Fluid dynamics applications play a crucial role in many real-world scenarios, as for instance industrial engineering and biomedical applications. From a mathematical standpoint, these phenomena are modeled by the Navier–Stokes equations, whose solution can be computationally prohibitive in many realistic settings. As a result, considerable effort has been devoted to reducing the computational cost of solving the Navier-Stokes equations without compromising accuracy. In this regard, the present thesis advances two innovative mathematical strategies, namely Hierarchical Model (HiMod) reduction and mesh adaptation. As a first mathematical tool, we extend the HiMod reduction by providing new theoretical results and applications. HiMod reduction is a mathematical technique that reduces the full-order model to a set of coupled one-dimensional equations through a separation of variables approach, by addressing differently the leading and the transverse dynamics, using a Finite Element or IsoGeometric discretization (HIGAMod) and a modal expansion, respectively. Specifically, we establish the inf-sup condition for the HiMod discretization of the Stokes problem, which leads to introduce an explicit requirement on the number of modal unknowns for the velocity and the pressure, respectively. After the successful extension of the HiMod/HIGAMod MATLAB library to include most capabilities required for fluid dynamics applications - namely, advection-diffusion reaction problem, Stokes problem and Navier-Stokes problem - we asses the methodology to a biomedical realistic scenario. Specifically, we address two patient-specific hemodynamics configurations: a coronary artery stenosis and an abdominal aortic aneurysm. The obtained results validate HiMod/HIGAMod as a viable tool for practical applications. As a second possible solution to efficiently solve the Navier-Stokes equation, we designed a spatial anisotropic mesh adaptation strategy in a 2D setting, targeting problems at high Reynolds numbers. The anisotropic adaptation of the spatial domain is driven by a recovery-based error estimator, namely an estimator relying on the recovered gradient error of a quantity of interest for the problem. This approach already yields a 46 % reduction in the CPU time associated with the methodology. To further improve computational efficiency, we also implement the temporal counterpart of the error estimator, which adapts the time-step size according to the features of the problem dynamics. We develop an algorithm coupling the two adaptive strategies and apply it to the VMS-Smagorinsky modeling for high Reynolds number flows. The proposed technique is validated on two benchmark problems to quantify the effectiveness of the proposed methodology. By jointly adapting the spatial mesh and the time step, the methodology achieves a substantial reduction in the computational cost, reaching a 98 % decrease in CPU time for the numerical solution.
Le applicazioni della fluidodinamica rivestono un ruolo di primaria importanza in numerosi contesti reali, quali, ad esempio, l'ingegneria industriale e il settore biomedico. Dal punto di vista matematico, tali fenomeni sono modellati dalle equazioni di Navier-Stokes, la cui risoluzione numerica può risultare computazionalmente proibitiva in molti scenari di interesse applicativo. Per questo motivo, negli ultimi decenni sono stati compiuti notevoli sforzi nella progettazione di metodologie volte a ridurre il costo computazionale della soluzione delle equazioni di Navier-Stokes, preservandone al contempo l'accuratezza. In questo contesto si colloca la presente tesi, che propone e analizza due strategie matematiche innovative: la riduzione di modello gerarchica (Hierarchical Model reduction - HiMod) e l'adattazione di mesh. Come primo strumento matematico, la tesi estende la metodologia HiMod attraverso lo sviluppo di nuovi risultati teorici e applicativi. La metodologia HiMod è una tecnica che consente di approssimare il modello ad ordine completo mediante un sistema di equazioni unidimensionali accoppiate, ottenuto tramite un approccio di separazione delle variabili. In tale contesto, le dinamiche lungo la direzione principale vengono trattate in modo differente rispetto a quelle trasversali: le prime sono risolte mediante una discretizzazione agli elementi finiti o isogeometrica (HIGAMod), mentre le seconde sono approssimate tramite espansione modale. In particolare, viene sviluppata una dimostrazione analitica della condizione di inf-sup per la discretizzazione HiMod del problema di Stokes, dalla quale deriva un vincolo esplicito sul numero di incognite modali associate alla velocità e alla pressione. Successivamente, in seguito all'estensione della libreria MATLAB HiMod/HIGAMod, finalizzata ad includere le principali funzionalità richieste in ambito fluidodinamico, quali i problemi di diffusione-trasporto-reazione, di Stokes e di Navier-Stokes, la metodologia viene valutata in uno scenario biomedico