Data-driven deep-learning methods for the accelerated simulation of Eulerian fluid dynamics

Deep-learning (DL) methods for the fast inference of the temporal evolution of fluid-dynamics systems, based on the previous recognition of features underlying large sets of fluid-dynamics data, have been studied. Specifically, models based on convolution neural networks (CNNs) and graph neural networks (GNNs) were proposed and discussed. A U-Net, a popular fully-convolutional architecture, was trained to infer wave dynamics on liquid surfaces surrounded by walls, given as input the system state at previous time-points. A term for penalising the error of the spatial derivatives was added to the loss function, which resulted in a suppression of spurious oscillations and a more accurate location and length of the predicted wavefronts. This model proved to accurately generalise to complex wall geometries not seen during training. As opposed to the image data-structures processed by CNNs, graphs offer higher freedom on how data is organised and processed. This motivated the use of graphs to represent the state of fluid-dynamic systems discretised by unstructured sets of nodes, and GNNs to process such graphs. Graphs have enabled more accurate representations of curvilinear geometries and higher resolution placement exclusively in areas where physics is more challenging to resolve. Two novel GNN architectures were designed for fluid-dynamics inference: the MuS-GNN, a multi-scale GNN, and the REMuS-GNN, a rotation-equivariant multi-scale GNN. Both architectures work by repeatedly passing messages from each node to its nearest nodes in the graph. Additionally, lower-resolutions graphs, with a reduced number of nodes, are defined from the original graph, and messages are also passed from finer to coarser graphs and vice-versa. The low-resolution graphs allowed for efficiently capturing physics encompassing a range of lengthscales. Advection and fluid flow, modelled by the incompressible Navier-Stokes equations, were the two types of problems used to assess the proposed GNNs. Whereas a single-scale GNN was sufficient to achieve high generalisation accuracy in advection simulations, flow simulation highly benefited from an increasing number of low-resolution graphs. The generalisation and long-term accuracy of these simulations were further improved by the REMuS-GNN architecture, which processes the system state independently of the orientation of the coordinate system thanks to a rotation-invariant representation and carefully designed components. To the best of the author’s knowledge, the REMuS-GNN architecture was the first rotation-equivariant and multi-scale GNN. The simulations were accelerated between one (in a CPU) and three (in a GPU) orders of magnitude with respect to a CPU-based numerical solver. Additionally, the parallelisation of multi-scale GNNs resulted in a close-to-linear speedup with the number of CPU cores or GPUs.Open Acces

Nonlinear physics of electrical wave propagation in the heart: a review

The beating of the heart is a synchronized contraction of muscle cells (myocytes) that are triggered by a periodic sequence of electrical waves (action potentials) originating in the sino-atrial node and propagating over the atria and the ventricles. Cardiac arrhythmias like atrial and ventricular fibrillation (AF,VF) or ventricular tachycardia (VT) are caused by disruptions and instabilities of these electrical excitations, that lead to the emergence of rotating waves (VT) and turbulent wave patterns (AF,VF). Numerous simulation and experimental studies during the last 20 years have addressed these topics. In this review we focus on the nonlinear dynamics of wave propagation in the heart with an emphasis on the theory of pulses, spirals and scroll waves and their instabilities in excitable media and their application to cardiac modeling. After an introduction into electrophysiological models for action potential propagation, the modeling and analysis of spatiotemporal alternans, spiral and scroll meandering, spiral breakup and scroll wave instabilities like negative line tension and sproing are reviewed in depth and discussed with emphasis on their impact in cardiac arrhythmias.Peer ReviewedPreprin

Integrated Heart - Coupling multiscale and multiphysics models for the simulation of the cardiac function

Mathematical modelling of the human heart and its function can expand our understanding of various cardiac diseases, which remain the most common cause of death in the developed world. Like other physiological systems, the heart can be understood as a complex multiscale system involving interacting phenomena at the molecular, cellular, tissue, and organ levels. This article addresses the numerical modelling of many aspects of heart function, including the interaction of the cardiac electrophysiology system with contractile muscle tissue, the sub-cellular activation-contraction mechanisms, as well as the hemodynamics inside the heart chambers. Resolution of each of these sub-systems requires separate mathematical analysis and specially developed numerical algorithms, which we review in detail. By using specific sub-systems as examples, we also look at systemic stability, and explain for example how physiological concepts such as microscopic force generation in cardiac muscle cells, translate to coupled systems of differential equations, and how their stability properties influence the choice of numerical coupling algorithms. Several numerical examples illustrate three fundamental challenges of developing multiphysics and multiscale numerical models for simulating heart function, namely: (i) the correct upscaling from single-cell models to the entire cardiac muscle, (ii) the proper coupling of electrophysiology and tissue mechanics to simulate electromechanical feedback, and (iii) the stable simulation of ventricular hemodynamics during rapid valve opening and closure

