Transient problem, one PDE

HydrOpTop considers a discrete adjoint.

Considering groundwater flow with a time-dependent pressure field \(P\) (i.e. head), a design parameter \(X\) and a general transient cost function

\[F(P(X),X) = \sum_i^n f(P_i(X),X,t_i)\]

Where \(P_i(X)\) is the pressure field at discretized time \(t_i\) solved by the equation

\[h(P_{i+1}(X), P_i(X), X, \Delta t_i) = 0\]

\(h\) represents the time integrator of the pressure. The initial pressure field could be the solution to another steady-state PDE (like solution of the Darcy equation)

\[g(P_0(X), X) = 0\]

the optimization problem is given by

\[\begin{split}&\min_X F(P(X),X) = \sum_i^n f(P_i(X),X,t_i) \\ &s.t. \begin{array} ( g(P_0(X), X) = 0 \\ h(P_{i+1}(X), P_i(X), X, \Delta t_i) = 0 \end{array}\end{split}\]

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

\[\mathcal{L} = f(P_0(X),X) + \mu^T g(P_0(X), X) + \sum_{i=1}^n \left[ f(P_i(X),X,t_i) + \lambda_{i}^T h(P_i(X), P_{i-1}(X), X, \Delta t_i) \right]\]

Taking the derivative of the lagrangian leads, after few ordering

\[\begin{split}\frac{d \mathcal{L}}{dX} = \frac{dF}{dX} = \left(\frac{\partial f}{\partial P_0} + \mu^T \frac{\partial g}{\partial P_0} + \lambda_1^T \frac{\partial h}{\partial P_0} \right) \frac{d P_0}{d X} + \mu^T \frac{\partial g}{\partial X} + \\ \sum_{i=1}^{n-1} \left[ \left(\frac{\partial f}{\partial P_i} + \lambda_i^T \frac{\partial h}{\partial P_i} \lambda_{i+1}^T \frac{\partial h}{\partial P_i} \right) \frac{\partial P_i}{\partial X} + \lambda_i^T \frac{\partial h}{\partial X} \right] + \\ \left(\frac{\partial f}{\partial P_n} + \lambda_n^T \frac{\partial h}{\partial P_n} \right) \frac{d P_n}{d X} + \lambda_n^T \frac{\partial h}{\partial X} + n \frac{\partial f}{\partial X}\end{split}\]

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

\[\begin{split}(\frac{\partial h}{\partial P_n})^T \lambda_n = - (\frac{\partial f}{\partial P_n})^T \\ (\frac{\partial h}{\partial P_i})^T (\lambda_i + \lambda_{i+1}) = - (\frac{\partial f}{\partial P_i})^T \\ (\frac{\partial g}{\partial P_0})^T \mu = - (\frac{\partial f}{\partial P_0})^T - (\frac{\partial h}{\partial P_0})^T \lambda_1\end{split}\]

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

\[\frac{dF}{dX} = n \frac{\partial f}{\partial X} + \mu^T \frac{\partial g}{\partial X} + \sum_{i=1}^{n} \lambda_i^T \frac{\partial h}{\partial X}\]

Yet, all of this was not yet tested…