矩阵束最佳逼近问题的数值算法
发布时间：2019-04-04 15:53
【摘要】：矩阵束的最佳逼近问题出现在结构系统的结构修改和有限元模型校正等领域。本文研究无阻尼结构系统同时修正有限元模型质量矩阵和刚度矩阵所导出的矩阵束最佳逼近问题。这类问题以矩阵束修正量的F-范数为目标函数,并以待修正矩阵束应具有的性质如满足特征方程、对称半正定性和稀疏性作为约束条件,形成带约束的优化问题,所取得的主要结果如下:利用矩阵的正逼近和矩阵对的标准相关分解,提出了求解矩阵束最佳逼近问题的交替投影法,说明了算法的收敛性;并应用松弛技术对该方法进行加速,给出了松弛交替投影法。基于部分Lagrange乘子法,把矩阵束最佳逼近问题转化为一个等价的线性矩阵变分不等式,将邻近点方法应用于求解线性矩阵变分不等式,导出了求解矩阵束最佳逼近问题的近似邻近点方法,分析了该方法的收敛性,并证明了该方法全局收敛且有O(1/t)的线性收敛速度(t为迭代步数)。结合交替投影和邻近点方法的优点,提出了求解矩阵束最佳逼近问题的APM-PPA方法,并说明了其收敛性。数值结果说明了理论结果的正确性和数值算法的有效性。
[Abstract]:The problem of optimal approximation of matrix beams occurs in the fields of structural modification and finite element model correction of structural systems. In this paper, the optimal approximation problem of matrix beams derived from the mass matrix and stiffness matrix of the finite element model is studied for undamped structural systems. This kind of problem takes the F-norm of the matrix bundle correction as the objective function, and takes the properties of the matrix bundle to be modified, such as satisfying the characteristic equation, symmetric semi-positive definiteness and sparsity as the constraint conditions, to form the constrained optimization problem. The main results obtained are as follows: by using the positive approximation of matrix and the standard correlation decomposition of matrix pairs, an alternating projection method is proposed to solve the optimal approximation problem of matrix bundles, and the convergence of the algorithm is illustrated. The relaxation technique is applied to accelerate the method, and the relaxation alternating projection method is given. Based on the partial Lagrange multiplier method, the optimal approximation problem of matrix bundle is transformed into an equivalent linear matrix variational inequality, and the adjacent point method is applied to solve the linear matrix variational inequality. The approximate neighbor point method for solving the optimal approximation problem of matrix bundles is derived. The convergence of the method is analyzed. It is proved that the method converges globally and has a linear convergence rate of O (1 t) (t is the iterative step). Combining the advantages of alternating projection and adjacent point method, the APM-PPA method for solving the optimal approximation problem of matrix bundles is proposed, and its convergence is explained. The numerical results show the correctness of the theoretical results and the effectiveness of the numerical algorithm.
【学位授予单位】：南京航空航天大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.6
本文编号：2453959
[Abstract]:The problem of optimal approximation of matrix beams occurs in the fields of structural modification and finite element model correction of structural systems. In this paper, the optimal approximation problem of matrix beams derived from the mass matrix and stiffness matrix of the finite element model is studied for undamped structural systems. This kind of problem takes the F-norm of the matrix bundle correction as the objective function, and takes the properties of the matrix bundle to be modified, such as satisfying the characteristic equation, symmetric semi-positive definiteness and sparsity as the constraint conditions, to form the constrained optimization problem. The main results obtained are as follows: by using the positive approximation of matrix and the standard correlation decomposition of matrix pairs, an alternating projection method is proposed to solve the optimal approximation problem of matrix bundles, and the convergence of the algorithm is illustrated. The relaxation technique is applied to accelerate the method, and the relaxation alternating projection method is given. Based on the partial Lagrange multiplier method, the optimal approximation problem of matrix bundle is transformed into an equivalent linear matrix variational inequality, and the adjacent point method is applied to solve the linear matrix variational inequality. The approximate neighbor point method for solving the optimal approximation problem of matrix bundles is derived. The convergence of the method is analyzed. It is proved that the method converges globally and has a linear convergence rate of O (1 t) (t is the iterative step). Combining the advantages of alternating projection and adjacent point method, the APM-PPA method for solving the optimal approximation problem of matrix bundles is proposed, and its convergence is explained. The numerical results show the correctness of the theoretical results and the effectiveness of the numerical algorithm.
【学位授予单位】：南京航空航天大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.6
【参考文献】
相关期刊论文 前3条
1 何炳生;申远;;求解凸规划及鞍点问题定制的PPA算法及其收敛速率[J];中国科学:数学;2012年05期
2 戴华;用试验数据修正刚度矩阵[J];航空学报;1994年09期
3 戴华;谱约束下实对称矩阵束的最隹逼近[J];高等学校计算数学学报;1990年02期
，本文编号：2453959
本文链接：https://www.wllwen.com/kejilunwen/yysx/2453959.html
