Partial Differential Equations describe a large number of phenomena of practical interest and their solution usually requires running huge simulations on supercomputing clusters.  Especially when dealing with turbulent flows, the cost of such simulations, if approached naively, makes them unfeasible, requiring modelling intervention.  This work is concerned with two main aspects in the field of Computational Sciences.  On the one hand we explore new directions in turbulence modelling and simulation of turbulent flows; we use an adaptive Finite Element Method and an \emph{infinite Reynolds number} model to reduce the computational cost of otherwise intractable simulations, showing that we are able to perform time-dependent computations of turbulent flows at very high Reynolds numbers, considered the main challenge in modern aerodynamics.  The other focus of this work is on biomedical applications.  We develop a computational model for (Cardiac) Radiofrequency Ablation, a popular clinical procedure administered to treat a variety of conditions, including arrhythmia.  Our model improves on the state of the art in several ways, most notably addressing the critical issue of accurately approximating the geometry of the configuration, which proves indispensable to correctly reproduce the physics of the phenomenon.

    Partiella differentialekvationer kan användas för att beskriva ett stort antal fenomen av praktiskt intresse.    Vanligtvis krävs enorma simuleringar på superdatorkluster för att hitta deras lösningar.    I synnerhet vid arbete med turbulent flöde.    Dessa simuleringar är så resurskrävande att utan specialbehandling så är de ohanterbara och kräver manuella modelleringsingrepp.    Denna avhandling består av två huvuddelar.    Först utforskar vi nya riktningar i turbulensmodellering och simulering av turbulent flöde.    Vi använder oss av en adaptiv finit elementmetod och en modell med  oändliga \emph{Reynoldstal} för att reducera beräkningskostnaden för annars ohanterbara simuleringar.    Avhandlingen visar att vi lyckats utföra tidsberoende beräkningar av turbulent flöde vid väldigt höga Reynoldstal, vilket är en av de stora utmaningarna i modern aerodynamik.    Den andra delen i denna avhandlingen fokuserar på biomedicinska tillämpningar.    Vi har utvecklat en modell för radiofrekvensablation, ett populärt medicinskt ingrepp som är del i behandlingen av ett flertal sjukdomar, inklusive arytmi.    Vår modell överträffar befintliga modeller på flera punkter.    Mest markant genom att noggrant approximera  konfigurationens geometri, vilket är väsentligt för att korrekt kunna reproducera fenomenets fysik.

    Las ecuaciones en derivadas parciales describen muchos fenómenos de interés práctico y sus soluciones suelen necesitar correr simulaciones muy costosas en clústers de cálculo.    En el ámbito de los flujos turbulentos, en particular, el coste de las simulaciones es demasiado grande si se utilizan métodos básicos, por eso es necesario modelizar el sistema.    Esta tesis doctoral trata principalmente de dos temas en Cálculo Científico.    Por un lado, estudiamos nuevos desarrollos en la modelización y simulación de flujos turbulentos; utilizamos un Método de Elementos Finitos adaptativo y un modelo de \emph{número de Reynolds infinito} para reducir el coste computacional de simulaciones que, sin estas modificaciones, serían demasiado costosas.    De esta manera conseguimos lograr simulaciones evolutivas de flujos turbulentos con número de Reynolds muy grande, lo cual se considera uno de los mayores retos en aerodinámica.    El otro pilar de esta tesis es una aplicación biomédica.    Desarrollamos un modelo computacional de Ablación (Cardiaca) por Radiofrecuencia, una terapia común para tratar varias enfermedades, por ejemplo algunas arritmias.    Nuestro modelo mejora los modelos existentes en varias maneras, y en particular en tratar de obtener una aproximación fiel de la geometría del sistema, lo cual se descubre ser crítico para simular correctamente la física del fenómeno.

Radiofrequency catheter ablation (RFCA) is an effective treatment for cardiac arrhythmias. Although generally safe, it is not completely exempt from the risk of complications. The great flexibility of computational models can be a major asset in optimizing interventional strategies if they can produce sufficiently precise estimations of the generated lesion for a given ablation protocol. This requires an accurate description of the catheter tip and the cardiac tissue. In particular, the deformation of the tissue under the catheter pressure during the ablation is an important aspect that is overlooked in the existing literature, which resorts to a sharp insertion of the catheter into an undeformed geometry. As the lesion size depends on the power dissipated in the tissue and the latter depends on the percentage of the electrode surface in contact with the tissue itself, the sharp insertion geometry has the tendency to overestimate the lesion obtained, which is a consequence of the tissue temperature rise overestimation. In this paper, we introduce a full 3D computational model that takes into account the tissue elasticity and is able to capture tissue deformation and realistic power dissipation in the tissue. Numerical results in FEniCS-HPC are provided to validate the model against experimental data and to compare the lesions obtained with the new model and with the classical ones featuring a sharp electrode insertion in the tissue.