realistico. Nello specifico, si analizzano due configurazioni geometriche reali, relative a una stenosi coronarica e a un aneurisma dell'aorta addominale. I risultati numerici ottenuti confermano la validità della metodologia HiMod/HIGAMod come strumento efficace per applicazioni pratiche. Come seconda strategia per la risoluzione efficiente delle equazioni di Navier-Stokes, viene sviluppata una metodologia di adattazione anisotropa della mesh spaziale in un contesto bidimensionale, specificamente progettata per problemi caratterizzati da elevati numeri di Reynolds. L'adattazione anisotropa del dominio spaziale è guidata da uno stimatore dell' errore di tipo recovery-based, fondato sull'errore del gradiente ricostruito e calcolato rispetto a una quantità di interesse del problema. Tale approccio consente una riduzione del 46 % del tempo di CPU richiesto. Al fine di incrementare ulteriormente l'efficienza computazionale, viene inoltre implementata la controparte temporale dello stimatore dell' errore, che permette di adattare dinamicamente la dimensione del passo temporale in funzione delle caratteristiche della soluzione. Viene quindi sviluppato un algoritmo che accoppia le due strategie adattive, applicato al modello VMS-Smagorinsky per flussi ad alto numero di Reynolds. La metodologia proposta è validata su due problemi di riferimento, con l'obiettivo di quantificarne l'efficacia. L'adattamento congiunto della mesh spaziale e del passo temporale consente una riduzione sostanziale del costo computazionale complessivo, raggiungendo una diminuzione del 98 % del tempo di CPU necessario alla soluzione numerica.
Advanced mathematical models and methods for the incompressible Navier-Stokes equations: Towards high Reynolds number regimes and realistic applications
Temellini, Erika
2025/2026
Abstract
Fluid dynamics applications play a crucial role in many real-world scenarios, as for instance industrial engineering and biomedical applications. From a mathematical standpoint, these phenomena are modeled by the Navier–Stokes equations, whose solution can be computationally prohibitive in many realistic settings. As a result, considerable effort has been devoted to reducing the computational cost of solving the Navier-Stokes equations without compromising accuracy. In this regard, the present thesis advances two innovative mathematical strategies, namely Hierarchical Model (HiMod) reduction and mesh adaptation. As a first mathematical tool, we extend the HiMod reduction by providing new theoretical results and applications. HiMod reduction is a mathematical technique that reduces the full-order model to a set of coupled one-dimensional equations through a separation of variables approach, by addressing differently the leading and the transverse dynamics, using a Finite Element or IsoGeometric discretization (HIGAMod) and a modal expansion, respectively. Specifically, we establish the inf-sup condition for the HiMod discretization of the Stokes problem, which leads to introduce an explicit requirement on the number of modal unknowns for the velocity and the pressure, respectively. After the successful extension of the HiMod/HIGAMod MATLAB library to include most capabilities required for fluid dynamics applications - namely, advection-diffusion reaction problem, Stokes problem and Navier-Stokes problem - we asses the methodology to a biomedical realistic scenario. Specifically, we address two patient-specific hemodynamics configurations: a coronary artery stenosis and an abdominal aortic aneurysm. The obtained results validate HiMod/HIGAMod as a viable tool for practical applications. As a second possible solution to efficiently solve the Navier-Stokes equation, we designed a spatial anisotropic mesh adaptation strategy in a 2D setting, targeting problems at high Reynolds numbers. The anisotropic adaptation of the spatial domain is driven by a recovery-based error estimator, namely an estimator relying on the recovered gradient error of a quantity of interest for the problem. This approach already yields a 46 % reduction in the CPU time associated with the methodology. To further improve computational efficiency, we also implement the temporal counterpart of the error estimator, which adapts the time-step size according to the features of the problem dynamics. We develop an algorithm coupling the two adaptive strategies and apply it to the VMS-Smagorinsky modeling for high Reynolds number flows. The proposed technique is validated on two benchmark problems to quantify the effectiveness of the proposed methodology. By jointly adapting the spatial mesh and the time step, the methodology achieves a substantial reduction in the computational cost, reaching a 98 % decrease in CPU time for the numerical solution.| File | Dimensione | Formato | |
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Descrizione: PhD Thesis Erika Temellini
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https://hdl.handle.net/10589/250257