Isogeometric approximation of cardiac electrophysiology models on surfaces: An accuracy study with application to the human left atrium

We consider Isogeometric Analysis in the framework of the Galerkin method for the spatial approximation of cardiac electrophysiology models defined on NURBS surfaces; specifically, we perform a numerical comparison between basis functions of degree p ≥ 1 and globally C k -continuous, with k = 0 or p − 1, to find the most accurate approximation of a propagating front with the minimal number of degrees of freedom. We show that B-spline basis functions of degree p ≥ 1, which are C p−1 -continuous capture accurately the front velocity of the transmembrane potential even with moderately refined meshes; similarly, we show that, for accurate tracking of curved fronts, high-order continuous B-spline basis functions should be used. Finally, we apply Isogeometric Analysis to an idealized human left atrial geometry described by NURBS with physiologically sound fiber directions and anisotropic conductivity tensor to demonstrate that the numerical scheme retains its favorable approximation properties also in a more realistic setting

Iterative Solvers for Physics-based Simulations and Displays

La génération d’images et de simulations réalistes requiert des modèles complexes pour capturer tous les détails d’un phénomène physique. Les équations mathématiques qui composent ces modèles sont compliquées et ne peuvent pas être résolues analytiquement. Des procédures numériques doivent donc être employées pour obtenir des solutions approximatives à ces modèles. Ces procédures sont souvent des algorithmes itératifs, qui calculent une suite convergente vers la solution désirée à partir d’un essai initial. Ces méthodes sont une façon pratique et efficace de calculer des solutions à des systèmes complexes, et sont au coeur de la plupart des méthodes de simulation modernes. Dans cette thèse par article, nous présentons trois projets où les algorithmes itératifs jouent un rôle majeur dans une méthode de simulation ou de rendu. Premièrement, nous présentons une méthode pour améliorer la qualité visuelle de simulations fluides. En créant une surface de haute résolution autour d’une simulation existante, stabilisée par une méthode itérative, nous ajoutons des détails additionels à la simulation. Deuxièmement, nous décrivons une méthode de simulation fluide basée sur la réduction de modèle. En construisant une nouvelle base de champ de vecteurs pour représenter la vélocité d’un fluide, nous obtenons une méthode spécifiquement adaptée pour améliorer les composantes itératives de la simulation. Finalement, nous présentons un algorithme pour générer des images de haute qualité sur des écrans multicouches dans un contexte de réalité virtuelle. Présenter des images sur plusieurs couches demande des calculs additionels à coût élevé, mais nous formulons le problème de décomposition des images afin de le résoudre efficacement avec une méthode itérative simple.Realistic computer-generated images and simulations require complex models to properly capture the many subtle behaviors of each physical phenomenon. The mathematical equations underlying these models are complicated, and cannot be solved analytically. Numerical procedures must thus be used to obtain approximate solutions. These procedures are often iterative algorithms, where an initial guess is progressively improved to converge to a desired solution. Iterative methods are a convenient and efficient way to compute solutions to complex systems, and are at the core of most modern simulation methods. In this thesis by publication, we present three papers where iterative algorithms play a major role in a simulation or rendering method. First, we propose a method to improve the visual quality of fluid simulations. By creating a high-resolution surface representation around an input fluid simulation, stabilized with iterative methods, we introduce additional details atop of the simulation. Second, we describe a method to compute fluid simulations using model reduction. We design a novel vector field basis to represent fluid velocity, creating a method specifically tailored to improve all iterative components of the simulation. Finally, we present an algorithm to compute high-quality images for multifocal displays in a virtual reality context. Displaying images on multiple display layers incurs significant additional costs, but we formulate the image decomposition problem so as to allow an efficient solution using a simple iterative algorithm

