二次特征值反問題的數(shù)值解法及其應用
發(fā)布時間：2019-01-12 12:39
【摘要】：代數(shù)特征值反問題的理論與方法是研究結構動力模型修正問題的主要方法之一。目前,如何同時保持結構矩陣的半正定性與稀疏性是結構動力模型修正問題中的一個重要研究課題。本文主要運用交替方向法與鄰近點方法,研究了二次特征值反問題,并討論了這些方法在阻尼振動系統(tǒng)、無阻尼陀螺結構系統(tǒng)的有限元模型修正中的應用,為代數(shù)特征值反問題以及有限元動力模型修正問題提供數(shù)學理論和有效的數(shù)值方法。本文主要包括如下內容:當質量矩陣為對角矩陣且足夠精確或固定時,基于不完備特征數(shù)據(jù),考慮了首一二次特征值反問題(MQIEP),要求修正的剛度矩陣、阻尼矩陣的對稱性、半正定性和稀疏性與初始系統(tǒng)保持一致。首先,利用約束條件的特殊結構,討論了MQIEP有解的條件。然后,將鄰近點方法與交替方向法結合,提出了一種求解MQIEP的乘子交替方向法,并給出該方法的收斂性分析。最后,將乘子交替方向法應用于帶阻尼振動系統(tǒng)的有限元模型修正問題,實驗結果表明該方法是可行的�；诓煌陚涮卣鲾�(shù)據(jù),考慮了結構化二次特征值反問題(SQIEP),要求修正的質量矩陣、阻尼矩陣與剛度矩陣的對稱性、半正定性和稀疏性與初始系統(tǒng)保持一致。首先,討論了SQIEP有解的條件。然后,利用拉格朗日函數(shù),給出SQIEP的單調變分不等式形式,提出了求解該不等式問題的定制鄰近點算法,并給出該算法的收斂性分析。最后,將該算法應用于阻尼振動系統(tǒng)的有限元模型修正問題,實驗結果表明該方法是可行的�；诓煌陚涮卣鲾�(shù)據(jù),考慮了無阻尼陀螺結構系統(tǒng)的結構化二次特征值反問題(GQIEP),要求修正的質量矩陣、陀螺矩陣與剛度矩陣的對稱性、反對稱性、半正定性以及稀疏性與初始系統(tǒng)保持一致。首先,討論了GQIEP有解的條件。然后利用約束條件的特殊結構,給出了求解GQIEP的定制鄰近點算法,并給出該算法的收斂性分析。實驗結果表明該算法是可行的。
[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.
【學位授予單位】：湖南大學
【學位級別】：博士
【學位授予年份】：2015
【分類號】：O241.6
本文編號：2407785
[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.
【學位授予單位】：湖南大學
【學位級別】：博士
【學位授予年份】：2015
【分類號】：O241.6
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