Optimal control of flow with discontinuities

Chris Homescu, I.M.Navon

Optimal control of the 1-D Riemann problem of Euler equations whose solution is characterized by discontinuities is carried out by both nonsmooth and smooth optimization methods. By matching a desired flow to the numerical model for a given time window we effectively change the location of discontinuities. The control parameters are chosen to be the initial values for pressure and density fields. Existence of solutions for the optimal control problem is proven. A high resolution model and a model with artificial viscosity, implementing two different numerical methods, are used to solve the forward model. The cost functional is the weighted difference between the numerical values and the observations for pressure, density and velocity. The observations are constructed from the analytical solution. We consider either distributed observations in time or observations calculated at the end of the assimilation window. We consider two different time horizons and two sets of observations. The gradient (respectively a subgradient) of the cost functional, obtained from the adjoint of the discrete forward model, are employed for the smooth minimization (respectively for the nonsmooth minimization) algorithm. Discontinuity detection improves the performance of the minimizer for the model with artificial viscosity by selecting the points where the shock occurs (and these points are then removed from the cost functional and its gradient). The numerical flow obtained with the optimal initial conditions obtained from the nonsmooth minimization matches very well the observations. The algorithm for smooth minimization converges for the shorter time horizon but fails to perform satisfactorily for the longer time horizon. Key words: optimal control of shocked flow, adjoint method, nonsmooth optimization, LBFGS unconstrained minimization, Euler equations, discontinuity detection, distributed observations