Toward affordable uncertainty quantification for industrial problems - Part I: Theory and validation

Non-intrusive Polynomial Chaos (NIPC) methods have become popular for uncertainty quantification, as they have the potential to achieve a significant reduction in computational cost (number of evaluations) with respect to traditional techniques such as the Monte Carlo approach, while allowing the model to be still treated as a black box. This work makes use of Least Squares Approximations (LSA) in the context of appropriately selected PC bases. An efficient technique based on QR column pivoting has been employed to reduce the number of evaluations required to construct the approximation, demonstrating the superiority of the method with respect to sparse grid quadratures and to LSA with randomly selected quadrature points. Orthogonal (or orthonormal) polynomials used for the PC expansion are calculated numerically based on the given uncertainty distribution, making the approach optimal for any type of input uncertainty. The benefits of the proposed techniques are verified on a number of analytical test functions of increasing complexity and on two engineering test problem (uncertainty quantification of the deflection of a 3- and a 10-bar structure with up to 15 uncertain parameters). The results demonstrate how an LSA approach within a PC framework can be an effective method for UQ, with a significant reduction in computational cost with respect to full tensor and sparse grid quadratures.