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August 1, 2011, 18:39 
[solved]differentiation in one direction

#1 
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Mieszko Młody
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Dear Foamers,
Does anyone of you know what is the best way to dicsretize following problem: we have: f(x,y)  scalar function we need: f_xxxx (fourth order) differentiation but only in one direction. i.e. laplacian(f) = f_xx + f_yy but I need only f_xx (in fact I need f_xxxx) Thanks ZM Last edited by ziemowitzima; August 4, 2011 at 14:16. Reason: solved 

August 2, 2011, 01:55 

#2 
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Fabian
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You should use a difference quoutient, like you have propably seen for f_x. You can perform the difference quotient "four" times or you can have a look here (they have already done it for you):
http://en.wikipedia.org/wiki/Difference_quotient Probably it should be the best for you to read the whole article, but you can also jump ahead to: http://en.wikipedia.org/wiki/Differe....C5.83th_Order 

August 2, 2011, 11:10 

#3 
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Mieszko Młody
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Thanks
This I know. I didn't put it explicit in my question but I need to know how to make it in OpenFOAM. Best 

August 2, 2011, 17:52 

#4 
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Mieszko Młody
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so far I found such explicit solution:
volVectorField gradT=fvc::grad(T); // (T_x, T_y) volScalarField Txxxx = gradT.component(0); // T_x gradT = fvc::grad(Txxxx); // (T_xx, T_xy) Txxxx = gradT.component(0); // T_xx gradT = fvc::grad(Txxxx); // (T_xxx, T_xxy) Txxxx = gradT.component(0); // T_xxx gradT = fvc::grad(Txxxx); // (T_xxxx, T_xxxy) Txxxx = gradT.component(0); // T_xxxx Maybe someone have some better idea ? Is it possible to descritize T_xxxx implicitly ? ZM 

August 4, 2011, 12:24 

#5 
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Mieszko Młody
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It seems that problem can be solved (after Akidess suggestion ):
"... it sounds like you want anisotropic diffusion  why not just pass a diffusion coefficient tensor to laplacian instead of a scalar?..." from: http://www.cfdonline.com/Forums/ope...operator.html So, x  direction diffusion: DTe1D DTe1D [ 0 2 1 0 0 0 0 ] (3e02 0 0 0 0 0 0 0 0 ); x,y and  directions diffusion: DTe3D DTe3D [ 0 2 1 0 0 0 0 ] (3e02 0 0 0 3e02 0 0 0 3e02 ); for f_xxxx it can be done as follows : suppose we have equation f_xxxx = g(x,y), substitution: h = f_xx so we have two "onedirectional" Poison equations to solve: h_xx = g(x,y) f_xx = h(x,y) in OF: fvm::laplacian(DTe1D, h) == g fvm::laplacian(DTe1D, f) == h 

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