Numerical methods for the discretization of random fields by means of the Karhunen-Loeve expansion
Numerical methods for the discretization of random fields by means of the Karhunen-Loeve expansion
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DOI：
10.1016/j.cma.2013.12.010
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发表时间：
2014-04-01
影响因子：
7.2
通讯作者：
Straub, Daniel
中科院分区：
文献类型：
--
作者：
Betz, Wolfgang;Papaioannou, Iason;Straub, Daniel
随机场表示与Karhunen-Loeve（KL）展开的计算效率依赖于Fredholm积分特征值问题的解决方案。本文比较了解决该问题的不同方法。重点放在适用于任意形状的域和任意自协方差函数的方法。这些方法包括Nystrom方法以及配置和Galerkin投影方法。在Galerkin方法中，我们研究了有限元方法，并提出了有限单元法的应用。这种方法是基于扩展的有限元，但避免网格生成域的复杂几何形状。FCM最初是在Parvizian等人（2007）[17]中提出的，用于解决椭圆边值问题。作为Galerkin方法中协方差函数的L-2投影的替代方案，本文研究了H-1/2投影和离散投影。结果表明，Li和Der Kiureghian（1993）[18]提出的扩展最优线性估计（EOLE）方法是Nystrom方法的一个特例。结果表明，EOLE方法对KL展开式的数值解是最有效的。FEM和FCM在评估随机场的实现方面比EOLE方法更有效，并且因此适用于其中在评估随机场实现中花费的时间对总体运行时间具有主要贡献的问题-例如，有限元可靠性分析。(C)2013爱思唯尔有限公司版权所有。
The computational efficiency of random field representations with the Karhunen-Loeve (KL) expansion relies on the solution of a Fredholm integral eigenvalue problem. This contribution compares different methods that solve this problem. Focus is put on methods that apply to arbitrary shaped domains and arbitrary autocovariance functions. These include the Nystrom method as well as collocation and Galerkin projection methods. Among the Galerkin methods, we investigate the finite element method (FEM) and propose the application of the finite cell method (FCM). This method is based on an extension to the FEM but avoids mesh generation on domains of complex geometric shape. The FCM was originally presented in Parvizian et al. (2007) [17] for the solution of elliptic boundary value problems. As an alternative to the L-2-projection of the covariance function used in the Galerkin method, H-1/2-projection and discrete projection are investigated. It is shown that the expansion optimal linear estimation (EOLE) method proposed in Li and Der Kiureghian (1993) [18] constitutes a special case of the Nystrom method. It is found that the EOLE method is most efficient for the numerical solution of the KL expansion. The FEM and the FCM are more efficient than the EOLE method in evaluating a realization of the random field and, therefore, are suitable for problems in which the time spent in the evaluation of random field realizations has a major contribution to the overall runtime - e.g., in finite element reliability analysis. (C) 2013 Elsevier B.V. All rights reserved.