An Adaptive Simplex Cut-Cell Method for Discontinuous Galerkin Discretizations of the Navier-Stokes Equations

Abstract

A cut-cell adaptive method is presented for high-order discontinuous Galerkin discretizations in two and three dimensions. The computational mesh is constructed by cutting a curved geometry out of a simplex background mesh that does not conform to the geometry boundary. The geometry is represented with cubic splines in two dimensions and with a tesselation of quadratic patches in three dimensions. High-order integration rules are derived for the arbitrarily-shaped areas and volumes that result from the cutting. These rules take the form of quadrature-like points and weights that are calculated in a pre-processing step. Accuracy of the cut-cell method is verified in both two and three dimensions by comparison to boundary-conforming cases. The cut-cell method is also tested in the context of output-based adaptation, in which an adjoint problem is solved to estimate the error in an engineering output. Two-dimensional adaptive results for the compressible Navier-Stokes equations illustrate automated anisotropic adaptation made possible by triangular cut-cell meshing. In three dimensions, adaptive results for the compressible Euler equations using isotropic refinement demonstrate the feasibility of automated meshing with tetrahedral cut cells and a curved geometry representation. In addition, both the two and three-dimensional results indicate that, for the cases tested, p = 2 and p = 3 solution approximation achieves the user-prescribed error tolerance more efficiently compared to p = 1 and p = 0.