A Convolutional Hierarchical Deep-learning Neural Network (C-HiDeNN) Framework for Non-linear Finite Element/Meshfree Analysis

Abstract: We present a framework for the Convolutional Hierarchical Deep Neural Network (C-HiDeNN) tailored for nonlinear finite element and meshfree analysis. Building upon the structured foundation of HiDeNN, which includes the evaluation of shape function derivatives, adaptivity, and material derivatives, C-HiDeNN introduces a convolution operator to enhance the HiDeNN approximation. A distinctive feature of C-HiDeNN is its expanded set of optimization parameters, such as the polynomial order 'p,' dilation parameter 'a,' patch size 's,' and nodal position 'X'. These parameters function as the weights and biases within the C-HiDeNN patch. In addition, C-HiDeNN can be prescribed in regions where high resolution is desired to adaptively improve prediction accuracy. To demonstrate the effectiveness of this framework, we provide numerical examples in the context of nonlinear finite element and meshfree analysis. The results show that our approach achieves significantly higher accuracy compared to the standard Finite Element Method (FEM) while substantially reducing computational costs.