Computational studies of high power nanosecond laser propagation in magnetised plasmas

The effects of magnetic fields on long-pulse (nanosecond) laser-plasma interactions have been a subject of research interest in recent years. Applied fields have been used for the formation and control of plasma waveguides (Froula 2009), for improving energy coupling under conditions relevant to indirect-drive ICF (Montgomery 2015) and have been observed to arise naturally in the gas-fill of hohlraums due to field generation by the Biermann battery mechanism at the wall (Li 2009). These systems are complicated by the range of coupled magnetised electron transport phenomena which can occur. For example, heat-flow across field lines is suppressed in a magnetised plasma and magnetic fields can rapidly advect along temperature gradients due to Nernst advection, an effect which is predominant at moderate magnetisations (wt ~ 1). This thesis addresses the question of how these phenomena, coupled with inverse bremsstrahlung heating, affect the hydrodynamic evolution of the plasma and in turn change laser self-focusing. This problem is investigated by means of theoretical and computational modelling. A paraxial wave solver has been developed and used in conjunction with the existing 2D plasma codes, CTC, an MHD code including a detailed model of Braginskii electron transport, and IMPACT, a Vlasov-Fokker-Planck code with fully implicit magnetic fields. Simulations of moderate intensity (~ 10^14 W/cm^2), 10 micron width infrared laser pulses propagating through under-dense (ne = 10^18 - 10^19 cm^-3) plasmas in the presence of 0 - 12 T applied fields demonstrate an inhibition to beam self-focusing and thermal pressure driven density channel formation resulting from Nernst advection over time-scales greater than ~ 200 ps. VFP simulations accounting for non-locality indicate that heat-flow and Nernst advection can be over-estimated however and result in a re-emergence of channeling phenomena under these conditions. Finally, the magnetothermal instability - the result of feedback between the Nernst effect and Righi-Leduc heat-flow - frequently arises, affecting temperature and field profiles and is considered in the context of such conditions.Open Acces

The role of fluctuations in ecological patterns and processes

Fluctuations are ubiquitous in nature and are relevant for nearly every ecological process. The main sources of fluctuations in population abundances are demographic and environmental stochasticity, whose effect on local population dynamics, metapopulations and metacommunities have attracted much interest in the ecological literature. A third source of stochasticity is demographic heterogeneity, which is the variability of demographic traits within a population. Despite the large body of literature dedicated to fluctuations in ecology, their role in some relevant ecological patterns and processes is still rather unexplored. For example, the effect of demographic and environmental stochasticity on species spread is poorly understood, mostly due to a scarcity of experimentation linking theoretical models with replicated experiments. Additionally, environmental stochasticity can induce population fluctuations and has been shown theoretically to determine the exponent of one of the most widespread scaling laws in nature, Taylor's law of fluctuation scaling. However, empirical observations point towards the existence of a single universal Taylorâs law exponent, in contrast with such model predictions. Here, experiments with protist microcosms and methods from statistical physics are used to investigate the role of fluctuations and heterogeneity on relevant ecological patterns and processes. The effect of demographic and environmental stochasticity on the propagation of biological invasions is studied in microcosm experiments with Tetrahymena sp. and Euglena gracilis and with stochastic generalizations of the Fisher-Kolmogorov equation. Demographic stochasticity is shown to induce fluctuations in the position of the propagating front and the statistical structure of the environmental heterogeneity is shown to cause a slowing-down of the invasion front at large autocorrelation lengths. The investigation of biological invasions in environments with heterogeneous distribution of resources is performed experimentally by manipulating light, the energy resource for photosynthetic organisms. Such experimental setup is further used to study phototaxis, the directed motion of phytoplankton towards or against light sources, a process that is important for relevant ecological phenomena such as diel vertical migration. A model for phototaxis is derived from the experiments in the generalized Keller-Segel framework. Large deviations theory is used to derive a generalized Taylor's law and to elucidate the origin of a universal scaling exponent as due to sampling rather than to the population growth process. The framework of finite-size scaling is used to characterize the demographic heterogeneity in a relevant ecological trait, the body size of individuals. Intra-specific body size distributions measured experimentally are shown to be described by a universal scaling distribution across different taxa and over four orders of magnitude in body size. Mathematical models of cell growth and division are shown to be compatible with the observed universal body size distribution