A multiresolution Vortex Particle-Mesh method

The Vortex Particle-Mesh (VPM) method, known for its accuracy properties and is able to capture the transport of vortical structures over long times and distances is one of the state-of-the-art method for large-scale simulation of wake flows. This class of method solves the velocity-vorticity form of the Navier-Stokes equations and combines the advantages of a particle method, i.e. low numerical dissipation and dispersion errors, with those of a mesh-based approach: highly efficient Poisson solvers and finite difference stencils. Many applications of the VPM approach have been proposed for wind energy and aircraft wake studies and the method has been proven to be a fast and accurate alternative to pseudospectral solvers to simulate vortical flows. However, those simulations using uniform grids often result in prohibitive computational costs, especially in three-dimensions. Adaptive Mesh Refinement (AMR) tackles the issue by adapting the local grid resolution to the flow characteristics but brings additional complexity in terms of algorithm and (parallel) implementation compared to their uniform counterpart. The AMR algorithm must identify the need for refinement or the opportunity to coarsen the resolution, then adapt the grid while preserving the conservation properties of the simulated physics. At the same time, the overhead in computational cost due to the adaptation machinery must be as low as possible to have a minimum im- pact on the gain made by reducing the number of computational elements. Bergdorf et al. were the first to implement a Lagrangian method combined with adaptative grids, yet they used the coarsest representation of the velocity to advect the parti- cles. Rossinelli et al. proposed a wavelet-based particle method but introduced a Courant–Friedrichs–Lewy (CFL)-like constraint, waiving one of the advantages of the particle method. Lonfils [8] was able to achieve patch-based grid refinement, high order accuracy, and CFL relaxation at the same time, the method suffers from load balancing and scalability issues. While lots of new AMR frameworks tar- getting exascale architectures have emerged in recent years, such as Basilisk [9], AMReX [10] or murphy [11], there is, to the authors’ knowledge, no Vortex Particle-Mesh method benefiting from adaptive grid refinement and tailored for the new exascale architectures currently being developped or maintained. This work aims at filling this gap and presents a scalable and efficient implementation of the VPM method capitaliz- ing on adaptive grid refinement. Our solver has been developped within murphy, a wavelet-based multiresolution framework for scientific computing on 3D block-structured collocated grids. We build a new approach to represent the particles, that allows their refinement or coarsening, thereby facilitating the interpolation between particles and mesh of varying resolutions. To solve the Poisson equation, we employ a Multigrid approach, with an FFT-based direct solver. The resulting software is designed to leverage parallel architectures, nevertheless further investigations are required to assess the scaling behavior of the particle method.

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Balty, P., Lambrechts, J., Duponcheel, M., Chatelain, P., & et al. (2024). A multiresolution Vortex Particle-Mesh method. DisCoVor 2024, Delft, Netherlands. https://hdl.handle.net/2078.5/261091