Nowadays, scientific computing pervades many fields of engineering research and practice. The identification of a convenient trade-off between computational effectiveness and reliability is a nontrivial task. An innovative technique known as Hierarchical Model Reduction (HiMod) matches computational efficiency with numerical accuracy. It is designed for problems featuring a main direction, coupled with minor transverse dynamics which may locally yield significant effects on the mainstream. According to this method, the transverse dynamics is represented in terms of (generalized) Fourier modal expansion, whose coefficients are discretized via a Finite Element Method. In such a way, thanks to the resulting separation of variables, the original problem is reformulated as a system of coupled one-dimensional problems. The power of this technique lies in its hierarchical nature, in fact the numerical 1D solution can be easily expanded towards the three-dimensional original domain thanks to the modal basis. In this work we focus on fluids motion in networks of pipes. Our purpose is to move a step forward to real medical applications through the device of HiMod for Navier-Stokes Equations on cylindrical domains, in the perspective of a further extension to modeling of the entire cardiovascular network. While the application of the idea is immediate in 2D and 3D slabs, where a Cartesian tensor product of the transverse directions can be advocated, the extension to 3D cylindrical domains is not trivial. In particular, the identification of a basis function set is a delicate issue. In this work we consider different options and their features in terms of quality of the associated numerical approximation. We perform an extensive numerical assessment. Elementary models for arterial stenosis and aneurysms are worked out through the introduction of non-standard geometries, which are described analytically via a dependence of the radius on the axial coordinate.
Diversi campi dell'ingegneria, sia per applicazioni sperimentali che di routine, richiedono strumenti all'avanguardia per il calcolo scientifico. L'identificazione di un opportuno trade-off tra efficienza computazionale e affidabilità costituisce, tuttavia, una questione non banale. La Riduzione Gerarchica di Modello, nota in letteratura come HiMod - Hierarchical Model Reduction, si propone come un metodo innovativo per coniugare il contenimento del costo computazionale e l'accuratezza numerica. Tale tecnica nasce nello specifico per problemi caratterizzati da una direzione dominante, associata a dinamiche trasversali minori, i cui effetti, tuttavia, possono risultare localmente non trascurabili. Essa prevede l'utilizzo di una classica discretizzazione elementi finiti lungo la direzione dominante, accoppiata ad una opportuna rappresentazione modale lungo le direzioni trasversali, sotto forma di serie (generalizzata) di Fourier. In tal modo, il problema originale viene riformulato come un sistema di problemi 1D accoppiati. L'efficacia di tale metodo è racchiusa nella sua natura gerarchica, infatti la soluzione numerica 1D può essere facilmente ricostruita sul dominio originario grazie all'impiego della base modale. In questa tesi di laurea ci proponiamo di sviluppare uno strumento utile per le applicazioni in campo medico, attraverso la riduzione gerarchica delle Equazioni di Navier-Stokes per domini cilindrici, nella prospettiva di una futura modellizzazione dell'intero sistema cardiovascolare. Se l'idea alla base del metodo è di immediata applicazione per domini rettangolari e parallelepipedi, la sua estensione a domini cilindrici 3D non è ovvia. In particolare, l'identificazione di un opportuno sistema di funzioni di base rappresenta una questione piuttosto delicata. In questo lavoro analizziamo diverse possibili alternative e le rispettive proprietà di approssimazione numerica. La trattazione è avallata da un'ampia casistica di valutazione numerica. Proponiamo, infine, alcuni modelli elementari di arterie stenotiche o affette da aneurisma, descritte analiticamente attraverso l'introduzione del raggio come funzione della coordinata assiale.
Hierarchical model reduction for incompressible flows in cylindrical domains
GUZZETTI, SOFIA
2013/2014
Abstract
Nowadays, scientific computing pervades many fields of engineering research and practice. The identification of a convenient trade-off between computational effectiveness and reliability is a nontrivial task. An innovative technique known as Hierarchical Model Reduction (HiMod) matches computational efficiency with numerical accuracy. It is designed for problems featuring a main direction, coupled with minor transverse dynamics which may locally yield significant effects on the mainstream. According to this method, the transverse dynamics is represented in terms of (generalized) Fourier modal expansion, whose coefficients are discretized via a Finite Element Method. In such a way, thanks to the resulting separation of variables, the original problem is reformulated as a system of coupled one-dimensional problems. The power of this technique lies in its hierarchical nature, in fact the numerical 1D solution can be easily expanded towards the three-dimensional original domain thanks to the modal basis. In this work we focus on fluids motion in networks of pipes. Our purpose is to move a step forward to real medical applications through the device of HiMod for Navier-Stokes Equations on cylindrical domains, in the perspective of a further extension to modeling of the entire cardiovascular network. While the application of the idea is immediate in 2D and 3D slabs, where a Cartesian tensor product of the transverse directions can be advocated, the extension to 3D cylindrical domains is not trivial. In particular, the identification of a basis function set is a delicate issue. In this work we consider different options and their features in terms of quality of the associated numerical approximation. We perform an extensive numerical assessment. Elementary models for arterial stenosis and aneurysms are worked out through the introduction of non-standard geometries, which are described analytically via a dependence of the radius on the axial coordinate.File | Dimensione | Formato | |
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https://hdl.handle.net/10589/94454