Implementation on Laplace-Beltrami operator

kuria · February 19, 2019, 03:16

Hey everyone

I was trying to implement Laplace-Beltrami operator of velocity
$\nabla_s u_i = \nabla u_i - (\Vec{m}\cdot\nabla u_i)\Vec{m}$
$\nabla^2_s u_i = \nabla\cdot\nabla_s u_i - (\Vec{m}\cdot\nabla)( \nabla_s u_i)\Vec{m}$

So this is what I tried
volVectorField GradUi= fvc::grad(phi) - (m & fvc::grad(phi))*m;
volScalarField LBUi= fvc::div(GradUi) - (m & fvc::grad( GradUi & m );

But the problem with is implementation is that
GradUi (a tensor)
fvc::div(GradUi) is a vector because divergence of tensor
(m & fvc::grad( GradUi & m ) is a scalar

Can some one help with implementing these equation?

Thanks in advance

massive_turbulence · February 19, 2019, 06:55

Quote:

Originally Posted by kuria

Hey everyone

I was trying to implement Laplace-Beltrami operator of velocity
$\nabla_s u_i = \nabla u_i - (\Vec{m}\cdot\nabla u_i)\Vec{m}$
$\nabla^2_s u_i = \nabla\cdot\nabla_s u_i - (\Vec{m}\cdot\nabla)( \nabla_s u_i)\Vec{m}$

So this is what I tried
volVectorField GradUi= fvc::grad(phi) - (m & fvc::grad(phi))*m;
volScalarField LBUi= fvc::div(GradUi) - (m & fvc::grad( GradUi & m );

But the problem with is implementation is that
GradUi (a tensor)
fvc::div(GradUi) is a vector because divergence of tensor
(m & fvc::grad( GradUi & m ) is a scalar

Can some one help with implementing these equation?

Thanks in advance

Which do you think is wrong , the openfoam implementation or the math behind the equations being solved? To me the openfoam part looks fine and the formula being solved is incorrect. The only operation between a scalar and a vector (that I'm familiar with) is multiplication so I don't see how the formula makes sense.

massive_turbulence · February 21, 2019, 04:46

Quote:

Originally Posted by kuria

Hey everyone

I was trying to implement Laplace-Beltrami operator of velocity
$\nabla_s u_i = \nabla u_i - (\Vec{m}\cdot\nabla u_i)\Vec{m}$
$\nabla^2_s u_i = \nabla\cdot\nabla_s u_i - (\Vec{m}\cdot\nabla)( \nabla_s u_i)\Vec{m}$

So this is what I tried
volVectorField GradUi= fvc::grad(phi) - (m & fvc::grad(phi))*m;
volScalarField LBUi= fvc::div(GradUi) - (m & fvc::grad( GradUi & m );

But the problem with is implementation is that
GradUi (a tensor)
fvc::div(GradUi) is a vector because divergence of tensor
(m & fvc::grad( GradUi & m ) is a scalar

Can some one help with implementing these equation?

Thanks in advance

There maybe a way to represent the solution of a scalar and vector commutative operation as a parametric scalar equation but as far as I can see openfoam doesn't have such a representation. You would have to overload the minus operator for your equation to take the scalar and vector types and then create a new type that takes the components of each scalar and vector as a parametric equation.

kuria · February 21, 2019, 05:57

I looked around a little bit and as far I can see the equation what I have wrote before is what people have used.

