Finite element-integrated neural network framework for elastic and elastoplastic solids

IF 6.9 1区 工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY
Ning Zhang , Kunpeng Xu , Zhen Yu Yin , Kai-Qi Li , Yin-Fu Jin
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引用次数: 0

Abstract

The Physics-informed neural network method (PINN) has shown promise in resolving unknown physical fields in solid mechanics, owing to its success in solving various partial differential equations. Nonetheless, effectively solving engineering-scale boundary value problems, particularly heterogeneity and path-dependent elastoplasticity, remains challenging for PINN. To address these issues, this study proposes a hybrid computational framework integrating finite element method (FEM) with PINN, known as FEINN. This framework employs finite elements for domain discretization instead of collocation points and utilizes the Gaussian integration scheme and strain-displacement matrix to establish the weak-form governing equation instead of the automatic differentiation operator. By harnessing the strengths of FEM and PINN, this framework exhibits inherent advantages in handling complex boundary conditions with heterogeneous materials. For addressing path-dependent elastoplasticity in material nonlinear boundary value problems, an incremental scheme is developed to accurately compute the stress. To validate the effectiveness of FEINN, five types of numerical experiments are conducted, involving homogenous and heterogeneous problems with various boundaries such as concentrated force, distributed force, and distributed displacement. Both linear elastic and elastoplastic (modified cam-clay) models are employed and evaluated. Using the solutions obtained from FEM as a reference, FEINN demonstrates exceptional accuracy and convergence rate in all experiments compared with previous PINNs. The mean absolute percentage errors between FEINN and FEM are consistently below 1%, and FEINN exhibits notably faster convergence rates than PINNs, highlighting its computational efficiency. Moreover, this study discusses the biases observed in regions of low stress and displacement, factors influencing FEINN's performance, and the potential applications of the FEINN framework.
弹性和弹塑性固体的有限元集成神经网络框架
物理信息神经网络法(PINN)在求解各种偏微分方程方面取得了成功,因此在解决固体力学中的未知物理场问题方面大有可为。然而,有效解决工程规模的边界值问题,尤其是异质性和路径依赖性弹塑性问题,对 PINN 来说仍然具有挑战性。为解决这些问题,本研究提出了一种将有限元法(FEM)与 PINN 相结合的混合计算框架,即 FEINN。该框架采用有限元法进行域离散化,而不是采用定位点,并利用高斯积分方案和应变-位移矩阵来建立弱式控制方程,而不是自动微分算子。通过利用有限元和 PINN 的优势,该框架在处理异质材料的复杂边界条件时表现出固有优势。为解决材料非线性边界值问题中的路径依赖弹塑性问题,开发了一种增量方案来精确计算应力。为了验证 FEINN 的有效性,进行了五种类型的数值实验,涉及具有各种边界条件(如集中力、分布力和分布位移)的同质和异质问题。采用并评估了线性弹性和弹塑性(改良凸轮粘土)模型。与以前的 PINN 相比,FEINN 以有限元求解为参考,在所有实验中都表现出了极高的精度和收敛速度。FEINN 与 FEM 之间的平均绝对百分比误差始终低于 1%,而且 FEINN 的收敛速度明显快于 PINN,突出了其计算效率。此外,本研究还讨论了在低应力和低位移区域观察到的偏差、影响 FEINN 性能的因素以及 FEINN 框架的潜在应用。
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来源期刊
CiteScore
12.70
自引率
15.30%
发文量
719
审稿时长
44 days
期刊介绍: Computer Methods in Applied Mechanics and Engineering stands as a cornerstone in the realm of computational science and engineering. With a history spanning over five decades, the journal has been a key platform for disseminating papers on advanced mathematical modeling and numerical solutions. Interdisciplinary in nature, these contributions encompass mechanics, mathematics, computer science, and various scientific disciplines. The journal welcomes a broad range of computational methods addressing the simulation, analysis, and design of complex physical problems, making it a vital resource for researchers in the field.
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