Orthogonal spline expansions for uncertainty quantification in linear dynamical systems

Abstract

This paper leverages recent progress on orthonormal splines for solving uncertainty quantification (UQ) problems from linear structural dynamics. The resulting methods, premised on spline chaos expansion (SCE) and spline dimensional decomposition (SDD), both construe Fourier-like expansion of a dynamic system response of interest with respect to measure-consistent orthonormalized basis splines in input random variables and standard least-squares regression for estimating the expansion coefficients. The SCE and SDD methods are capable of capturing high nonlinearity and non-smoothness, if they exist, in a stochastic dynamic response markedly better than the polynomial chaos expansion (PCE) method. However, due to the tensor-product structure, SCE, like PCE, also suffers from the curse of dimensionality. In contrast, SDD, equipped with a desirable dimensional hierarchy of input variables, deflates the curse of dimensionality to a great extent. Numerical results from frequency response analysis of a two-degree-of-freedom dynamic system indicate that a low-order SCE with fewer basis functions removes or markedly reduces the spurious oscillations generated by high-order PCE in estimating the response statistics. Finally, a high-dimensional modal analysis of a fighter jet comprising 110 random variables was conducted, demonstrating the ability of SDD in solving large-scale UQ problems.