rp-adaptation for compressible flows

J. Marcon, G. Castiglioni, D. Moxey, S. J. Sherwin, J. Peiró

Int. J. Numer. Meth. Eng., vol. 121, pp. 5405–5425 (2020)

@article{marcon-2020,
  title = {$rp$-adaptation for compressible flows},
  author = {Marcon, J. and Castiglioni, G. and Moxey, D. and Sherwin, S. J. and Peir\'o, J.},
  volume = {121},
  number = {23},
  pages = {5405--5425},
  journal = ijnme,
  year = {2020},
  url = {https://onlinelibrary.wiley.com/doi/10.1002/nme.6529},
  doi = {10.1002/nme.6529},
  abstract = {We present an $rp$-adaptation strategy for the high-fidelity simulation of compressible inviscid flows with shocks.  The mesh resolution in regions of flow discontinuities is increased by using a variational optimiser to $r$-adapt the mesh and cluster degrees of freedom there.  In regions of smooth flow, we locally increase or decrease the local resolution through increasing or decreasing the polynomial order of the elements.  This dual approach allows us to take advantage of the strengths of both methods for best computational performance, thereby reducing overall cost of the simulation.  The adaptation workflow uses a sensor for both discontinuities and smooth regions that is cheap to calculate, but the framework is general and could be used in conjunction with other feature-based sensors or error estimators.  We demonstrate this proof-of-concept using two geometries at transonic and supersonic flow regimes.  The method was implemented in the open-source spectral/$hp$ element framework {\em Nektar++}, and its dedicated high-order mesh generation tool {\em NekMesh}.  The results show that the proposed $rp$-adaptation methodology is a reasonably cost-effective way of improving accuracy.  }
}

A flow with shocks needs fine resolution at the discontinuities and can manage with less elsewhere. This paper supplies it two ways at once: a variational optimiser moves the mesh to cluster degrees of freedom at the shocks, while the polynomial order is raised or lowered locally in the smooth regions. The approach is demonstrated on two geometries in transonic and supersonic flow, using Nektar++ and NekMesh.