Showing papers on "Eulerian path published in 1988"

Hamiltonian fluid mechanics

Abstract: This paper reviews the relatively recent application of the methods of Hamiltonian mechanics to problems in fluid dynamics. By Hamiltonian mechanics I mean all of what is often called classical mechanics-the subject of the textbooks by Lanczos ( 1970), Goldstein ( 1 980), and Arnol'd (1978). Since the advent of quantum mechanics, Hamiltonian methods have played an increasingly important role in both the classical and quan­ tum mechanics of particles and fields. By comparison, the introduction of Hamiltonian methods into fluid mechanics has been tardy. Why is this so? In general mechanical systems, the Lagrangian or Hamiltonian equa­ tions of motion are coupled equations governing the locations and veloc­ ities of massive particles or rigid bodies. These coupled equations cannot generally be solved for any subset of the dependent variables without also finding all of the other dependent variables. By contrast, the conventional Eulerian fluid equations are closed equations in the velocity, density, and entropy (regarding pressure as a prescribed function of the density and entropy) that can (in principle) be solved without also finding the trajectory of every fluid particle. Once the velocity field is known, the particle tra­ jectories can always be reconstructed by solving the equations for three independent, passively advected tracers (such as the initial Cartesian com­ ponents), but these extra computations are not required if only the Eulerian fields are sought. In the special case of constant-density flow, the Eulerian equations are dramatically simpler than the general Lagrangian or Hamil­ tonian equations for the fluid. From the Hamiltonian perspective, the extraordinary simplicity of the Eulerian description derives from a symmetry property of the fluid

A reduction method to find spanning Eulerian subgraphs

Abstract: We ask, When does a graph G have a subgraph Γ such that the vertices of odd degree in Γ form a specified set S ⊆ V(G), such that G - E(Γ) is connected? If such a subgraph can be found for a suitable choice of S, then this can be applied to problems such as finding a spanning eulerian subgraph of G. We provide a general method, with applications.

Approximate parallel scheduling. Part I: the basic technique with applications to optimal parallel list ranking in logarithmic time

Abstract: We define a novel scheduling problem; it is solved in parallel by repeated, rapid, approximate reschedulings. This leads to the first optimal logarithmic time PRAM algorithm for list ranking. Companion papers show how to apply these results to obtain improved PRAM upper bounds for a variety of problems on graphs, including the following: connectivity, biconnectivity, Euler tour and $st$-numbering, and a number of problems on trees.

A Two-Time-Level Semi-Lagrangian Semi-implicit Scheme for Spectral Models

Abstract: Recently, it has been demonstrated that the semi-implicit semi-Lagrangian technique can be successfully coupled with a three-time-level spectral discretization of the barotropic shallow-water equations. This permits the use of time steps that are much larger than those permitted by the Courant-Friedrichs-Lewy (CFL) stability criterion for the corresponding Eulerian model, without loss of accuracy. In this paper we show that it is possible to further quadruple the efficiency of semi-implicit semi-Lagrangian spectral models beyond that already demonstrated. A doubling of efficiency accrues from the use of the stable and accurate two-time-level scheme described herein. For semi-implicit semi-Lagrangian spectral models a further doubling of efficiency can be achieved by using a smaller computational Gaussian grid than the usual one, without incurring the significant loss of stability and accuracy that is observed for the corresponding Eulerian spectral model in analogous circumstances.

Gravity wave heat fluxes - A Lagrangian approach

Abstract: The effect of a vertically propagating, internal gravity wave on the vertical flux of potential temperature (heat) is considered by averaging the local heat flux vector over a potential temperature surface. This approach gives the wave heat flux a simple physical picture which is not readily apparent from the more common Eulerian formulation. This method also allows the eddy diffusion coefficient to be a function of the phase of the wave. Such a phase dependent eddy diffusion has been previously considered from an Eulerian viewpoint as a model of a convectively unstable gravity wave. Here, the Lagrangian method confirms and corrects the Eulerian results. Earlier work is extended by modeling a constant amplitude “breaking” wave, as well as by considering eddy diffusion coefficients that are asymmetric with respect to the wave breaking region. In all cases studied, 1ocalizing the eddy diffusion to the region of wayebreaking decreases the average heat flux.

