  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 method[1] 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 FE$^2$ 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.
