Radioactive waste repositories, designed in accordance with the current UK concept, would be required to provide containment for thousands of years beneath hundreds of metres of rock. The physical processes, both geological and other processes, that might lead to migration of radionuclides are slow in comparison to human timescales — it is impractical to make an experiment of the whole system and so these systems are typically investigated through the use of numerical models. Predictive models are based on combinations of: assumptions, mathematical formulations and parameter values derived from experimental observations. The Ventilation Experiment in the Opalinus Clay at Mont Terri, Switzerland, was designed to involve geological and other physical processes that would be active during the excavation and construction phases of a repository, and with consequences for the repository performance during the operational phase. The experiment consisted of a 10m long tunnel of 1:3m diameter through which air of known relative humidity was circulated in order to force drying and re-saturation through the tunnel wall. Two such cycles over four years have been observed via installed instrumentation. Several numerical models have been constructed of the ventilation experiment by different international teams under the decovalex project using different approaches for cross-validation. Through participation in this project, a 1D model using Richards’ Equation was developed that effectively reproduces the hydrodynamic, mechanical and conservative mass transport results. During the course of developing that model, many other domains, meshes, formulations and software versions were investigated. Now that the field scale Ventilation Experiment can be reproduced with numerical models, the findings (assumptions, formulations, parameter values, computational methods and software) would be transferable to other argillaceous formations to enable predictive modelling of similar scenarios and contribute to the safe disposal of nuclear waste and other problems involving similar geological processes. Work of this type fills the gap between laboratory scale experiments and regional scale modelling of geological systems. The gap is especially wide for low-permeability formations because the size and time-scale limitations effect the ability to make direct observations and measurements. Two particular problems were also addressed in this work: that of the use relative permeability functions and also the computational treatment of the physical interface between the tunnel domain and the rock domain. A sensitive component in many models of unsaturated flow through porous media and covering a wide variety of applications, including reservoir engineering, is the representation of permeability at an unsaturated point (kx) as a scaling of the saturated permeability (ksat) by introducing some function of the pressure head, or saturation as the relative permeability (krel) in the relation kx = ksatkrel. The choice of the particular function and its parameter values adds little to our understanding of the physical parameters. A solution is proposed to the second problem, of how to computationally represent, implement and manage the interface between two physical (i.e. spatial) domains. The scheme maps every part of the boundary of one domain onto the corresponding part of the boundary of the other domain, storing the state variables in shared memory and converting between physical components.

The effect of building geometry on the wind environment of cities is such that it can cause problems like wind danger, discomfort and poor ventilation of airborne pollutants. Computational fluid dynamics (CFD) can play a role in assessing changes in wind environment caused by building projects before realisation at little cost. However, the current state-of-the-art methods, RANS and LES, force a steep trade-off between accuracy and computational cost, and neither method is truly predictive. Time-resolved adaptive direct finite element simulation (DFS) is a method for CFD that is predictive and automatically optimises the mesh for a goal quantity, making it both efficient and accurate. In this thesis, DFS was implemented in FEniCS and used on basic validation cases to provide a proof of concept for the use of this method in the building aerodynamics, on resources freely available to anyone. The results show that the method is accurate to within 10% of the validation data with respect to the goal quantity. Visually, the expected flow features are clearly identifiable. DFS was successfully applied to a relatively complicated building geometry, with a total computation time of about 120 core-hours. We conclude that DFS has significant potential as a method for evaluating urban wind environments. Furthermore, because of its ease of use and lack of parameters, DFS can play an important role in helping architects, designers and students understand the effect of urban geometries on the wind environment. This report provides a basis for further research on DFS for building aerodynamics, as validation on more diverse urban geometries is still necessary. / Effekten av byggnaders form och geometri är så viktig att den kan ge problem för ventilation av t.ex. föroreningar, för energieffektivitet, och för vindfaror med t.ex. hög vindhastihet som kan vara farligt eller skapa obehag. Beräkningsströmningsdynamik (CFD) kan ha en roll i bedömningen av byggnadsprojekt i ett tidigt skede till liten kostnad. Dock är de etablerade och ledande metodikerna, RANS och LES, inte prediktiva och tvingar fram en kompromiss mellan beräkningskosnad och noggrannhet. Vår metodik “Time-resolved adaptive direct finite element simulation” (DFS) är en metod för CFD som är prediktiv och automatiskt optimerar beräkningsnätet (och därmed beräkningskostnaden) för en given målkvantitet, som ger både effektivitet och noggrannhet. I denna avhandling implementerades DFS i FEniCS och användes i grundläggande valideringsfall för att ge ett proof of conceptför användning av denna metod i byggnadsaerodynamik, på resurser som är fritt tillgängliga för alla. Resultaten visar att metoden är korrekt inom 10% av valideringsdata med avseende på målkvantiteten. Visuellt är de förväntade flödesfunktionerna tydligt identifierbara. DFS applicerades framgångsrikt på en relativt komplicerad byggnadsgeometri med en total beräkningstid på cirka 120 kärntimmar, vilket är en försumbar kostnad. Vi drar slutsatsen att DFS har en betydande potential som metod för utvärdering av stadsvindmiljöer. Dessutom, på grund av dess användarvänlighet och frihet från parametrar, kan DFS spela en viktig roll för att hjälpa arkitekter, designers och studenter att förstå effekterna av stadsgeometrier på vindmiljön. Denna rapport ger en grund för vidare forskning om DFS för aerodynamik, eftersom validering av mer olika stadsgeometrier fortfarande är nödvändig.

