Efficient uncertainty propagation across continuum length scales for reliability estimates

Abstract Microstructural variabilities are among the predominant sources of uncertainty in structural performance and reliability. In engineering applications, it is computationally infeasible to include the important fine-scale details throughout the domain of interest. Hence, multiscale numerical methods that couple fine-scale models within a subdomain of the engineering-scale models have been a focus of research. These methods remain computationally expensive, mostly due to the complexity and grid size associated with the fine-scale models. Moreover, merely coupling scales does not address the fine-scale uncertainties that motivated their inclusion at the outset. The multiscale method is necessary but not sufficient to make microstructurally informed estimates of reliability. The primary goal of this research is to develop theory and tools for efficient uncertainty propagation (UP) across length scales using finite elements for tractable multiscale structural reliability calculations and uncertainty quantification. We propose a hierarchical solution that uses low-fidelity estimates to improve efficiency and adds higher-fidelity, concurrent multiscale models for improved accuracy. The low-fidelity estimates are used to identify hotspots which highlight subdomains of interest and compute the marginal probabilities associated with failure occurring in those locations. Subsequently, the high-fidelity models concurrently couple fine-scale models in the subdomains of interest to refine the calculation. We represent the uncertain variables with stochastic reduced–order models and construct surrogate models to expedite Monte Carlo simulation for each length scale considered. Even though the written focus is on continuum length scales, there is nothing inherent in the hierarchy to preclude coupling other models if there is a method available to do so. We conclude with three application examples. In the first, we use a low-fidelity geometric representation of threaded fasteners to identify hotspots in a bolted component. The predictions are subsequently refined by coupling higher-fidelity geometry models of the screws, which includes the geometry of the helical threads. The second application illustrates UP at the microstructural length scale of a polycrystalline alloy. We develop a reduced representation of the crystallographic orientation and use it for estimates of the probability law describing the apparent modulus of elasticity. In the final application, our hierarchy couples a dual-phase polycrystalline aluminum alloy with the engineering scale to make probabilistic estimates of the fracture load in a fine-scale particle.