source term modification in poisson equation does not affect results

babakflame · August 17, 2016, 17:58

Dear Fellows

I have written a simple poissonFoam code which contains a source term in form of a

function.

The code solves this equation:
$\nabla^2 \varphi = \beta sinh (\alpha \varphi)$ .
I have attached the code and a sample case. I have assigned different values for $\alpha$ and $\beta$ , however, I get the following distribution every time. Do u have any idea why this happens and how would I be able to see the effect of $\alpha$ and $\beta$ on the results?

PS: in the attached picture, phi is nondimensionalized by the value at the left boundary of sample case (alpha3.tar.gz) and x is non-dimensionalized by 30e-09.

babakflame · August 17, 2016, 23:13

Dear Fellows

I have posted the text of my poissonFoam.c and createFields.H files below. Probably, I have a problem in clearly defining my source term:

poissonFoam.C :

Code:

#include "fvCFD.H"

// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //

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

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

    // * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //

    Info<< "\nStarting iteration loop\n" << endl;

    while (runTime.loop())
    {
        Info<< "Iteration = " << runTime.timeName() << nl << endl;

        solve
        (
            fvm::laplacian(phi) - beta*sinh(alpha*phi)
        );

        runTime.write();

        Info<< "ExecutionTime = " << runTime.elapsedCpuTime() << " s"
            << "  ClockTime = " << runTime.elapsedClockTime() << " s"
            << nl << endl;
    }

    Info<< "End\n" << endl;

    return 0;
}

createFields.H :

Code:

   Info<< "Reading physicalProperties\n" << endl;

    IOdictionary physicalProperties
    (
        IOobject
        (
            "physicalProperties",
            runTime.constant(),
            mesh,
            IOobject::MUST_READ_IF_MODIFIED,
            IOobject::NO_WRITE
        )
    );

    dimensionedScalar alpha
    (
        physicalProperties.lookup("alpha")
    );

    dimensionedScalar beta
    (
        physicalProperties.lookup("beta")
    );


    Info<< "Reading field phi\n" << endl;
    volScalarField phi
    (
        IOobject
        (
            "phi",
            runTime.timeName(),
            mesh,
            IOobject::MUST_READ,
            IOobject::AUTO_WRITE
        ),
        mesh
    );

Any hint is kindly appreciated.

babakflame · August 19, 2016, 22:22

Dear Fellows

I figured out my problem. It was on the value of Phi which actually made the hyperbolic function be in the location near zero.

BTW: the poisson solver works perfectly. feel free to use it.

Tobi · August 22, 2016, 09:19

Yes, phi is in radian

babakflame · September 5, 2016, 21:18

Dear Fellows

I am experiencing a weird issue in dealing with poisson equation in OpenFoam. In the first post of this thread you can see that I have written a poissonFoam code that works with openFoam 2.1.x.
This code solves the following relation:
$\nabla^2 \varphi = \beta sinh (\alpha \varphi)$

Actually, I am interested in solving above equation in one dimensional which in my grid would be from left wall to right wall.

My issue is with the grids. If I use a rectangular shaped grid with known and fixed values for $\varphi$ on left and right boundaries, the result is a linear distribution for $\varphi$ . The linear distribution mathematically happens merely when the source term in poisson equation is zero (in my case $\beta sinh (\alpha \varphi)$ ). But my source term is not zero.
However, if the grid be a sector with the angle of $1^\circ$ , then the resulting distribution for $\varphi$ is an exponential distribution.

I have attached the grid types with the achieved distribution for $\varphi$ .

As you can see from the figures the value of $\varphi$ at left wall is 25.4e-03 and on the right wall is 0.

Actually, I am interested in solving the above poisson equation in 1 dimensional. So, using a sector (even with 1 degree) is somehow unphysical for me.

Has some body else experienced sth similar to me? or any hint on this weird experience?

Is there any way to solve above equation on rectangular-shaped grid?

