Uncertainty quantification for complex systems with very high dimensional response using Grassmann manifold variations

摘要

This paper addresses uncertainty quantification (UQ) for problems wherescalar (or low-dimensional vector) response quantities are insufficient and,instead, full-field (very high-dimensional) responses are of interest. To doso, an adaptive stochastic simulation-based methodology is introduced thatrefines the probability space based on Grassmann manifold variations. Theproposed method has a multi-element character discretizing the probabilityspace into simplex elements using a Delaunay triangulation. For every simplex,the high-dimensional solutions corresponding to its vertices (sample points)are projected onto the Grassmann manifold. The pairwise distances between thesepoints are calculated using appropriately defined metrics and the elements withlarge total distance are sub-sampled and refined. As a result, regions of theprobability space that produce significant changes in the full-field solutionare accurately resolved. An added benefit is that an approximation of thesolution within each element can be obtained by interpolation on the Grassmannmanifold without the need to develop a mathematical surrogate model. The methodis applied to study the probability of shear band formation in a bulk metallicglass using the shear transformation zone theory.