A general‐purpose computational homogenization framework is proposed for the nonlinear dynamic analysis of membranes exhibiting complex microscale and/or mesoscale heterogeneity characterized by in‐plane periodicity that cannot be effectively treated by a conventional method, such as woven fabrics. The framework is a generalization of the “finite element squared” (or FE2) method in which a localized portion of the periodic subscale structure is modeled using finite elements. The numerical solution of displacement driven problems involving this model can be adapted to the context of membranes by a variant of the Klinkel–Govindjee method1 originally proposed for using finite strain, three‐dimensional material models in beam and shell elements. This approach relies on numerical enforcement of the plane stress constraint and is enabled by the principle of frame invariance. Computational tractability is achieved by introducing a regression‐based surrogate model informed by a physics‐inspired training regimen in which FE2 is utilized to simulate a variety of numerical experiments including uniaxial, biaxial and shear straining of a material coupon. Several alternative surrogate models are evaluated including an artificial neural network. The framework is demonstrated and validated for a realistic Mars landing application involving supersonic inflation of a parachute canopy made of woven fabric.

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.

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