An implicit shock tracking method for simulation of shock-dominated flows over complex domains using mesh-based parametrizations

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 to: (1) give concrete examples of the formulation, (2) show that accurate parametrizations can be obtained despite the surrogate surfaces only being C 0 , (3) show they integrate seemlessly with implicit shock tracking and can be used to parametrize surfaces without explicit expressions, and (4) effectively parametrize complex geometries and prevent nodes from moving off their original boundaries.