3

I would like to know if there is anything available in order to discretize a 3D curve given by parametric equations in order to apply FEM analysis, e.g. to solve the wave equation on a thin wire with the shape of the given curve.

SpaceChild
  • 353
  • 1
  • 6

2 Answers2

3

It can be done easy in MMA 11, just define the Region-wire by built-in functions

ir = ParametricRegion[{Cos@t, (Sin@t)^2, 10/(t + 0.1)^0.5}, {{t, 0, 6 π}}]
DiscretizeRegion[ir]

enter image description here

Rom38
  • 5,129
  • 13
  • 28
  • When I run it to MA 11 the following error message appears : "DiscretizeRegion did not find any sample points in the region ParametricRegion[<<2>>] with bounds {{-1.,1.},{0.,1.},{2.29721,31.6228}}. If this is not correct, different bounds may help" – SpaceChild Jul 25 '17 at 09:50
  • 1
    Also the depicted curve is 2D while in the parametricregion command you used a 3D representation – SpaceChild Jul 25 '17 at 10:39
  • In my case (MMA 11.1), it is working without error-messages. This figure is namely 3D figure which can be rotated as you wish. And the region is in common the 3D-region. – Rom38 Jul 26 '17 at 04:14
  • Ok the 11.1 version is capable for this discretization, but the NDSolve and NDSolveValue commands cannot solve a PDE on this curve. It returns "The current version of NDSolve cannot solve equations over boundaries or surfaces. Please specify a region where the embedding dimension is the same as the dimension."... Any idea? – SpaceChild Jul 26 '17 at 11:13
  • Guys, what was an initial question? I guess, "How to discretize parametric curves for FEM analysis" means that answer have to be about discretization, or not? There is not a problem with discretization, it is easy. The main problem of topic starter (as it is evident now) is about how to use the obtained mesh. Unfortunately, Wolfram have failed the realization of the FEM.. – Rom38 Jul 27 '17 at 03:46
  • @user21 you're right, I accepted the answer before trying to use NDSolve. Of course the main problem is how to use the obtained mesh, but I thought that once obtained, the utilization would be simpler. Unfortunately this is not the case when using Mathematica. – SpaceChild Jul 27 '17 at 12:50
  • @DK13, probably the easiest is to do full 3D analysis - would this be prohibitive? – user21 Jul 27 '17 at 12:53
  • @Rom38 sorry, but since FEM analysis isn't applicable I think that user21 is right! – SpaceChild Jul 27 '17 at 12:54
  • @Rom38, I would not go as far as saying Wolfram have failed the realization of the FEM - as the FEM in Mathematica does work in 1/2/3D - what the question asks is if a 1D object embedded in 3D can be used for FEM analysis. If you know of a FEM tool that can do that, I'd appreciate if you could share a link. Thanks. – user21 Jul 27 '17 at 12:57
  • @user21 yes I tried 3D analysis and it works. However the 3D case depends on the cross sectional shape of the wire or tube, something which limits the arbitrariness. My goal was to study the propagation of waves, some kind of modal analysis on torus knots and generalized helices. I'm not sure if an analytic approach is possible, so I thought that I could start with a numerical analysis. Do you have any idea? – SpaceChild Jul 27 '17 at 12:59
  • @DK13, my understanding is that an embedding of a 1D object to in 3D is also a limitation. Modes do behave differently depending on the boundary shape. I am not sure modal analysis without taking the boundary into consideration is possible. Maybe this is useful. – user21 Jul 27 '17 at 13:35
  • @user21 yes indeed this is very useful. Now I would like to turn this problem into a time dependent one. I'll try to do it... – SpaceChild Jul 27 '17 at 13:38
  • @DK13 a modal analysis is closely related to a transient (time dependent) analysis. See the details section of NDEigensystem that has some information. – user21 Jul 27 '17 at 16:19
2

No, you have to discretize the wire fully; alternatively you could find a mapping of your wire to 1D and go from there.

user21
  • 39,710
  • 8
  • 110
  • 167