Analysis and numerical approximation of shape optimization problems governed by the Navier-Stokes and the Boussinesq equations

Viac o knihe

Shape optimization problems related to complex fluids are significant in mathematics, physics, and engineering. These problems involve partial differential equations and represent an active research area in mathematical science. Notable contributions to the mathematical study of shape optimization in fluid dynamics have been made by researchers such as Pironneau, Simon, and Bello et al. This thesis examines shape optimization problems governed by the Navier-Stokes and Boussinesq equations, which model the time-dependent flow of particles and their temperature distribution. Given the vast number of optimization variables, modern gradient-based optimization methods utilize the adjoint approach, which necessitates the differentiability of the control-to-state operator linked to the differential equation in a strong topology. This work addresses the differentiability of this operator through the method of mapping by Murat and Simon applied to the incompressible, unsteady Boussinesq equations. Additionally, the thesis explores the analytic properties of the adjoint linearized Boussinesq operator for rigorous application of the adjoint approach, validating theoretical findings through an adjoint-based optimization framework. The method of mapping is also applied to the discrete system from a P1/P1 discretization of the Navier-Stokes equations, focusing on the differentiability of the discrete control-to-state operator while ensuring