You have my solver above, you can try this by yourself and see what happens.

Tobi · September 6, 2016, 03:27

Hello Bobi,

I am not 100% sure but if we would solve the simple laplace equation for a disc (or wedge) we would observe the same result. The analytical solution for the temperature equation within a cylinder is a logarithmic law (not linear). So I think everything is okay with your calculation.

babakflame · September 6, 2016, 13:13

Hey Tobi

Many thanks for your reply. Let me explain this a little further.

I am interested in solving equation
$\nabla^2 \varphi^ {\ast}= \beta sinh (\alpha \varphi \ast)$ in 1 dimensional situation with non-zero value at left side and zero value at right hand side. In one dimensional I have:
$\frac{d^2 \varphi^ \ast}{d \eta ^2}= \beta sinh (\alpha \varphi \ast)$

Here $\eta = \frac {y}{h}$ is the non-dimensional coordinate and $\varphi^ \ast = \frac{\varphi}{\zeta}$ is the non-dimensional variable present in poisson equation.

Here, $\zeta$ is the value of the left wall used to non-dimesionalize the $\varphi$ variable.

Since the right boundary is zero, by integrating twice, the above non-dimesional equation gives the following solution:
$\varphi ^ \ast = \frac{4}{\alpha} tanh^{-1} \left[tanh (\frac {\alpha}{4}) exp (-\sqrt{\alpha \beta}\eta)\right]$

In the attached picture, you can observe the analytic solution:

I am interested in solving above equation with
$\alpha =1 , \beta = 10000, h=30e-09 , \zeta= 25.4e-03$ .
AFA I know, first of all, I need to make this equation dimensioanl in 1D and solve it with my solver. How ever, as I described in previous post, the results are based on grid shape which is weird for me. I mean If I use rectangular grid, result is linear and with curved left and right walls, result are exponential, however, I want the above solution seen in my results without grid shape dependancy

Am I missing sth here?

By the way, the dimensional equation which I am solving becomes as follows:

$\frac{d^2 (\varphi / \zeta)}{d (y / h)^2}= \beta sinh (\alpha \varphi / \zeta)$

That is so weird for me

babakflame · September 9, 2016, 14:14

Dear Fellows

The last thing that comes to my mind is that "probably there is a bug in O.F.2.1.x since It can not solve the Poisson equation (which is actually a Laplacian equation with source term) correctly on rectangle grids.

My equation is $\nabla^2 \varphi^ {\ast}= \beta sinh (\alpha \varphi \ast)$
with left boundary condition 1 and right boundary condition zero would give the following exponential solution:
$\varphi ^ \ast = \frac{4}{\alpha} tanh^{-1} \left[tanh (\frac {\alpha}{4}) exp (-\sqrt{\alpha \beta}\eta)\right]$

However OF. 2.1.x gives linear disribution on roughly 1D rectangle grids

Tobi · September 11, 2016, 04:13

What came into mind is, what is alpha in your graph? Degrees? If yes, you have to transform this into Radian because foam work's with radian.

babakflame · September 11, 2016, 14:11

Dear Tobi

Many thanks for your reply and your time.

Here, $\alpha$ is just a coefficient that helps us to achieve different exponential distributions. Otherwise, all results are the same.

As it is seen in the attached figure, the following distributions are achieved by assuming different values for $\alpha$ from 1 to 10 and $\beta$ is just another constant considered to be equal to 10000.

