On the usefulness of gradient information in surrogate modeling: Application to uncertainty propagation in composite material models

In this work, the performance of non-gradient as well as gradient-enhanced versions of two different classes of surrogate modeling approaches, polynomial least squares regression and kernel based radial basis function interpolation, are compared in the context of a composite mechanics problem. Sequential space filling random designs are used for selecting the training points. The primary goal is to investigate whether additional gradient information obtained at a relatively small cost helps in generating surrogates of better quality compared to those obtained without any gradient information. It is found from the study that if the gradient and/or function evaluations are noisy, then the quality of the surrogate approximation is similar for both the gradient enhanced and the non-gradient based surrogate models. However, if the gradient and function evaluations are accurate, the gradient-enhanced surrogate models consistently perform better than the non-gradient based surrogate models, indicating that the gradient information enhances the quality of the surrogates. Low dimensional analytical test functions are used to demonstrate this behavior. As an application problem, we consider a multi-fiber reinforced composite model with a different interfacial damage parameter assigned to each fiber∕matrix interface. In particular, the surrogate describes the variation of the homogenized stress at a given input strain as a function of the interface damage parameters. The Interface-Enriched Generalized Finite Element Method (IGFEM) is used in this case to solve for the stress as well as the gradients of the stress with respect to the damage parameters. Thus the goal of this study is two-fold: (1) to compare the error convergence properties in surrogate modeling using different sequential random space filled designs, with and without gradient information; (2) to identify the circumstances in which additional gradient information is beneficial for surrogate modeling.