was developed to reduce the computational time and number of degrees of freedom needed to solve vibration and wave propagation problems. This element is used in a dynamic test case, and the results are compared with the results using a standard, constant strain, rod element. The results show a 5:1 reduction in required number of degrees of freedom and in the computational time needed for equivalent accuracy. ONVENTIONAL finite elements are formed using the Rayleigh-Ritz method wherein polynomials are used as ap- proximation functions. In most applications, constant strain or poly- nomial strain elements are used.1 To model a complex subject, addi- tional elements are combined with the first, resulting in an approx- imation of the required solution. This paper discusses a different type of finite element, the wavelet finite element, in which a wavelet approximation function is used. The wavelet finite element was developed specifically for vibra- tion problems and wave propagation analysis. The concept is that the frequency response of the wavelet's associated scaling function models displacements from 0 Hz to a certain cutoff frequency and thus models vibrations more accurately and with less CPU time than conventional elements. The finite element model of a vibrating object requires a certain density of elements to accurately simulate vibratory response. This element density is usually given in terms of the wavelength of the highest frequency. The goal of this study is to derive a finite element that will attain a high level of accuracy for a low mesh density as compared with conventional finite elements. Wavelets were used to solve the one-dimensional wave equation by Bacry et al.2 Wavelets are used in solving a two-point boundary problem by Beylkin. 3 Sarkar et al. present an investigation of the application of wavelet-like triangular functions to finite elements.4 The interpolation functions exhibit the translation and dilation prop- erties of wavelets but are not wavelets. Finite elements were derived in Kurdila et al. based on wavelets generated using affine, fractal interpolation functions and were ap- plied to two-dimension al elasticity problems.5 A wavelet-based method for solving one-dimensional displacement problems was introduced in Bertoluzza et al.6