| 一种基于Neumann级数的区间有限元方法 |
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| 引用本文： | 伍鹏革,倪冰雨,姜潮.一种基于Neumann级数的区间有限元方法[J].力学学报,2020,52(5):1431-1442. |
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| 作者姓名： | 伍鹏革 倪冰雨 姜潮 |
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| 作者单位： | 湖南大学机械与运载工程学院, 长沙410006 |
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| 基金项目： | 国家自然科学基金;国家自然科学基金 |
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| 摘 要： | 实际工程问题中通常存在大量的不确定参数, 区间有限元方法是一种结合有限元数值计算工具对结构进行不确定性分析的区间方法. 区间有限元的目的是获得在含有区间不确定性参数条件下的结构响应上下边界, 其关键问题在于区间平衡方程组的求解, 而这属于一类往往很难求解的NP-hard问题. 本文归纳了一类工程实际中常见的结构不确定性问题, 即可线性分解式区间有限元问题, 并针对此提出一种基于Neumann级数的区间有限元方法. 在区间有限元分析中, 当区间不确定参数表示为一组独立区间变量线性叠加时, 若结构的刚度矩阵也可表示为这些独立区间变量的线性叠加形式, 则称此类区间有限元问题为可线性分解式区间有限元问题. 对于此类问题, 采用Neumann级数对其刚度矩阵的逆矩阵进行表示, 可获得结构响应关于区间变量的显式表达式, 从而可高效求解结构响应的上下边界. 最后通过两个算例验证了本文所提方法的有效性.
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| 关 键 词： | 区间有限元分析 线性分解式 Neumann级数 上下边界 |
| 收稿时间： | 2020-05-08 |
AN INTERVAL FINITE ELEMENT METHOD BASED ON THE NEUMANN SERIES EXPANSION |
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| Affiliation: | College of Mechanical and Vehicle Engineering, Hunan University, Changsha 410006, China |
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| Abstract: | Uncertainty is common in the practical engineering. The interval finite element method is an interval method which introduces the numerical computational method of finite element to structural uncertainty analysis. The aim of the interval finite element analysis is to obtain the upper and lower response bounds of the structure with interval uncertain parameters, where solving the interval finite element equilibrium equations is the key issue. But the solution of interval linear equations belongs to a class of NP-hard problems which are often difficult to solve. This paper classifies and defines a type of linearly decomposable interval finite element problems, which exist commonly in practical engineering. To solve this type of problems, an interval finite element method based on Neumann series is proposed. It is named as the linearly decomposable interval finite element problem if the stiffness matrix in the interval finite element analysis formulation can be expressed as a linear superposition of a set of independent interval variables when the interval uncertain parameter is expressed as a linear superposition form of the independent interval variables. For this kind of problems, the inverse of the stiffness matrix can be represented by its Neumann series expansion. Thus the explicit expressions of structural responses with interval variables can be then obtained, with which the upper and lower bounds of the structural response can be solved efficiently. Finally, two numerical examples show the effectiveness and accuracy of the proposed method. |
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