In numerical fluid dynamics there are field methods like finite-volume, finite-element, etc. and particle methods like Smoothed-Particle-Hydrodynamics – SPH and others. Both approaches have advantages and disadvantages depending on the application. Both, in principal completely different approaches, can be combined using a Voronoi-Diagram. These Voronoi-Cells which behave like particles fill out the complete regarded space and are moving according to the physical laws (conservation of momentum and energy; see also http://ivancic.de/cfd2k/WhatIsCFD2k.html). These particles also can interact with each other (exchange of momentum, energy, etc. ==> the Voronoi cells/particles obey the Navier-Stokes Equations). Such a Voronoi approach can combine the advantages of field and particle methods and therefore lead to better results in the numerical fluid dynamics (e.g. for turbulent flows which cannot be predicted accurately up to now).

Unfortunately I do not know any numerical method to apply it for such Voronoi-Particles in order to describe physical flows correctly. Does anybody know such a numerical method or is anyone interested in developing such a method together with me? I am an aerospace engineer very familiar with fluid dynamics, turbulence and their physical and thermodynamic laws but sadly I am not an expert in numerical mathematics which is necessary to derive such a new approach.


3 Answers 3


There are some interesting numerical schemes for incompressible Euler equations that involve power diagrams (a generalization of Voronoi diagrams), there is a survey paper by Jean-Marie Mirebeau there:


The idea is based on a connection that was discovered by Yann Brenier between :

  • the projection operator onto the set of volume-preserving vector fields

  • the problem of optimal transport

More details on Yann Brenier's homepage: http://www.cmls.polytechnique.fr/perso/brenier/ see in particular:

  • polar decomposition of vector fields

  • a computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem

The problem of optimal transport has a certain form (semi-discrete) that naturally makes Voronoi diagram (or power diagrams) appear. The nice point is that the theory of optimal transport manipulates a very general class of mathematical objects (probablity measures), that can be both continous functions and sums of Dirac masses. This is when using a continuous function on one side and a sum of Dirac masses on the other side that Voronoi diagrams appear. More importantly, since the theory encompasses both cases, this is not a discretization, this is an application of the theory in a special case. Therefore, all the mathematical properties are preserved by the discrete version.

Recent numerical schemes were developed for some forms of Euler fluids, including:

  • Merigot and Mirebeau's paper: http://arxiv.org/abs/1505.03306 (where initial and final boundary conditions are fixed, i.e. the goal is to compute a certain geodesic that describes the motion of the fluid)
  • Power particles (a computer graphics paper that uses very similar notions, but with only initial boundary conditions fixed): http://dl.acm.org/citation.cfm?id=2766901

I think that these methods are interesting, because they can be directly derived from the principle of least action, and they can embed some conservation laws directly in their formulation, without needing to enforce additional constraint. I think this is very elegant.

The basic components for these numerical solution mechanisms is solving (some form of) the Monge-Ampere equation. I published a method to do that in 3D (my method is a 3D extension of a 2D multi-level solver by Merigot), the tech report is here:

(I published later a version in Mathematical Modeling and Analysis J.)

The source-code is part of my GEOGRAM open-source library:

Other references: there is also an interesting simulator for astrophysics (AREPO) that uses similar notions, reference is there:

A version of the algorithm that works for free-surface fluids (partial optimal transport) that I published in Journal of Computational Physics:

Applications in cosmology, that we published in Physical Review Letters:


Besides the PIC methods that Wolfgang Bangerth refers to there are some approaches in computer graphics or more specifically in computer animation that explicitly make use of Voronoi cell discretizations to solve the Navier-Stokes or Euler equations (without the viscous terms). In contrast to the PIC approaches, there the computational mesh is not constant but it changes during the simulation as the particles move.

The two methods I have in mind are are published by Brochu et al. (http://dl.acm.org/citation.cfm?doid=1778765.1778784) and Sin et al. (http://dl.acm.org/citation.cfm?doid=1599470.1599502). There, the particles carry all information and a Voronoi diagram is constructed in each time step to compute pressure corrections. Both methods are able to compute single-phase and pseudo-two-phase flows. It is important to note that these approaches are for computer animation, i.e., for visually plausible simulations and are not designed for highly accurate flow prediction. Maybe these two references might help in designing a method that you have in mind.

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    $\begingroup$ There is also this paper on a voronoi tesselation for relativistic hydrodynamics: arxiv.org/abs/1104.3562 $\endgroup$
    – Hydro Guy
    Mar 26, 2016 at 19:57

I don't know very much about the application you have in mind, but combinations of finite element or finite volume methods with particle methods typically go by the name "Particle-In-Cell" (or, for short, PIC) methods. You may want to do a literature search. These methods are quite popular in geodynamics.


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