The method of constructing any scale wavelet finite element (WFE) based on the one-dimensional or two-dimensional Daubechies scaling functions was presented, and the corresponding WFE adaptive lifting algorithm was given. In order to obtain the nested increasing approximate subspaces of multiscale finite element, the Daubechies scaling functions with the properties of multi-resolution analysis were employed as the finite element interpolating functions. Thus, the WFE could adaptively mesh the singularity domain caused by local cracks, which resulted in better approximate solutions than the traditional finite element methods. The calculations of natural frequencies of cracked beam were used to check the accuracy of given methods. In addition, the results of cracked cantilever beam and engineering application were satisfied. So, the current methods can provide effective tools in the numerical modeling of the fault prognosis of incipient crack.

The two-dimensional wavelet nite element (WFE) is constructed, in which the scaling functions of Daubechies wavelets are adopted as trial functions. In order to overcome the integral di:culty caused by lack of the explicit Daubechies scaling function expression, a new and e:cient integral method for stiffness and load matrices is presented. Then the bending characters of a thin plate are studied based on WFE. Numerical tests indicate that the WFE is highly accurate and effective. For engineering application, the internal temperature distribution of office paper used in office machines such as printers and copiers is analyzed by WFE. The results show that WFE has desirable calculation precision while dealing with singularity problems

为了解决机电设备早期故障难以正确识别的问题，有效地提高分类的准确率，提出了一种基于提升小波包变换和集成支持矢量机的早期故障智能诊断新方法。首先，该方法采用提升策略构造基于冲击故障信号特征的双正交小波，借助提升小波包变换提取信号的敏感频带特征，从而通过对敏感频带中的小波包系数进行包络解调分析检测出故障特征频率。其次，通过距离评估技术从原始信号和小波包系数的统计特征中选取最优特征集。最后，将最优特征输入到集成支持矢量机中，实现对不同故障类型的识别。将该方法应用于滚动轴承的早期故障诊断中，测试结果表明，同常规的小波包变换相比，该方法能够有效地提取故障特征；同时，该方法具有比单一支持矢量机更好的分类性能，故障诊断准确率高。

A new time-domain analysis method that uses second generation wavelet transform (SGWT) for weak fault feature extraction is proposed. To extrata incipient fault feature, a biorthogonal wavelet with the characteristics of impact is constructed by using SGWT. Processing detail signal of SGWT with a sliding window devised on the basis of rotating operation cycle, and extracting modulus maxi-mum from each window, fault features in time-domain are highlighted. To make fitrther analysis on the reason of the fault, wavelet package transform based on SGWT is used to process vibration data again. Calculating the energy of each frequency-band, the energy distribution features of die signal are at- tained. Then taking account of the fault features and the energy distribution, the reason of die fault is worked out. An early impact-rob fault caused by axis misalignment and rotor imbalance is successfully detected by using this method in an oil refinery.

An important property of wavelet multiresolution analysis is the capability to represent functions in a dynamic multiscale manner, so the solution in the wavelet domain enables a hierarchical approximation to the exact solution. The typical problem that arises when using Daubechies wavelets in numerical analysis, especially in finite element analysis, is how to calculate the connection coefficients, an integral of products of wavelet scaling functions or derivative operators associated with these. The method to calculate multiscale connection coefficients for stiffness matrices and load vectors is presented for the first time. And the algorithm of multiscale lifting computation is developed. The numerical examples are given to verify the effectiveness of such a method.