【作者】 孙亮；

【导师】 吴柏生；

【作者基本信息】 吉林大学 ， 固体力学， 2004， 硕士

【摘要】 随着科学技术的发展，人们对工程结构的要求越来越高，有大批复杂结构需要进行优化设计。在这种优化设计中，为了获得令人满意的性能，往往需要对结构进行几十次甚至上百次的修改，即需要反复进行修改设计—再分析—修改设计的过程，所以计算成本是相当大的。为了减少计算成本，以不直接求解结构修改后的隐式方程，而根据原始结构的计算结果，高效、高精度为目标的重分析方法，日益受到人们的重视，并得到飞速的发展。本文用向量值有理逼近方法研究结构动力重分析问题，主要做了以下两方面的工作:一、将孤立特征值的矩阵摄动方法用MATLAB程序实现。我们知道，借助幂级数来研究函数或向量的性质或直接用它的部分和逼近该函数或向量，不仅是纯数学领域中经常使用的手段，也是数值计算中非常有效的方法，矩阵摄动方法就是很好的例子。二、但有时矩阵摄动方法的应用显露出某些缺陷，主要为收敛速度较慢和收敛半径较窄，不适合表示变化较大的情况。如果采用有理函数作为逼近工具，不仅能改善逼近精度，而且还能扩大其逼近范围。为提高矩阵摄动方法的精度，本文提出了向量值有理逼近方法。即用向量值有理逼近方法对结构进行动力重分析，改进矩阵摄动方法的精度。向量值有理逼近方法的步骤如下: <WP=33>1、利用原始结构的精确主振型和特征值计算出各阶摄动的振型列向量和特征值：， ， ；计算出摄动解； (1) (2) 3、应用向量值有理逼近方法计算出振型列向量； (3) 式中的为截断幂级数法的第个振型的前项部分和 (4) 方程（3）中的是标量，满足如下线性方程组 (5) 式中的是内积 . (6)方程（5）是如下最小二乘问题的解所满足的方程 式中 表示为欧氏模。 <WP=34>数值例子表明，同样是利用幂级数的系数向量，利用向量值有理逼近方法比单纯利用幂级数求和的摄动方法的精度高很多，因此向量值有理逼近方法是一种对结构进行动力重分析的高效算法。更多还原

【Abstract】 With the development of science and technology, many large-complicated structures need to be designed. In structural design or optimization, the procedures are generally iterative and require repeated analysis as the structures are progressively modified. Each resign involves extensive calculations. This difficulty motivates extensive studies on reanalysis techniques. The object of reanalysis is to evaluate the structural response for successive modifications in the design without solving the set of the modified implicit equations so that the computational cost is significantly reduced. In order to avoid a fresh analysis after each modification, many reanalysis techniques have been devised. In this thesis, a vector-valued rational approximate method in structural dynamic reanalysis problems is developed. The thesis is composed of two aspects. In the first part, we program the perturbation method for the matrix eigenvalue problems by using MATLAB language. It is well known that if a power series of converges slowly or has a finite radius of convergence, it is not suitable to represent the function in a large range of . Therefore, one need to develop efficient summation methods to achieve, or improve the convergence of power series. In the second part of the thesis, we present a vector-valued rational approximate method. This method is based on combining the power-series expansion with a rational approximation. This method provides high quality approximations of the structural behavior for large changes in the design variables. The vector-valued rational approximate method is as follows: <WP=36>Compute the perturbative vibration modes and eigenvalues of modified structure by using vibration modes and eigenvalues of original structure: ， ， 2、Compute matrix perturbation results: (1) (2)3、Compute vibration modes of modified structure by applying vector-valued rational approximate method: (3)where is the partial sum of truncated power series by using the first coefficients (4)Here, are scalars and satisfy a linear system of equations of the form (5) where are defined as the dinner product (6) <WP=37>The equations in (5) involving are typical equation in least-squares problems of the following form where denotes the norm. Numerical examples have shown that high quality approximations of the structural behavior for large changes in the design variables can be achieved. In addition, the overall computational effort is considerably reduced. Therefore it is an efficient method to solve structural dynamic reanalysis problems.更多还原