A Generalized Iterative Approach to Improve Reduced-Order Model Accuracy for Inverse Problem Applications

摘要

A generally applicable algorithm for the iterative generation of data ensembles to efficiently create accurate computational mechanics reduced-order models (ROMs) for use in computational approaches to approximate inverse problem solutions is presented and numerically evaluated. The ROM approach considered is based on identifying the optimal low-dimensional basis to be used within a Galerkin weak-form finite-element (FE) method to provide substantially reduced computational cost while maintaining accuracy relative to that of a (traditional) full-order FE model. Furthermore, proper orthogonal decomposition is used to derive the ROM basis from a set of response fields (i.e.,snapshots) generated a priori with full-order FE analyses. Therefore, the set of full-order FE analyses used to create the ROM directly affects the accuracy/generalization of the ROM. The core hypothesis of the algorithm presented is that maximizing the diversity, as defined in a measurable sense, of the full-order models (FOM) used to create the ROM will improve the accuracy of the ROM over a range of input system parameters. Based on an initial (small) set of snapshots, the algorithm uses snapshot correlation to quantify the snapshot diversity with respect to the system input parameters. Then the algorithm iteratively applies surrogate-model optimization to identify the next set(s) of system input parameters to be evaluated with full-order analyses to create additional so-called optimal snapshots. Although generally applicable to a variety of physical processes, the ROM approach with the iterative snapshot generation algorithm is presented within the context of the steady-state dynamic solid mechanics of heterogeneous media. Two simulated case studies are then presented involving forward analysis and inverse characterization of semilocalized Young's modulus distributions in structural components as could be relevant to nondestructive evaluation problems. The iterative snapshot generation algorithm is shown to produce ROMs that can accurately estimate displacement response fields over a wide range of material parameters and that are substantially more accurate than ROMs created from randomly generated snapshot sets. Moreover, the accurate generalization of the iteratively generated ROMs is shown to be sufficient to consistently produce accurate inverse characterization solution estimates with a fraction of the computational expense that would be required to do so with full-order analyses.