My research focus is vortex particle methods for flow simulation. In the last two decades I have created a number of Eulerian, Lagrangian, and hybrid vortex methods in both two- and three-dimensions. In addition, I am interested in GPU and parallel programming, computational geometry, cellular automata, and high-resolution image rendering and display. See my research on ORCID
Most fluid dynamics algorithms use Eulerian methods, in which flow moves through fixed, regular triangular or rectangular elements. In addition, they use velocity-pressure coordinates, which allows simulation of compressible flows, but imposes severe limitations on time step size and algorithmic stability. A cure to that is Stam's Stable Fluids method, which makes a fluid simulation stable, but at the cost of large numerical dissipation. My solution to these problems is a novel backward-timestep Semi-Lagrangian vortex method. Here are CPU and GPU versions of that method, first implemented in 2004.
Digital Ebru, a 2D real-time fluid simulator with extremely low diffusion, is written in Lua and OpenGL. Vorticity is advected using a 4th order Runge-Kutta method with 4th order M4' interpolation kernels. Velocity is computed from vorticity using a multigrid solver implemented in OpenGL compute shaders - which I believe is the first time that has been done. The code is open-source, and uses J. Susinno's OpenGL With Luajit framework.
vic2d is the original, C-based, 2D fluid simulator from 2004. Similarly, vorticity is advected using a 4th order Runge-Kutta method with 4th order M4' interpolation kernels, but velocity is computed using a multithreaded multigrid solver for the CPU. The code is open-source, and has a minimum of dependencies.
Recently, Applied Scientific Research has released an open source project for 2D flows to github. It is part of an ongoing NIH-funded project to create a new tool to improve heart valve designers' design toolbox. The first release of the software and a relevant paper follow.
Omega2D is a new, real-time 2D fluid simulator with GUI written in C++. It merges a standard Lagrangian Vortex Particle Method with a panel-based Boundary Element Method to make a complete tool. The code is open-source, borrows from a number of header-only libraries, and should compile on Linux, Windows, and MacOS.
M. Stock and A. Gharakhani, Open Source Accelerated Vortex Particle Methods for Unsteady Flow Simulation [PDF paper], ASME 2020 Fluids Engineering Division Summer Meeting, Jul 12-16, 2020, Orlando, FL.
The most capable vortex methods stem from discretization of vorticity as Lagrangian vortex blobs, with the vorticity-velocity inversion calculated using a multipole-accelerated treecode, Fast Multipole Method, or Vortex-in-Cell method. Applied Scientific Research is currently conducting advanced research in vortex particle methods, including Large-Eddy Simulation, efficient parallel treecodes, and GPGPU for vortex simulation. Publications that I have been involved with at ASR follow.
M. Stock, A. Gharakhani, and C. Stone, Modeling Rotor Wakes with a Hybrid OVERFLOW-Vortex Method on a GPU Cluster [PDF paper] [PDF slides] [Movie at YouTube], AIAA 28th AIAA Applied Aerodynamics Conference, 28 Jun-1 Jul 2010, Chicago, IL.
M. Stock and A. Gharakhani, A GPU-accelerated Boundary Element Method and Vortex Particle Method [PDF paper] [PDF slides], AIAA 40th Fluid Dynamics Conference and Exhibit, 28 Jun-1 Jul 2010, Chicago, IL.
M. Stock and A. Gharakhani, Toward efficient GPU-accelerated N-body simulations [PDF paper] [PDF slides], 46th AIAA Aerospace Sciences Meeting, 7-10 January 2008, Reno, NV.
A. Gharakhani and M. Stock, 3-D Vortex simulation of flow over a circular disk at an angle of attack, 17th AIAA Computational Fluid Dynamics Conference, 6-9 June 2005, Toronto, Canada.
A. Gharakhani, J. Sitaraman, M. Stock, A Lagrangian vortex method for simulating flow over 3-D objects, 2005 ASME Fluids Engineering Division Summer Meeting, June 19-23, Houston, TX.
My dissertation research concerned three-dimensional vortex sheet methods for inviscid simulation. At the time, the only three-dimensional vortex sheet methods could not track flows for very long times because they were limited either by rectangular or curvature-dependent discretization methods. My donation to the field was a front-tracking vortex sheet method that used edge splitting and node merging to accomplish long-time simulation of inviscid vortex sheet flows. The research spawned a paper appearing in the Journal of Computational Physics.
M. Stock, W.J.A. Dahm, G. Tryggvason, Impact of a vortex ring on a density interface using a regularized inviscid vortex sheet method, Journal of Computational Physics, 227/21, pp. 9021-9043, Nov. 2008, 2.5 MB PDF
M. Stock, A Regularized Inviscid Vortex Sheet Method for Three Dimensional Flows With Density Interfaces, PhD thesis, University of Michigan, Ann Arbor, 2006, 7.0 MB PDF
In the process of conducting this research, I assembled a summary of the major aspects of vortex methods research as well as 415 literature references. It is a living document, and thus contains not only references and questions, but my throughts regarding various lines of research. I am making it available in the hopes that newcomers to the field might find it useful.
M. Stock, Summary of Vortex Methods Literature, unpublished, 2007, 1 MB PDF