Showing papers on "Eulerian path published in 2006"

An Eulerian approach to fluid–structure interaction and goal‐oriented mesh adaptation

Abstract: We propose an Eulerian framework for modelling fluid-structure interaction (FSI) of incompressible fluids and elastic structures. The model is based on an Eulerian approach for describing structural dynamics. This is achieved by tracking the movement of the initial positions of all 'material' points. In this approach the displacement appears as a primary variable in an Eulerian framework. Our approach uses a technique which is similar to the level set method in so far that it also tracks initial data, in our case the set of initial positions (IP), and from this determines to which 'phase' a point belongs. To avoid the occasional reinitialization of the initial position set we employ the harmonic continuation of the structure velocity field into the fluid domain. By using the IP set for tracking the structure displacement, we can ensure that corners and edges of the fluid-structure interface are preserved well. Based on this monolythic model of the FSI we apply the Dual Weighted Residual (DWR) method for goal-oriented a posteriori error estimation to stationary FSI problems. Examples are presented based on the model and for the goal-oriented local mesh adaptation.

An Improvement of a Recent Eulerian Method for Solving PDEs on General Geometries

Abstract: We improve upon a method introduced in Bertalmio et al. [4] for solving evolution PDEs on codimension-one surfaces in $$\mathbb{R}^N.$$ As in the original method, by representing the surface as a level set of a smooth function, we use only finite differences on a Cartesian mesh to solve an Eulerian representation of the surface PDE in a neighborhood of the surface. We modify the original method by changing the Eulerian representation to include effects due to surface curvature. This modified PDE has the very useful property that any solution which is initially constant perpendicular to the surface remains so at later times. The change remedies many of problems facing the original method, including a need to frequently extend data off of the surface, uncertain boundary conditions, and terribly degenerate parabolic PDEs. We present numerical examples that include convergence tests in neighborhoods of the surface that shrink with the grid size

Adjoint-based control of a new eulerian network model of air traffic flow

Abstract: An Eulerian network model for air traffic flow in the National Airspace System is developed and used to design flow control schemes which could be used by Air Traffic Controllers to optimize traffic flow. The model relies on a modified version of the Lighthill-Whitham-Richards (LWR) partial differential equation (PDE), which contains a velocity control term inside the divergence operator. This PDE can be related to aircraft count, which is a key metric in air traffic control. An analytical solution to the LWR PDE is constructed for a benchmark problem, to assess the gridsize required to compute a numerical solution at a prescribed accuracy. The Jameson-Schmidt-Turkel (JST) scheme is selected among other numerical schemes to perform simulations, and evidence of numerical convergence is assessed against this analytical solution. Linear numerical schemes are discarded because of their poor performance. The model is validated against actual air traffic data (ETMS data), by showing that the Eulerian description enables good aircraft count predictions, provided a good choice of numerical parameters is made. This model is then embedded as the key constraint in an optimization problem, that of maximizing the throughput at a destination airport while maintaining aircraft density below a legal threshold in a set of sectors of the airspace. The optimization problem is solved by constructing the adjoint problem of the linearized network control problem, which provides an explicit formula for the gradient. Constraints are enforced using a logarithmic barrier. Simulations of actual air traffic data and control scenarios involving several airports between Chicago and the U.S. East Coast demonstrate the feasibility of the method

An explicit material point finite element method for hyper-velocity impact

Abstract: In this paper, an explicit material point finite element (FE) method is proposed and a computer code EMPFE-3D is developed for simulating hyper-velocity impact. The material domain is discretized by a mesh of finite elements. The momentum equations are solved on a predefined computational grid (like the material point method) in the large deformation zone, and on the FE mesh (like the traditional FE method) elsewhere. The grid may be fixed in space or moved in a predefined way. The nodes covered by the grid are treated as material particles, and the remaining nodes are treated as FE nodes. The proposed method yields the same results as the traditional FE method if the grid vanishes. On the other hand, it yields the same results as the material point method if the grid covers the entire material domain at all time steps. The method combines the advantages of Eulerian and Lagrangian descriptions of motion while eliminates their drawbacks due to element entanglement and numerical dissipation. The method is computationally efficient and can be easily implemented in an existing explicit FE code like DYNA3D. Copyright © 2005 John Wiley & Sons, Ltd.

