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Old   August 3, 2006, 18:41
Default diffusion on surface of a sphere
  #1
Vena Pearl Bongolan
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Does anyone have simple code for this (e.g., MatLab), assuming there is no flux in nor out of the sphere, ie, it is only TWO-dimensional!

This can be done in terms of spherical harmonics, but what I really want to do is solve the Fokker-Planck (diffusion/convection) on the sphere, so I would rather 'modify' existing code for this.

Also, I intend to add stochastic terms to it; all the more reason I don't want to start from scratch!

Thanks in advance!
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Old   August 4, 2006, 15:14
Default Re: diffusion on surface of a sphere
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Jonas Holdeman
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I have a Matlab code using divergence-free finite elements that I think would be almost trivial to modify for the surface of a sphere. I know what the spherical surface elements look like. What would a test problem look like?
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Old   August 4, 2006, 15:52
Default Re: diffusion on surface of a sphere
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Vena Pearl Bongolan
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Thank you very much, Jonas!

Indeed, my problem is simple: I just want to solve the diffusion equation on the surface (2D problem) of a sphere. There is no flux in nor out, i.e., it is a 2-manifold.

On a plane, the equation is u_t = u_xx + u_yy or

u_t = laplacian u

This has been worked on by many people (although I find the approach of spherical harmonics most intuitive), but what I ultimately want to do is solve Fokker-Planck on the surface of the sphere. This is just a diffusion/convection/reaction equation, so I will just add the other terms.

Then I would like to add small perturbations to the radius of the sphere, to simulate a cell that is growing.

The Fokker-Planck on this growing cell has theoretical results from Paris 6, and they need numerical work done on this.

This is where I come in. If you have the code for simple diffusion on the unit sphere, and are willing to share it, that would be great. Do you have a paper I can cite?
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Old   August 4, 2006, 16:24
Default Re: diffusion on surface of a sphere
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Hrvoje Jasak
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Hi,

Well, I've got an option for you, but it's not trivial. A student of mine did a series of simulations of free rising bubbles of air in water with mesh deformation and surfactant effects in OpenFOAM. In order to handle the surfactant behaviour, he/we implemented a 2-D Finite Volume method on a moving surface in 3-D (I call it Finite Area).

The code is freely available and shipped as a part of OpenFOAM - I can also give you some examples + there is a series of comparisons in the PhD Thesis.

The reason I say this is not trivial is that mesh generation, post-processing etc is not as slick as the rest of OpenFOAM. For examples of this in action, have a look at the 3-D rising bubble simulations in some of my slides, e.g.

http://www.h.jasak.dial.pipex.com/pr...t_7Apr2005.pdf

on page 23.

Hrv
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Old   August 9, 2006, 20:02
Default Re: diffusion on surface of a sphere
  #5
Vena Pearl Bongolan
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Thank you very much, Hrv!

Just snooped around your presentation, and indeed, your application is not trivial, but might very well be the proper approach to one application I have in mind, viz, a growing cell (start with a sphere, and randomly perturb the radius in space and time). I know people who have theoretical results for this.

It does not matter that the post-processing is not 'slick', right now, I am in the 'study' phase, so that is the least of my problems.

Will appreciate links to downloads, and examples you can provide.

Thank you very much for taking the time.

Pearl
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Old   August 22, 2006, 08:03
Default Divergence-free finite element code
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Yubin Yan
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Dear Jonas, I am now working a project for the stabilization of the Navier-Stiokes equation. I need to solve the linearized Navier-Stokes equation by using the divergence-free finite element space. Could you please share your matlab code? Thank you very much. This is the first time I found this message place. I am very appreciated that you could write to my email address Y.Yan@sheffield.ac.uk for some information of the code. Best regards, Yubin Yan
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Old   August 22, 2006, 08:07
Default Re: diffusion on surface of a sphere
  #7
Yubin Yan
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thanks
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