Μέθοδοι Μειωμένης Βάσης για Μ.Δ.Ε. και Ψηφιακά Δίδυμα

This thesis presents a study on numerical ways for the efficient visualization of fluid problems through the prism of Partial Differential Equations. This study focuses mainly on the review and comparison between the Finite Element Methods, Reduced Basis Methods, and also on the construction and function of Digital Twins such as Physics Informed Neural Networks (PINNs), Deep Learning Operator Networks (DeepONet), etc. as recent and very useful applications of technology. The thesis starts with an introduction to the generalized category of Fluid problems. In the first part we investigate the Finite Elements Method as a classic method for the approximation to the solutions of the partial differential equations. We look into the creation of the mesh, the discretization of the function spaces, as well as the algorithm that is implemented in order to approximate the solutions. In the second part we introduce the Reduced Basis Methods which provide us with a more efficient way for the reduction of the computational complexity by constructing models of a lower dimension. The study here investigates the construction and implementation of these models, which is presented within the second chapter. In the third part we explore the structure, the function, and the construction of Digital Twins. The latter are known to the world of science as a newfound technology that is mainly used in solving complex problems and optimizing procedures. The thesis analyzes the methods used by the Digital Twins as well as the procedures followed to accomplish the desired result in the best time possible. Lastly, we state the importance of Digital Twins in modern daily life, and a conversation is held around the targets that can be actualized through them in the following years.