Unsteady Metric-Based Adaptation via Koopman Expansion

Unsteady flowfields are integral to high-speed applications, demanding precise modeling to accurately characterize their dynamic features. The simulation of unsteady supersonic and hypersonic flows is inherently computationally expensive, necessitating a highly refined mesh to capture these dynamic effects. While anisotropic metric-based adaptive mesh refinement has proven effective in achieving accuracy with much less complexity, current algorithms are primarily tailored for steady flowfields. This paper presents a novel approach to address the challenges of anisotropic grid adaptation of unsteady flows by leveraging a data-driven technique called dynamic mode decomposition (DMD). DMD has proven to be a powerful tool to model complex nonlinear flows, given its links to the Koopman operator and also its easy mathematical implementation. This research proposes the integration of DMD into the process of anisotropic grid adaptation to dynamically adjust the mesh in response to evolving flow features. The effectiveness of the proposed approach is demonstrated through numerical experiments on representative unsteady flow configurations, such as a cylinder in a subsonic flow and an oscillating cylinder in a supersonic channel flow. Results indicate that the incorporation of DMD enables an accurate representation of unsteady flow dynamics independently of the remeshing interval. Computational fluid dynamics results obtained with the dynamic anisotropic mesh adaptation achieved a fourfold reduction in drag error compared to static meshing methods.