This work presents a framework for predicting the unknown probability distributions of input parameters, starting from scarce experimental measurements of other input parameters and the Quantity of Interest (QoI), as well as a computational model of the system. This problem is relevant to aeronautics, an example being the calculation of the material properties of carbon fibre composites, which are often inferred from experimental measurements of the full-field response. The method presented here builds a probability distribution for the missing inputs with an approach based on probabilistic equivalence. The missing inputs are represented with a multi-modal Polynomial Chaos Expansion (mmPCE), a formulation which enables the algorithm to efficiently handle multi-modal experimental data. The parameters of the mmPCE are found through an optimisation process. The mmPCE is used to produce a dataset for the missing inputs, the input uncertainties are then propagated through the computational model of the system using arbitrary Polynomial Chaos (aPC) in order to produce a probability distribution for the QoI. This is in addition to an estimate of the QoI’s probability distribution arising from experimental measurements. The coefficients of the mmPCE are adjusted such that the statistical distance between the two estimates of the probability distribution of the QoI is minimised. The algorithm has two key aspects: the metric used to quantify the statistical distance between distributions and the aPC formulation used to propagate the input uncertainties. In this work the Kolmogorov–Smirnov (KS) distance was used to quantify the distance between probability distributions for the QoI as it allowed high order statistical moments to be matched and is non-parametric.

The framework for back-calculating unknown input distributions was demonstrated using a dataset comprising scarce experimental measurements of the material properties of a batch of carbon fibre coupons. The ability of the algorithm to back-calculate a distribution for the shear and compression strength of the composite, based on limited experimental data, was demonstrated. It was found that it was possible to recover reasonably accurate probability distributions for the missing material properties, even when an extremely scarce data set with a fairly simplistic computational model was used.

这项工作提供了一个用于预测输入参数未知概率分布的框架，该框架从对其他输入参数和感兴趣量（QoI）的稀缺实验测量开始，以及系统的计算模型。这个问题与航空业有关，例如碳纤维复合材料的材料性能计算，这通常是从对全场响应的实验测量中得出的。此处介绍的方法使用基于概率等价的方法为缺少的输入建立概率分布。缺少的输入用多模态多项式混沌扩展（mmPCE）表示，该公式使算法能够有效处理多模态实验数据。通过优化过程找到mmPCE的参数。mmPCE用于生成缺少输入的数据集，然后使用任意多项式混沌（aPC）将输入不确定性传播到系统的计算模型中，以生成QoI的概率分布。这是对来自实验测量的QoI概率分布的估计的补充。调整mmPCE的系数，以使QoI概率分布的两个估计之间的统计距离最小。该算法具有两个关键方面：用于量化分布之间统计距离的度量，以及用于传播输入不确定性的aPC公式。

使用一个数据集演示了用于反向计算未知输入分布的框架，该数据集包括一批碳纤维试样的材料特性的稀缺实验测量值。基于有限的实验数据，证明了该算法能够反算复合材料的剪切强度和抗压强度的分布。发现即使使用极其简单的计算模型的极其稀缺的数据集，也有可能针对缺失的材料属性恢复合理准确的概率分布。