基于多变量小波有限元的一维结构分析

MULTIVARIABLE WAVELET FINITE ELEMENT METHOD FOR 1D STRUCTURAL ANALYSIS

基于区间B样条小波(B-Spline Wavelet on the Interval,BSWI)和多变量广义势能函数,该文构造了二类变量小波有限单元,并用于一维结构的弯曲与振动分析。基于广义变分原理,从多变量广义势能函数出发,推导得到多变量有限元列式,并以区间B样条小波尺度函数作为插值函数对两类广义场变量进行离散。此单元的优势在于可以提高广义力的求解精度,因为在传统有限元中,只有一类广义位移场函数,所以广义力通常是通过对位移的求导得到,而多变量单元中,广义位移和广义力都是作为独立变量处理的,避免了求导运算。此外,区间B样条小波是现有小波中数值逼近性能非常好的小波函数,以它作为插值函数可进一步保证求解精度。转换矩阵的应用,可以将无任何明确物理意义的小波系数转换到相应的物理空间,方便了问题的处理。最后,通过数值算例对Euler梁和平面刚架的分析,验证了此单元的正确性和有效性。

Based on B-spline wavelet on the interval （BSWI） and multivariable generalized potential energy functional, the wavelet finite elements with two kinds of variables for 1D structural analysis are constructed. The formulations are derived from generalized potential energy functional based on potential variational principle and BSWI is selected as trial function to discrete the field functions. The advantage of the elements proposed in this paper is that it can improve the solving accuracy of generalized stress. Because there is only one kind of field function in traditional method, generalized stress should be calculated through differentiation of generalized displacement. However, to multivariable element, generalized displacement and stress are treated as independent variables, differentiation is avoided. Besides, BSWI has very good numerical approximation property among all existing wavelets, which can further guarantee the precision. The meaningless wavelet coefficients are translated to physical space by transformation matrix, which makes the solving process more convenient. In the end, several numerical examples of Euler beam and Plane frame are provided to verify the correctness and efficiency.