Soft, porous mechanical metamaterials exhibit pattern transformations that may have important applications in soft robotics, sound reduction and biomedicine. To design these innovative materials, it is important to be able to simulate them accurately and quickly, in order to tune their mechanical properties. Since conventional simulations using the finite element method entail a high computational cost, in this article we aim to develop a machine learning-based approach that scales favorably to serve as a surrogate model. To ensure that the model is also able to handle various microstructures, including those not encountered during training, we include the microstructure as part of the network input. Therefore, we introduce a graph neural network that predicts global quantities (energy, stress, stiffness) as well as the pattern transformations that occur (the kinematics) in hyperelastic, two-dimensional, microporous materials. Predicting these pattern transformations means predicting the displacement field. To make our model as accurate and data-efficient as possible, various symmetries are incorporated into the model. The starting point is an E(n)-equivariant graph neural network (which respects translation, rotation and reflection) that has periodic boundary conditions (i.e., it is in-/equivariant with respect to the choice of RVE), is scale in-/equivariant, can simulate large deformations, and can predict scalars, vectors as well as second and fourth order tensors (specifically energy, stress and stiffness). The incorporation of scale equivariance makes the model equivariant with respect to the similarities group, of which the Euclidean group E(n) is a subgroup. We show that this network is more accurate and data-efficient than graph neural networks with fewer symmetries. To create an efficient graph representation of the finite element discretization, we use only the internal geometrical hole boundaries from the finite element mesh to achieve a better speed-up and scaling with the mesh size.

中文翻译：

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用于超材料均质化的相似性等变图神经网络


柔软、多孔的机械超材料表现出图案转变，可能在软机器人、降噪和生物医学中具有重要应用。为了设计这些创新材料，能够准确快速地模拟它们，以调整它们的机械性能非常重要。由于使用有限元方法的传统模拟需要高计算成本，因此在本文中，我们的目标是开发一种基于机器学习的方法，该方法可以很好地扩展以用作替代模型。为了确保模型也能够处理各种微结构，包括训练过程中没有遇到的微结构，我们将微结构作为网络输入的一部分。因此，我们引入了一个图神经网络，用于预测超弹性二维微孔材料中的全局量（能量、应力、刚度）以及发生的模式变换（运动学）。预测这些模式变换意味着预测位移场。为了使我们的模型尽可能准确和数据效率，模型中加入了各种对称性。起点是一个 E（n） 等变图神经网络（尊重平移、旋转和反射），它具有周期性边界条件（即，就 RVE 的选择而言，它是内/等变的），是按比例/等变的，可以模拟大变形，并且可以预测标量、向量以及二阶和四阶张量（特别是能量、 应力和刚度）。尺度等方差的合并使模型相对于相似性组是等变的，其中欧几里得群 E（n） 是其中的一个子组。 我们表明，这个网络比对称性较少的图神经网络更准确、数据效率更高。为了创建有限元离散化的有效图形表示，我们只使用有限元网格中的内部几何孔边界，以实现更好的加速和网格大小的缩放。