I just want to solve the following Poisson equation in 1 dimensional. This direction is assumed to be y direction between two boundaries. The value at left boundary is considered to be $\zeta$ (could be any value) and right boundary is zero.
The non-dimensional equation is:
$\frac{d^2 \varphi^ \ast}{d \eta ^2}= \beta sinh (\alpha \varphi^ \ast)$

using these scales: $\eta = \frac {y}{h} \quad$ , $\varphi^ \ast = \frac{\varphi}{\zeta}$

I found the dimensional equation as:
$\frac{d^2 (\varphi / \zeta)}{d (y / h)^2}= \beta sinh (\alpha \varphi / \zeta)$

which finally becomes:

$\frac{d^2 (\varphi )}{d (y^2)}= \frac{\beta \zeta}{h^2} sinh (\frac{\alpha}{\zeta} \varphi)$

Here, I have used the value of

and $\beta=1000 ,\quad \zeta= 25.4e-03$ and $\alpha$ from 1 to 10.

I have uploaded my results from the above curved grid. However, a roughly 1 D rectangular grid just gives linear lines. Even , my results from a curved grid are not fitted completely.

Here, in my results, the distance from the wall has been divided by 30e-09.

PS: the distance between left and right boundary is considered to be 120e-09 m in my simulation. So $\chi$ is from 0 to 4.

Tobi · September 12, 2016, 04:49

Dear Bobi,

I am sorry for the last replay. I did not read everything and I saw that I already mentioned the stuff with the radian. Well I am not so familiar with non-dimensionalizing equations and using them in OpenFOAM. If the solution (integral) is correct, you should be able to get the same or closely the same values.

In my opinion you will always have differences between the wedge and linear mesh. For me it is simply because of the fact that if we would solve only the laplace equation, we would also observe different behaviors. Wedge - curved (the analytical solution is a logarithmic one); 1D mesh, linear solution. This is related to the non linear distribution from the inner ring to the outer ring. Like, inner ring: area = x; outer ring: area = y while x << y.

I checked your source term:

$\mathrm{sinh}{(\phi \alpha)}$

To do that, I made a new field and stored it. The result of the source term is as expected (like the distribution you showed). The behavior is like a 1/x function but based on the small grid, the source term become like 1e-8 which will not decrease your quantity phi very much and that is the reason for the linear distribution. By the way, I was implementing the following equation:

$\frac{\partial \phi}{\partial t} - \frac{\partial^2 \phi}{\partial x^2} = \beta \mathrm{sinh}{(\phi \alpha)}$

Maybe the fact that I did not non-dimens. this equation lead to the behavior I got. In any case I have to say that I never did something like you are doing at the moment and therefore I am sorry not to provide you more information. Maybe Sergei (Zappo) or other more advanced and skilled people can help you in that topic.

Good luck Bobi.

babakflame · September 12, 2016, 10:04

Thanks Tobi.

I will try to add transient term and check it that way too.

Zeppo · September 16, 2016, 13:00

But your plots "numeric/analytical" are qualitatively very similar. Unfortunatelly, I don't know why they don't match completely.

babakflame · September 16, 2016, 15:18

@Tobi

many thanks for your time Tobi. Actually, I made an elliptic code (without time derivative) that now applies the Poisson equation with

source term on rectangular grids correctly.

But still a problem exists.

@Sergei
Thanks Sergei for looking at my question.

Actually, I made a code that solves
$\frac{d^2 \varphi}{d x ^2}= \beta sinh (\varphi)$ on rectangular grids

Right Now, I am capable of achieving results up to the value of $\beta = 9$ and the results seem fine. However, further increasing of $\beta$ will result in the following error:

Code:

0  Foam::error::printStack(Foam::Ostream&) at ??:?
#1  Foam::sigFpe::sigHandler(int) at ??:?
#2   in "/lib/x86_64-linux-gnu/libc.so.6"
#3  __sinh_finite in "/lib/x86_64-linux-gnu/libm.so.6"
#4  sinh in "/lib/x86_64-linux-gnu/libm.so.6"
#5  Foam::sinh(Foam::Field<double>&, Foam::UList<double> const&) at ??:?
#6  
 at ??:?
#7  
 at ??:?
#8  
 at ??:?
#9  __libc_start_main in "/lib/x86_64-linux-gnu/libc.so.6"
#10  
 at ??:?
Floating point exception (core dumped)

