Hailing from the scicomp community recently getting into computer-graphics, I have noticed that the scicomp communities talk about mesh-based methods like FDM, FVM, FEM, etc, vs meshfree or meshless methods like SPH (though I like to call them virtual particle methods), but I've never heard them talk about Eulerian approach vs Lagrangian approach, something I hear a lot in computer-graphics communities involved in computational-physics/mechanics, and they never talk about the terms meshfree or meshless, and rarely mention finite-elements.

Would it be correct to say that Eulerian approach is another name for mesh-based simulation, while Lagrangian approach is a synonym for meshfree/meshless?

If not, can anyone clarify, or provide a good reading on this topic? (No need to invoke the boat analogy in which Lagrangian is like sitting in a boat in a river, and Eulerian is like watching the boat from the river bank. I've heard/read/watched that analogy about a dozen times and it has zero comparison to FEM, meshless, meshfree and is therefore useless in answering my question).


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    $\begingroup$ Langrangian/Eulerian refers to the frame of reference, not whether a mesh is used or not. $\endgroup$ – Paul Jul 7 '17 at 1:41
  • $\begingroup$ I do not wish to be pedant here, but many mesh-based method can be Lagrangian or particle based method can sometimes be Eulerian. As @Paul mentioned, it is solely the frame of reference in which the equations are solved. For instance, for rotating domains in CFD, we often do rotating geometries with a static mesh, but by solving in a Lagrangian (in this case rotating) frame of reference we are able to obtain the rotating geometry. On a more practical point of view, you will sometimes see meshlesh particle-based methods (PIC, SPH) being referred to as Lagrangian methods. $\endgroup$ – BlaB Jul 10 '17 at 11:43
  • $\begingroup$ @BlaisB For a CFD problem with rotating symmetry, I would use polar/cylindrical/spherical coordinate system, whichever is best, instead of cartesian, and that's it. I don't see how I need to make any other change, or why I should claim to have switched from an Eulerian to a Lagrangian frame of reference. I hope you don't mind, but your example simply gives me the impression that these are redundant new terminologies for well-known and older concepts. $\endgroup$ – Fi Zixer Jul 13 '17 at 9:27
  • $\begingroup$ @Paul and yet the Wikipedia page on Eulerian/Lagrangian CFD says these are 'specifications of the flow field' and not 'frames of reference' (which it calls loose notation), and that either specification could be used in any observer's frame of reference, and any coordinate system. $\endgroup$ – Fi Zixer Jul 13 '17 at 9:45
  • $\begingroup$ But what if you have no rotating symmetry. If you have an impeller or a turbine without baffle, then you often use a change of reference frame. There are multiple variation of this ( single rotating frame, multiple rotating frame and sliding mesh ). See Blais et al 2016 in computer and chemical engineering for a thorough explanation and images. $\endgroup$ – BlaB Jul 13 '17 at 10:28

Well, I do not claim to be an expert on the subject as I am still an early grad student, but I do have experience completing long-term projects in both areas, so I think I can probably frame an answer here in the context of fluid mechanics.

As Paul pointed out in the comment, Lagrangian/Eulerian refers to the frame of reference, the boat analogy :P. Strictly speaking, you cannot directly relate them to meshed/meshless methods. FEM, FVM are usually seen as meshed methods because they rely on the assumption of infinite, tiny control volumes making up a larger volume; normally people rely on the continuum assumption to hold and the fluid to have uniform properties. But resolve the interior of the particles. In this case, it is still a meshfree method, but since we are tracking the particle trajectories, it is no longer a purely Eulerian problem. But if, the particle is significantly deformable, and FEM is used to figure out its internal stresses, we are in the realm of Eulerian tracking.

As another example, molecular dynamics simulations-which I think helped form the part of the fundamental technology behind computer graphics simulations-are purely Lagrangian approaches, tracking particles through space. But use MD on couette/poiseuille flow to figure out the entry/exit velocities of the channel as a whole and voila! you are suddenly in the Eulerian regime(tracking averaged properties over a control volume)

My point is, that while these methods have developed differentiation based on the enormous applications in a wide range of study, in computer graphics I am principally concerned about making it look the way I want it to. So the discussion might usually boil droplet/particle tracking down to Lagrangian and water surface animations and stuff to Eulerian. But they are not interchangeable.


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