Optimal Control Of Viscous Flow PDF Download

The present research is concerned with the application of optimal control theory to time-dependent viscous flows. Some specific designs of artificial hearts are described briefly, followed by a review on common hemodynamic problems created by artificial hearts and heart valves. Some mathematical models for certain types of artificial hearts are formulated. Optimal control theory has been applied to some of these models to obtain the necessary conditions of optimality. The optimization problem consists of determining the optimal control policy in order to minimize the value of an appropriate cost functional, which accounts for differences from a desired flow rate and the mean squared shear stresses and vorticity. The necessary conditions of optimality include the Navier-Stokes equations, the associated adjoint equations and an inequality. For the purpose of the numerical solution of the governing equations, a numerical code has been developed, which solves the Navier-Stokes and the corresponding adjoint equations by the Finite Element Method with penalty function formulation. The Navier-Stokes and the adjoint equations were solved iteratively. Each system of equations was solved separately but the two systems were iteratively coupled in the control problem context. At the end of each iteration a new control was computed using the inequality, and the process was repeated until convergence was achieved. Numerical solution of the optimization algorithm was achieved for a two-dimensional model with an initial velocity distribution as the control variable, and a two-dimensional model with a time-dependent boundary velocity applied at some part of the boundary as the control. The latter case was further extended to an approximate three-dimensional model of an artificial heart. In all cases, it was observed that, in the absence of shear stress and vorticity in the cost functional, the desired flow rate was almost achieved, while including these parameters resulted in a decreased optimal flow rate but with smoother velocity distributions. In the case of two- and three-dimensional boundary control, it was found that, when the stress and vorticity levels were minimized, the optimization algorithm could successfully rearrange the flow distribution, smoothen the sharp gradients and remove the vortices that would occur otherwise in the velocity field. This study demonstrates that optimal control of flow systems is practical and can be used as a design tool.