The code is so simple, that other than refining grid and reducing the PCG solver tolerance into lower than 10e-11 nothing comes to my mind. The following parts show the createFields and poisson.C routines:

Code:

 volScalarField beta
    (
        IOobject
        (
            "beta",
            runTime.timeName(),
            mesh,
            IOobject::NO_READ,
            IOobject::NO_WRITE
        ),
        mesh,
    dimensionedScalar("beta", dimensionSet(0, -2, 0, 0, 0, 0, 0), scalar(12.0))
    );


    Info<< "Reading field phi\n" << endl;
    volScalarField phi
    (
        IOobject
        (
            "phi",
            runTime.timeName(),
            mesh,
            IOobject::MUST_READ,
            IOobject::AUTO_WRITE
        ),
        mesh
    );

    volScalarField tmp1
    (
        IOobject
        (
            "tmp1",
            runTime.timeName(),
            mesh,
            IOobject::NO_READ,
            IOobject::AUTO_WRITE
        ),
         // Foam::sinh(alpha * phi)
           
        1 *  Foam::sinh(phi)
    );

Code:

 while (runTime.loop())
    {
        Info<< "Iteration = " << runTime.timeName() << nl << endl;

        solve
        (
              fvm::laplacian(phi) -  beta * sinh(phi)
//          fvm::laplacian(phi) - beta * (phi + (1/6)*pow(phi,3) + (1/120)*pow(phi,5) + (1/362880)*pow(phi,9) + (1/39916800)*pow(phi,11))
        );

        runTime.write();

        Info<< "ExecutionTime = " << runTime.elapsedCpuTime() << " s"
            << "  ClockTime = " << runTime.elapsedClockTime() << " s"
            << nl << endl;
    }

I even tried to implement the

term as a summation of polynomials, but this did not remove the above error.

The following is my fvSolution and fvSchemes:

Code:

laplacianSchemes
{
    default         Gauss linear corrected;
    laplacian(phi)  Gauss linear corrected;
}

Code:

 phi
    {
        solver          PCG;
        preconditioner  DIC;
        tolerance       1e-11;
        relTol          0.0;
    }

If you could help me figure out this error, I would really appreciate it.

Zeppo · September 16, 2016, 15:40

Quote:

Originally Posted by babakflame

However, further increasing of $\beta$ will result in the following error:

Does the error show up before even iteration process starts running?

babakflame · September 16, 2016, 15:47

Hi Sergei

Actually, it pops up after around 100 iterations, I monitored the value of $\varphi$ , it seems that by increasing the vaue of $\beta$ over 9, during the simulation the $\varphi$ value exceeds 1e22 and leads into the above error.

However, this deosn't happen for lower values of $\beta$

Tobi · September 16, 2016, 16:22

Hi,

based on the fact that you do not have a time derivation (this is somehow a limiter), you should underrelax your equation. For that you have to build the matrix first, relax it and then solve it. I am not sure if it will help but I think it should. If you increase the value of the quantity beta, your problem gets more stiff and your solution procedure has to be improved. That is my experience but I think Sergei can make a better statement.

Good luck.

babakflame · September 16, 2016, 16:27

Thanks Tobi

Your suggestion makes sense to me too.
I will try it and provide the feedback.

Regards

babakflame · September 16, 2016, 16:55

Hey Tobi,

I built the matrix and did the relaxation as follows inside poisson.C file as:

Code:

fvScalarMatrix phiEqn

        
        (
              fvm::laplacian(phi) -  beta * sinh(phi)
//          fvm::laplacian(phi) - beta * (phi + (1/6)*pow(phi,3) + (1/120)*pow(phi,5) + (1/362880)*pow(phi,9) + (1/39916800)*pow(phi,11))
        );

         phiEqn.relax();
         phiEqn.solve();

