Multimodal ellipsoid model for non-probabilistic structural uncertainty quantification and propagation

Jie Liu,Zhongbo Yu,Dequan Zhang,Xu Han,Hao Liu

doi:10.1007/s10999-021-09551-z

Abstract

The traditional ellipsoid convex set is a kind of basic non-probabilistic model to measure uncertainties. However, it is difficult or inaccurate to quantify the uncertainties of variables with multimodal distributed samples. In this paper, a more generalized non-probabilistic ellipsoid model named multimodal ellipsoid model is proposed to effectively deal with the multimodal distributed samples. The samples with one or more similar properties are clustered together, and the principal directions of the samples and characteristic matrix are appropriately found through the Gaussian mixture model. Then, the multimodal ellipsoid model can be constructed by using the elliptical contour features of the Gaussian model to measure the uncertainties of variables. The proposed multimodal ellipsoid model can not only establish traditional ellipsoid model, but also establish multi-ellipsoid model for uncertain variables with multimodal samples. Furthermore, combining with the multimodal ellipsoid model and performance measure approach, the uncertain propagation results of system are obtained accurately. Three numerical examples and one engineering application are provided to demonstrate the effectiveness and accuracy of the proposed multimodal ellipsoid model.

Highlights

There are various inherent uncertainties in practical structures, which are commonly associated with material properties, loads, manufacturing errors and boundary conditions, etc
The uncertainty modeling approaches based on the non-probabilistic convex set theory have achieved considerable developments, most of the above-mentioned ellipsoid models only use single convex model to surround the limited samples of uncertain variables as well as a lack of consideration regarding on the multimodal status of samples
We find that the characteristic matrix of the ellipsoid model in Eq (6) has the same formula with exponential term of the Gaussian mixture model (GMM), so it is promising to introduce it for the ellipsoid modeling

Summary

Introduction

There are various inherent uncertainties in practical structures, which are commonly associated with material properties, loads, manufacturing errors and boundary conditions, etc. Using a quantified measure for nonprobabilistic reliability based on the ellipsoid convex model, a topology optimization of continuum structures (Luo et al 2009) was investigated considering uncertain-but-bounded variables. The uncertainty modeling approaches based on the non-probabilistic convex set theory have achieved considerable developments, most of the above-mentioned ellipsoid models only use single convex model to surround the limited samples of uncertain variables as well as a lack of consideration regarding on the multimodal status of samples. The proposed MEM is a more generalized nonprobabilistic ellipsoid model for uncertainty measurement, which can realize the traditional single ellipsoid modeling, and can effectively establish a multi-ellipsoid for the uncertain variables with multimodal samples. If the dimension of uncertain variables and sample numbers are large enough, it seems hard to establish a minimum volume model (MVM) for uncertain variables

Correlation approximation method

Gaussian clustering

Non-probabilistic uncertainty propagation

Numerical example 2

Numerical example 3

Conclusions