【作者】 李素华；

【导师】 李耀堂；

【作者基本信息】 云南大学 ， 计算数学， 2020， 博士

【摘要】 张量在科学和工程领域具有许多应用.作为一类特殊的张量,四阶部分对称张量在弹性力学中弹性材料的强椭圆性的判定中起着重要的作用.2009年,文[L.Q.Qi,H.H.Dai,D.R.Han.Conditions for strong ellipticity and M-eigenvalues.Frontiers of Mathematics in China,4(2009)349-364]借助四阶部分对称张量及其M-特征值给出了弹性材料具有强椭圆性的一个判定条件.然而,当张量的维数大于3时,该条件的验证是困难的.本文在对四阶部分对称张量M-特征值深入研究的基础上,围绕弹性材料强椭圆性条件的判定问题开展研究,主要做了如下四方面工作.首先,应用矩阵和张量特征值定位理论给出了四阶部分对称张量M-特征值的几个包含区间,再将这些区间应用于四阶部分对称张量M-谱半径的估计,得到了其M-谱半径的几个界;第二,应用张量的矩阵展开技术和对两类具有特殊结构四阶部分对称张量M-特征值的研究给出了具强椭圆性四阶部分对称张量几个充分条件,然后,将这些条件算法化,得到了具强椭圆性四阶部分对称张量的几个判定算法,数值例子表明所获判定算法是有效的;第三,研究了四阶部分对称张量的扰动问题,通过引入四阶部分对称张量ε-伪-M-谱的概念并讨论其性质和定位,给出了四阶部分对称张量保持强椭圆性的扰动界,文中数值例子说明所获结果是有意义的;最后,将四阶部分对称张量及其M-特征值的概念推广到任意偶数阶部分对称张量,并讨论了其相关性质.更多还原

【Abstract】 Tensors have many applications in science and engineering.Fourth-order partially symmetric tensors,as a special kind of tensor,play an important role in identifying the strong ellipticity of elastic materials in elasticity.In 2009,a criterion for identifying the strong ellipticity of elastic materials is given by using the fourth-order partially symmetric tensors and its M-eigenvalues in [L.Q.Qi,H.H.Dai,D.R.Han.Conditions for strong ellipticity and M-eigenvalues.Frontiers of Mathematics in China,4(2009)349-364].However,it is difficult to verify this condition when the dimension of the tensor is greater than 3.Based on the further study for M-eigenvalues of fourth-order partially symmetric tensors,in this dissertation,we focus on the identification of the condition of strong ellipticity of elastic materials and do the following four aspect works.Firstly,by using the eigenvalue localization theory of matrices and tensors,several inclusion intervals for the M-eigenvalues of fourth-order partially symmetric tensors are given,then these intervals are used to estimate the M-spectral radius of fourth-order partially symmetric tensors,and some bounds for the M-spectral radius are obtained;secondly,by using the matrix unfolding technique of tensor and studying the M-eigenvalues of two classes of fourth-order partially symmetric tensors with special structures,some sufficient conditions for the fourth-order partially symmetric tensor with strong ellipticity are given,and by algorithmizing these conditions,several judgment algorithms for the fourth-order partially symmetric tensor with strong ellipticity are gained,furthermore,numerical examples show that the proposed algorithms are effective;thirdly,the perturbation problem of fourth-order partially symmetric tensor is studied,by presenting the definition of ε-pseudo-M-spectrum of fourth-order partially symmetric tensors and discussing its properties and localization,the perturbation bounds for fourth-order partially symmetric tensor to maintain strong ellipticity are given,and numerical examples show that the proposed results are meaningful.Finally,we generalize the concepts of the fourth-order partially symmetric tensor and its M-eigenvalue to any even-order partially symmetric tensor,and the related properties are discussed.更多还原