利用物理信息神经网络解决接触力学的正向和反向问题

Tarik Sahin, Max von Danwitz, Alexander Popp
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引用次数: 0

摘要

本文探讨了物理信息神经网络(PINN)解决小变形弹性接触力学正演和反演问题的能力。我们在混合变量公式中部署了 PINNs,并通过输出变换将 Dirichlet 和 Neumann 边界条件作为硬约束强制执行。接触问题的不等式约束条件,即 Karush-Kuhn-Tucker (KKT) 类型条件,通过在网络训练过程中将其纳入损失函数作为软约束条件来执行。为了制定 KKT 约束的损失函数,我们研究了应用于弹塑性问题的现有方法,并探索了一种非线性互补问题(NCP)函数,即 Fischer-Burmeister 函数,它在优化方面具有优势特点。基于赫兹接触问题,我们证明了 PINN 可作为纯偏微分方程 (PDE) 求解器、数据增强前向模型、参数识别逆求解器和快速评估代用模型。此外,我们还证明了选择适当的超参数(如损失权重)以及亚当和 L-BFGS-B 优化器组合的重要性,目的是在精度和训练时间方面获得更好的结果。
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Solving forward and inverse problems of contact mechanics using physics-informed neural networks
This paper explores the ability of physics-informed neural networks (PINNs) to solve forward and inverse problems of contact mechanics for small deformation elasticity. We deploy PINNs in a mixed-variable formulation enhanced by output transformation to enforce Dirichlet and Neumann boundary conditions as hard constraints. Inequality constraints of contact problems, namely Karush–Kuhn–Tucker (KKT) type conditions, are enforced as soft constraints by incorporating them into the loss function during network training. To formulate the loss function contribution of KKT constraints, existing approaches applied to elastoplasticity problems are investigated and we explore a nonlinear complementarity problem (NCP) function, namely Fischer–Burmeister, which possesses advantageous characteristics in terms of optimization. Based on the Hertzian contact problem, we show that PINNs can serve as pure partial differential equation (PDE) solver, as data-enhanced forward model, as inverse solver for parameter identification, and as fast-to-evaluate surrogate model. Furthermore, we demonstrate the importance of choosing proper hyperparameters, e.g. loss weights, and a combination of Adam and L-BFGS-B optimizers aiming for better results in terms of accuracy and training time.
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来源期刊
Advanced Modeling and Simulation in Engineering Sciences
Advanced Modeling and Simulation in Engineering Sciences Engineering-Engineering (miscellaneous)
CiteScore
6.80
自引率
0.00%
发文量
22
审稿时长
30 weeks
期刊介绍: The research topics addressed by Advanced Modeling and Simulation in Engineering Sciences (AMSES) cover the vast domain of the advanced modeling and simulation of materials, processes and structures governed by the laws of mechanics. The emphasis is on advanced and innovative modeling approaches and numerical strategies. The main objective is to describe the actual physics of large mechanical systems with complicated geometries as accurately as possible using complex, highly nonlinear and coupled multiphysics and multiscale models, and then to carry out simulations with these complex models as rapidly as possible. In other words, this research revolves around efficient numerical modeling along with model verification and validation. Therefore, the corresponding papers deal with advanced modeling and simulation, efficient optimization, inverse analysis, data-driven computation and simulation-based control. These challenging issues require multidisciplinary efforts – particularly in modeling, numerical analysis and computer science – which are treated in this journal.
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