【摘要】：矩陣束的最佳逼近問題出現(xiàn)在結(jié)構(gòu)系統(tǒng)的結(jié)構(gòu)修改和有限元模型校正等領域。本文研究無阻尼結(jié)構(gòu)系統(tǒng)同時修正有限元模型質(zhì)量矩陣和剛度矩陣所導出的矩陣束最佳逼近問題。這類問題以矩陣束修正量的F-范數(shù)為目標函數(shù),并以待修正矩陣束應具有的性質(zhì)如滿足特征方程、對稱半正定性和稀疏性作為約束條件,形成帶約束的優(yōu)化問題,所取得的主要結(jié)果如下:利用矩陣的正逼近和矩陣對的標準相關分解,提出了求解矩陣束最佳逼近問題的交替投影法,說明了算法的收斂性;并應用松弛技術對該方法進行加速,給出了松弛交替投影法�；诓糠諰agrange乘子法,把矩陣束最佳逼近問題轉(zhuǎn)化為一個等價的線性矩陣變分不等式,將鄰近點方法應用于求解線性矩陣變分不等式,導出了求解矩陣束最佳逼近問題的近似鄰近點方法,分析了該方法的收斂性,并證明了該方法全局收斂且有O(1/t)的線性收斂速度(t為迭代步數(shù))。結(jié)合交替投影和鄰近點方法的優(yōu)點,提出了求解矩陣束最佳逼近問題的APM-PPA方法,并說明了其收斂性。數(shù)值結(jié)果說明了理論結(jié)果的正確性和數(shù)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.
【學位授予單位】：南京航空航天大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O241.6

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本文編號：2453959