A numerical study of viscous vortex rings using a spectral method

Viscous, axisymmetric vortex rings are investigated numerically by solving the incompressible Navier-Stokes equations using a spectral method designed for this type of flow. The results presented are axisymmetric, but the method is developed to be naturally extended to three dimensions. The spectral method relies on divergence-free basis functions. The basis functions are formed in spherical coordinates using Vector Spherical Harmonics in the angular directions, and Jacobi polynomials together with a mapping in the radial direction. Simulations are performed of a single ring over a wide range of Reynolds numbers (Re approximately equal gamma/nu), 0.001 less than or equal to 1000, and of two interacting rings. At large times, regardless of the early history of the vortex ring, it is observed that the flow approaches a Stokes solution that depends only on the total hydrodynamic impulse, which is conserved for all time. At small times, from an infinitely thin ring, the propagation speeds of vortex rings of varying Re are computed and comparisons are made with the asymptotic theory by Saffman. The results are in agreement with the theory; furthermore, the error is found to be smaller than Saffman's own estimate by a factor square root ((nu x t)/R squared) (at least for Re=0). The error also decreases with increasing Re at fixed core-to-ring radius ratio, and appears to be independent of Re as Re approaches infinity). Following a single ring, with Re=500, the vorticity contours indicate shedding of vorticity into the wake and a settling of an initially circular core to a more elliptical shape, similar to Norbury's steady inviscid vortices. Finally, we consider the case of leapfrogging vortex rings with Re=1000. The results show severe straining of the inner vortex core in the first pass and merging of the two cores during the second pass

Atrial conduction velocity mapping: clinical tools, algorithms and approaches for understanding the arrhythmogenic substrate

Characterizing patient-specific atrial conduction properties is important for understanding arrhythmia drivers, for predicting potential arrhythmia pathways, and for personalising treatment approaches. One metric that characterizes the health of the myocardial substrate is atrial conduction velocity, which describes the speed and direction of propagation of the electrical wavefront through the myocardium. Atrial conduction velocity mapping algorithms are under continuous development in research laboratories and in industry. In this review article, we give a broad overview of different categories of currently published methods for calculating CV, and give insight into their different advantages and disadvantages overall. We classify techniques into local, global, and inverse methods, and discuss these techniques with respect to their faithfulness to the biophysics, incorporation of uncertainty quantification, and their ability to take account of the atrial manifold

Approximating the Solution of Surface Wave Propagation Using Deep Neural Networks

Partial differential equations formalise the understanding of the behaviour of the physical world that humans acquire through experience and observation. Through their numerical solution, such equations are used to model and predict the evolution of dynamical systems. However, such techniques require extensive computational resources and assume the physics are prescribed \textit{a priori}. Here, we propose a neural network capable of predicting the evolution of a specific physical phenomenon: propagation of surface waves enclosed in a tank, which, mathematically, can be described by the Saint-Venant equations. The existence of reflections and interference makes this problem non-trivial. Forecasting of future states (i.e. spatial patterns of rendered wave amplitude) is achieved from a relatively small set of initial observations. Using a network to make approximate but rapid predictions would enable the active, real-time control of physical systems, often required for engineering design. We used a deep neural network comprising of three main blocks: an encoder, a propagator with three parallel Long Short-Term Memory layers, and a decoder. Results on a novel, custom dataset of simulated sequences produced by a numerical solver show reasonable predictions for as long as 80 time steps into the future on a hold-out dataset. Furthermore, we show that the network is capable of generalising to two other initial conditions that are qualitatively different from those seen at training time

Towards fast simulation of environmental fluid mechanics with multi-scale graph neural networks

Numerical simulators are essential tools in the study of natural fluid-systems, but their performance often limits application in practice. Recent machine-learning approaches have demonstrated their ability to accelerate spatio-temporal predictions, although, with only moderate accuracy in comparison. Here we introduce MultiScaleGNN, a novel multi-scale graph neural network model for learning to infer unsteady continuum mechanics in problems encompassing a range of length scales and complex boundary geometries. We demonstrate this method on advection problems and incompressible fluid dynamics, both fundamental phenomena in oceanic and atmospheric processes. Our results show good extrapolation to new domain geometries and parameters for long-term temporal simulations. Simulations obtained with MultiScaleGNN are between two and four orders of magnitude faster than those on which it was trained

Combined CG-HDG Method for Elliptic Problems: Performance Model

We combine continuous and discontinuous Galerkin methods in the setting of a model diffusion problem. Starting from a hybrid discontinuous formulation, we replace element interiors by more general subsets of the computational domain - groups of elements that support a piecewise-polynomial continuous expansion. This step allows us to identify a~new weak formulation of Dirichlet boundary condition in the continuous framework. We examine the expected performance of a Galerkin solver that would use continuous Galerkin method with weak Dirichlet boundary conditions in each mesh partition and connect partitions weakly using trace variable as in HDG method

On weak Dirichlet boundary conditions for elliptic problems in the continuous Galerkin method

This is the author accepted manuscript. The final version is available from Elsevier via the DOI in this record.We combine continuous and discontinuous Galerkin methods in the setting of a model diffusion problem. Starting from a hybrid discontinuous formulation, we replace element interiors by more general subsets of the computational domain – groups of elements that support a piecewisepolynomial continuous expansion. This step allows us to identify a new weak formulation of Dirichlet boundary condition in the continuous framework. We show that the boundary condition leads to a stable discretization with a single parameter insensitive to mesh size and polynomial order of the expansion. The robustness of the approach is demonstrated on several numerical examples.European Union Horizon 2020US National Science Foundatio

Developing a cohesive theme for a programmatic behaviour change strategy

While it is commonly accepted that water, sanitation, and hygiene (WASH) programming should include both hardware and software components, these two elements are usually unintegrated because they are not conceived as a whole. Program implementers often choose to prioritize the hardware component as the sole deliverable of a program, and fail to see the uptake of software-related practices as critical for sustained health impact. Through a participatory and iterative process, Samaritan’s Purse and Clear Cambodia have developed a single cohesive theme that encapsulates the three key messages of their household water program: drinking treated water, hand-washing with soap, and practicing safe sanitation. This focused intentional effort to clarify the messages of the program forms the basis of a programmatic behaviour change strategy. This approach is applicable and adaptable for a variety of behaviour change interventions and across numerous geographic contexts