Vortex particle-mesh methods: accurate and efficient handling of solid boundaries

Vortex methods have been through many developments over the past decades, yet the efficient and accurate treatment of solid boundaries has remained a challenge. In this talk, we will discuss recent efforts aimed towards that goal, and in the context of a vortex particle-mesh (VPM) method. The VPM method is a state-of-the-art variant, which relies on a dual discretization: the particles handle the advection of vorticity, while the mesh is used, not only for the “remeshing” (i.e., the “particle redistribution” step, required in Lagrangian methods to maintain time accuracy), but also for the efficient evaluation of the differential operators (diffusion, vortex stretching) and the efficient solution of the elliptic problems (Biot- Savart law, re-projection step when in 3D). Specifically, this presentation will focus on the handling of solid boundaries, first through penalization methods, then through Immersed Interface techniques. We discuss the porting of such techniques to the context of vortex methods and some specific developments to ensure their efficiency [3, 5, 4]. Additional results for the handling of fluid-structure interaction problems will also be presented [2, 1]. References [1] Caroline Bernier, Mattia Gazzola, Renaud Ronsse, and Philippe Chatelain. Coupling a vortex particle-mesh method to a multi-body system solver for the simulation of articulated swimmers. In 7th International Conference on Vortex Flows and Vortex Models (ICVFM 2016), Sep 2016. [2] Mattia Gazzola, Philippe Chatelain, Wim M. van Rees, and Petros Koumoutsakos. Simulations of single and multiple swimmers with non-divergence free deforming geometries. Journal of Computa- tional Physics, 230(19):7093–7114, 8 2011. [3] Thomas Gillis, Gr ́egoire Winckelmans, and Philippe Chatelain. An efficient iterative penalization method using recycled krylov subspaces and its application to impulsively started flows. Journal of Computational Physics, 347:490–505, 2017. [4] Thomas Gillis, Gr ́egoire Winckelmans, and Philippe Chatelain. Fast immersed interface poisson solver for 3d unbounded problems around arbitrary geometries. Journal of Computational Physics, 354:403–416, 2018. [5] Yves Marichal, Philippe Chatelain, and Gr ́egoire Winckelmans. Immersed interface interpolation schemes for particle-mesh methods. Journal of Computational Physics, 326:947–972, Dec 2016