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.

中文翻译：

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用于弹性和弹塑性固体的有限元集成神经网络框架


物理信息神经网络方法 （PINN） 在求解各种偏微分方程方面取得了成功，因此在解析固体力学中的未知物理场方面显示出前景。尽管如此，有效解决工程尺度的边界值问题，特别是异质性和路径依赖性弹塑性，对 PINN 来说仍然具有挑战性。为了解决这些问题，本研究提出了一种将有限元法 （FEM） 与 PINN 集成的混合计算框架，称为 FEINN。该框架采用有限元而不是搭配点进行域离散化，并利用高斯积分方案和应变-位移矩阵来建立弱形式控制方程，而不是自动微分运算符。通过利用 FEM 和 PINN 的优势，该框架在处理异质材料的复杂边界条件方面表现出先天优势。为了解决材料非线性边界值问题中的路径依赖性弹塑性问题，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 5 种类型的数值实验，涉及具有聚力、分布力和分布位移等各种边界的齐质和非均匀问题。采用和评估了线弹性和弹塑性（改性 cam-clay）模型。使用从 FEM 获得的解作为参考，与以前的 PINN 相比，FEINN 在所有实验中都表现出卓越的准确性和收敛率。FEINN 和 FEM 之间的平均绝对百分比误差始终低于 1%，并且 FEINN 表现出明显比 PINN 更快的收敛速率，凸显了其计算效率。 此外，本研究讨论了在低应力和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。