The main purpose of engineering applications for fluid with natural and mixed convection is to control or enhance the flow motion and the heat transfer. In this paper, we use mathematical tools based on optimal control theory to show the possibility of systematically controlling natural and mixed convection flows. We consider different control mechanisms such as distributed, Dirichlet, and Neumann boundary controls. We introduce mathematical tools such as functional spaces and their norms together with bilinear and trilinear forms that are used to express the weak formulation of the partial differential equations. For each of the three different control mechanisms, we aim to study the optimal control problem from a mathematical and numerical point of view. To do so, we present the weak form of the boundary value problem in order to assure the existence of solutions. We state the optimization problem using the method of Lagrange multipliers. In this paper, we show and compare the numerical results obtained by considering these different control mechanisms with different objectives.

Fluid-structure interaction (FSI) problems are of great interest, due to their applicability in science and engineering. However, the coupling between large fluid domains and small moving solid walls presents numerous numerical difficulties and, in some configurations, where the thickness of the solid wall can be neglected, one can consider membrane models, which are derived from the Koiter shell equations with a reduction of the computational cost of the algorithm. With this assumption, the FSI simulation is reduced to the fluid equations on a moving mesh together with a Robin boundary condition that is imposed on the moving solid surface. In this manuscript, we are interested in the study of inverse FSI problems that aim to achieve an objective by changing some design parameters, such as forces, boundary conditions, or geometrical domain shapes. We study the inverse FSI membrane model by using an optimal control approach that is based on Lagrange multipliers and adjoint variables. In particular, we propose a pressure boundary optimal control with the purpose to control the solid deformation by changing the pressure on a fluid boundary. We report the results of some numerical tests for two-dimensional domains to demonstrate the feasibility and robustness of our method.

This work presents a stabilized Nitsche-type Cut Finite Element Method (CutFEM) for simulating two-phase flows with complex interfaces. The method addresses the challenges of capturing discontinuities in material properties and governing equations that arise from implicitly defined interfaces. By employing a Continuous Interior Penalty (CIP) method, Nitsche’s method for weak interface coupling, and Ghost Penalty (GP) terms for stability, the formulation enables an accurate representation of abrupt changes in physical properties across cut elements. A stability analysis and a priori error estimation, utilizing Oseen’s formulation, demonstrate the method’s robustness. At the same time, a numerical convergence study incorporating adaptivity and a best-fit quadratic level-set interpolation validates its accuracy. Finally, the method’s efficacy in mitigating spurious currents is confirmed through the Spurious Current Test, demonstrating its potential for reliable simulation of multi-phase flow phenomena.