On Connectivity Properties of Eulerian Digraphs

Abstract: Directed counterparts of theorems of Rothschild and Whinston and of Lovasz concerning Eulerian graphs are proved. As a consequence, a polynomial time algorithm is presented to solve the integral two-commodity flow problem in directed graphs in the case where all capacities are ‘Eulerian‘.

Some remarks on arc-connectivity, vertex splitting, and orientation in graphs and digraphs

Abstract: We apply proof techniques developed by L. Lovasz and A. Frank to obtain several results on the arc-connectivity of graphs and digraphs. The first results concern the operation of splitting two arcs from a vertex of an Eulerian graph or digraph in such a way as to preserve local connectivity conditions. The final result is concerned with orienting the edges of a mixed graph (consisting of vertices, undirected edges, and directed arcs) in such a way that the resulting digraph is as arc-connected as possible.

Dynamic crack propagation analysis using Eulerian-Lagrangian kinematic descriptions

Abstract: A dynamic version of the Eulerian-Lagrangian kinematic description (ELD) (Koh and Haber (1986 a) is applied to the analysis of elastodynamic fracture problems. The ELD formulation leads to moving finite element procedures which adjust the mesh to changes in the structural geometry due to crack extension. The use of quarter-point isoparametric finite elements with the dynamic ELD ensures correct modeling of the singular forms in both the strain field and the material velocity field. A two-level mapping is used to describe the Eulerian mesh motion based on the crack-tip motion. This greatly reduces, or eliminates, the need to remesh and interpolate field variables to the new node locations, as is required in transient analyses based on Lagrangian models. Stress intensity factors are computed from numerical evaluations of either the actual dynamic energy release rate for running cracks or the instantaneous virtual energy release rate for stationary cracks. Numerical examples indicate that the ELD accurately models geometric changes due to crack growth and their effect on material motion. Three-dimensional color computer visualization techniques are used to interpret the transient field solutions.

Hamiltonian formulation of inviscid flows with free boundaries

Abstract: The formulation of the Hamiltonian structures for inviscid fluid flows with material free surfaces is presented in both the Lagrangian specification, where the fundamental Poisson brackets are canonical, and in the Eulerian specification, where the dynamics is given in noncanonical form. The noncanonical Eulerian brackets are derived explicitly from the canonical Lagrangian brackets. The Eulerian brackets are, with the exception of a single term at each material free surface separating flows in different phases, identical to those for isentropic flow of a compressible, inviscid fluid. The dynamics of the free surface is located in the Hamiltonian and in the definition of the Eulerian variables of mass density, ρ(x, t), momentum density, M(x,t) [which is ρ times the fluid velocity v(x,t)], and the specific entropy, σ(x,t). The boundary conditions for the Eulerian variables and the evolution equations for the free surfaces come from the Euler equations of the flow. This construction provides a unified treatment of inviscid flows with any number of free surfaces.

The statistics of the organized vortical structure in turbulent mixing layers

Abstract: The statistics of the large scale vortex structure in turbulent mixing layers have been investigated theoretically. It is shown that similarity in the fully developed flow results in a common description of the Eulerian and Lagrangian statistics. In the Eulerian frame of reference, a conservation equation is derived and solved to show that the distribution of vortex circulation is lognormal. It is also shown that the standard deviation normalized by the mean value of the distribution depends only on the amalgamation mechanism. The value for pairing is in good agreement with experimental measurements. These results are used to calculate the life span and survival probabilities of the vortices in the Lagrangian frame of reference. These distributions are in good agreement with direct measurements of the life span probability and with space‐time correlation measurements, respectively. Some implications of these results on the dynamics of the large scale vortices in the fully developed turbulent flow are discussed.

