Geometry-aware framework for deep energy method: an application to structural mechanics with hyperelastic materials
Universite Paris-Saclay, ENS Paris-Saclay, CNRS, Centre Borelli（巴黎-萨克莱大学、巴黎-萨克莱高等学院、国家科学研究中心、Borelli中心）；Michelin, Centre de Recherche de Ladoux（米其林公司、Ladoux研究中心）；CEA, DAM, DIF（法国原子能委员会、DAM、DIF）；ENSIIE, Évry-Courcouronnes（巴黎-萨克莱高等工程师学院、Évry-Courcouronnes）
物理信息神经网络（Physics-Informed Neural Networks, PINNs）因能够将物理定律整合到深度学习模型中，已在多个工程领域引发广泛关注。近期，研究人员开发了基于几何感知PINN的方法，该类方法采用底层物理系统方程的强形式，旨在将几何信息整合到PINNs中。尽管相关研究仍在推进，但评估PINNs在不同几何构型问题中的表现仍是一个活跃的研究方向。\n本研究提出了一种名为几何感知深度能量法（Geometry-Aware Deep Energy Method, GADEM）的新型物理信息框架，用于求解不同几何构型下的结构力学问题。由于物理系统方程的弱形式（即基于能量的方法）在求解固体力学问题时相比强形式展现出明显优势，GADEM采用弱形式，旨在推断多种几何形状下的解。将几何感知框架整合到基于能量的方法中，可得到在精度和计算成本方面表现优异的物理信息深度学习模型。\n本研究探究了表示几何信息和编码几何潜向量的不同方式，提出了GADEM的损失函数，该函数基于所有考虑的几何构型的势能进行最小化，同时还采用了自适应学习方法对配点进行采样，以提升GADEM的性能。研究展示了GADEM在求解固体力学问题中的若干应用，包括涉及接触力学和大变形超弹性的玩具轮胎加载仿真。数值结果表明，仅需一个训练好的模型，GADEM就具备出色的能力，可推断多种及全新几何形状下的解。
英文摘要：
Physics-Informed Neural Networks (PINNs) have gained considerable interest in diverse engineering domains thanks to their capacity to integrate physical laws into deep learning models. Recently, geometry-aware PINN-based approaches that employ the strong form of underlying physical system equations have been developed with the aim of integrating geometric information into PINNs. Despite ongoing research, the assessment of PINNs in problems with various geometries remains an active area of investigation. In this work, we introduce a novel physics-informed framework named the Geometry-Aware Deep Energy Method (GADEM) for solving structural mechanics problems on different geometries. As the weak form of the physical system equation (or the energy-based approach) has demonstrated clear advantages compared to the strong form for solving solid mechanics problems, GADEM employs the weak form and aims to infer the solution on multiple shapes of geometries. Integrating a geometry-aware framework into an energy-based method results in an effective physics-informed deep learning model in terms of accuracy and computational cost. Different ways to represent the geometric information and to encode the geometric latent vectors are investigated in this work. We introduce a loss function of GADEM which is minimized based on the potential energy of all considered geometries. An adaptive learning method is also employed for the sampling of collocation points to enhance the performance of GADEM. We present some applications of GADEM to solve solid mechanics problems, including a loading simulation of a toy tire involving contact mechanics and large deformation hyperelasticity. The numerical results of this work demonstrate the remarkable capability of GADEM to infer the solution on various and new shapes of geometries using only one trained model.
Fully data-driven inverse hyperelasticity with hyper-network neural ODE fields
Department of Mechanical Engineering, Purdue University（机械工程系，普渡大学）；Department of Aerospace Engineering and Engineering Mechanics, The University of Texas at Austin（航空航天工程与工程力学系，德克萨斯大学奥斯汀分校）；Department of Mechanical and Metallurgical Engineering and Institute for Biological and Medical Engineering, Pontificia Universidad Catolica de Chile（机械与冶金工程系和生物医学工程研究所，智利天主教大学）；Weldon School of Biomedical Engineering, Purdue University（生物医学工程学院，普渡大学）
A deep learning energy method for hyperelasticity and viscoelasticity
National Center for Supercomputing Applications（国家超级计算应用中心）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Advanced Digital & Additive Manufacturing Center（先进数字与增材制造中心）；Khalifa University of Science and Technology（哈利法科学技术大学）
Constitutive parameterized deep energy method for solid mechanics problems with random material parameters
National Center for Applied Mathematics, Tianjin University（天津大学国家应用数学中心）；School of Mechanical and Aerospace Engineering, Jilin University（吉林大学机械与航空航天工程学院）
