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fvm::laplacian on a regular tetrahedral mesh

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Old   May 30, 2017, 08:29
Default fvm::laplacian on a regular tetrahedral mesh
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Benjamin Kalloch
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Hello everyone!

I have a seemingly simple issue for which I cannot find a solution, probably because I am a beginner with OpenFOAM and FVM in general.

My test-case:
- an ideal capacitor with two infinitely large capacitor plates and a homogeneous conductive medium in between as a steady-state case

The Geometry:
- box domain 5x5x5m on a regular tetrahedral grid
- left and right boundaries of the box have a value of +5/-5 Volts
- all other boundaries (top&bottom, front&back) have a zeroGradient BC
(see the attached images 'domain_1' & 'domain_2' for a depiction of the geometry and the BCs)
domain_1.jpg
domain_2.jpg

The solver setup:
I want to solve the electrical potential between the capacitor plates using the equation
Code:
0 = ∇( sigma ∇(ElPot) )
using this simple custom solver application
Code:
#include "fvCFD.H"

int main( int argc, char *argv[] )
{
    #include "setRootCase.H"

    #include "createTime.H"      
    #include "createMesh.H"      
    #include "createFields.H"    

// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
    solve( fvm::laplacian( sigma, ElPot) );

    ElPot.write();
    
    return(0);
}
which should result in a perfect gradient from +5 to -5 Volts between the capacitor plates. In more detail: values sampled from a line parallel to the capacitor planes should be all equal and values sampled perpendicular to the capacitor planes should change linearly.
(see image 'expected_result'; aside from the [expected] jumps from one cell to another the value remains the same)
expected_result.jpg
This works perfectly fine for a hexahedral grid.

The issue:
However the result is impaired by the structure of the regular tetrahedral grid. Values sampled from a line parallel to the capacitor plates are not the same but slowly decrease.
(see image 'calculated_result'; here the values decrease over the sampling-line)
calculated_result.jpg

As a discretization scheme I use:
Code:
divSchemes
{
  default       Gauss linear limited 1.0;
} 
laplacianSchemes
{
    laplacian(sigma,ElPot)  Gauss linear limited 1.0;   
}
I tried to choose the discretization schemes according to:
http://www.tfd.chalmers.se/~hani/kur...stPractice.pdf

The solver settings are are:
Code:
ElPot
    {
        solver            PCG;
        preconditioner    DIC;
        tolerance        1e-15;
        relTol            0;        
        maxIter         50000; 
    };
My question:
Is this a inevitable issue with this kind of grid and the way OpenFOAM solves the laplacian or did I miss a vital scheme / setting? How could I solve this issue?

Thanks a lot in advance!

Last edited by ben_k; June 7, 2017 at 09:52. Reason: For a better referencing, I have inserted the attachements directly into the text.
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