2025 Theses Doctoral

Nonlinear dimension reduction with neural kernel for path-dependent solid mechanics problems

Xiong, Zeyu

Data-driven modeling of the path-dependent behavior of materials with complex internal structures enables us to predict the mechanical response under time-dependent deformation through the homogenized internal failure mechanism. However, these materials present a significant modeling challenge because they require a large number of variables to capture their internal distortions, leading to higher dimensionality and increased computational costs. Hence, this dissertation aims to achieve efficient and accurate material modeling through dimension reduction. To address this challenge, we introduce the data- or node-dependent neural kernel (NK) to train a reduced number of NK basis functions that efficiently span a lower-dimensional latent space for the higher-dimensional model.

One data-driven modeling approach explored is to train a surrogate micromorphic constitutive law. This approach introduces additional variables to describe the higher-order deformation of sampled materials and then examines the mechanical responses of these materials under such deformation. In this context, NK is employed to train the plastic yield function based on sampled yield stress data. With the trained yield function, the return mapping algorithm captures the path-dependent relationship between higher-order stress and deformation, or micromorphic mechanical response. Results demonstrate that the NK method surpasses the conventional multi-layer perceptron (MLP) in extrapolating sparsely distributed yield stress data, which is achieved by leveraging the span of kernel basis functions to learn data patterns more effectively.

Another approach involves modeling cracked materials using fine-mesh finite element methods (FEM). Although fine-mesh FEM can approximate discontinuities caused by internal cracks, it generates large matrix equations, which makes the approach computationally intensive. To address this, a hierarchical neural kernel (HiNK) architecture is introduced, combining hierarchical deep neural networks for FEM (HiDeNN-FEM) and NK enhancement to efficiently solve discontinuous boundary value problems (BVPs). In this architecture, HiDeNN-FEM generates trainable, coarse-mesh finite element shape functions (piecewise polynomials) to approximate the continuous solution. NK enhancement is then allocated to the region of discontinuity, accurately capturing these complex features. Results indicate that HiNK basis functions efficiently characterize discontinuous BVPs, providing a powerful tool for solving high-dimensional and computationally demanding FEM models.