An efficient discrete physics-informed neural networks for geometrically nonlinear topology optimization

The application of geometrically nonlinear topology optimization (GNTO) poses a substantial challenge due to the extensive memory requirements and prohibitive computational demands involved. To tackle this challenge, a discrete physics-informed neural network (dPINN) is suggested as a promising approach to alleviate computational demands and enhance the applicability to large-scale problems. In comparison to collocation point-based PINNs, the most distinctive characteristic of dPINN is its mesh-based local interpolation for the evaluation of the system energy. This approach not only circumvents the issue of material mapping between elements and collocation points, but also provides improved robustness. Moreover, the partial differential equation (PDE) that corresponds to the adjoint equations lacks explicit expressions. The dPINN is capable of naturally evaluating equivalent energy through discrete expressions, a capability that collocation point-based PINNs lack. Furthermore, the activation state of sub-networks in series is determined in accordance with the density variation, thereby saving computational costs by dynamically incorporating each sub-network to reduce the trainable parameters in certain optimization steps, while conserving computational resources. The dPINN demonstrates exceptional accuracy and efficiency, along with enhanced resilience against mesh distortion compared to the finite element method (FEM), thereby enabling the application of larger loads. The dPINN-based GNTO is validated to be robust with regard to different geometries, loads, and volume fractions through several examples, and the outcomes are largely consistent with those of the FEM-based approach. Of greater significance is the fact that dPINN is capable of solving a million-DOFs 3D GNTO problem, which represents a notable advantage.