参数化深度Ritz方法的非线性椭圆均匀化
Nonlinear elliptic homogenization with the parametric Deep Ritz method
- Smead Aerospace Engineering Sciences（斯米德航空航天工程科学学院）
- University of Colorado Boulder（科罗拉多大学博尔德分校）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文提出参数化深度Ritz方法求解非线性椭圆均匀化中的参数化单元问题，提供连续可微的单元响应表示，显著加速宏观尺度求解。
中文摘要 AI 辅助
椭圆均匀化用于确定具有小尺度特征材料的粗粒化性质。当这些微小特征具有快速周期波动时，对应于均匀化本构关系的解场与基于非均质材料的真实解非常接近。材料的这种均匀化行为通过单元问题计算，其中单元定义为波动材料的一个周期。在线性椭圆偏微分方程的情况下，均匀化本构关系仅由常系数张量定义，但对于非线性问题，均匀化响应取决于宏观状态和/或其梯度，因此需要求解参数化单元问题。当使用均匀化本构关系计算数值解时，拥有单元问题解的可微表示是有用的，因为在宏观状态场的牛顿迭代中需要均匀化本构关系的导数。在本工作中，我们使用深度Ritz方法求解由非线性均匀化产生的参数化单元问题。首先，我们利用单元问题的变分结构，然后使用神经网络对单元响应在空间和宏观状态上的依赖性进行离散化。在强加单元响应的边界条件后，我们接下来使用参数化深度Ritz方法同时求解一系列宏观状态下的单元问题。我们证明该方法准确、高效，并提供单元响应在宏观状态和梯度上的连续且可微的表示。然后我们证明，与传统的$\ ext{FE}^2$方案相比，我们的参数化单元响应表示显著加速了宏观尺度的求解。
英文摘要
Elliptic homogenization is used to determine coarse-grained properties of materials with features on small scales. When these small scale features have rapid, periodic fluctuations, the solution field corresponding to a homogenized constitutive relation closely resembles the true solution based on the heterogeneous material. This homogenized behavior of the material is computed from a cell problem, where a cell is defined to be one period of the fluctuating material. In the context of linear elliptic partial differential equations, the homogenized constitutive relation is defined simply by a constant coefficient tensor, but for nonlinear problems, the homogenized response depends on the macroscopic state and/or its gradient, thus requiring solutions to parametric cell problems. When computing a numerical solution with the homogenized constitutive relation, it is useful to have a differentiable representation of the solution to the cell problem, as derivatives of the homogenized constitutive relation are required in Newton iterations for the macroscopic state field. In this work, we use the Deep Ritz method to solve the parametric cell problems that arise from nonlinear homogenization. First, we exploit the variational structure of the cell problem, then we discretize the dependence of the cell response on both space and the macroscopic state with a neural network. Enforcing boundary conditions on the cell response strongly, we next use the parametric Deep Ritz method to simultaneously solve the cell problem over a range of macroscopic states. We show that this method is accurate, efficient, and offers a continuous and differentiable representation of the cell response over the macroscopic state and gradient. We then show that our parametric representation of the cell response significantly expedites macroscale solutions when compared to a traditional $\text{FE}^2$ scheme.
