Uncertainty Quantification Modeling of Structures and Materials Using the Hypercomplex Differentiation Method

The advent of the use of complex and hypercomplex algebras for arbitrary order sensitivity computations provide a new and perhaps more efficient approach to estimate uncertainty. The basic concept is to explore probabilistic moment approximations of multivariable structure and material models, using high order, below sensitivity results from the hypercomplex finite-element method. The utilization of hypercomplex derivatives allow probabilistic moments to be estimated in a single simulation run, compared to Monte Carlo sampling, which can be computationally expensive for complex models. These moment approximations, specifically expected value, variance, skewness, and kurtosis, are used to estimate the probability density function of structure and material models using the generalized lambda distribution. Criteria to determine optimal order of approximation and parameter importance will be proposed and examined. These moment approximations can also be an efficient proxy for variance based Sobol indices. Sobol indices provide an estimate to the individual contribution of each input parameter to the output variance and identify non-important variables that can be fixed at their nominal values to reduce model complexity. For high dimensional models, the direct application of traditional variance-based global sensitivity analysis measures can be time consuming and impractical, whereas a sensitivity-based estimation can provide a low-cost meaningful substitute. An uncertainty quantification analysis is conducted on a fracture mechanics related effective modulus application.