On the relationship between the Eulerian and Lagrangian descriptions of finite rigid plasticity

Abstract: The motivation for the present paper is to clarify certain unresolved issues pertaining to the relation between the Lagrangian (or referential) and Eulerian (or spatial) strain-space formulations of finite plasticity. For conceptual simplicity, attention is confined to rigid-plastic materials. It is shown first that for constitutive equations in which the hardening parameter is a scalar, the Lagrangian and Eulerian descriptions are equivalent; and that, additionally, the choice of objective stress rate is immaterial. In the light of these developments, the role of objective rates is further explored in connection with more general (“anisotropic”) hardening laws which contain a shift tensor. A form of the constitutive equation for the rate of the shift tensor is motivated in which the choice of objective rate is arbitrary. It is then demonstrated that the structure of the constitutive equations of the theory — in both the Eulerian and Lagrangian descriptions — is form-invariant under arbitrary transformations of objective rate. The approach taken here contrasts with that adopted in a number of recent papers in which preference is given to one particular objective rate or another.

Contractions of graphs with no spanning eulerian subgraphs

Abstract: Letp≧2 be a fixed integer, and letG be a connected graph onn vertices. Ifδ(G)≧2, ifd(u)+d(v)>2n/p−2 holds wheneveruv∉E(G), and ifn is sufficiently large compared top, then eitherG has a spanning eulerian subgraph, orG is contractible to a graphG1 of order less thenp and with no spanning eulerian subgraph. The casep=2 was proved by Lesniak-Foster and Williamson. The casep=5 was conjectured by Benhocine, Clark, Kohler, and Veldman, when they proved virtually the casep=3. The inequality is best-possible.

Physical Modeling With Eulerian Frames and Bond Graphs

Abstract: In this paper it is shown that open systems, in which matter can enter and leave, are not incompatible with bond graph topology, as commonly reported. Effective modeling of these systems requires that attention be paid to the convective coupling between systems and environment which exchange matter. The convection models proposed in this work do not require the use of active bonds (other than modulation signals), controlled sources, ad hoc elements, or any other special bond graph artifacts.

Regular and chaotic motion of fluid particles in a two-dimensional fluid

Abstract: The authors study the chaotic behaviour exhibited by particles which move in a two-dimensional fluid. The connection of this Lagrangian chaos with the velocity field behaviour is discussed both in the Lorenz model and in truncated Navier-Stokes equations. They indicate a possible method for the onset of Lagrangian chaos which seems to be rather generic. Lagrangian chaos appears when the Eulerian equation passes from a steady solution to a periodic one via Hopf bifurcation. It is also shown that the transition to chaos for the velocity field ('Eulerian chaos') does not affect the particle motion properties in some typical cases.

A consistent Eulerian formulation for large deformation analysis with reference to metal-extrusion process

Abstract: A critical discussion of the different formulation methods for the finite element analysis of non-linear problems is given. The discussion is concerned mainly with the differences between updated Lagrangian and Eulerian formulations, and with the specific nature and basic characteristics of each. A consistent Eulerian formulation is developed by means of the virtual work principle in an explicit form. Differences between the presented formulation and similar formulations as well as the attempted Eulerian formulations are discussed according to the classification of formulation methods. To demonstrate the applicability and the effectiveness of the Eulerian finite element analysis using a fixed mesh in space, a metal-extrusion problem has been solved. In this approach, the mesh is maintained fixed in space and the increment of stress tensors for a forward incremental step are added to a set of interpolated stress tensors. Then, these stresses are interpolated back to obtain the state of stress of the body-points momentarily occupying the fixed integration points of the mesh.

Computation of two-phase shear-layer flow using an Eulerian-Lagrangian analysis

Abstract: A combined Eulerian-Lagrangian analysis which combines a linearized block implicit Navier-Stokes analysis for the continuous phase with a Lagrangian analysis for the discrete phase has been utilized to simulate a two-phase shear-layer flow in a two-dimensional channel flow configuration. The motion of the particles is tracked in computational coordinate space, resulting in improved computational efficiency, and interphase coupling terms for the Eulerian analysis are computed from the instantaneous distribution of the particles rather than the trajectory information. The use of the implicit Navier-Stokes analysis for the continuous phase has made it possible to efficiently use a highly stretched mesh. A low Reynolds number form of the k-epsilon turbulence model was used with near-wall resolution, thus eliminating the need to use the 'wall-function' approach. The streamwise velocity profiles for the continuous and the discrete phases have been compared with experimental data for two test cases.

