Eulerian Grid

[Gaynelle Alnutt]

Thegraph for the 8 x 9 grid depicted in the photo is Eulerian and solved with a braiding algorithm which for an N x M grid only works if N and M are relatively prime. A general algorithm like Hierholzer could be used but its regularity implies the existence of a deterministic algorithm to traverse the (2N+1) x (2M +1) verticies of the graph. I'm stuggling to find this algorithm but I'm sure it's there.

The motivation for this question is that continous extrusion with a 3D printer relies on Eulerian circuits to traverse all edges of the graph once and once only which is desirable for creating strong structures in plastic, clay or cement.

The solution I arrived at was to modify the Hierholzer algorithm slightly. Rather than choose the next adjacent vertex at random, if possible choose the vertex which lies on the same diagonal direction. This means weighting each edge with a code for the diagonal direction it is on and maintaining the direction as the algorithm proceeds. The adjacency structure is an array of lists of weighted edges.

Photochemical air quality models have become widely recognized and routinely utilized tools for regulatory analysis and attainment demonstrations by assessing the effectiveness of control strategies. These photochemical models are large-scale air quality models that simulate the changes of pollutant concentrations in the atmosphere using a set of mathematical equations characterizing the chemical and physical processes in the atmosphere. These models are applied at multiple spatial scales from local, regional, national, and global.

There are two types of photochemical air quality models commonly used in air quality assessments: the Lagrangian trajectory model that employs a moving frame of reference, and the Eulerian grid model that uses a fixed coordinate system with respect to the ground. Earlier generation modeling efforts often adopted the Lagrangian approach to simulate the pollutants formation because of its computational simplicity. The disadvantage of Lagrangian approach, however, is that the physical processes it can describe are somewhat incomplete. Most of the current operational photochemical air quality models have adopted the three-dimensional Eulerian grid modeling mainly because of its ability to better and more fully characterize physical processes in the atmosphere and predict the species concentrations throughout the entire model domain. This site provides links to several photochemical air quality models as follows:

Comprehensive Air quality Model with extensions (CAMx)- The CAMx model simulates air quality over many geographic scales. The model treats a wide variety of inert and chemically active pollutants, including ozone, particulate matter, inorganic and organic PM2.5/PM10, and mercury and other toxics. CAMx also has plume-in-grid and source apportionment capabilities.

Over the last two decades, EPA has devoted significant efforts to developing photochemical air quality models for the assessment of air pollution issues and evaluation of control strategies. The EPA's Air Quality Modeling Group has used photochemical models as part of its modeling analyses to support policy and regulatory decisions in OAR for which information can be found at Modeling Applications & Tools and provides guidance on the use of these models for attainment demonstrations available at Modeling Guidance & Support. Additional information about photochemical models can be found at Related Links.

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We present a coupled Eulerian-Lagrangian method to simulate cloud cavitation in a compressible liquid. The method is designed to capture the strong, volumetric oscillations of each bubble and the bubble-scattered acoustics. The dynamics of the bubbly mixture is formulated using volume-averaged equations of motion. The continuous phase is discretized on an Eulerian grid and integrated using a high-order, finite-volume weighted essentially non-oscillatory (WENO) scheme, while the gas phase is modeled as spherical, Lagrangian point-bubbles at the sub-grid scale, each of whose radial evolution is tracked by solving the Keller-Miksis equation. The volume of bubbles is mapped onto the Eulerian grid as the void fraction by using a regularization (smearing) kernel. In the most general case, where the bubble distribution is arbitrary, three-dimensional Cartesian grids are used for spatial discretization. In order to reduce the computational cost for problems possessing translational or rotational homogeneities, we spatially average the governing equations along the direction of symmetry and discretize the continuous phase on two-dimensional or axi-symmetric grids, respectively. We specify a regularization kernel that maps the three-dimensional distribution of bubbles onto the field of an averaged two-dimensional or axi-symmetric void fraction. A closure is developed to model the pressure fluctuations at the sub-grid scale as synthetic noise. For the examples considered here, modeling the sub-grid pressure fluctuations as white noise agrees a priori with computed distributions from three-dimensional simulations, and suffices, a posteriori, to accurately reproduce the statistics of the bubble dynamics. The numerical method and its verification are described by considering test cases of the dynamics of a single bubble and cloud cavitaiton induced by ultrasound fields.

