提升多小波在有限元法中的应用研究

发布时间：2018-06-29 23:00

本文选题：提升方法 + Hermite插值　；参考：《西安建筑科技大学》2014年硕士论文

【摘要】：近几十年来，在第一代小波基础上发展起来的第二代小波，继承了第一代小波所具有的多分辨特点，但不再是某个给定小波函数的伸缩和平移，而是采用柔性化提升方法构造，使用户可以对已选定初始小波的性能进行改善，得到具有优良特性的新小波函数，从而满足某些特殊的工程问题的求解需要。多小波拥有多个尺度函数和小波函数，基于提升框架的多小波运行速度快，求解精度高。本文基于Hermite插值，构造了提升多小波，利用多小波的多分辨分析的特点，将多小波理论和有限元法结合起来，利用多小波函数或尺度函数构造有限元法中的位移函数，并推导出小波有限元列式，可方便求解传统有限元法难以解决的大梯度、奇异性突变等工程实际问题。针对一维问题，本文主要研究了一维梁问题，构造了基于三次Hermite函数的一维三次Hermite小波梁单元，并通过数值算例对等截面梁弯曲问题和自由振动问题进行了分析，与理论解和ANSYS软件仿真解进行对比，求解的效果甚佳，精度较高。针对二维问题，基于一维三次Hermite插值尺度函数，运用张量积构造了二维三次Hermite提升多小波尺度函数。构造了四节点矩形单元，研究了二维薄板、斜板的弯曲问题和自由振动问题，与采用多项式插值的传统有限元法和大型有限元分析软件ANSYS的仿真解相比，本文方法简便快捷，且占用内存少，运算速度快，求解精度高。最后，针对工程实际问题，利用该方法结合ANSYS软件三维实体建模对单裂纹梁的定量故障诊断问题进行了研究，预测梁类结构中裂纹的存在并计算单裂纹梁的裂纹存在的位置和深度。对双裂纹梁的故障诊断问题进行了定性分析，，为梁类结构的多裂纹的早期故障诊断做了铺垫，也为进一步研究更为复杂的工程结构中的多裂纹问题及其它复杂问题提供了一种行之有效的方法。
[Abstract]:In recent decades, the second generation wavelet, which is developed on the basis of the first generation wavelet, inherits the multi-resolution characteristic of the first generation wavelet, but is no longer the expansion and translation of a given wavelet function, but is constructed by flexible lifting method. The new wavelet function with excellent characteristics can be obtained by improving the performance of the selected initial wavelet, thus satisfying the need of solving some special engineering problems. Multiwavelets have multiple scale functions and wavelet functions. The multi-wavelets based on lifting frame have high speed and high accuracy. In this paper, based on Hermite interpolation, lifting multiwavelets are constructed. The multi-wavelet theory and finite element method are combined to construct the displacement function of finite element method by using multi-wavelet function or scale function. The wavelet finite element formula is derived, which can easily solve the large gradient, singularity mutation and other engineering practical problems which are difficult to be solved by the traditional finite element method. Aiming at the one-dimensional problem, this paper mainly studies the one-dimensional beam problem, constructs the one-dimensional cubic Hermite wavelet beam element based on the cubic Hermite function, and analyzes the bending problem and the free vibration problem of the equal-section beam by numerical examples. Compared with the theoretical solution and ANSYS software simulation solution, the result of the solution is very good and the precision is high. Based on the one-dimensional cubic Hermite interpolation scaling function, a two-dimensional cubic Hermite lifting multiwavelet scaling function is constructed by using the tensor product. A four-node rectangular element is constructed, and the bending problem and free vibration problem of two-dimensional thin plate and inclined plate are studied. Compared with the traditional finite element method using polynomial interpolation and the simulation solution of ANSYS, the method in this paper is simple and fast. Moreover, it occupies less memory, has the advantages of fast operation speed and high accuracy. Finally, aiming at the practical engineering problems, the quantitative fault diagnosis problem of single crack beam is studied by using the method and 3D solid modeling of ANSYS software. The existence of cracks in beam structures is predicted and the location and depth of cracks in single crack beams are calculated. The fault diagnosis of double-crack beam is analyzed qualitatively, which lays the foundation for the early fault diagnosis of multi-crack beam structure. It also provides an effective method for further study of multi-crack problems and other complex problems in more complex engineering structures.
【学位授予单位】：西安建筑科技大学
【学位级别】：硕士
【学位授予年份】：2014
【分类号】：O174.2;TB115

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