Based on the generalized variational principle and B-spline wavelet on the interval （BSWI）, the multivariable BSWI elements with two kinds of variables （TBSWI） for hyperboloidal shell and open cylindrical shell are constructed in this paper. Different from the traditional method, the present one treats the generalized displacement and stress as independent variables. So differentiation and integration are avoided in calculating generalized stress and thus the precision is improved. Furthermore, compared with commonly used Daubechies wavelet, BSWI has explicit expression and excellent approximation property and thus further guarantee satisfactory results. Finally, the efficiency of the constructed multivariable shell elements is validated through several numerical examples.
A new wavelet-based finite element method is proposed for solving the Poisson equation. The wavelet bases of Hermite cubic splines on the interval are employed as the multi-scale interpolation basis in the finite element analysis. The lifting scheme of the wavelet-based finite element method is discussed in detail. For the orthogonal characteristics of the wavelet bases with respect to the given inner product, the corresponding multi-scale finite element equation can be decoupled across scales, totally or partially, and suited for nesting approximation. Numerical examples indicate that the proposed method has the higher efficiency and precision in solving the Poisson equation.
The presence of cracks in the rotor is one of the most dangerous and critical defects for rotating machinery. Defect of fatigue cracks may lead to long out-of-service periods, heavy damages of machines and severe economic consequences. With the method of finite element, vibration behavior of cracked rotors and crack detection was received considerable attention in the academic and engineering field. Various researchers studied the response of a cracked rotor and most of them are focused on the crack detection based on vibration behavior of cracked rotors. But it is often difficult to identify the crack parameters quantitatively. Second generation wavelets （SGW） finite element has good ability in modal analysis for singularity problems like a cracked rotor. Based on the fact that the feature of SGW could be designed depending on applications, a multiresolution finite element method is presented. The new model of SGW beam element is constructed. The first three natural frequencies of the rotor with different crack location and size were solved with SGW beam elements, and the database for crack diagnosis is obtained. The first three metrical natural frequencies are employed as inputs of the database and the intersection of the three frequencies contour lines predicted the normalized crack location and size. With the Bently RK4 rotor test rig, rotors with different crack location and size are tested and diagnosed. The experimental results denote the cracks quantitative identification method has higher identification precision. With SGW finite element method, a novel method is presented that has higher precision and faster computing speed to identify the crack location and size.
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。
