Multiscale Uncertainty Quantification with Arbitrary Polynomial Chaos

Abstract

This work presents a framework for upscaling uncertainty in multiscale models. The problem is relevant to aerospace applications where it is necessary to estimate the reliability of a complete part such as an aeroplane wing from experimental data on coupons. A particular aspect relevant to aerospace is the scarcity of data available.The framework needs two main aspects: an upscaling equivalence in a probabilistic sense and an efficient (sparse) Non-Intrusive Polynomial Chaos formulation able to deal with scarce data. The upscaling equivalence is defined by a Probability Density Function (PDF) matching approach. By representing the inputs of a coarse-scale model with a generalised Polynomial Chaos Expansion (gPCE) the stochastic upscaling problem can be recast as an optimisation problem. In order to define a data driven framework able to deal with scarce data a Sparse Approximation for Moment Based Arbitrary Polynomial Chaos is used. Sparsity allows the solution of this optimisation problem to be made less computationally intensive than upscaling methods relying on Monte Carlo sampling. Moreover this makes the PDF matching method more viable for industrial applications where individual simulation runs may be computationally expensive. Arbitrary Polynomial Chaos is used to allow the framework to use directly experimental data. Finally, the difference between the distributions is quantified using the Kolmogorov–Smirnov (KS) distance and the method of moments in the case of a multi-objective optimisation. It is shown that filtering of dynamical information contained in the fine-scale by the coarse model may be avoided through the construction of a low-fidelity, high-order model.