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A Galerkin isogeometric method for Karhunen-Loeve approximation of random fields

机译：随机场的Karhunen-Loeve逼近的Galerkin等几何方法

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This paper marks the debut of a Galerkin isogeometric method for solving a Fredholm integral eigenvalue problem, enabling random field discretization by means of the Karhunen-Loeve expansion. The method involves a Galerkin projection onto a finite-dimensional subspace of a Hilbert space, basis splines (B-splines) and non-uniform rational B-splines (NURBS) spanning the subspace, and standard methods of eigensolutions. Compared with the existing Galerkin methods, such as the finite-element and mesh-free methods, the NURBS-based isogeometric method upholds exact geometrical representation of the physical or computational domain and exploits regularity of basis functions delivering globally smooth eigensolutions. Therefore, the introduction of the isogeometric method for random field discretization is not only new; it also offers a few computational advantages over existing methods. In the big picture, the use of NURBS for random field discretization enriches the isogeometric paradigm. As a result, an uncertainty quantification pipeline of the future can be envisioned where geometric modeling, stress analysis, and stochastic simulation are all integrated using the same building blocks of NURBS. Three numerical examples, including a three-dimensional random field discretization problem, illustrate the accuracy and convergence properties of the isogeometric method for obtaining eigensolutions. (C) 2018 Elsevier B.V. All rights reserved.
机译：本文标志着用于解决Fredholm积分特征值问题的Galerkin等几何方法的首次亮相，该方法可通过Karhunen-Loeve展开实现随机场离散化。该方法涉及到在希尔伯特空间的有限维子空间上的Galerkin投影，跨越子空间的基础样条（B样条）和非均匀有理B样条（NURBS）以及本征解的标准方法。与现有的Galerkin方法（例如有限元方法和无网格方法）相比，基于NURBS的等几何方法可保持物理或计算域的精确几何表示，并利用基本函数的规律性来提供全局光滑的本征解。因此，引入等几何法进行随机场离散化不仅是新的，而且还具有许多优点。与现有方法相比，它还提供了一些计算优势。总体而言，将NURBS用于随机场离散化可以丰富等几何范式。结果，可以预见未来的不确定性量化流水线，其中使用相同的NURBS构建模块将几何建模，应力分析和随机模拟都集成在一起。三个数值示例，包括三维随机场离散问题，说明了用于获得特征解的等几何方法的准确性和收敛性。 （C）2018 Elsevier B.V.保留所有权利。

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