Random fields representation over manifolds via isometric feature mapping-based dimension reduction

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.

Automated summary

This paper proposes a two-stage strategy to map manifolds to 2D Euclidean space using Isometric feature mapping, allowing for the generation of random fields over irregular geometries. The stochastic harmonic function method is used to generate random fields, and case studies on stochastic finite element analysis of different structures are performed to demonstrate applicability and efficiency of the proposed method.