Parametric extended physics-informed neural networks for solid mechanics with complex mixed boundary conditions

Continuum solid mechanics form the foundation of numerous theoretical studies and engineering applications. Distinguished from traditional mesh-based numerical solutions, the rapidly developing field of scientific machine learning, exemplified by methods such as physics-informed neural networks (PINNs), shows great promise for the study of forward and inverse problems in mechanics. However, accurately imposing boundary conditions (BCs) in the training and prediction of neural networks (NNs) has long been a significant challenge in the application and research of PINNs. This paper integrates the concept of isogeometric analysis (IGA) by parameterizing the physical model of the structure with spline basis functions to form analytical distance functions (DFs) for arbitrary structural boundaries. Meanwhile, by means of the energy approach to circumvent the solution of boundary stress components, the accurate imposition of both Dirichlet and Neumann BCs is ultimately achieved in PINNs. Additionally, to accommodate the complex mixed BCs often encountered in engineering applications, where Dirichlet and Neumann BCs simultaneously appear on adjacent irregular boundary segments, structural subdomain decomposition and multi-subdomain stitching strategies are introduced. The effectiveness and accuracy of the proposed method are verified through two numerical experiments with various cases.