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

Highlights: • Performs uncertainty quantification directly on high-dimensional system response. • Proposes a multi-element approach where elements are refined based on Grassmann distances between solution snapshots. • Adaptively resolves regions of the probability space corresponding to significant changes in system behavior. • Once converged, enables direct interpolation of high-dimensional solutions for any point in the probability space. • The method is applied to modeling shear localization in amorphous solids with stochastic initial conditions. This paper addresses uncertainty quantification (UQ) for problems where scalar (or low-dimensional vector) response quantities are insufficient and, instead, full-field (very high-dimensional) responses are of interest. To do so, an adaptive stochastic simulation-based methodology is introduced that refines the probability space based on Grassmann manifold variations. The proposed method has a multi-element character discretizing the probability space 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 these points are calculated using appropriately defined metrics and the elements with large total distance are sub-sampled and refined. As a result, regions of the probability space that produce significant changes in the full-field solution are accurately resolved. An added benefit is that an approximation of the solution within each element can be obtained by interpolation on the Grassmann manifold. The method is applied to study the probability of shear band formation in a bulk metallic glass using the shear transformation zone theory.