Generating random fields over irregular geometries (e.g., two-dimensional (2D) manifolds embedded in the three-dimensional (3D) Euclidean space) is a great challenge because the geometry structure is complex and the correlation function can hardly be derived, thus the traditional methods, for example, spatial discretization methods or series expansion methods, cannot be directly adopted. To solve this issue, the present paper develops a two-stage strategy to simulate random fields over manifolds. The core idea is to map the manifolds into the 2D Euclidean space through Isometric feature mapping (Isomap), with which the geodesic distance between points in the mapped 2D Euclidean space and the original manifold space is kept as the same. Therefore, the correlation function can be readily derived and the conventional methods can be directly employed to generate random fields. To validate the proposed method, several different types of manifolds are dimensionally reduced to 2D planar domains, and the stochastic harmonic function method is used to generate the random fields. Finally, case studies on the stochastic finite element analysis of two different structures are also performed to demonstrate the applicability and efficiency of the proposed method.

中文翻译：

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通过基于等距特征映射的降维在流形上表示随机场

在不规则几何（例如，嵌入三维（3D）欧几里得空间中的二维（2D）流形）上生成随机场是一个巨大的挑战，因为几何结构复杂且难以导出相关函数，因此传统的不能直接采用空间离散化方法或级数展开法等方法。为了解决这个问题，本文开发了一种两阶段策略来模拟流形上的随机场。其核心思想是通过等距特征映射（Isomap）将流形映射到二维欧几里得空间，映射的二维欧几里得空间中的点与原始流形空间中的点之间的测地线距离保持不变。所以，相关函数可以很容易地导出，并且可以直接使用常规方法来生成随机场。为了验证所提出的方法，几种不同类型的流形被降维为二维平面域，并使用随机调和函数方法生成随机场。最后，还对两种不同结构的随机有限元分析进行了案例研究，以证明所提出方法的适用性和效率。