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Add a constraint equation

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Old   July 10, 2013, 09:28
Default Add a constraint equation
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sébastien cadalen
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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, V is the volume of the fluid domain, and T_0 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 C is a row vector composed of V_i/V (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
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Old   April 2, 2019, 15:23
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Hello,

Were you able to figure out the solution to a problem. I am trying to add a constraint equation in OpenFOAM
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Old   September 21, 2021, 07:26
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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.
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