PENMN: A physics-encoded numerical manifold network for unified continuous-discontinuous fracture analysis

In this paper, a novel Physics-encoded Numerical Manifold Network (PENMN) is proposed. This method deeply integrates the dual cover system of the Numerical Manifold Method (NMM) with the discretized differentiable solution architecture of physics-encoded numerical networks. It constructs the shape function matrix through mathematical covers and physical covers, and establishes the explicit topological connection of the degrees of freedom (DOFs) of cover nodes, which effectively mitigates the core limitation of classical coordinate-based Physics-Informed Neural Networks (PINNs) that they cannot assign multiple sets of independent DOFs to the same spatial coordinate. A global scalar loss function is constructed based on the principle of minimum potential energy and the weighted residual formulation, and the l -BFGS second-order optimizer is adopted to complete the solution. The discontinuous representation of the displacement field across crack surfaces is naturally realized via the dual-cover cutting mechanism, with no need for additional signed distance functions, domain decomposition, or interface continuity constraints. The solution accuracy, convergence characteristics and applicability of the proposed method are systematically verified through multiple sets of numerical examples of static discontinuity problems and quasi-static crack propagation. The results show that the displacement and stress calculated by PENMN are in high agreement with those obtained by NMM, and the method maintains excellent numerical stability under different mesh sizes, crack inclination angles and complex crack configurations. The predicted overall crack propagation path agrees well with the reference results, with no pronounced mesh orientation bias. The transfer learning strategy can effectively optimize the initial loss in the incremental analysis of crack propagation.