Vortex particle methods (VPM) have historically faced computational bottlenecks caused by the pairwise interactions of particles in solving the vorticity-velocity coupling. Several algorithms have been developed to address this bottleneck, including the Fast Multipole Method (FMM). This work describes the development of a FMM solver which is integrated within an in-house developed open-source VPM solver. The FMM improves the computational efficiency of particle velocities and velocity gradients from order of N^2 to order of N. The formulation uses a full octree structure and a Cartesian formulation of multipole and local expansions based on Taylor series.
Inline with the increasing adoption of GPUs in performing scientific computations, the solver developed assigns computationally heavy tasks to parallelised execution on the CPU or GPU, using the taichi compiler. The accompanying introduction of reduced floating point accuracy introduces new considerations for the accuracy of the method when performed on the GPU. Error quantification is performed on a single timestep for a range of flow scenarios, with an emphasis on the effect of main FMM parameters on accuracy. Results indicate a positive impact of increasing the Taylor series truncation order on FMM accuracy, up to a plateau caused by the accumulation of rounding errors. The octree depth was found to have low impact on the accuracy. Validation of the solver is performed through underresolved direct numerical simulation (DNS) of Lamb-Oseen vortex, vortex ring, and colliding vortex ring test cases. The impact of FMM on particle velocities and positions throughout simulations is measured, and key flow diagnostics are provided, indicating minimal FMM impact on the physical validity of simulations.
More chaotic flows exhibited an unbounded nature of FMM error growth while low Reynolds number flows counteracted the FMM error. The performance of the main operations of the solver was profiled, highlighting trade-offs between speed and accuracy, and indicating areas for future performance improvements. ... Read more