Regularized vortex particle methods offer an appealing alternative to common mesh-based numerical methods for simulating vortex-driven fluid flows. While inherently mesh-free and adaptive, a stable implementation using particles for discretizing the vorticity field must provide a scheme for treating the overlap condition, which is required for convergent regularized vortex particle methods. Moreover, the use of particles leads to an N-body problem. By the means of fast, multipole-based summation techniques, the unfavorable yet intrinsic O(N2)-complexity of these problems can be reduced to at least O(N log N). However, this approach requires a thorough and challenging analysis of the underlying regularized smoothing kernels. In his project, we introduce a novel class of algebraic kernels, analyze its properties and formulate a decomposition theorem, which radically simplifies the theory of multipole expansions for this case. This decomposition is of great help for the convergence analysis of the multipole series and an in-depth error estimation of the remainder. We use these results to implement a massively parallel Barnes-Hut tree code with O(N log N )-complexity, based on the hybrid MPI/Pthreads code PEPC, which is developed at Jülich Supercomputing Centre (JSC). A thorough investigation shows excellent scalability up to the full IBM Blue Gene/P system JUGENE at JSC for suitable problem sizes. We demonstrate the code’s capabilities along different numerical examples, including the dynamics of two merging vortex rings. In addition, we extend the tree code to account for the overlap condition using the concept of remeshing, thus providing a promising and mathematically well-grounded alternative to standard mesh-based algorithms. In cooperation with Jülich Supercomputing Centre.
Prof. Dr. Rolf Krause; PI; ICS Institute of Computational Science