【作者】 林福泳；

【导师】 王太勇；

【作者基本信息】 天津大学 ， 机械制造及其自动化， 2003， 博士

【摘要】 正交函数由于其相互正交使得计算量大大地减少，历来为人们所关注，在连续函数中有著名的调和函数（三角函数和三角级数），离散分析中有正弦变换和余弦变换，半离散分析近年来有正交小波函数。它们都有良好的正交性，对于半离散分析除了正交小波外是否还有更好性质的正交有限元函数。本文应用群（变换群）的正则表示构造群上空间，利用自共轭算子作用群上空间。用求特征向量的方法求出该自共轭算子的特征向量，由于自共轭算子的特征向量是相互正交的，因此所求的特征向量将群上空间分解为正交的子空间。找出一般周期空间的对称变换群，并将已知的有限元的单元基函数转化为节点基函数，将变换群的特征子空间作用有限元的节点基函数得出一系列的正交有限元函数。这些函数是相互正交或是某几个函数与另外几个函数正交。将所构造的函数用于对称结构分析，这样问题只要在正交子空间中求解，因此、大大降低了计算量。应用迦辽金方法给出一般微分方程的求解方法，与传统的求解方法相比计算量可以降一维。由于所构造的有限元是相互正交的，除了计算量降低外还有一个好处就是可以很容易实现并行运算。本文给出板弯曲问题的具体例子，计算结果表明应用正交有限元法有良好的精度并且计算量小。利用合成群列给出粗剖分有限元和细剖有限元逼近的递推关系，并将其应用于信号分析与图像处理，结果表明所提出的方法具有良好的实用性，它具有小波分析的局域性又有余弦变换的良好的滤波性能。更多还原

【Abstract】 A series of orthogonal finite element function is presented by group method and finite element method. In continuous analysis there are famous function called consonance function ( the sine and cosine function), for disperse analysis there are sinusoid transform and cosine transform to be used, in continuous-disperse analysis recently there is wavelet transform to be applied to. Is there still other function can be used in continuous-disperse analysis? The paper first used normal representation theory of group to construct a group space and used self-cognation operator to get eigenvector of the operator, the eigenvector is orthogonal to each other. From the existed finite element the node bases of element is constructed. Consider symmetry of periodic area we get the symmetry transform group, act the eigenvector of self-cognation operator on the node bases element function, a series of orthogonal function is constructed. The functions is applied to make analysis of symmetry structure. By applied Γалёркин method the function is used to solute partial differential equation, the method greatly reduce the computation. An example is presented in the paper, the result show the accuracy of this method. According to the rudely divided element space included in the finely divided element space, the recurrence equation of relation between the two element space image is also presented in the paper. Used the recurrence equation to proceed signal, the result show the potential application of proposed method. The orthogonal finite element method have both advantage of wavelet analysis and cosine transform.更多还原