An entropy stable discontinuous Galerkin method for the shallow water equations on curvilinear meshes with wet/dry fronts accelerated by GPUs

We extend the entropy stable high order nodal discontinuous Galerkin spectral element approximation for the non-linear two dimensional shallow water equations presented by Wintermeyer et al. [N. Wintermeyer, A. R. Winters, G. J. Gassner, and D. A. Kopriva. An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry. Journal of Computational Physics, 340:200-242, 2017] with a shock capturing technique and a positivity preservation capability to handle dry areas. The scheme preserves the entropy inequality, is well-balanced and works on unstructured, possibly curved, quadrilateral meshes. For the shock capturing, we introduce an artificial viscosity to the equations and prove that the numerical scheme remains entropy stable. We add a positivity preserving limiter to guarantee non-negative water heights as long as the mean water height is non-negative. We prove that non-negative mean water heights are guaranteed under a certain additional time step restriction for the entropy stable numerical interface flux. We implement the method on GPU architectures using the abstract language OCCA, a unified approach to multi-threading languages. We show that the entropy stable scheme is well suited to GPUs as the necessary extra calculations do not negatively impact the runtime up to reasonably high polynomial degrees (around $N=7$). We provide numerical examples that challenge the shock capturing and positivity properties of our scheme to verify our theoretical findings

An Entropy Stable Nodal Discontinuous Galerkin Method for the Two Dimensional Shallow Water Equations on Unstructured Curvilinear Meshes with Discontinuous Bathymetry

We design an arbitrary high-order accurate nodal discontinuous Galerkin spectral element approximation for the nonlinear two dimensional shallow water equations with non-constant, possibly discontinuous, bathymetry on unstructured, possibly curved, quadrilateral meshes. The scheme is derived from an equivalent flux differencing formulation of the split form of the equations. We prove that this discretisation exactly preserves the local mass and momentum. Furthermore, combined with a special numerical interface flux function, the method exactly preserves the mathematical entropy, which is the total energy for the shallow water equations. By adding a specific form of interface dissipation to the baseline entropy conserving scheme we create a provably entropy stable scheme. That is, the numerical scheme discretely satisfies the second law of thermodynamics. Finally, with a particular discretisation of the bathymetry source term we prove that the numerical approximation is well-balanced. We provide numerical examples that verify the theoretical findings and furthermore provide an application of the scheme for a partial break of a curved dam test problem

Leonardo's rule, self-similarity and wind-induced stresses in trees

Examining botanical trees, Leonardo da Vinci noted that the total cross-section of branches is conserved across branching nodes. In this Letter, it is proposed that this rule is a consequence of the tree skeleton having a self-similar structure and the branch diameters being adjusted to resist wind-induced loads

Relationship between casting modulus and grain size in cast A356 aluminium alloys

Microstructure of Al-Si alloy castings depends most generally on melt preparation and on the cooling rate imposed by the thermal modulus of the component. In the case of Al-Si alloys, emphasis is put during melt preparation on refinement of pro-eutectic (Al) grains and on modification of the Al-Si eutectic. Thermal analysis has been used since long to check melt preparation before casting, i.e. by analysis of the cooling curve during solidification of a sample cast in an instrumented cup. The conclusions drawn from such analysis are however valid for the particular cooling conditions of the cups. It thus appeared of interest to investigate how these conclusions could extrapolate to predict microstructure in complicated cast parts showing local changes in the solidification conditions. For that purpose, thermal analysis cups and instrumented sand and die castings with different thermal moduli and thus cooling rates have been made, and the whole set of cooling curves thus recorded has been analysed. A statistical analysis of the characteristic features of the cooling curves related to grain refinement in sand and die castings allowed determining the most significant parameters and expressing the cube of grain size as a polynomial of these parameters. After introduction of a further parameter quantifying melt refining an excellent correlation, with a R2 factor of 0.99 was obtained

The Integrated Polarization of Spiral Galaxy Disks

We present integrated polarization properties of nearby spiral galaxies at 4.8 GHz, and models for the integrated polarization of spiral galaxy disks as a function of inclination. Spiral galaxies in our sample have observed integrated fractional polarization in the range < 1% to 17.6%. At inclinations less than 50 degrees, the fractional polarization depends mostly on the ratio of random to regular magnetic field strength. At higher inclinations, Faraday depolarization associated with the regular magnetic field becomes more important. The observed degree of polarization is lower (<4%) for more luminous galaxies, in particular those with L_{4.8} > 2 x 10^{21} W/Hz. The polarization angle of the integrated emission is aligned with the apparent minor axis of the disk for galaxies without a bar. In our axially symmetric models, the polarization angle of the integrated emission is independent of wavelength. Simulated distributions of fractional polarization for randomly oriented spiral galaxies at 4.8 GHz and 1.4 GHz are presented. We conclude that polarization measurements, e.g. with the SKA, of unresolved spiral galaxies allow statistical studies of the magnetic field in disk galaxies using large samples in the local universe and at high redshift. As these galaxies behave as idealized background sources without internal Faraday rotation, they can be used to detect large-scale magnetic fields in the intergalactic medium.Comment: 13 pages, 6 figures; Accepted for publication in The Astrophysical Journa

Wine industry perceptions and reactions to the COVID-19 crisis in the Old and New Worlds: Do business models make a difference?

