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Old   February 13, 2008, 13:00
Default Smooth reconstruction of flowsolution
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Can anyone give me any references about smooth reconstruction of different order of accuracy of numerical flow solutions for postprocessing? I need to compute the result of applying the differential Euler or Navier-Stokes operators to a continuous flowfield obtained by smooth reconstruction of a numerical solution obtained with a node-centered 3D finite-volume code. Any clues? Thanks!
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Old   February 13, 2008, 14:01
Default Re: Smooth reconstruction of flowsolution
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there are variaty of methods for this purpose,

your field is formally C0 space, because in formal numerical method we just attend to continuty of field not its derivative (it is very obvious in FEM)

a very general method (not computationally cheap) is to assume a desired smooth space (C^n, n>=1) with its basis (e.g. high order spline basis) and then solve a least square problem to find unknown interpolation coefficient, this system is not essentially wel-posed (and could be over/under specify) and maybe methods like SVD are needed.

if you search computer graphics literature you find several alternatives, your problem is common there.

if you look for a toolbox have a look at this:

as a very simple and easy to implement solution:

assume your filed is member of L2 space (space filled by square integrable function, Hilbert space, L2-norm = \int (.)^2 ), so there is not garantee to have continuous derivative.

then you want to have a filedwith continuty of first derivative, so we should map from L2 to H1 (H1 is Sobolev space filled with function has continuous first derivative, H1-norm = \int (.)^ + \nabla(.)^2)

assume g is original and G is smoothed filed, for this mapping you should solve this poisson equation (weak solution):

- \nabla \cdot (\nabla G) + G = g

to control lenght scale of smoothing, it could be modified to:

- \nabla \cdot (D \nabla G) + G = g

where D is diffusion coefficent control smoothing, D = 0, could give original field.

Hope this helps.

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Old   February 14, 2008, 12:25
Default Re: Smooth reconstruction of flowsolution
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Thanks rt! I'll take a look at your suggestions.
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