Add a constraint equation

sebastien.cadalen · July 10, 2013, 09:28

Dear all,

This post is about the way to impose an integral relation on the problem variables. As an example, let us consider a steady-state thermal transport problem with the following constraint:
$\frac{1}{V} \int_{\mathcal{V}} T \, d\mathcal{V} = T_0$
where $\mathcal{V}$ is the fluid domain,

is the volume of the fluid domain, and

a given temperature.

From a mathematical point of view this is equivalent to introduce a constant source term in the domain to satisfy the constraint. If the unconstrain problem is writen in the discrete space $A \cdot X = B$ hence the constrained problem is:
$\left( \begin{array}{cc} A & C^t \\ C & 0 \end{array} \right) \cdot \left( \begin{array}{c} X \\ \lambda \end{array} \right) = \left( \begin{array}{c} B \\ T_0 \end{array} \right)$
where

is a row vector composed of

(the cell volume divided by the total volume) and $\lambda$ is the Lagrange multiplier associated to the constraint.

This way the constraint is implicitly solved. We can also imagine the same kind of contraint on a part of the domain, on a surface of the domain, or involving more than one unknown, etc.

Is there any way to solve such a problem with implicit or explicit method?

If yes, how behaves the linear solvers with such a matrix to invert?

Best regards,

Sébastien

charles4allme · April 2, 2019, 15:23

Hello,

Were you able to figure out the solution to a problem. I am trying to add a constraint equation in OpenFOAM

fanny · September 21, 2021, 07:26

Hi all,

I would also want to impose a constraint as an integral relation over the whole fluid domain, but I do not know how to to do it..

Do you have some ideas ?
May I use " fvOptions type vectorDomainIntegral; " ? How it works ? Is there somewhere any tutorial ?

Thank you
Fanny

PS: I am using OpenFOAM v7.

July 10, 2013, 09:28	Add a constraint equation	#1
sebastien.cadalen New Member sébastien cadalen Join Date: Jul 2013 Posts: 1 Rep Power: 0	Dear all, This post is about the way to impose an integral relation on the problem variables. As an example, let us consider a steady-state thermal transport problem with the following constraint: $\frac{1}{V} \int_{\mathcal{V}} T \, d\mathcal{V} = T_0$ where $\mathcal{V}$ is the fluid domain, is the volume of the fluid domain, and a given temperature. From a mathematical point of view this is equivalent to introduce a constant source term in the domain to satisfy the constraint. If the unconstrain problem is writen in the discrete space $A \cdot X = B$ hence the constrained problem is: $\left( \begin{array}{cc} A & C^t \\ C & 0 \end{array} \right) \cdot \left( \begin{array}{c} X \\ \lambda \end{array} \right) = \left( \begin{array}{c} B \\ T_0 \end{array} \right)$ where is a row vector composed of (the cell volume divided by the total volume) and $\lambda$ is the Lagrange multiplier associated to the constraint. This way the constraint is implicitly solved. We can also imagine the same kind of contraint on a part of the domain, on a surface of the domain, or involving more than one unknown, etc. Is there any way to solve such a problem with implicit or explicit method? If yes, how behaves the linear solvers with such a matrix to invert? Best regards, Sébastien RGUP likes this.

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April 2, 2019, 15:23		#2
charles4allme Member Join Date: Feb 2018 Posts: 91 Rep Power: 8	Hello, Were you able to figure out the solution to a problem. I am trying to add a constraint equation in OpenFOAM

September 21, 2021, 07:26		#3
fanny New Member Join Date: Feb 2016 Posts: 13 Rep Power: 10	Hi all, I would also want to impose a constraint as an integral relation over the whole fluid domain, but I do not know how to to do it.. Do you have some ideas ? May I use " fvOptions type vectorDomainIntegral; " ? How it works ? Is there somewhere any tutorial ? Thank you Fanny PS: I am using OpenFOAM v7.