[论文解读] Neural networks meet hyperelasticity: A guide to enforcing physics

Lennart Linden, Dominik K. Klein+4 more|arXiv (Cornell University)|Feb 5, 2023

Elasticity and Material Modeling被引用 7

一句话总结

本文提出了一种用于超弹性材料的物理增强神经网络模型，通过模型架构本身强制满足所有基本本构条件——客观性、材料对称性、多凸性、热力学一致性以及无应力未变形状态。通过使用基于不变量的输入和解析归一化项，该模型确保了非负且物理解释合理的应力响应，并在有限元模拟中实现了对干净数据和噪声数据的精确标定与外推。

ABSTRACT

In the present work, a hyperelastic constitutive model based on neural networks is proposed which fulfills all common constitutive conditions by construction, and in particular, is applicable to compressible material behavior. Using different sets of invariants as inputs, a hyperelastic potential is formulated as a convex neural network, thus fulfilling symmetry of the stress tensor, objectivity, material symmetry, polyconvexity, and thermodynamic consistency. In addition, a physically sensible stress behavior of the model is ensured by using analytical growth terms, as well as normalization terms which ensure the undeformed state to be stress free and with zero energy. In particular, polyconvex, invariant-based stress normalization terms are formulated for both isotropic and transversely isotropic material behavior. By fulfilling all of these conditions in an exact way, the proposed physics-augmented model combines a sound mechanical basis with the extraordinary flexibility that neural networks offer. Thus, it harmonizes the theory of hyperelasticity developed in the last decades with the up-to-date techniques of machine learning. Furthermore, the non-negativity of the hyperelastic neural network-based potentials is numerically examined by sampling the space of admissible deformations states, which, to the best of the authors' knowledge, is the only possibility for the considered nonlinear compressible models. For the isotropic neural network model, the sampling space required for that is reduced by analytical considerations. In addition, a proof for the non-negativity of the compressible Neo-Hooke potential is presented. The applicability of the model is demonstrated by calibrating it on data generated with analytical potentials, which is followed by an application of the model to finite element simulations. In addition, an adaption of the model to noisy data is shown and its [...]

研究动机与目标

• 开发一种基于神经网络的超弹性本构模型，同时满足所有关键物理约束。
• 通过将物理原理直接嵌入网络架构，克服黑箱神经网络在本构建模中的局限性。
• 在各向同性和横观各向同性超弹性行为中，确保热力学一致性、客观性和材料对称性。
• 实现在稀疏或噪声实验数据上的可靠外推与鲁棒标定。
• 通过变形空间采样，提供数值验证的、非负的可压缩材料超弹性势能。

提出的方法

• 将超弹性势能表述为使用形变不变量（I1, I2, I3）作为输入的凸神经网络。
• 通过凸网络架构和解析增长项，强制实现多凸性和热力学一致性。
• 应用基于不变量的应力归一化项，以强制实现无应力参考构型和零形变时的零能量。
• 采用Sobolev训练方法，直接在应力-应变数据上标定网络，提升泛化能力与物理一致性。
• 通过解析方法减少采样空间，以验证各向同性情况下的非负性。
• 通过数值采样可接受形变状态验证非负性，并为可压缩Neo-Hookean势能提供证明。

实验结果

研究问题

• RQ1 基于神经网络的超弹性模型能否同时强制实现所有基本本构条件——客观性、材料对称性、多凸性和热力学一致性？
• RQ2 如何解析设计归一化项与增长项，以确保未变形状态为无应力且无能量状态？
• RQ3 当在噪声或稀疏实验数据上训练时，该模型在多大程度上保持物理一致性和准确性？
• RQ4 能否通过数值采样严格验证可压缩、非线性模型中超弹性势能的非负性？
• RQ5 在有限元模拟和外推任务中，该模型的性能与经典解析模型相比如何？

主要发现

• 所提出的模型通过架构设计强制实现所有标准本构条件（包括多凸性和客观性），确保热力学一致性。
• 引入解析归一化项可确保无应力且无能量的参考状态，这对物理真实性至关重要。
• 该模型在基于解析势能生成的数据上实现了精确标定，并在有限元模拟中表现出优异的预测性能。
• 在噪声数据上，物理增强模型优于具有较少物理约束的标准神经网络，展现出更优的外推能力。
• 通过在可接受形变状态空间中的数值采样，验证了超弹性势能的非负性，且通过解析对称性考虑显著减少了采样空间。
• 为可压缩Neo-Hookean势能的非负性提供了正式证明，验证了该框架的关键理论组成部分。