Eulerian backtracking of atmospheric tracers. I: Adjoint derivation and parametrization of subgrid‐scale transport

Abstract: The problem of identification of sources of atmospheric tracers is most classically addressed through either Lagrangian backtracking or adjoint integration. On the basis of physical considerations, the retro-transport equation, which is at the basis of Lagrangian backtracking, can be derived in a Eulerian framework as well. Because of a fundamental time symmetry of fluid transport, Lagrangian or Eulerian backtracking can be used for inverting measurements of the concentration of an atmospheric tracer. The retro-transport equation turns out to be the adjoint of the direct transport equation, with respect to the scalar product defined by integration with respect to air mass. In the present paper, the exact equivalence between the physically-derived retro-transport and adjoint equations is proved. The transformation from the direct to the retro-transport equation requires only simple transformations. The sign of terms describing explicit advection is changed. Terms describing linear sources or sinks of tracers are kept unchanged. Terms representing diffusion by unresolved time-symmetric motions of the transporting air are also unchanged. This is rigorously shown for turbulent eddy-diffusion or mixing length theory. The case of subgrid-scale vertical transport by non-time-symmetric motions of air is studied using the example of the Tiedtke mass-flux scheme for cumulus convection. The retro-transport equation is then obtained by simply inverting the roles of updraughts and downdraughts, as well as of entrainment and detrainment. Conservation of mass of the transporting air is critical for all those properties to hold. Copyright © 2006 Royal Meteorological Society

Adaptive Finite Element Approximation of Fluid-Structure Interaction Based on an Eulerian Variational Formulation

Abstract: We propose a general variational framework for the adaptive finite element approximation of fluid-structure interaction problems. The modeling is based on an Eulerian description of the (incompressible) fluid as well as the (elastic) structure dynamics. This is achieved by tracking the movement of the initial positions of all `material' points. In this approach the deformation appears as a primary variable in an Eulerian framework. Our approach uses a technique which is similar to the 'Level Set' method in so far that it also tracks initial data, in our case the set of 'Initial Positions', and from this determines to which `phase' a point belongs. To avoid the need for reinitialization of the initial position set, we employ the harmonic continuation of the structure velocity field into the fluid domain. Based on this monolithic model of the fluid-structure interaction we apply the 'dual weighted residual method' for goal-oriented a posteriori error estimation and mesh adaptation to fluid-structure interaction problems. Results from nonstationary examples are presented.

Computer-Aided Eulerian Air Traffic Flow Modeling and Predictive Control

Abstract: Eulerian models are used to represent the air traffic environment as traffic flows between interconnected control volumes representing the airspace system. While these models can be manually derived for simple air traffic patterns, computer-based approaches are essential for modeling realistic airspaces involving multiple traffic streams. A computer- aided methodology for deriving large-dimensional Eulerian models of air traffic flow is described here. Starting from the specification of a few airspace parameters, and traffic data, the modeling technique can automatically construct Eulerian models of the airspace. The synthesis of air traffic flow control algorithms using the model predictive control technique in conjunction with these models is given. It is shown that the flow control logic synthesis can be cast as a linear programming problem. The flow control methodology is illustrated using air traffic data over two regions in U.S. airspace.

Eulerian transported probability density function sub-filter model for large-eddy simulations of turbulent combustion

Abstract: Reactive flow simulations using large-eddy simulations (LES) require modelling of sub-filter fluctuations. Although conserved scalars like mixture fraction can be represented using a beta-function, the reactive scalar probability density function (PDF) does not follow an universal shape. A one-point one-time joint composition PDF transport equation can be used to describe the evolution of the scalar PDF. The high-dimensional nature of this PDF transport equation requires the use of a statistical ensemble of notional particles and is directly coupled to the LES flow solver. However, the large grid sizes used in LES simulations will make such Lagrangian simulations computationally intractable. Here we propose the use of a Eulerian version of the transported-PDF scheme for simulating turbulent reactive flows. The direct quadrature method of moments (DQMOM) uses scalar-type equations with appropriate source terms to evolve the sub-filter PDF in terms of a finite number of delta-functions. Each delta-peak is c...

