有限实验数据下的复合材料属性随机识别与不确定性建模
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"小样本数据处理在复合材料特性识别中的应用：不确定性建模"
这篇论文主要探讨了在有限的实验数据库中，如何通过随机识别方法来确定复合材料的宏观性质，特别是面对小样本数据时的不确定性建模问题。作者包括Loujaine Mehrez、Alireza Doostan、David Mossa和Dirk Vandepitte，他们分别来自比利时鲁汶大学机械工程学院和美国科罗拉多大学航空航天工程科学系。
文章的关键点包括：
1. **复合材料**：复合材料由两种或多种不同的材料组成，具有独特的性能，如高强度、轻量化等。它们在航空、航天、汽车和建筑等领域有着广泛应用。
2. **随机识别（Stochastic Identification）**：这是一种统计方法，用于从有限的数据中推断出材料属性的随机性或不确定性。在本文中，它被用来估计复合材料的宏观力学性质，如杨氏模量，这些性质可能因材料的不均匀性而变化。
3. **有限实验数据库**：由于实验成本高、时间长或者技术限制，获取大量实验数据通常是困难的，因此必须在小样本数据集上进行分析和建模。
4. **Karhunen-Loève变换（KLT）**：这是一种统计工具，常用于降维和特征提取，特别适合处理高维随机过程。在这里，KLT可能被用来将复杂的材料属性变化转换成一组相互独立的随机变量，简化不确定性建模。
5. **多项式混沌（Polynomial Chaos）**：这是一种数值方法，用于模拟随机输入下的系统行为。在本文中，它可能被用来构建基于实验数据的复合材料性质的不确定性模型。
6. **不确定性量化（Uncertainty Quantification, UQ）**：UQ是评估和传播模型参数不确定性到预测输出的过程。在复合材料的特性识别中，UQ可以帮助理解测量数据的局限性和对结果的影响。
7. **先验不确定性（Epistemic Uncertainty）**：这是由于知识不足或信息缺乏导致的不确定性，可以通过收集更多数据或改进模型来减少。
在第一篇论文（Part I）中，建立了观察到的不均匀杨氏模量场的数据库。第二篇论文（Part II，即本文）则聚焦于如何处理和建模这些测量数据中的不确定性，这对于实际工程应用中的材料性能预测和决策至关重要。
这篇研究工作提供了一种处理有限实验数据的方法，以应对复合材料性质的随机性和不确定性，这对于提高工程设计的安全性和可靠性具有重要意义。
approach, the macro-scale material uncertainty is directly identified based on the measurements of the macro-scale,
e.g. coupon level, observables [30,31]. The interested reader is referred to [18,32] for more comprehensive reviews on the
uncertainty representation of composite material properties.
In the present study, a probabilistic approach is developed to reconstruct stochastic macro-scale material properties
from direct macro-scale measurements. The study is presented in a sequence of two papers. The first paper [33] is
concerned with the construction of the experimental database. While in the second paper, the present manuscript,
the experimental information is assimilated to reconstruct a random field characterising the Young’s modulus.
The experimental measurements [33] consists of an ensemble of mobility frequency response functions measured at
several fixed points along a number of geometrically identical cantilever beams obtained using a linear laser Doppler
vibrometer. For each realisation of the experiment, a deterministic inverse analysis based on a forward finite element
model of the cantilever beam is performed in [33] to infer the corresponding equivalent Young’s modulus of the composite
material. In the present manuscript, based on a limited number of vibration tests, the work in [34,35] is extended to model
the heterogeneous Young’s modulus of a composite cantilever beam in the form of a one-dimensional spatial random field.
In addition to the characterisation of the aleatory uncertainty associated with sample-to-sample variability of the Young’s
modulus, the proposed approach also enables the quantification of the epistemic uncertainty due to the insufficiency of the
number of experiments. In order to enable a finite-dimensional representation of uncertainty, the Young’s modulus
random field is discretised using the Karhunen–Lo
eve expansion [36] based on the empirical sample covariance matrix.
The Karhunen–Lo
eve random variables are then expanded into the Hermite Polynomial Chaos (PC) basis [1]. Point
estimates of the PC expansion coefficients are obtained using Bayesian inference techniques. Additionally, to quantify the
uncertainty due to the small-size ensemble measurements, the coefficients of the PC expansion are themselves considered
as random variables whose statistics are obtained from the asymptotic distribution of the Bayesian estimates. Clearly,
most identification problems are limited to the insufficiency of available data, a problem usually addressed by quantifying
the associated epistemic uncertainty. The validity of the present uncertainty representation approach is investigated using
a standard cross-validation approach. It is worth noting that although the methodology used in this work has been applied
to identify the Young’s modulus field, it could be applied to any general identification problem, i.e., identification of other
properties as well as other materials or fabrics, provided that enough experimental information is available. The proposed
construction has several advantages : (i) it does not assume any particular form, e.g. Gaussian, log-normal, or Weibull, for
the marginal probability distribution of the Young’s modulus, (ii) it preserves the first and second order statistics of the
experimental samples and approximates their marginal distribution, (iii) it can be readily integrated into uncertainty
propagation solvers such as the Spectral Stochastic Finite Element (SSFEM) [1], perturbation-based techniques [2,11], and
the Monte Carlo simulation, and (iv) it is applicable to representation of other spatial stochastic material properties such as
the shear modulus.
In this manuscript, the proposed approach to reconstruct the Young’s modulus random field is introduced first in
Sections 2 and 3. The work in these sections extends the probabilistic identification approach of [34,35] to reconstruct the
random field E given the discrete ensemble of realisations ½
~
E. Synthesised realisations of the reconstructed random field
are used to quantify the aleatory and the epistemic uncertainties associated with the experimental database of the field in
Section 4. Finally, in Section 5, an attempt is made to validate the identified stochastic Young’s modulus field before
concluding the work in Section 6.
2. Spectral decomposition of the Young’s modulus field
Corresponding to each of M composite beam specimen, optimal realisations of Young’s modulus ½
~
E at certain spatial
locations have been identified in Part I [33], Fig. 1. Based on an ensemble of measured frequency response functions on
each composite specimen, the realisations of the Young’s modulus have been acquired by solving M deterministic inverse
problems. Each of these spatially variable Young’s modules fields can be considered as independent realisations of an
underlying continuous Young’s modulus random field.
2.1. Karhunen–Lo
eve expansion of the Young’s modulus
The Young’s modulus is assumed to be a second-order random field (RF) and is denoted by E; therefore, it admits a
Karhunen–Lo
eve (KL) expansion [36]. The expansion is based on the spectral decomposition of the covariance matrix [C
E
]
of the discrete (nodal) values of the Young’s modulus. In the absence of a complete statistical description of E, ½C
~
E
can be
P5
P10
P2
P6
P1
P3
P4
P7
P8
P9
P13
P12
P11
P0
Fig. 1. A composite beam specimen.
L. Mehrez et al. / Mechanical Systems and Signal Processing 27 (2012) 484–498486
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