Two-dimensional adaptive Eulerian-Lagrangian method for mass transport with spatial velocity distribution

Abstract: A Eulerian-Lagrangian scheme is used to solve the two-dimensional advection-dispersion equation. Concentration and its partial differential operator are decomposed into advection and dispersion terms. Thus, advection is formally decoupled from dispersion and solved by continuous forward particle tracking. Dispersion is handled by implicit finite elements on a fixed Eulerian grid. Translation of steep gradients of concentration in advection-dominated flow regimes, is done without numerical distortion. Continuous spatial distribution of velocities are evaluated by using Galerkin's approach in conjunction with Darcy's law based on hydraulic input data from each element. The method was implemented on coarse FE grid with linear shape functions, demonstrating no over/under shooting and practically no numerical dispersion. Simulations, covering a wide range of Peclet numbers, yield high agreement with analytic and practical results.

On dual Eulerian paths and circuits in plane graphs

Abstract: Given a nonseparable plane graph G, a path or circuit is called dual if it is also a path or circuit, respectively, in the geometric dual of G. Motivated by a layout design problem of CMOS integrated circuits, the authors consider some problems of partitioning the edges of G into the minimum number of dual paths or circuits. The results include a constructive proof of the fact that the following problems are solvable in polynomial time: determining whether G has a dual circuit; finding a dual Eulerian path or circuit if one exists; and finding the minimum set of dual paths that partitions the edges of G when G has no dual circuits. >

A fast heuristics for optimal CMOS functional cell layout generation

Abstract: The problem of generating minimal-area CMOS functional cell layout can be converted to that of decomposing the transistor connection graph into a minimum number of subgraphs, each having a pair of Euler paths, with the same sequence of input labels on the N-graph and P-graph, which are portions of the graph corresponding to NMOS and PMOS parts, respectively. The authors propose a heuristic algorithm which yields a nearly minimum number of Euler paths from the path representation formulation which represents the given logic function. Subpath merging is done through a list processing scheme where the pairs of paths that results in the lowest cost are successively merged from all candidate merge pairs until no further path merging and no further reduction of the number of subgraphs are possible. >

Tricritical points of trails, their Euler digraphs and graphs: exact results on the Sierpinski gasket

Abstract: The collapse transitions of trails, their oriented graphs and silhouettes (Eulerian digraphs and graphs, respectively), as the fugacity for crossings is increased, are investigated by exact decimation on the 2D Sierpinski gasket. Recursion relations between the generating functions for the three basic configurations on consecutive levels are derived. For all models the authors find tricritical points which move along a line in a four-dimensional parameter space, as the fugacity is varied, and terminate at a decoupled first- or second-order multicritical point. It suggests these models belong to distinct universality classes which differ from that of self-attracting polymer chains which do not undergo a collapse Theta transition on this fractal lattice.

Method for tying and untying path access in a CPU-based, layered communications system

Abstract: A method for maintaining the integrity of ties and their associated tie groups in a CPU-based, layered communications subsystem in which the connection endpoints in each layer are denoted by a connection control block (CCB), the relationship between CCBs being denoted by ties. Ties and their CCBs can be mapped onto an edge-oriented graph of tie group relations. The arbitrary removal of an edge (tie) in the graph compromises graph integrity by possible formation of unenumerated subgraphs or independent graphs. The solution involves enumerating those edges having vertices which no longer reference CCBs within the tie group, removing them, and forming a second tie group. The enumeration is conducted over a Eulerian traverse of the remaining n edges of the graph. A Eulerian traverse of a graph is one which traverses each edge exactly once. Such a traverse reduces the number of comparisons M to a range N

Finite element eulerian-lagrangian method for time dependent flow of multimode fluids

Abstract: This work is directed towards the application of numerical methods to the solution of time-dependent flows of viscoelastic fluids. An Eulerian-Lagrangian method has been developed in which time differences are carried out in the Lagrangian sense on a fixed mass of material occupying the control volume during a single time step. After transformation of the Lagrangian difference equations to the Eulerian description, the spatial dependence of the unknown fields is calculated by the finite element method. The backward Lagrangian time difference automatically builds exact upwinding into the scheme, and reduces the asymmetry of the matrix of coefficients. An important feature of this technique is its applicability to multi-mode fluids whose properties are characterized by a spectrum of relaxation times. In this paper the method is demonstrated with the problem of startup of Poiseuille flow for the Oldroyd-B fluid and for a multimode, rubberlike fluid.

