Akademik Veri Yönetim Sistemi
Tezin Türü: Doktora
Tezin Yürütüldüğü Kurum: Orta Doğu Teknik Üniversitesi, Fen Edebiyat Fakültesi, Matematik Bölümü, Türkiye
Tezin Onay Tarihi: 2018
Öğrenci: FATMA GÜLER EROĞLU
Danışman: SONGÜL KAYA MERDAN
Özet:Proper orthogonal decomposition (POD), as one of the most commonly used tools to generate reduced order models, has been utilized in many engineering and scientific applications. The idea of POD consists of extracting the dominant features of a data set, which are naturally assumed to represent Galerkin finite element solution of a partial differential equation. In this way, POD reduces the complexity of systems. Despite the widespread use of POD, it can perform quite poorly for turbulence flows. Projection-based variational multiscale (VMS) method is one of the best approaches that increase the numerical stability. The basic idea in VMS is adding artificial viscosity only to smallest resolved scales instead of all resolved scales to eliminate small scale oscillations. The usual finite element discretization sorting of scales is complicated, but in POD, basis functions are sorted in descending order with respect to their kinetic energy. Thus, the POD is suitable to the VMS methodology. First, we propose, analyze and test a post-processing implementation of a projectionbased VMS method with POD for the incompressible Navier-Stokes equations. We present a theoretical analysis of the method, and give results for several numerical tests on benchmark problems which both illustrate the theory and show the proposed method’s effectiveness. Second, we extend POD reduced order modeling to flows governed by double diffusive convection, which models flow driven by two potentials with different rates of diffusion. We present a stability and convergence analysis for it, and give results for numerical tests. In the last part of the thesis, we present a VMS reduced order model based on POD for the Darcy Brinkman equations. The proposed scheme uses VMS type stabilization in POD. For the temporal discretization of the system, Crank Nicholson is utilized. The numerical analysis for the VMS-POD is carried out and numerical studies are performed to verify the theoretical findings.