In this problem I am trying to solve, I have a messy set of points distributed in 3D space, each with a defined temperature. If I would want to calculate the heat transfer scenario in this system, how would you recommend me an approach (numerically and with each computational tool)? I thought about triangulating the points and using a unstructured grid Finite Volume Method. Considering the maths around this is a bit harder than that of a structured grid, I assumed the possibility of interpolating the unstructured grid into a structured grid, solving the discretized diffusion model and then interpolating back to the unstructured grid.

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    $\begingroup$ What do you mean by "calculate the heat transfer scenario"? $\endgroup$ – nicoguaro Apr 24 '18 at 16:03
  • $\begingroup$ Solving the transient heat equation $\endgroup$ – Vinícius Godim Apr 24 '18 at 17:18
  • $\begingroup$ What I understand from your first sentence is that you already knew the temperature at the points. $\endgroup$ – nicoguaro Apr 24 '18 at 17:22
  • $\begingroup$ Yes, but I want to calculate the temperature field (at each point) in the next instant in time considering thermal diffusion. $\endgroup$ – Vinícius Godim Apr 24 '18 at 17:26
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    $\begingroup$ But we don't know how heat is transferred. Are the points connected by little rods that transport heat? Are the points part of a homogenous medium? An imhomogenous medium? If it is part of a homogenous medium, what do you know about the initial temperature between points? Without saying what you want to do, i.e., what the exact model is, there is no correct answer to your question. $\endgroup$ – Wolfgang Bangerth Apr 25 '18 at 4:32

If you have a cloud of points and you don't want to use mesh-based methods like FEM or FVM, a possibility is to use a mesh-free method like the Finite Point Method. For instance, you could have a look at this article:

Tatari, M., Kamranian, M., & Dehghan, M. (2011). The finite point method for the p-Laplace equation. Computational Mechanics, 48(6), 689-697.

There are many other mesh-free methods, Finite Point Method is only one of them.


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