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

Abstract

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.

Three‐stage non‐Gaussian homogeneous random field representation over manifolds

“…According to Equation (), the distance error is $$e_{D} = 0.87\%$$ for the Canopy shell. This result is consistent with the investigation (Feng et al., 2021) that the distance error is less than 1% for general manifolds.…”

“…For the first part, the error may come from the first and third stages. In the first stage, the correlation distance may not be preserved completely for undeveloped manifolds, especially for the Hemisphere manifold (Feng et al., 2021). Then, the simulated ACF and the target ACF may be with the same random values for different distances, that is, the same ACF value for different g .…”

“…The basic information of the full cylinder and the FE model of the half‐cylinder are shown in Figure 7. In the FE model, a symmetric boundary condition is assigned according to the study (Feng et al., 2021), the shell element S4R is adopted with about 5 mm mesh size, and the uniform pressure on the top edge is loaded by the shell edge load 1 N/mm. By the buckling analysis of the FE model, the axial buckling load is 5354 N. Although only a half‐cylinder is adopted, the buckling load is almost equal to 5350 N obtained from the full cylinder.…”

“…According to the proposed method, the original geometry is first reduced to a two‐dimensional Euclidean domain. The distance error related to Isomap can be evaluated by the residual variance (Feng et al., 2021), which is expressed as $$$$\begin{equation} {e}_{D}=1-{C}_{r}^{2}({D}_{G}-{D}_{\hat{X}}) \end{equation}$$$$where $$C_r$$ is the standard linear correlation coefficient, taken over all elements of the matrices; $${D}_{G}=[{g}_{\textit{ij}}]$$ and $${D}_{\hat{X}}=[{d}_{\textit{ij}}]$$ are the geodesic distance matrix and the Euclidean distance matrix of the two‐dimensional domain, respectively.…”

“…With the rapid development of machine learning techniques and artificial intelligence (Adeli, 2001; Alam et al., 2019; Chen et al., 2021; Dong et al., 2021; Pereira et al., 2019; Rafiei & Adeli, 2017b), more and more machine learning techniques are used in civil engineering applications (Jiang & Adeli, 2005; Luo & Kareem, 2019; Panakkat & Adeli, 2009; Rafiei & Adeli, 2017c; Rafiei et al., 2017; Rafiei & Adeli, 2017a, 2018). Based on the manifold learning method, a two‐stage method (Feng et al., 2021) was proposed by authors to simulate Gaussian random fields over manifolds, and it has been used in the stochastic nonlinear analysis of an irregular wall (Liang et al., 2021). In this method, a typical manifold learning method, the isometric feature mapping (Isomap) (Tenenbaum et al., 2000), is adopted to map the manifold to a regular Euclidean domain.…”