The Snapping Phenomenon and Boundary Enforcement in Model Reduction: Mechanisms and Mitigation
Abstract: Projection-based model order reduction mitigates the computational cost of high-fidelity parametric simulations but can suffer from the "snapping phenomenon", whereby a projection-based reduced-order model (PROM) rigidly latches onto a training snapshot, impairing parametric interpolation. Standard methods also struggle to enforce parameter-dependent boundary conditions without costly constraints or penalty schemes. This work proposes a tripartite taxonomy of snapping, distinguishing intrinsic, extrinsic, and numerical mechanisms. Focusing on intrinsic and numerical snapping, we introduce a unified framework based on a parameter-dependent shift vector. By embedding spatial transport into the state ansatz, the method enforces boundary conditions exactly, restores continuous parametric sensitivity, and eliminates snapshot latching in affine and regression-augmented PROMs without increasing online complexity. We establish the framework's mathematical foundations in the linear regime, showing how it overcomes test-space orthogonality blind spots. In nonlinear computational fluid dynamics, we identify the roles of boundary conditions, mesh grading, and discrete equation scaling in numerical snapping. We also show that formulation-toggling is an unreliable heuristic that can compromise near-field accuracy. Finally, we develop a field-theoretic realization using a spatially varying shift vector governed by a scaled Laplace equation. The approach is validated on structural and aerodynamic benchmarks, including viscous flow around a NACA airfoil and a three-dimensional NASA Common Research Model.