An implicit shock tracking method for simulation of shock-dominated flows over complex domains using mesh-based parametrizations
Alexander M. Perez Reyes, Matthew J. Zahr
TL;DR
This paper addresses the challenge of simulating shock-dominated flows on complex geometries by introducing mesh-based parametrizations that generate boundary-preserving, elementwise mappings from a high-order mesh. These parametrizations are integrated into the High-Order Implicit Shock Tracking (HOIST) framework, enabling a fully coupled optimization that moves mesh nodes to align mesh faces with shocks while keeping each node on its original boundaries. The approach combines a formal definition of mesh-based parametrizations, an efficient high-order element-search algorithm, and a PDE-constrained optimization with explicit derivatives, demonstrated across surface geometry tests and shock-dominated flows (linear advection and Euler). The results show accurate surface representations, robust shock alignment, and preserved geometry on complex domains, offering a practical path to applying HOIST to vehicles and geometries lacking simple analytical parametrizations.
Abstract
A mesh-based parametrization is a parametrization of a geometric object that is defined solely from a mesh of the object, e.g., without an analytical expression or computer-aided design (CAD) representation of the object. In this work, we propose a mesh-based parametrization of an arbitrary $d'$-dimensional object embedded in a $d$-dimensional space using tools from high-order finite elements. Using mesh-based parametrizations, we construct a boundary-preserving parametrization of the nodal coordinates of a computational mesh that ensures all nodes remain on all their original boundaries. These boundary-preseving parametrizations allow the nodes of the mesh to move only in ways that will not change the computational domain. They also ensure nodes will not move between boundaries, which would cause issues assigning boundary conditions for partial differential equation simulations and lead to inaccurate geometry representations for non-smooth boundary transitions. Finally, we integrate boundary-preserving, mesh-based parametrizations into high-order implicit shock tracking, an optimization-based discontinuous Galerkin method that moves nodes to align mesh faces with non-smooth flow features to represent them perfectly with inter-element jumps, leaving the intra-element polynomial basis to represent smooth regions of the flow with high-order accuracy. Mesh-based parametrizations enable implicit shock tracking simulations of shock-dominated flows over geometries without simple analytical parametrizations. Several demonstrations of mesh-based parametrizations are provided.