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'' 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. The approach involves the numerical enforcement of the plane stress constraint. 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.

中文翻译：

---

用于膜织物非线性动态多尺度建模的计算上易于处理的框架

提出了一种通用计算均质化框架，用于对具有复杂微尺度和/或中尺度异质性的膜进行非线性动力学分析，这些膜具有面内周期性，无法通过常规方法（例如机织织物）进行有效处理。提议的框架是“有限元平方”方法的推广，其中周期性子尺度结构的局部部分 - 通常称为代表性体积元素（RVE） - 使用有限元建模。使用该模型的位移驱动问题的数值解提供了变形梯度和第一 Piola-Kirchhoff 应力张量之间的映射。该方法涉及平面应力约束的数值实施。最后，当数据足以拟合能够提供足够近似值的模型时，通过引入基于回归的替代模型来避免进一步求解 RVE 模型，从而实现计算易处理性。为此，提出了一种受物理学启发的训练方案，该方案涉及利用我们的广义 FE$^2$ 方法来模拟各种数值实验——包括但不限于材料试样的单轴、双轴和剪切应变——被提出作为一种实用的数据收集方法。提议的框架用于火星着陆应用程序，该应用程序涉及大气空气动力学减速器系统的超音速充气，该系统包括由机织织物制成的降落伞罩。评估了几种替代代理模型，包括神经网络。