The introduction of Physics-informed Neural Networks (PINNs) has led to an increased interest in deep neural networks as universal approximators of PDEs in the solid mechanics community. Recently, the Deep Energy Method (DEM) has been proposed. DEM is based on energy minimization principles, contrary to PINN which is based on the residual of the PDEs. A significant advantage of DEM, is that it requires the approximation of lower order derivatives compared to formulations that are based on strong form residuals. However both DEM and classical PINN formulations struggle to resolve fine features of the stress and displacement fields, for example concentration features in solid mechanics applications. We propose an extension to the Deep Energy Method (DEM) to resolve these features for finite strain hyperelasticity. The developed framework termed mixed Deep Energy Method (mDEM) introduces stress measures as an additional output of the NN to the recently introduced pure displacement formulation. Using this approach, Neumann boundary conditions are approximated more accurately and the accuracy around spatial features which are typically responsible for high concentrations is increased. In order to make the proposed approach more versatile, we introduce a numerical integration scheme based on Delaunay integration, which enables the mDEM framework to be used for random training point position sets commonly needed for computational domains with stress concentrations. We highlight the advantages of the proposed approach while showing the shortcomings of classical PINN and DEM formulations. The method is offering comparable results to Finite-Element Method (FEM) on the forward calculation of challenging computational experiments involving domains with fine geometric features and concentrated loads.
翻译:采用物理-知情神经网络(PINN)已导致人们更加关注深神经网络,将其作为固体机械界中PDE的普遍接近点;最近,提出了深能源方法(DEM),该方法以能源最小化原则为基础,而PINN则以PDE的剩余部分为基础;德国马克的一个重大优势是,它要求将低顺序衍生物与基于强质形式残留的配方相比近似。然而,德国马克和古典PINN的配方都努力解决压力和迁移场的细微特征,例如固体机械应用中的集中特征。我们提议扩大深能源方法(DEM),以解决这些特性的极限性超弹性。开发后的称为混合深能源方法(MDEM),提出了压力措施,作为NN与最近采用的纯度配方相比的额外产出。使用这一方法,神经边界条件的近似更准确,通常对高浓度负责的空间特征的准确度增加。为了使拟议的方法更具有弹性,例如固体机械工艺的集中性特点。我们提议扩大深能源方法,以便用可比较的精度计算法的精度计算。我们提出了用于DEM的精度的精度计算方法,而我们提出了用于计算法系的精度的精度的精度的精度的精度。