Detection of Lagrangian Coherent Structures in 3D Turbulence

Abstract: We use Direct Lyapunov Exponents (DLE) to identify Lagrangian coherent structures in two different three-dimensional flows, including a single isolated hairpin vortex, and a fully developed turbulent flow These results are compared with commonly used Eulerian criteria for coherent vortices We find that, despite additional computational cost, the DLE method has several advantages over Eulerian methods, including greater detail and the ability to define structure boundaries without relying on a preselected threshold As a further advantage, the DLE method requires no velocity derivatives, which are often too noisy to be useful in the study of a turbulent flow We study the evolution of a single hairpin vortex into a packet of similar structures, and show that the birth of a secondary vortex corresponds to a loss of hyperbolicity of the Lagrangian coherent structures

Eulerian versus Lagrangian Approaches to the Wave-Induced Transport in the Upper Ocean

Abstract: It is demonstrated that the Eulerian and the Lagrangian descriptions of fluid motion yield the same form for the mean wave-induced volume fluxes in the surface layer of a viscous rotating ocean. In the Eulerian case, the volume fluxes are obtained in the familiar way by integrating the horizontal components of the Navier–Stokes equation in the vertical direction, as seen, for example, in the book by Phillips. In the direct Lagrangian approach, the perturbation equations for the second-order mean drift are integrated in the vertical direction. This yields the advantage that the form drag, which is a source term for the wave-induced transports, can be related to the virtual wave stress that acts to transfer dissipated mean wave momentum into mean currents. In particular, for waves that are periodic in space and time, comparisons between empirical and theoretical relations for the form drag yield an estimate for the wave-induced bulk turbulent eddy viscosity in the surface layer. A simplistic approa...

q-Eulerian Polynomials: Excedance Number and Major index

Abstract: In this research announcement we present a new q-analog of a classical formula for the exponential generating function of the Eulerian polynomials. The Eulerian polynomials enumerate permutations according to their number of descents or their number of excedances. Our q-Eulerian polynomials are the enumerators for the joint distribution of the excedance statistic and the major index. There is a vast literature on q-Eulerian polynomials which involve other combinations of Mahonian and Eulerian permutation statistics, but the combination of major index and excedance number seems to have been completely overlooked until now. We use symmetric function theory to prove our formula. In particular, we prove a symmetric function version of our formula, which involves an intriguing new class of symmetric functions. We also present connections with representations of the symmetric group on the homology of a poset recently introduced by Bj\"orner and Welker and on the cohomology of the toric variety associated with the Coxeter complex of the symmetric group, studied by Procesi, Stanley, Stembridge, Dolgachev and Lunts.

Simulation of Pollutant Dispersion in the Atmosphere by the Laplace Transform: The ADMM Approach

Abstract: Transport and diffusion models of air pollution are based either on simple tech-nique, such as the Gaussian approach, or on more complex algorithms, such as thenumerical solution of air dispersion differential equation, based on K-theory. TheGaussianequationisaneasyandfastmethodwhich,however,cannotproperlysim-ulate complex nonhomogeneous conditions. The K-theory can accept virtually anycomplex meteorological input, but generally requires numerical integration whichis computationally expensive and is often affected by large numerical advectionerrors. Conversely, Gaussian models are fast, simple, do not require complex me-teorological input, and describe the diffusive transport in an Eulerian framework,making easy use of the Eulerian nature of measurements.For these reasons they are still widely used by the environmental agencies allover the world for regulatory applications. However, because of their well knownintrinsic limits, the reliability of a Gaussian model strongly depends on the waythe dispersion parameters are determined on the basis of the turbulence structure ofthe Planetary Boundary Layer (PBL) and the model’s ability to reproduce exper-imental diffusion data. The Gaussian model has to completed by empirically de-termined standard deviations (the so called “sigmas”) while some commonly mea-surable turbulent exchange coefficient has to introduce in the advection-diffusionequation.Analytical solutions to the complete advection-diffusion equation cannot begiven but in a few specialized cases (Tirabassi, 2003), and numerical solutions areexpansive and cannot be easily “interpreted” as the simple Gaussian model. Asa consequence, the major part of applications to practical problems are currentlydone by using the Gaussian model, and great deal of empirical work has been donedo determinate the “sigmas” appropriate to the PBL under various meteorologi-cal conditions and to extend the basic formulation of this model and its range ofapplicability (Zannetti, 1990).