The International Energy Agency Technology Collaboration Programme for Ocean Energy Systems (OES) initiated the OES Wave Energy Conversion Modelling Task, which focused on the verification and validation of numerical models for simulating wave energy converters (WECs). The long-term goal is to assess the accuracy of and establish confidence in the use of numerical models used in design as well as power performance assessment of WECs. To establish this confidence, the authors used different existing computational modelling tools to simulate given tasks to identify uncertainties related to simulation methodologies: (i) linear potential flow methods; (ii) weakly nonlinear Froude-Krylov methods; and (iii) fully nonlinear methods (fully nonlinear potential flow and Navier-Stokes models). This article summarizes the code-to-code task and code-to-experiment task that have been performed so far in this project, with a focus on investigating the impact of different levels of nonlinearities in the numerical models. Two different WECs were studied and simulated. The first was a heaving semi-submerged sphere, where free-decay tests and both regular and irregular wave cases were investigated in a code-to-code comparison. The second case was a heaving float corresponding to a physical model tested in a wave tank. We considered radiation, diffraction, and regular wave cases and compared quantities, such as the WEC motion, power output and hydrodynamic loading.

The numerical simulation of the diffusion MRI signal arising from complex tissue micro-structures is helpful for understanding and interpreting imaging data as well as for designing and optimizing MRI sequences. The discretization of the Bloch-Torrey equation by finite elements is a more recently developed approach for this purpose, in contrast to random walk simulations, which has a longer history. While finite elements discretization is more difficult to implement than random walk simulations, the approach benefits from a long history of theoretical and numerical developments by the mathematical and engineering communities. In particular, software packages for the automated solutions of partial differential equations using finite elements discretization, such as FEniCS, are undergoing active support and development. However, because diffusion MRI simulation is a relatively new application area, there is still a gap between the simulation needs of the MRI community and the available tools provided by finite elements software packages. In this paper, we address two potential difficulties in using FEniCS for diffusion MRI simulation. First, we simplified software installation by the use of FEniCS containers that are completely portable across multiple platforms. Second, we provide a portable simulation framework based on Python and whose code is open source. This simulation framework can be seamlessly integrated with cloud computing resources such as Google Colaboratory notebooks working on a web browser or with Google Cloud Platform with MPI parallelization. We show examples illustrating the accuracy, the computational times, and parallel computing capabilities. The framework contributes to reproducible science and open-source software in computational diffusion MRI with the hope that it will help to speed up method developments and stimulate research collaborations.

Radiofrequency catheter ablation (RFCA) is widely used for the treatment of various types of cardiac arrhythmias. Typically, the efficacy and the safety of the ablation protocols used in the clinics are derived from tests carried out on animal specimens, including swines. However, these experimental findings cannot be immediately translated to clinical practice on human patients, due to the difference in the physical properties of the types of tissue. Computational models can assist in the quantification of this variability and can provide insights in the results of the RFCA for different species. In this work, we consider a standard ablation protocol of 10 g force, 30 W power for 30 s. We simulate its application on a porcine cardiac tissue, a human ventricle and a human atrium. Using a recently developed computational model that accounts for the mechanical properties of the tissue, we explore the onset and the growth of the lesion along time by tracking its depth and width, and we compare the lesion size and dimensions at the end of the ablation.

Radiofrequency catheter ablation (RFCA) is an effective treatment for different types of cardiac arrhythmias. However, major complications can occur, including thrombus formation and steam pops. We present a full 3D mathematical model for the radiofrequency ablation process that uses an open-irrigated catheter and accounts for the tissue deformation, an aspect overlooked by the existing literature. An axisymmetric Boussinesq solution for spherical punch is used to model the deformation of the tissue due to the pressure of the catheter tip at the tissue-catheter contact point. We compare the effect of the tissue deformation in the RFCA model against the use of a standard sharp insertion of the catheter in the tissue that other state-of-the-art RFCA computational models use.

The goal of this work is to study the dynamics of floating platforms that are designed for marine energy generation. This work is done in collaboration with Tecnalia R&I, a company settled in the Basque Country which designs this kind of platforms. To our purpose we present a method for the simulation of two-phase flow with the presence of floating bodies. We consider the variable density incompressible Navier-Stokes equations and discretize them by the finite element method with a variational multiscale stabilization. A level-set type method is adopted to model the interphase between the two fluids. The mixing or smearing in the interphase is prevented with a compression technique. Turbulence is implicitly modeled by the numerical stabilization. The floating device simulation is done by a rigid body motion scheme where a deforming mesh approach is used. The mesh deforms elastically following the movement of the body. Simulation of a decay test on a cube is performed and the results are presented in this paper. All the simulations are done with the open source finite elements parallel software FEniCS-HPC. 

We present an adaptive finite element method for time-resolved simulation of aerodynamics without any turbulence-model parameters, which is applied to a benchmark problem from the HiLiftPW-3workshop to compute the flowpast a JAXA Standard Model (JSM) aircraft model at realistic Reynolds numbers. The mesh is automatically constructed by the method as part of an adaptive algorithm based on a posteriori error estimation using adjoint techniques. No explicit turbulence model is used, and the effect of unresolved turbulent boundary layers is modeled by a simple parametrization of the wall shear stress in terms of a skin friction. In the case of very high Reynolds numbers, we approximate the small skin friction by zero skin friction, corresponding to a free-slip boundary condition, which results in a computational model without any model parameter to be tuned, and without the need for costly boundary-layer resolution. We introduce a numerical tripping-noise term to act as a seed for growth of perturbations; the results support that this triggers the correct physical separation at stall and has no significant pre-stall effect. We show that the methodology quantitavely and qualitatively captures the main features of the JSM experiment-aerodynamic forces and the stall mechanism-with a much coarser mesh resolution and lower computational cost than the state-of-the-art methods in the field, with convergence under mesh refinement by the adaptive method. Thus, the simulation methodology appears to be a possible answer to the challenge of reliably predicting turbulent-separated flows for a complete air vehicle.