# 二次特征值反问题的数值解法及其应用

【摘要】：代数特征值反问题的理论与方法是研究结构动力模型修正问题的主要方法之一。目前,如何同时保持结构矩阵的半正定性与稀疏性是结构动力模型修正问题中的一个重要研究课题。本文主要运用交替方向法与邻近点方法,研究了二次特征值反问题,并讨论了这些方法在阻尼振动系统、无阻尼陀螺结构系统的有限元模型修正中的应用,为代数特征值反问题以及有限元动力模型修正问题提供数学理论和有效的数值方法。本文主要包括如下内容:当质量矩阵为对角矩阵且足够精确或固定时,基于不完备特征数据,考虑了首一二次特征值反问题(MQIEP),要求修正的刚度矩阵、阻尼矩阵的对称性、半正定性和稀疏性与初始系统保持一致。首先,利用约束条件的特殊结构,讨论了MQIEP有解的条件。然后,将邻近点方法与交替方向法结合,提出了一种求解MQIEP的乘子交替方向法,并给出该方法的收敛性分析。最后,将乘子交替方向法应用于带阻尼振动系统的有限元模型修正问题,实验结果表明该方法是可行的。基于不完备特征数据,考虑了结构化二次特征值反问题(SQIEP),要求修正的质量矩阵、阻尼矩阵与刚度矩阵的对称性、半正定性和稀疏性与初始系统保持一致。首先,讨论了SQIEP有解的条件。然后,利用拉格朗日函数,给出SQIEP的单调变分不等式形式,提出了求解该不等式问题的定制邻近点算法,并给出该算法的收敛性分析。最后,将该算法应用于阻尼振动系统的有限元模型修正问题,实验结果表明该方法是可行的。基于不完备特征数据,考虑了无阻尼陀螺结构系统的结构化二次特征值反问题(GQIEP),要求修正的质量矩阵、陀螺矩阵与刚度矩阵的对称性、反对称性、半正定性以及稀疏性与初始系统保持一致。首先,讨论了GQIEP有解的条件。然后利用约束条件的特殊结构,给出了求解GQIEP的定制邻近点算法,并给出该算法的收敛性分析。实验结果表明该算法是可行的。
[Abstract]:The theory and method of algebraic inverse eigenvalue problem is one of the main methods to study the problem of structural dynamic model modification. At present, how to maintain the positive semidefinite and sparsity of structural matrix simultaneously is an important research topic in the problem of structural dynamic model modification. In this paper, the inverse problem of quadratic eigenvalue is studied by means of alternating direction method and adjacent point method, and the application of these methods in the finite element model modification of damping vibration system and undamped gyroscope structure system is discussed. It provides mathematical theory and effective numerical method for algebraic inverse eigenvalue problem and finite element dynamic model modification problem. The main contents of this paper are as follows: when the mass matrix is diagonal matrix and sufficiently accurate or fixed, based on incomplete characteristic data, the stiffness matrix and the symmetry of damping matrix, which are required by the inverse problem of first-order eigenvalue (MQIEP), are considered. The semi-positive definiteness and sparsity are consistent with the initial system. Firstly, by using the special structure of constraint conditions, we discuss the conditions under which MQIEP has solutions. Then, by combining the adjacent point method with the alternating direction method, a multiplier alternating direction method for solving MQIEP is proposed, and the convergence analysis of the method is given. Finally, the multiplier alternating direction method is applied to the finite element model modification problem of damped vibration system. The experimental results show that the method is feasible. Based on incomplete characteristic data, the modified mass matrix required by (SQIEP), for the inverse problem of structured quadratic eigenvalue is considered. The symmetry, semi-positive definiteness and sparsity of damping matrix and stiffness matrix are consistent with the initial system. First, we discuss the conditions under which SQIEP has solutions. Then, using Lagrangian function, we give the form of SQIEP's monotone variational inequality, propose a custom adjacent point algorithm for solving the inequality problem, and give the convergence analysis of the algorithm. Finally, the algorithm is applied to the finite element model modification problem of damped vibration system. The experimental results show that the method is feasible. Based on incomplete characteristic data, the inverse problem of structured quadratic eigenvalue of undamped gyroscope system is considered. The mass matrix, symmetry and antisymmetry of gyroscope matrix and stiffness matrix, which are required by (GQIEP), are modified. The positive semidefinite and sparsity are consistent with the initial system. First, we discuss the conditions under which GQIEP has solutions. Then, by using the special structure of the constraint conditions, a custom neighborhood algorithm for solving GQIEP is presented, and the convergence analysis of the algorithm is given. Experimental results show that the algorithm is feasible.
【学位授予单位】：湖南大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O241.6

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