针对强噪声背景下机械信号故障特征的提取问题,构造了一种提取该类信号时域特征的冗余第2代小波方法。该方法通过对初始预测算子和更新算子插值补0，来获得不同分解层上的预测算子和更新算子。冗余第2代小波不需要剖分运算,直接利用构造的算子对逼近信号进行对称预测和更新，可使逼近信号和细节信号的数据点数保持不变，并根据每层细节信号的噪声特点选取降噪阈值门限。实验和工程振动信号分析表明，冗余第2代小波的降噪效果优于其他类型的小波方法，较理想地提取出了滚动轴承内圈剥落和汽轮发电机组高压缸蒸汽激振的时域故障特征。

小波有限元是一类新的有限元逼近方法，将信号处理领域中小波函数的多分辨思想引入有限元法中，以小波函数作为插值函数，构造出嵌套递进的多尺度广义有限元逼近空间，使得求解问题可以先用较粗的网格分析，特定奇异区域通过自适应多分辨剖分获得更好的逼近，该算法数值稳定性好、适宜求解奇异性问题。从小波加权残值法、小波有限元理论以及自适应小波有限元三个方面，综述了小波有限元国内外研究现状，并介绍了小波有。限元在大梯度非线性、裂纹定量预示等方面的工程应用进展，指出了其关键技术、存在问题以及工程实用前景。

提出了任意尺度Daubechies小波一维二维单元构造方法以及小波有限元自适应提升算法，采用小波尺度函数作为插值函数，利用小波多分辨分析了具有变尺度逐层逼近函数的特性，获得了嵌套递进的多尺度有限元逼近空间。

The model-based forward and inverse problems in the diagnosis of structural crack location and size by using the finite element method of a B-spline wavelet on the interval (FEM BSWI) were studied. First the crack and uncracked elements of BSWI were built to solve the forward problem. The first three frequencies influencing functions of normalized crack location and size are approximated by means of surface-fitting techniques. Then the first three measured natural frequencies are employed as inputs of the functions. The intersection of the three frequencies contour lines predicted the normalized crack location and size. Both the numerical and experimental studies verified the validity of the BSWI elements in solving crack singular problems with high performance.

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