杂交元本征应力模式和应力子空间的性质研究

详细讨论了有限元本征应力模式和应力子空间的性质，并着重讨论和进一步完善了与杂交应力有限元应力子空间有关的一些定理，为提出新方法提供了理论基础，主要包括：（1）证明了杂交元特征值不大于对应位移元的特征值；（2）证明了矩阵H非奇异的充分必要条件是假设应力模式线性无关；（3）证明了杂交元所对应位移元的本征应力模式形成的杂交元与该位移元相同；（4）证明了等价假设应力模式形成相同的杂交元；（5）证明了确定杂交元本征应力模式的充分必要条件是其范数平方等于所形成杂交元的变形模态特征值；（6）证明了杂交元假设应力模式与变形模态的能量一一对应的充分必要条件是假设应力模式彼此正交且与所对应位移元的本征应力模式除了一一对应者之外都正交。

A Characteristic Study of Natural Stress Modes and Stress Subspace in Hybrid Finite Element Method

The characteristics of natural stress modes and stress subspace in finite element were discussed in detail. The stress subspace of hybrid finite element was focused on and some of the theorems were developed and improved so that the new methods can be put forward. The main results are as follows. (1) The eigenvalues of hybrid element cannot be larger than those of the displacement counterpart. (2) The linear independence of the assumed stress modes is the sufficient and necessary condition for the nonsingular matrix H. (3) The hybrid element is the same as its displacement counterpart when the assumed stress modes are the natural stress modes from the displacement counterpart. (4) The same hybrid elements are established with the equivalent assumed stress modes. (5) The sufficient and necessary condition for the identification of the natural stress modes is that their norm squares are equal to the deformation eigenvalues. (6) The sufficient and necssary condition for the one-one correspondence of the assumed stress modes to the deformation modal energy of the hybrid element is that they are orthogonal to each other and to the natural stress modes of the displacement counterpart except their one-one correspondents.