Flow and transport problems through a porous medium are crucial in several fields of engineering and natural sciences. The complex microstructure of these materials makes the complete solution of problems defined on them tricky to calculate. This difficulty lead to the formulation of multiscale analysis techniques, like homogenisation or volume average, apt to find macroscopic approximations of the problem’s solution, without solving it at the microscopic level. Homogenisation adopt the separation of scales in order to define a new problem describing the macroscopic behaviour of the solution. At the same time, in the field of computational science, diverse numerical methods, which take advantage of the geometry or of peculiar features of the problem by including them inside the definition of the numerical method itself, have been developed. One of those is HiMod (Hierarchical Model reduction), which focuses on problems with a clear main direction, like the flow in arteries or in pipes. This numerical method splits the solution in an axial and a transverse component, thus reducing it to an enriched 1D problem, which takes into account also the transverse contributions. The separation of variables employed by HiMod bears a resemblance with the results of the scales separation used by the homogenisation. Exploiting this similarity, it is possible to define a new numerical model called HiPhom" (Hierarchical Perturbation-based Model reduction), which includes the results of homogenisation inside HiMod. This method has more information about the problem at hand and, thanks to this close relationship, proves to be efficient and able to capture the complex feature of the solution.
I problemi di flusso attraverso un materiale poroso sono di fondamentale importanza in svariati campi dell’ingegneria e delle scienze naturali. La struttura microscopica complessa di questi materiali rende proibitiva la risoluzione locale dei problemi su essi definiti. Questo ostacolo ha portato alla creazione di tecniche di analisi multiscala, quali il volume average o l’omogeneizzazione, atte a ricavare approsimazioni macroscopiche della soluzione del problema, senza doverlo risolvere a livello microscopico. L’omogeinizzazione ricorre ad una separazione delle scale per ottenere un nuovo problema che descriva il comportamento macroscopico della soluzione. Allo stesso tempo, nel campo del calcolo numerico sono stati sviluppati svariati metodi che prendendo spunto dalla geometria o da caratteristiche peculiari del problema in esame, incorporandole all’interno del metodo stesso e rendendolo così meno computazionalmente dispendioso. Tra questi, il metodo HiMod (Hierarchical Model reduction) si focalizza su domini caratterizzati da una direzione principale di flusso, quali arterie o canali. Questo metodo numerico decompone la soluzione lungo l’asse principale e le direzioni trasversali, in modo da ridurlo ad un probema 1D arricchito, che tenga conto dei contributi trasversali alla soluzione. La separazione di variabili utilizzata da HiMod ha una notevole somiglianza con i risultati ottenuti dall’omogeinizzazione. Sfruttando questa peculiarità, si può definire un nuovo metodo numerico, chiamato HiPhom" (Hierarchical Perturbation-based Model reduction), che incorpora direttamente i risultati dell’omogeneizzazione all’interno di HiMod. Questo possiede maggiori informazioni riguardo al problema, e, grazie a questo stretto legame, si dimostra efficiente e capace di catturare caratteristiche complesse della sua soluzione dove HiMod incorre in maggiori difficoltà.
Hierarchical perturbation-based model reduction : applications to transport problems in porous media
CONNI, GIOVANNI
2017/2018
Abstract
Flow and transport problems through a porous medium are crucial in several fields of engineering and natural sciences. The complex microstructure of these materials makes the complete solution of problems defined on them tricky to calculate. This difficulty lead to the formulation of multiscale analysis techniques, like homogenisation or volume average, apt to find macroscopic approximations of the problem’s solution, without solving it at the microscopic level. Homogenisation adopt the separation of scales in order to define a new problem describing the macroscopic behaviour of the solution. At the same time, in the field of computational science, diverse numerical methods, which take advantage of the geometry or of peculiar features of the problem by including them inside the definition of the numerical method itself, have been developed. One of those is HiMod (Hierarchical Model reduction), which focuses on problems with a clear main direction, like the flow in arteries or in pipes. This numerical method splits the solution in an axial and a transverse component, thus reducing it to an enriched 1D problem, which takes into account also the transverse contributions. The separation of variables employed by HiMod bears a resemblance with the results of the scales separation used by the homogenisation. Exploiting this similarity, it is possible to define a new numerical model called HiPhom" (Hierarchical Perturbation-based Model reduction), which includes the results of homogenisation inside HiMod. This method has more information about the problem at hand and, thanks to this close relationship, proves to be efficient and able to capture the complex feature of the solution.File | Dimensione | Formato | |
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https://hdl.handle.net/10589/146074