Développement de stratégies d’adaptation de maillage pour les problèmes à paramètres variables

In numerical simulations of physical models, Adaptive Mesh Refinement (AMR) iteratively adjusts local mesh resolution to solution features, achieving meshes that meet prescribed accuracy. Beyond computing the final solution, AMR introduces overhead from repeated solution evaluations and mesh updates, which becomes significant when multiple simulations under varying operating conditions are needed, such as in parametric or sensitivity analyses. Strategies that avoid repeated full AMR are thus of practical interest.This work addresses simultaneous control of cost and accuracy under variable operating conditions through three mesh adaptation paradigms:Mean Mesh Adaptation (MMA): Builds a single adapted mesh minimizing the average error over a range of conditions, using Monte Carlo estimates of local error at each iteration. Applied to an industrial supersonic scramjet case, MMA outperforms state-of-the-art methods, but becomes costly for low mean errors or highly condition-dependent features.Error-Based Mesh Selection (EMS): For solutions with highly sensitive topologies, EMS selects the mesh from a precomputed library that minimizes error for the chosen condition. A fast approximate selection method is proposed, and tests on simplified shock flows demonstrate near-optimal mesh selection and error control.Transport-Based Mesh Interpolation (TMI): EMS relies on meshes adapted to specific conditions, which limits its robustness. TMI constructs meshes tailored to new conditions via interpolation between adapted meshes represented as point clouds, using Optimal Transport Wasserstein barycenters. A key contribution of TMI is the reconstruction of the interpolated point clouds into meshes. With similar libraries, this approach outperforms EMS error levels with minimal overhead.