Για την επίλυση προβλημάτων πολλαπλών πεδίων / πολλαπλών φυσικών έχουν προταθεί διάφορες τεχνικές στη βιβλιογραφία. Οι μέθοδοι χαλάρωσης στις διεπαφές είναι μια ενδιαφέρουσα προσέγγιση για την επίλυση αυτών των προβλημάτων. Υποθέτοντας κάποιες αρχικές τιμές στις διεπαφές του προβλήματος, οι μέθοδοι χαλάρωσης στις διεπαφές επιλύουν επαναληπτικά τα υποπροβλήματα και χαλαρώνουν τις τιμές στις διεπαφές έως ότου επιτευχθεί σύγκλιση. Τα κυριότερα πλεονεκτήματα των μεθόδων αυτών είναι ότι οι ρυθμοί σύγκλισής τους εξαρτώνται μόνο από τις παραμέτρους των ίδιων των προβλημάτων, τις παραμέτρους που σχετίζονται με την ανάλυση των προβλημάτων σε υποπροβλήματα και τις παραμέτρους που σχετίζονται με τον τελεστή που εφαρμόζεται στις διεπαφές. Στην παρούσα μεταπτυχιακή εργασία παρουσιάζεται μια νέα υλοποίηση μιας μεθόδου χαλάρωσης στις διεπαφές, της μεθόδου GEO. Η GEO βασίζεται σε έναν απλό μηχανισμό γεωμετρικής διόρθωσης και δρα επαναληπτικά, ώστε να χαλαρώσει τις τιμές στις διεπαφές. Πιο συγκεκριμένα, προσθέτει στις παλιές τιμές των διεπαφών έναν γεωμετρικά σταθμισμένο συνδυασμό των κανονικών παραγώγων των οριακών σημείων των γειτονικών πεδίων. Η υλοποίηση της μεθόδου GEO γίνεται στην πλατφόρμα FEniCS, η οποία είναι μια συλλογή ελεύθερου λογισμικού για την αυτοματοποιημένη, αποδοτική υλοποίηση διαφορικών εξισώσεων. Η υλοποίηση πραγματοποιείται με τρόπο τέτοιο που ο χρήστης να μπορεί εύκολα να ορίζει τις ιδιότητες των πεδίων του προβλήματος (όπως, γεωμετρία του προβλήματος, τελεστής μερικής διαφορικής εξίσωσης, οριακές συνθήκες, συνθήκες διεπαφών). Στο παρασκήνιο γίνεται η δημιουργία ή / και η βελτιστοποίηση των πλεγμάτων (τριγωνικά στοιχεία) για κάθε πεδίο του προβλήματος, η επίλυση των τοπικών προβλημάτων μερικών διαφορικών εξισώσεων και η εμφάνιση των υπολογισμένων τιμών για το πρόβλημα συνολικά και για τις διεπαφές. Η κυριότερη πρόκληση της υλοποίησης είναι η πρόσβαση στις τιμές των λύσεων στις διεπαφές και η εισαγωγή των χαλαρωμένων τιμών πίσω στα υποπροβλήματα για να αποτελέσουν τις οριακές συνθήκες των υποπροβλημάτων. Πραγματοποιούνται πειράματα σε προβλήματα-μοντέλα, ώστε να ελεγχθεί η εφαρμοσιμότητα και η σύγκλιση της μεθόδου. Στη συνέχεια, υλοποιείται μια παράλληλη έκδοση της μεθόδου και πραγματοποιούνται συγκριτικά πειράματα απόδοσης. Η παράλληλη υλοποίησή μας αποδεικνύεται πολύ χρήσιμη, ειδικά για μεγάλα προβλήματα. / Several approaches have been suggested in the literature for the solution of multiphysics / multidomain problems. Interface Relaxation (IR) methods are an interesting approach for the solution of these problems. Assuming initial guesses on the interfaces of the original problem, IR methods iteratively solve the subproblems and relax for new values on the interfaces until convergence is succeeded. Their main advantages are that their rates of convergence only depend on the parameters of the problem itself, the parameters related to its decomposition into subproblems and the parameters related to the operator imposed on the interfaces. In the present master thesis a new implementation of an IR method named GEO is presented. GEO is based on a simple geometric correction mechanism and acts iteratively so as to relax the values of the solution on the interfaces. In particular, it adds to the old interface values a geometrically weighted combination of the normal boundary derivatives of the adjacent subdomains. In this thesis GEO is implemented in FEniCS. The FEniCS project is a collection of free software for automated, efficient solution of differential equations. In order to evaluate the GEO implementation, it is applied on two different PDE problems with the same differential equation and boundary conditions and different domains. FEniCS methods are used to specify the problem's subdomains properties (i.e. geometry, PDE operator and boundary/interface conditions). They are also used to generate and/or refine meshes (triangular elements) for each subdomain, solve the local PDE problems and show the computed results in the global domain and on the interfaces. Getting values of the solutions on the interface (boundaries of the subproblems) and passing the new relaxed values back to the subproblems as updated values for the boundary conditions is the main challenge of the IR methodology implementation. Experiments are performed for 2-dimensional elliptic partial differential model problems with partitions in multiple subdomains and the results are examined in terms of the method's applicability and convergence. A parallel implementation of the GEO method using FEniCS is also presented, as well as its performance comparison to the serial implementation. The parallel implementation proves to be really useful, especially for large problems.