A Method to Reduce Cost of Mesh Deformation in Eulerian-Lagrangian Formulation

Abstract: In the past few years, the Eulerian-Lagrangian formulation in simulating forming processes has been develloped by several authors in order to overcome problems met by using purely Eulerian or purely Lagrangian formulation. The principal drawback of this new method is the computational time required in handling the moving finite element mesh. In the present paper, we present an explicit, low computer cost procedure for mesh adaptation based on the transfinite mapping method.

Bigeodetic graphs

Abstract: Bigeodetic graphs, a generalization of geodetic and interval-regular graphs, are defined as graphs in which each pair of vertices has at most two paths of minimum length between them. The block cut-vertex incidence pattern of bigeodetic separable graphs are discussed. Two characterizations of bigeodetic graphs are given and some properties of these graphs are studied. Construction of planar bigeodetic blocks with given girth and diameter, and construction of hamiltonian and eulerian/nonhamiltonian and noneulerian, perfect bigeodetic blocks are discussed. The extremal bigeodetic graph of diameterd onp ? d + 1 vertices is constructed.

Wave Propagation in Elastic Media

Abstract: In previous chapters we investigated the properties of waves propagating in water and in inviscid and viscous fluids. It was pointed out that the conservation law that distinguishes one medium from another is the energy equation. It is the conservation law that contains the appropriate equation of state or constitutive equation which defines the medium. For example, an adiabatic equation of state was used to define a fluid such as air and a different equation of state for water. The adiabatic condition cannot be used across a shock wave since we must allow for a jump in entropy, etc. It was further pointed out in previous chapters that the approach used in mathematically describing the conservation laws was the Euler representation. This representation is more useful in dealing with large particle motions of the fluid. It was also stated that the conservation laws contain the fundamental physics of a given situation in the sense that from these conservation equations, which are called the field equations, we can obtain the velocity, pressure fields, etc., which give the wave properties. Since the field equations were couched in the Eulerian coordinates, the various fields that were derived from their solutions were also expressed as functions of these Eulerian coordinates. In principle, we can map back into the Lagrangian coordinates and thereby obtain the particle trajectories.

A Characteristic Alternating Direction Implicit Scheme for the Advection-Dispersion Equation

Abstract: Publisher Summary This chapter describes a numerical scheme based on combining the utility of a fixed grid in Eulerian coordinates with the computational power of the Lagrangian method. This is followed by a detailed comparison of the simulated concentrations with the analytical solutions. An analytical solution of the three-dimensional advection–dispersion equation is developed in this connection. A new method for the numerical solution of the convection–diffusion equation in one, two, and three dimensions is presented. The method is employed to obtain the numerical solution of some solute transfer and heat-transfer problems. The numerical results presented demonstrate that the method is capable of solving advection–dispersion problems without generating significant numerical diffusion when Peclet number is not too large, and oscillations. Numerical diffusion is mainly caused by the interpolation between nodes. Also, due to the increasing of interpolation and computation, the results of two- and three-dimensional problems are not as good as one-dimensional one.

Elastic-plastic flow using operator splitting and particle-in-cell methods in a two-dimensional Eulerian hydrodynamic code

Abstract: The particle-in-cell method has been used for computing compressible, multimaterial problems for more than 20 years. Current work extends the same numerical approximation with operator splitting to hydro-elastic-plastic flow problems in two-dimensional Eulerian coordinates. In the operator splitting method, the basic set of cylindrical equations is split in radial (r) and axial (z) directions. The calculations, performed in each direction separately, are alternated for each time advancement to maintain the accuracy of one-dimensional procedure. A shaped charge problem is treated with the present code, and the results are compared with the experimental data and with results from other codes. 12 refs., 22 figs.