Eulerian backtracking of atmospheric tracers. II: Numerical aspects

Abstract: In Part I of this paper, a mathematical equivalence was established between Eulerian backtracking or retro- transport, on the one hand, and adjoint transport with respect to an air-mass-weighted scalar product, on the other. The time symmetry which lies at the basis of this mathematical equivalence can however be lost through discretization. That question is studied, and conditions are explicitly identified under which discretization schemes possess the property of time symmetry. Particular consideration is given to the case of the LMDZ model. The linear schemes used for turbulent diffusion and subgrid-scale convection are symmetric. For the Van Leer advection scheme used in LMDZ, which is nonlinear, the question of time symmetry does not even make sense. Those facts are illustrated by numerical simulations performed in the conditions of the European Transport EXperiment (ETEX). For a model that is not time-symmetric, the question arises as to whether it is preferable, in practical applications, to use the exact numerical adjoint, or the retro-transport model. Numerical results obtained in the context of one-dimensional advection show that the presence of slope limiters in the Van Leer advection scheme can produce in some circumstances unrealistic (in particular, negative) adjoint sensitivities. The retro-transport equation, on the other hand, generally produces robust and realistic results, and always preserves the positivity of sensitivities. Retro-transport may therefore be preferable in sensitivity computations, even in the context of variational assimilation. Copyright © 2006 Royal Meteorological Society

New asymptotic description of nonlinear water waves in Lagrangian coordinates

Abstract: A new description of two-dimensional continuous free-surface flows in Lagrangian coordinates is proposed. It is shown that the position of a fluid particle in such flows can be represented as a fixed point of a transformation in R 2 . Components of the transformation function satisfy the linear Euler-type continuity equation and can be expressed via a single function analogous to an Eulerian stream function. Fixed-point iterations lead to a simple recursive representation of a solution satisfying the Lagrangian continuity equation. Expanding the unknown function in a small-perturbation asymptotic expansion we obtain the complete asymptotic formulation of the problem in a fixed domain of Lagrangian labels. The method is then applied to the classical problem of a regular wave travelling in deep water, and the fifth-order Lagrangian asymptotic solution is constructed, which provides a much better approximation of steep waves than the corresponding Eulerian Stokes expansion. In contrast with early attempts at Lagrangian regular-wave expansions, the asymptotic solution presented is uniformly valid at large times.

A comparison of three Lagrangian approaches for extending near field mixing calculations

Abstract: Lagrangian techniques have previously been employed to extend initial mixing calculations beyond the near field, either alone or in combination with Eulerian models. Computational efficiency and accuracy are of prime importance in designing these ‘hybrid’ approaches to simulating a pollutant discharge, and we characterize three relatively simple Lagrangian techniques in this regard: random walk particle tracking (RWPT), forward Gaussian puff tracking (FGPT), and backward Gaussian puff tracking (BGPT). RWPT is generally the most accurate, capable of handling complexities in the flow field and domain geometry. It is also the most computationally expensive, as a large number of particles are generally required to generate a smooth concentration distribution. FGPT and BGPT offer dramatic savings in computational expense, but their applicability is limited by accuracy concerns in the presence of spatially variable flow or diffusivity fields or complex no-flux or open boundary conditions. For long simulations, particle and/or puff methods can transition to an Eulerian model if appropriate, since the relative computational expense of Lagrangian methods increases with time for continuous sources. Although we focus on simple Lagrangian models that are not suitable to all environmental applications, many of the implementation and computational efficiency concerns outlined herein would also be relevant to using higher order particle and puff methods to extend the near field.

