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sebastien.cadalen July 10, 2013 08:28

Add a constraint equation
 
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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