I found this implementation to atleast compile:
surfaceVectorField GradUi= fvc::interpolate(fvc::grad(phi)) - (m & fvc::interpolate(fvc::grad(phi)) )*m;

surfaceScalarField LBUi1= fvc::interpolate(fvc::div(GradUi)) & mesh.Sf();
surfaceScalarField LBUi2= (m & ( fvc::interpolate(fvc::grad(GradUi)) ) ) & m;
LBUi = (LBUi1+LBUi2)/mesh.magSf();

I used the normal surface vector to the cell (mesh.Sf()) to ensure that the terms are compatible. I found a similar approach while calculating the curvature of interface where curvature which is defined as
$k = - \nabla \cdot \nabla \frac{\nabla\alpha}{|\nabla\alpha|}$
and implemented as
surfaceVectorField nHatfv(gradAlphaf/(mag(gradAlphaf) + deltaN));

surfaceScalarField nHatf = nHatfv & Sf;
volScalarField K = -fvc::div(nHatf);

What do you think of the the way I implemented it?

massive_turbulence · February 25, 2019, 10:00

Quote:

Originally Posted by kuria

I looked around a little bit and as far I can see the equation what I have wrote before is what people have used.

I found this implementation to atleast compile:
surfaceVectorField GradUi= fvc::interpolate(fvc::grad(phi)) - (m & fvc::interpolate(fvc::grad(phi)) )*m;

surfaceScalarField LBUi1= fvc::interpolate(fvc::div(GradUi)) & mesh.Sf();
surfaceScalarField LBUi2= (m & ( fvc::interpolate(fvc::grad(GradUi)) ) ) & m;
LBUi = (LBUi1+LBUi2)/mesh.magSf();

I used the normal surface vector to the cell (mesh.Sf()) to ensure that the terms are compatible. I found a similar approach while calculating the curvature of interface where curvature which is defined as
$k = - \nabla \cdot \nabla \frac{\nabla\alpha}{|\nabla\alpha|}$
and implemented as
surfaceVectorField nHatfv(gradAlphaf/(mag(gradAlphaf) + deltaN));

surfaceScalarField nHatf = nHatfv & Sf;
volScalarField K = -fvc::div(nHatf);

What do you think of the the way I implemented it?

It's nice!

February 19, 2019, 03:16	Implementation on Laplace-Beltrami operator	#1
kuria Member K Join Date: Mar 2018 Posts: 34 Rep Power: 8	Hey everyone I was trying to implement Laplace-Beltrami operator of velocity $\nabla_s u_i = \nabla u_i - (\Vec{m}\cdot\nabla u_i)\Vec{m}$ $\nabla^2_s u_i = \nabla\cdot\nabla_s u_i - (\Vec{m}\cdot\nabla)( \nabla_s u_i)\Vec{m}$ So this is what I tried volVectorField GradUi= fvc::grad(phi) - (m & fvc::grad(phi))*m; volScalarField LBUi= fvc::div(GradUi) - (m & fvc::grad( GradUi & m ); But the problem with is implementation is that GradUi (a tensor) fvc::div(GradUi) is a vector because divergence of tensor (m & fvc::grad( GradUi & m ) is a scalar Can some one help with implementing these equation? Thanks in advance

February 21, 2019, 05:57		#4
kuria Member K Join Date: Mar 2018 Posts: 34 Rep Power: 8	I looked around a little bit and as far I can see the equation what I have wrote before is what people have used. I found this implementation to atleast compile: surfaceVectorField GradUi= fvc::interpolate(fvc::grad(phi)) - (m & fvc::interpolate(fvc::grad(phi)) )*m; surfaceScalarField LBUi1= fvc::interpolate(fvc::div(GradUi)) & mesh.Sf(); surfaceScalarField LBUi2= (m & ( fvc::interpolate(fvc::grad(GradUi)) ) ) & m; LBUi = (LBUi1+LBUi2)/mesh.magSf(); I used the normal surface vector to the cell (mesh.Sf()) to ensure that the terms are compatible. I found a similar approach while calculating the curvature of interface where curvature which is defined as $k = - \nabla \cdot \nabla \frac{\nabla\alpha}{\|\nabla\alpha\|}$ and implemented as surfaceVectorField nHatfv(gradAlphaf/(mag(gradAlphaf) + deltaN)); surfaceScalarField nHatf = nHatfv & Sf; volScalarField K = -fvc::div(nHatf); What do you think of the the way I implemented it? massive_turbulence likes this.

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