The COVID-19 crisis has severely impacted the wine industry, with producers in different countries affected differently and, therefore, differing in their perceptions toward it. These differing perceptions are assumed to be due to different business models, mainly linked to the distribution system adopted and resulting in varying distances of producers to distributors and consumers. While upstream integration characterizes the Old World, the New World applies a downstream business model, being more closely linked to distributors and consumers and, therefore, more vulnerable to shocks, which should lead to higher perceived impacts of the COVID-19 crisis. This study analyzes 542 surveys collected from wineries in nine countries, divided into New World, historical Old World, and emerging Old World. Econometric results show statistically significant differences in both the perceived impact of COVID-19 and wineries'' responses in terms of planned investments, with the New World being more affected. A common desire by wineries to direct future investments towards direct-to-consumer sales and communication was found, to the detriment of investments in vineyards and cellars. This desire is particularly strong in the New World, in line with their focus on the downstream part of the value chain, underlying their greater reactivity to shocks and capacity to innovate. EconLit Citations: [D22; L21; L66; Q13]

Conformal regularization of Einstein's field equations

To study asymptotic structures, we regularize Einstein's field equations by means of conformal transformations. The conformal factor is chosen so that it carries a dimensional scale that captures crucial asymptotic features. By choosing a conformal orthonormal frame we obtain a coupled system of differential equations for a set of dimensionless variables, associated with the conformal dimensionless metric, where the variables describe ratios with respect to the chosen asymptotic scale structure. As examples, we describe some explicit choices of conformal factors and coordinates appropriate for the situation of a timelike congruence approaching a singularity. One choice is shown to just slightly modify the so-called Hubble-normalized approach, and one leads to dimensionless first order symmetric hyperbolic equations. We also discuss differences and similarities with other conformal approaches in the literature, as regards, e.g., isotropic singularities.Comment: New title plus corrections and text added. To appear in CQ

Two-Photon Na+ Imaging Reports Somatically Evoked Action Potentials in Rat Olfactory Bulb Mitral and Granule Cell Neurites

Dendrodendritic synaptic interactions are a hallmark of neuronal processing in the vertebrate olfactory bulb. Many classes of olfactory bulb neurons including the principal mitral cells (MCs) and the axonless granule cells (GCs) dispose of highly efficient propagation of action potentials (AP) within their dendrites, from where they can release transmitter onto each other. So far, backpropagation in GC dendrites has been investigated indirectly via Ca2+ imaging. Here, we used two-photon Na+ imaging to directly report opening of voltage-gated sodium channels due to AP propagation in both cell types. To this end, neurons in acute slices from juvenile rat bulbs were filled with 1 mM SBFI via whole-cell patch-clamp. Calibration of SBFI signals revealed that a change in fluorescence Delta F/F by 10% corresponded to a Delta[Na+](i) of similar to 22 mM. We then imaged proximal axon segments of MCs during somatically evoked APs (sAP). While single sAPs were detectable in similar to 50% of axons, trains of 20 sAPs at 50 Hz always resulted in substantial Delta F/F of similar to 15% (similar to 33 mM Delta[Na+](i)). Delta F/F was significantly larger for 80 Hz vs. 50 Hz trains, and decayed with half-durations tau(1/2) similar to 0.6 s for both frequencies. In MC lateral dendrites, AP trains yielded small Delta F/F of similar to 3% (similar to 7 mM Delta[Na+](i)). In GC apical dendrites and adjacent spines, single sAPs were not detectable. Trains resulted in an average dendritic Delta F/F of 7% (16 mM Delta[Na+](i)) with tau(1/2) similar to 1 s, similar for 50 and 80 Hz. Na+ transients were indistinguishable between large GC spines and their adjacent dendrites. Cell-wise analysis revealed two classes of GCs with the first showing a decrease in Delta F/F along the dendrite with distance from the soma and the second an increase. These classes clustered with morphological parameters. Simulations of Delta[Na+](i) replicated these behaviors via negative and positive gradients in Na+ current density, assuming faithful AP backpropagation. Such specializations of dendritic excitability might confer specific temporal processing capabilities to bulbar principal cell-GC subnetworks. In conclusion, we show that Na+ imaging provides a valuable tool for characterizing AP invasion of MC axons and GC dendrites and spines