Steady state, two sequentially-coupled PDEs

Considering the transport of a dissolved species at concentration \(C\) in groundwater with a pressure field \(P\) (i.e. head), a design parameter \(X\) and a general cost function

\[f(C(P,X),P(X),X)\]

\(C(P,X)\) is solved by the discretized transport equation

\[h(C,P(X),X) = 0\]

and \(P(X)\) solved by the Darcy / Richards equation

\[g(P(X),X) = 0\]

the optimization problem is given by

\[\begin{split}&\min_X f(C(P,X),X) \\ &s.t. \begin{array} ( g(P(X),X) = 0 \\ h(C(P,X),X) = 0 \end{array}\end{split}\]

The derivative of the cost function relative to the material property \(X\) could be obtained considering the Lagrangian

\[\mathcal{L} = f(C(P,X),P(X),X) + \lambda^T g(P,X) + \mu^T h(C,P(X),X)\]

Taking the derivative of the lagrangian leads

\[\frac{d \mathcal{L}}{dX} = \frac{df}{dX} = \frac{\partial f}{\partial C} \frac{\partial C}{\partial X} + \frac{\partial f}{\partial X} + \lambda^T (\frac{\partial g}{\partial P} \frac{\partial P}{\partial X} + \frac{\partial g}{\partial X}) + \mu^T (\frac{\partial h}{\partial C} \frac{\partial C}{\partial X} + \frac{\partial h}{\partial P}\frac{\partial P}{\partial X} + \frac{\partial h}{\partial X})\]

Further development leads

\[\frac{df}{dX} = (\frac{\partial f}{\partial C} + \mu^T \frac{\partial h}{\partial C}) \frac{\partial C}{\partial X} + (\lambda^T \frac{\partial g}{\partial P} + \mu^T \frac{\partial h}{\partial P}) \frac{\partial P}{\partial X} + \lambda^T \frac{\partial g}{\partial X} + \mu^T \frac{\partial h}{\partial X} + \frac{\partial f}{\partial X}\]

The unknown terms \(\frac{\partial P}{\partial X}\) and \(\frac{\partial C}{\partial X}\) could be withdraw by considering the adjoint equations

\[\begin{split}(\frac{\partial h}{\partial C})^T \mu = - (\frac{\partial f}{\partial C})^T \\ (\frac{\partial g}{\partial P})^T \lambda = - (\frac{\partial h}{\partial P})^T \mu\end{split}\]

And finally, the total derivative of the cost function relative to the material property \(X\) is

\[\frac{d f}{d X} = \lambda^T \frac{\partial g}{\partial X} + \mu^T \frac{\partial h}{\partial X} + \frac{\partial f}{\partial X}\]