Reduced-order homogenization (ROH) and related methods are important computational tools for simulating the material behavior of composites. These methods generally sacrifice accuracy in exchange for superior computational efficiency, relative to methods such as classical computational homogenization (CCH). In this study, building on the recently developed solver-free CCH, we propose a fine-scale solver-free reduced order homogenization approach that avoids solving the fine-scale equilibrium equations and approximates the phase-average eigenstrains by sampling the fine-scale eigenstrain at a small number of points. The proposed method, which we call solver-free ROH, works by pre-computing history-dependent eigenstrain influence function tensors and sampling point contribution factors based on training data from a small set of CCH simulations. Then, during the online stage of the computation, phase-average eigenstrains are computed from sampling point eigenstrains and used to compute homogenized stresses and strains. Focusing on small-deformation problems, this paper formulates and verifies the solver-free ROH approach for nonlinear periodic heterogeneous media. First, in the formulation, we delineate a sampling point approximation of the phase eigenstrains and describe the use of this approximation within the coarse-scale stress update. Next, we verify the solver-free ROH using loading cases outside the training data set. Finally, we use the proposed method to simulate a multilayer composite plate in three point bending (3pt-bend) and open hole tension (OHT), demonstrating the method’s efficiency and accuracy relative to the CCH.

中文翻译：

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非线性周期性异质介质的无求解器降阶均质化

降阶均质化（ROH）和相关方法是模拟复合材料材料行为的重要计算工具。相对于经典计算均质化 (CCH) 等方法，这些方法通常会牺牲准确性来换取卓越的计算效率。在本研究中，基于最近开发的无求解器 CCH，我们提出了一种精细尺度无求解器降阶均质化方法，该方法避免求解精细尺度平衡方程，并通过对精细尺度特征应变进行采样来近似相位平均特征应变在少数点上。我们将所提出的方法称为无求解器 ROH，其工作原理是基于一小组 CCH 模拟的训练数据来预先计算历史相关的特征应变影响函数张量和采样点贡献因子。然后，在计算的在线阶段，根据采样点特征应变计算相位平均特征应变，并用于计算均匀应力和应变。本文针对小变形问题，制定并验证了非线性周期性异质介质的免求解器ROH方法。首先，在公式中，我们描绘了相位特征应变的采样点近似，并描述了该近似在粗尺度应力更新中的使用。接下来，我们使用训练数据集之外的加载案例验证无求解器的 ROH。最后，我们使用所提出的方法来模拟多层复合材料板的三点弯曲 (3pt-bend) 和开孔拉伸 (OHT)，证明了该方法相对于 CCH 的效率和准确性。