Buch, Englisch, 249 Seiten, PB, Format (B × H): 170 mm x 240 mm, Gewicht: 512 g
Reihe: Mathematik
Buch, Englisch, 249 Seiten, PB, Format (B × H): 170 mm x 240 mm, Gewicht: 512 g
Reihe: Mathematik
ISBN: 978-3-8439-3268-4
Verlag: Dr. Hut
Due to the huge amount of optimization variables in shape optimization problems with partial differential equations, modern gradient based optimization methods make use of the well-known adjoint approach for optimal control problems and require the differentiability of the control-to-state operator associated with the differential equation in a sufficiently strong topology. Therefore, the question of differentiability of this operator resulting from an application of the method of mapping by Murat and Simon to the incompressible instationary Boussinesq equations is addressed in this thesis. Furthermore, for a rigorous application of the adjoint approach we investigate the analytic properties of the adjoint linearized Boussinesq operator. Those theoretical results are also validated by implementing an adjoint based optimization framework in order to practically solve a shape optimization problem associated with the Boussinesq equations.
In this thesis we furthermore apply the method of mapping to the discrete system resulting from an application of a P1/P1 discretization to the Navier-Stokes equations and investigate the differentiability properties of the corresponding discrete control-to-state operator. Here, the main focus is put on the independence of the spatial and time discretization sizes in the differentiability results, such that the results remain stable for decreasing discretization sizes and a convergence process through the infinite dimensional solution can be considered.