Lattice-Boltzmann Method on Quadtree-Type Grids for Fluid-Structure Interaction

Abstract: In this work a Lattice Boltzmann (LB) fluid flow solver based on unstructured quadtree/octree type Eulerian grids is coupled with a spectral Finite Element (p-FEM) structural mechanics solver based on a Lagrangian description to predict bidirectional fluid-structure interaction (FSI). The methods and algorithms are described in detail. Benchmark computations of a coupled transient problem of a 2D beam-like structure in a channel as defined by the DFG-Research Unit 493 are presented.

Coupling the Level Set Method and the Topological Gradient in Structural Optimization

Abstract: A numerical coupling of two recent methods in shape and topology optimization of structures is proposed. On the one hand, the level set method, based on the shape derivative, is known to easily handle boundary propagation with topological changes. However, in practice it does not allow for the nucleation of new holes. On the other hand, the bubble or topological gradient method is precisely designed for introducing new holes in the optimization process. Therefore, the coupling of these two methods yields an efficient algorithm which can escape from local minima. It have a low CPU cost since it captures a shape on a fixed Eulerian mesh. The main advantage of our coupled algorithm is to make the resulting optimal design more independent of the initial guess.

Connection between two statistical approaches for the modelling of particle velocity and concentration distributions in turbulent flow: The mesoscopic Eulerian formalism and the two-point probability density function method

Abstract: The objective of the paper is to elucidate a connection between two approaches that have been separately proposed for modelling the statistical spatial properties of inertial particles in turbulent fluid flows One of the approaches proposed recently by Fevrier, Simonin, and Squires [J Fluid Mech 533, 1 (2005)] is based on the partitioning of particle turbulent velocity field into spatially correlated (mesoscopic Eulerian) and random-uncorrelated (quasi-Brownian) components The other approach stems from a kinetic equation for the two-point probability density function of the velocity distributions of two particles [Zaichik and Alipchenkov, Phys Fluids 15, 1776 (2003)] Comparisons between these approaches are performed for isotropic homogeneous turbulence and demonstrate encouraging agreement

On the diameter of Eulerian orientations of graphs

Abstract: We compare the diameter of a graph with the directed diameter of its Eulerian orientations. We obtain positive results under certain symmetry conditions.An Eulerian orientation of a graph is an orientation such that each vertex has the same indegree and outdegree. A graph is vertex-transitive if its vertices are equivalent under automorphisms.We show that the directed diameter of an Eulerian orientation of a finite vertex-transitive graph cannot be much larger than the undirected diameter; our bound on the directed diameter is O (dΔ ln n) where d is the undirected diameter, Δ is the (out)degree of the vertices, and n is the number of vertices. This implies that for Eulerian orientations of vertex-transitive graphs-of bounded degree, the gap between the two diameters is at most quadratic.As a consequence, we are able to compare the word length and the positive word length of elements of a finite group in terms of a given set of generators; we show that the gap is at most nearly quadratic, where the term "nearly" refers to a factor, polylogarithmic in the order of the group.It follows that recent polynomial bounds on the diameter of certain large classes of Cayley graphs of the symmetric group and certain linear groups automatically extend to directed Cayley graphs. The result also shows that the directed and undirected versions of long standing conjectures regarding the diameter of Cayley graphs of various classes of groups, including transitive permutation groups and finite simple groups, are equivalent.We also show that for edge-transitive digraphs, the directed diameter is O(d ln n).On the other hand, if we weaken the condition of vertex-transitivity to regularity (all vertices have the same degree), then the directed diameter is no longer polynomially bounded in terms of the undirected diameter and the maximum degree (and In n = O(d ln Δ)).Our upper bounds on the diameter raise the algorithmic challenge to find paths of the length guaranteed by these results. While for undirected graphs, most (but not all) relevant proofs are algorithmic, our bounds for the directed diameter are obtained via a pigeon-hole argument based on expansion and yield existence only.