I think these were the required modifications. Is there anything else that I forgot?
Now, the code runs for $\beta =9$ , however, for $\beta = 12$ gives the same error:

Code:

Iteration = 0.0005

DICPCG:  Solving for phi, Initial residual = 1, Final residual = 7.04378610773e-12, No Iterations 79
ExecutionTime = 0.07 s  ClockTime = 1 s

Iteration = 0.00055

#0  Foam::error::printStack(Foam::Ostream&) at ??:?
#1  Foam::sigFpe::sigHandler(int) at ??:?
#2   in "/lib/x86_64-linux-gnu/libc.so.6"
#3  __sinh_finite in "/lib/x86_64-linux-gnu/libm.so.6"
#4  sinh in "/lib/x86_64-linux-gnu/libm.so.6"
#5  Foam::sinh(Foam::Field<double>&, Foam::UList<double> const&) at ??:?
#6  
 at ??:?
#7  
 at ??:?
#8  __libc_start_main in "/lib/x86_64-linux-gnu/libc.so.6"
#9  
 at ??:?
Floating point exception (core dumped)

I will test the code with time derivative as well and produce the feedback.

Regards

Tobi · September 16, 2016, 16:57

I am not sure if my assumptions were correct. It was just a guess. Well, do you have the necessary entries in the fvSolutions? I think so, or not?

Code:

relaxationFactors
{
    equations
    {
         phi       0.3;
         phiFinal   1;
     }
}

September 12, 2016, 04:49		#11
Tobi Super Moderator Tobias Holzmann Join Date: Oct 2010 Location: Tussenhausen Posts: 2,708 Blog Entries: 6 Rep Power: 51	Dear Bobi, I am sorry for the last replay. I did not read everything and I saw that I already mentioned the stuff with the radian. Well I am not so familiar with non-dimensionalizing equations and using them in OpenFOAM. If the solution (integral) is correct, you should be able to get the same or closely the same values. In my opinion you will always have differences between the wedge and linear mesh. For me it is simply because of the fact that if we would solve only the laplace equation, we would also observe different behaviors. Wedge - curved (the analytical solution is a logarithmic one); 1D mesh, linear solution. This is related to the non linear distribution from the inner ring to the outer ring. Like, inner ring: area = x; outer ring: area = y while x << y. I checked your source term: $\mathrm{sinh}{(\phi \alpha)}$ To do that, I made a new field and stored it. The result of the source term is as expected (like the distribution you showed). The behavior is like a 1/x function but based on the small grid, the source term become like 1e-8 which will not decrease your quantity phi very much and that is the reason for the linear distribution. By the way, I was implementing the following equation: $\frac{\partial \phi}{\partial t} - \frac{\partial^2 \phi}{\partial x^2} = \beta \mathrm{sinh}{(\phi \alpha)}$ Maybe the fact that I did not non-dimens. this equation lead to the behavior I got. In any case I have to say that I never did something like you are doing at the moment and therefore I am sorry not to provide you more information. Maybe Sergei (Zappo) or other more advanced and skilled people can help you in that topic. Good luck Bobi. babakflame likes this. __________________ Keep foaming, Tobias Holzmann

September 16, 2016, 16:22		#17
Tobi Super Moderator Tobias Holzmann Join Date: Oct 2010 Location: Tussenhausen Posts: 2,708 Blog Entries: 6 Rep Power: 51	Hi, based on the fact that you do not have a time derivation (this is somehow a limiter), you should underrelax your equation. For that you have to build the matrix first, relax it and then solve it. I am not sure if it will help but I think it should. If you increase the value of the quantity beta, your problem gets more stiff and your solution procedure has to be improved. That is my experience but I think Sergei can make a better statement. Good luck. babakflame likes this. __________________ Keep foaming, Tobias Holzmann

