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.
