Eulerian vs Lagrangian vs Mesh-based vs Meshfree/Meshless methods

Question

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).

Thanks.

Answer 1

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.