September 16, 2016, 16:57		#20
Tobi Super Moderator Tobias Holzmann Join Date: Oct 2010 Location: Tussenhausen Posts: 2,708 Blog Entries: 6 Rep Power: 51	I am not sure if my assumptions were correct. It was just a guess. Well, do you have the necessary entries in the fvSolutions? I think so, or not? Code: relaxationFactors { equations { phi 0.3; phiFinal 1; } } babakflame likes this. __________________ Keep foaming, Tobias Holzmann

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August 19, 2016, 22:22		#3
babakflame Senior Member Bobby Join Date: Oct 2012 Location: Michigan Posts: 454 Rep Power: 15	Dear Fellows I figured out my problem. It was on the value of Phi which actually made the hyperbolic function be in the location near zero. BTW: the poisson solver works perfectly. feel free to use it. Tobi likes this.

August 22, 2016, 09:19		#4
Tobi Super Moderator Tobias Holzmann Join Date: Oct 2010 Location: Tussenhausen Posts: 2,708 Blog Entries: 6 Rep Power: 51	Yes, phi is in radian __________________ Keep foaming, Tobias Holzmann

September 6, 2016, 03:27		#6
Tobi Super Moderator Tobias Holzmann Join Date: Oct 2010 Location: Tussenhausen Posts: 2,708 Blog Entries: 6 Rep Power: 51	Hello Bobi, I am not 100% sure but if we would solve the simple laplace equation for a disc (or wedge) we would observe the same result. The analytical solution for the temperature equation within a cylinder is a logarithmic law (not linear). So I think everything is okay with your calculation. __________________ Keep foaming, Tobias Holzmann

September 9, 2016, 14:14		#8
babakflame Senior Member Bobby Join Date: Oct 2012 Location: Michigan Posts: 454 Rep Power: 15	Dear Fellows The last thing that comes to my mind is that "probably there is a bug in O.F.2.1.x since It can not solve the Poisson equation (which is actually a Laplacian equation with source term) correctly on rectangle grids. My equation is $\nabla^2 \varphi^ {\ast}= \beta sinh (\alpha \varphi \ast)$ with left boundary condition 1 and right boundary condition zero would give the following exponential solution: $\varphi ^ \ast = \frac{4}{\alpha} tanh^{-1} \left[tanh (\frac {\alpha}{4}) exp (-\sqrt{\alpha \beta}\eta)\right]$ However OF. 2.1.x gives linear disribution on roughly 1D rectangle grids

September 11, 2016, 04:13		#9
Tobi Super Moderator Tobias Holzmann Join Date: Oct 2010 Location: Tussenhausen Posts: 2,708 Blog Entries: 6 Rep Power: 51	What came into mind is, what is alpha in your graph? Degrees? If yes, you have to transform this into Radian because foam work's with radian. __________________ Keep foaming, Tobias Holzmann

September 12, 2016, 10:04		#12
babakflame Senior Member Bobby Join Date: Oct 2012 Location: Michigan Posts: 454 Rep Power: 15	Thanks Tobi. I will try to add transient term and check it that way too.

September 16, 2016, 13:00		#13
Zeppo Senior Member Sergei Join Date: Dec 2009 Posts: 261 Rep Power: 21	But your plots "numeric/analytical" are qualitatively very similar. Unfortunatelly, I don't know why they don't match completely.

September 16, 2016, 15:47		#16
babakflame Senior Member Bobby Join Date: Oct 2012 Location: Michigan Posts: 454 Rep Power: 15	Hi Sergei Actually, it pops up after around 100 iterations, I monitored the value of $\varphi$ , it seems that by increasing the vaue of $\beta$ over 9, during the simulation the $\varphi$ value exceeds 1e22 and leads into the above error. However, this deosn't happen for lower values of $\beta$

September 16, 2016, 16:27		#18
babakflame Senior Member Bobby Join Date: Oct 2012 Location: Michigan Posts: 454 Rep Power: 15	Thanks Tobi Your suggestion makes sense to me too. I will try it and provide the feedback. Regards