Above: "Party" scene: Three hyper-elastic and two elasto-plastic objects are squashed into a complex contact configuration, all while fully two-way coupled with the surrounding fluid. All of the objects and the fluid are represented on a 200x180x200 Eulerian grid.

We present a new method that achieves a two-way coupling between deformable solids and an incompressible fluid where the underlying geometric representation is entirely Eulerian. Using the recently developed Eulerian Solids approach [Levin et al. 2011], we are able to simulate multiple solids undergoing complex, frictional contact while simultaneously interacting with a fluid. The complexity of the scenarios we are able to simulate surpasses those that we have seen from any previous method. Eulerian Solids have previously been integrated using explicit schemes, but we develop an implicit scheme that allows large time steps to be taken. The incompressibility condition is satisfied in both the solid and the fluid, which has the added benefit of simplifying collision handling.

In this paper an approach to multidimensional magnetohydrodynamics (MHD) which correctly handles shocks but does not use an approximate Riemann solver is proposed. This approach is simple and is based on control volume averaging with a staggered grid. The method builds on the older and often overlooked technique of on each step taking a fully 3-D Lagrangian step and then conservatively remapping onto the original grid. At the remap step gradient limiters are applied so that the scheme is monotonicity-preserving. For Euler's equations this technique, combined with an appropriately staggered grid and Wilkins artificial viscosity, can give results comparable to those from approximate Riemann solvers. We show how this can be extended to include a magnetic field, maintaining the divergence-free condition and pressure positivity and then present numerical test results. Where possible a comparison with other shock capturing techniques is presented and the advantages and disadvantages of the proposed scheme are clearly explained.

An axis-free Yin-Yang grid configuration plotted on a spherical surface. Both the Yin (red) and Yang (blue) grid are the low latitude part of the normal spherical polar grid and are identical in geometry. The Yang grid is obtained from the Yin grid by two rotations, and vice versa.

There are two main schools of thought for simulating fluids. The Lagrangianapproach simulates a large number of particles to approximate fluid moleculeswhereas the Eulerian approach divides the simulation space into grid cellsat which we keep track of different fluid properties such as velocity, pressure,and temperature.

The Lagrangian approach can easily simulate small-scale effects since itmeasures properties of the fluid at each individual particle. To mimic thislevel of detail, the Eulerian approach would need to increase the resolution ofthe grid as fluid effects smaller than the size of a grid cell cannot beaccurately represented.

In contrast, the Eulerian approach can simulate large-scale effectssignificantly faster as it only needs to maintain the one value per property ineach grid cell whereas the Lagrangian approach would need to simulate a largequantity of particles to fill the each grid cell in order to produce the sameeffect.

As a midground, there are hybrid models that use Eulerian grids overall withLagrangian particles near the surface where there many small-scale fluideffects. Alternatively, non-uniform adaptively sampled grids can similarlyreduce the computational workload while maintaining high levels of resolutionwhere needed.

For the remainder of this article, we'll be focusing on the Eulerian approachfor incompressible fluids. This article can be seen as my abridged version ofRobert Bridson's notes / textbook1 and draws some intuition andideas from Foster and Metaxas.2

Given that we're measuring quantities on the grid, we need to know how theychange in order to update their values. Namely, the question we want to answeris what is the rate of change of a fluid property at a given location over time?

Naively, we would think that this would simply be the time derivative of thefluid property, but this does not account for fluid movement. Imagine that at agrid point \(x\), we have some fluid particles \(f^0\). After we advance onetime step in the simulation, barring zero velocity at \(x\), those fluidparticles should have moved to a new location and other fluid particles\(f^1\) may have taken its place at \(x\). As a result, the value of themeasured fluid property at \(x\) is now that of the particles \(f^1\).

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