A Computationally Tractable Framework for Nonlinear Dynamic Multiscale Modeling of Membrane Fabric

Abstract

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 proposed framework is a generalization of the "Finite Element squared" (or FE 2 ) method in which a localized portion of the periodic subscale structure -typically referred to as a Representative Volume Element (RVE) -is modeled using finite elements. The numerical solution of displacement-driven problems using this model furnishes a mapping between the deformation gradient and the first Piola-Kirchhoff stress tensor. This unconventional material model can be readily applied in the context of membranes by using a variant of the approach proposed by Klinkel and Govindjee 1 for using conventional, finite strain, three-dimensional material models in beam and shell elements. The approach involves the numerical enforcement of the plane stress constraint, which is typically performed on out-of-plane components of the second Piola-Kirchhoff stress tensor ( ). Observing the principal of frame invariance, the RVE solution is reinterpreted as a mapping between the right stretch tensor and the symmetric Biot stress tensor conjugate pair. This facilitates the development of a drop-in replacement for any conventional finite strain plane stress material model formulated in terms of the in-plane components of the Green-Lagrange strain tensor and . Finally, computational tractability is achieved by introducing a regression-based surrogate model to avoid further solution of the RVE model when data sufficient to fit a model capable of delivering adequate approximations is available. For this purpose, a physics-inspired training regimen involving the utilization of our generalized FE 2 method to simulate a variety of numerical experiments -including but not limited to uniaxial, biaxial and shear straining of a material coupon -is proposed as a practical method for data collection. The proposed framework is demonstrated for a Mars landing application involving the supersonic inflation of an atmospheric aerodynamic decelerator system that includes a parachute canopy made of a woven fabric. Several alternative surrogate models are evaluated including a neural network.

Improve Unscented Kalman Inversion With Low-Rank Approximation and Reduced-Order Model

“…Consider the algorithm defined by Eqs. ( 4) to (7). By utilizing the aforementioned quadrature rule, we obtain the following UKI algorithm:…”

“…Inverse problems are ubiquitous in engineering and scientific applications. These include, to name only a few, global climate model calibration [2,3,4], material constitutive relation calibration [5,6,7], seismic inversion in geophysics [8,9], and medical tomography [10,11]. The associated forward problems, which may feature multiple scales or include chaotic and turbulent phenomena, can be very expensive.…”