Location
MC 6460
Candidate
Kunal Ghosh | Applied Mathematics, University of Waterloo
Title
Provably Stable and Conservative Mesh Adaptation Driven by Lagrangian Coherent Structures
Abstract
Adaptive mesh refinement can reduce the computational cost of flow simulations by concentrating resolution in regions containing dynamically important structures. A central challenge, however, is determining where refinement should be applied before those structures become under-resolved. Refinement criteria based only on the instantaneous flow field are generally reactive, whereas Lagrangian information can provide a more anticipatory view of the evolving flow by identifying regions associated with future transport and coherent motion. Lagrangian coherent structures (LCS) are distinguished material structures in a flow that organize fluid trajectories and transport over finite time intervals, providing a physics-based means of identifying dynamically significant regions. Finite-time Lyapunov exponent (FTLE) fields characterize regions of strong finite-time material stretching and can reveal attracting or repelling transport structures, while Lagrangian-averaged vorticity deviation (LAVD) identifies rotationally coherent regions associated with vortical motion. These complementary Lagrangian indicators therefore provide information about both transport and vortex dynamics that can be used to refine the mesh in anticipation of where dynamically important flow features will develop. Using this information for adaptive refinement requires modifying the computational mesh as the flow evolves, which in turn requires transferring the numerical solution between different discrete spaces. If this transfer is not carefully controlled, it can introduce interpolation errors, violate conservation, or artificially increase the discrete energy.
This seminar presents an adaptive framework in which one-step FTLE and LAVD indicators are combined to guide budget-constrained mesh refinement for a two-dimensional cylinder wake. The Lagrangian indicators are used to provide forward-looking information for the mesh-adaptation step so that additional resolution can be introduced before the corresponding flow features are encountered at the subsequent state. To make the resulting mesh-change step mathematically reliable, a complementary transfer framework is developed using summation-by-parts (SBP) discretizations. Conditions governing the existence of a transfer operator that is constructed to reproduce a prescribed accuracy space while remaining non-expansive in the SBP energy norm are investigated, together with the role of consistent representation of constants in maintaining conservation. If an adapted mesh does not satisfy the resulting transfer-feasibility criterion, a matrix-level first-order node-relocation problem is used to determine an admissible correction subject to geometric validity, after which the transfer-feasibility condition is re-evaluated before remapping. Together, these components provide a framework for LCS-driven mesh adaptation in which refinement is guided proactively by Lagrangian flow information while stability and conservation are explicitly considered during the solution-transfer step.