In situ adaptive reduction of nonlinear multiscale structural dynamics models

Abstract

Conventional offline training of reduced-order bases in a predetermined region of a parameter space leads to parametric reduced-order models that are vulnerable to extrapolation. This vulnerability manifests itself whenever a queried parameter point lies in an unexplored region of the parameter space. This article addresses this issue by presenting an in situ, adaptive framework for nonlinear model reduction where computations are performed by default online and shifted offline as needed. The framework is based on the concept of a database of local reduced-order bases (ROBs), where locality is defined in the parameter space of interest. It achieves accuracy by updating on-the-fly a precomputed ROB and approximating the solution of a dynamical system along its trajectory using a sequence of most appropriate local ROBs. It achieves efficiency by managing the dimension of a local ROB and incorporating hyperreduction in the process. While sufficiently comprehensive, the framework is described in the context of dynamic multiscale computations in solid mechanics. In this context, even in a nonparametric setting of the macroscale problem and when all offline, online, and adaptation overhead costs are accounted for, the proposed computational framework can accelerate a single three-dimensional, nonlinear, multiscale computation by an order of magnitude, without compromising accuracy.

A computationally tractable framework for nonlinear dynamic multiscale modeling of membrane woven fabrics

“…PMOR continues to be an active and fertile area of research that can be leveraged to extend and improve this framework. In particular, the recent emergence of in‐situ training methodologies 4 presents an attractive option to streamline and enhance PMOR utilization by eliminating the conventional and potentially cumbersome offline‐online decomposition of computational effort and vulnerabilities associated with extrapolation.…”

“…Numerical methods that attempt to resolve all relevant scales typically lead to massive discretized problems. However, recent developments using a variety of alternative surrogate modeling techniques—including nonlinear, projection‐based model order reduction (PMOR), 2‐4 kriging, 5 and artificial neural networks (NNs) 6,7 —to accelerate the solution of one or more scales within the context of a computational homogenization framework present a coherent methodology by which a computationally tractable approximation can be attained without resorting to ad hoc approximations. Notably, thin shell and membrane discretizations have not been considered in this context prior to this work, although several frameworks for multiscale modeling of shells without emphasis on computational efficiency have been proposed 8‐10 .…”

Uncertainty quantification for industrial numerical simulation using dictionaries of reduced order models

“…-Online-adaptive model reduction methods update the ROM in the exploitation phase by collecting new information online as explained in [26], in order to limit extrapolation errors when solving the parametrized governing equations in a region of the parameter space that was not explored in the training phase. The ROM can be updated for example by querying the high-fidelity model when necessary for basis enrichment [14,[27][28][29][30]. -ROM interpolation methods [31][32][33][34][35][36][37][38][39][40][41][42] use interpolation techniques on Grassmann manifolds or matrix manifolds to adapt the ROM to the parameters considered in the exploitation phase by interpolating between two precomputed ROMs.…”

Mesh sampling and weighting for the hyperreduction of nonlinear Petrov–Galerkin reduced‐order models with local reduced‐order bases

“…However, these methods differ by how they specifically partition into subsets of solution snapshots. For example, snapshot partitioning has been performed by simply partitioning the time 34 or parameter 35,36 domain. Alternatively, state space (or HDM‐based solution manifold) partitioning has been advocated and realized by clustering and compressing the solution snapshots: 32 this enables the construction of local, nonlinear PROMs capable of capturing the different regimes and features (e.g., discontinuities and fronts) that may be experienced by the solution of an HDM such as (1), 32 as well as capturing the effects on this solution of variations in the parameters of such an HDM 37 .…”