Identification of missing input distributions with an inverse multi-modal Polynomial Chaos approach based on scarce data

Abstract

Abstract 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.