Mark Stock


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

Eulerian Velocity-Vorticity Methods

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.

2D Open-Source Lagrangian Vortex Particle Methods

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.

3D Lagrangian Vortex Particle Methods

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.

3D Vortex Sheet Method

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

More to come...