The 3D vortex particle method in parallel computations on many GPUs Andrzej Kosior ⇑ , Henryk Kudela Wroclaw University of Technology, Wybrzeze Wyspianskiego 27, 50-537 Wroclaw, Poland article info Article history: Received 6 May 2013 Received in revised form 18 September 2013 Accepted 9 October 2013 Available online xxxx Keywords: Leapfrogging Head-on collision Vortex in cell Parallel computations Graphics cards abstract Parallel implementation of the Vortex-in-Cell (VIC) method for 3D flow on many graphics cards was pre- sented. As test problems it was chosen the leapfrogging and head-on collision of two vortex rings for which a well documented visualization exists in the literature. Our aim was to show the great potential of the VIC method for solution of 3D flow problems and that it is very well suited for parallel computation. Ó 2013 Elsevier Ltd. All rights reserved. 1. Introduction Numerical solution of the 3D Navier – Stokes equations for high Reynolds number, using any method, is a very time consuming process. Recently there is not much increase in the computational power of a single processor. Instead we are forced to use multicore architectures and parallel computation. But to take advantage of their potential one needs to use proper numerical algorithms. One of the multiprocessor hardware that can be efficiently used in scientific computations is the graphics processing unit (GPU). It is built of hundreds of simple streaming processors which alto- gether give a great computational power. GPUs are relatively cheap and commonly available. Our first implementation used only one graphics card for computation. We were able to use the computational grid of 128  128  128 nodes (on a newer GPU we were able to fit com- putational grid of 224  224  224 nodes). Our results can be found in [11]. Very quickly we found the limitation of the RAM memory of a single GPU. We were forced to use many graphics cards. For communication we used MPI library. As a numerical method we chose the 3D Vortex-in-Cell (VIC) method that become more and more important method in numer- ical investigation of the fluid dynamics phenomena [4–6,20,18]. In this method particles carry information about vorticity. It is well known that the velocity may be calculated from the vorticity distri- bution. Next the vortex particles are displaced according to local velocity field. Particles intensity is then interpolated back to the grid nodes. To simulate the effect of the viscosity the viscous split- ting was used and the diffusion equation was solved in each time step. The VIC method is very well suited for parallel computation [11,10,19,7]. The displacement and redistribution processes, which have to be done at each time step, have a local character and the computations for each particle can be done independently. So the whole set of particles can be divided into independent groups and operations over these groups can be done concurrently. In the paper it was presented the numerical results of the inter- action between two vortex rings. The phenomena of leapfrogging and head-on collision of vortex rings were simulated. Experimental results for both cases are well documented in the literature [12,13]. The structure of the article is as follows: in the next section a short description of the VIC method is given, in Section 3 it was presented the numerical test cases and its results and the last sec- tion are closing remarks. 2. Equations of the motion and description of the vortex particle method Equations of incompressible and viscous fluid motion have the following form: @u @t þðu  rÞu ¼ 1 q rp þ mDu ð1Þ r  u ¼ 0 ð2Þ where u =(u, v, w) is velocity vector, q is fluid density, p is pressure, m is kinematic viscosity. Eq. (1) can be transformed to the Helmholtz equation for vorticity evolution [21]: 0045-7930/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compfluid.2013.10.011 ⇑ Corresponding author. Tel.: +48 713203553; fax: 48 713417708. E-mail addresses: andrzej.kosior@pwr.wroc.pl (A. Kosior), henryk.kudela@pwr. wroc.pl (H. Kudela). Computers & Fluids xxx (2013) xxx–xxx Contents lists available at ScienceDirect Computers & Fluids journal homepage: www.elsevier.com/locate/compfluid Please cite this article in press as: Kosior A, Kudela H. The 3D vortex particle method in parallel computations on many GPUs. Comput Fluids (2013), http:// dx.doi.org/10.1016/j.compfluid.2013.10.011