Extended Method Of Moments Research Articles

Determination of the diffusivity, dispersion, skewness and kurtosis in heterogeneous porous flow. Part II: Lattice Boltzmann schemes with implicit interface

Abstract A simple local two-relaxation-time Lattice Boltzmann numerical formulation (TRT-EMM) of the extended method of moments (EMM) is proposed for analysis of the spatial and temporal Taylor dispersion in d-dimensional streamwise-periodic stationary mesoscopic velocity field resolved in a piecewise-continuous porous media. The method provides an effective diffusivity, dispersion, skewness and kurtosis of the mean concentration profile and residence time distribution. The TRT-EMM solves a chain of steady-state heterogeneous advection–diffusion equations with the pre-computed space-variable mass-source and automatically undergoes diffusion-flux jump on the abrupt-porosity streamwise-normal interface. The temporal and spatial systems of moments are computed within the same run; the symmetric dispersion tensor can be restored from independent computations performed for each periodic mean-velocity axis; the numerical algorithm recursively extends for any order moment. We derive an exact form of the bulk equation and implicit closure relations, construct symbolic TRT-EMM solutions and determine specific relation between the equilibrium and the collision degrees of freedom viewing an exact parameterization by the physical non-dimensional numbers in two alternate situations: “parallel” fracture/matrix flow and “perpendicular” Darcy flow through porous blocks in “series”. Two-dimensional simulations in linear Brinkman flow around solid and through porous obstacles validate the method in comparison with the two heterogeneous direct LBM-ADE schemes with different anti-numerical-diffusion treatment which are proposed and examined in parallel. On the coarse grid, accuracy of the three moments is essentially determined by the free-tunable collision rate in all schemes, and especially TRT-EMM. However, operated within a single periodic cell, the TRT-EMM is many orders of magnitude faster than the direct solvers, numerical-diffusion free, more robust and much more capable for accuracy improving, high Peclet range and free-parameter influence reduction with the mesh refinement. The method is an efficient predicting tool for the Taylor dispersion, asymmetry and peakedness; moreover, it allows for an optimal analysis between the mutual effect of the flow regime, Peclet number, porosity, permeability and obstruction geometry.

Structural estimation of behavioral heterogeneity

SummaryWe develop a behavioral asset pricing model in which agents trade in a market with information friction. Profit‐maximizing agents switch between trading strategies in response to dynamic market conditions. Owing to noisy private information about the fundamental value, the agents form different evaluations about heterogeneous strategies. We exploit a thin set—a small sub‐population—to point identify this nonlinear model, and estimate the structural parameters using extended method of moments. Based on the estimated parameters, the model produces return time series that emulate the moments of the real data. These results are robust across different sample periods and estimation methods.

Determination of the diffusivity, dispersion, skewness and kurtosis in heterogeneous porous flow. Part I: Analytical solutions with the extended method of moments.

Abstract The extended method of moments (EMM) is elaborated in recursive algorithmic form for the prediction of the effective diffusivity, the Taylor dispersion dyadic and the associated longitudinal high-order coefficients in mean-concentration profiles and residence-time distributions. The method applies in any streamwise-periodic stationary d-dimensional velocity field resolved in the piecewise continuous heterogeneous porosity field. It is demonstrated that EMM reduces to the method of moments and the volume-averaging formulation in microscopic velocity field and homogeneous soil, respectively. The EMM simultaneously constructs two systems of moments, the spatial and the temporal, without resorting to solving of the high-order upscaled PDE. At the same time, the EMM is supported with the reconstruction of distribution from its moments, allowing to visualize the deviation from the classical ADE solution. The EMM can be handled by any linear advection-diffusion solver with explicit mass-source and diffusive-flux jump condition on the solid boundary and permeable interface. The prediction of the first four moments is decisive in the optimization of the dispersion, asymmetry, peakedness and heavy-tails of the solute distributions, through an adequate design of the composite materials, wetlands, chemical devices or oil recovery. The symbolic solutions for dispersion, skewness and kurtosis are constructed in basic configurations: diffusion process and Darcy flow through two porous blocks in “series”, straight and radial Poiseuille flow, porous flow governed by the Stokes–Brinkman–Darcy channel equation and a fracture surrounded by penetrable diffusive matrix or embedded in porous flow. We examine the moments dependency upon porosity contrast, aspect ratio, Peclet and Darcy numbers, but also for their response on the effective Brinkman viscosity applied in flow modeling. Two numerical Lattice Boltzmann algorithms, a direct solver of the microscopic ADE in heterogeneous structure and a novel scheme for EMM numerical formulation, are called for validation of the constructed analytical predictions.

Prediction of the moments in advection-diffusion lattice Boltzmann method. I. Truncation dispersion, skewness, and kurtosis.

The effect of the heterogeneity in the soil structure or the nonuniformity of the velocity field on the modeled resident time distribution (RTD) and breakthrough curves is quantified by their moments. While the first moment provides the effective velocity, the second moment is related to the longitudinal dispersion coefficient (k_{T}) in the developed Taylor regime; the third and fourth moments are characterized by their normalized values skewness (Sk) and kurtosis (Ku), respectively. The purpose of this investigation is to examine the role of the truncation corrections of the numerical scheme in k_{T}, Sk, and Ku because of their interference with the second moment, in the form of the numerical dispersion, and in the higher-order moments, by their definition. Our symbolic procedure is based on the recently proposed extended method of moments (EMM). Originally, the EMM restores any-order physical moments of the RTD or averaged distributions assuming that the solute concentration obeys the advection-diffusion equation in multidimensional steady-state velocity field, in streamwise-periodic heterogeneous structure. In our work, the EMM is generalized to the fourth-order-accurate apparent mass-conservation equation in two- and three-dimensional duct flows. The method looks for the solution of the transport equation as the product of a long harmonic wave and a spatially periodic oscillating component; the moments of the given numerical scheme are derived from a chain of the steady-state fourth-order equations at a single cell. This mathematical technique is exemplified for the truncation terms of the two-relaxation-time lattice Boltzmann scheme, using plug and parabolic flow in straight channel and cylindrical capillary with the d2Q9 and d3Q15 discrete velocity sets as simple but illustrative examples. The derived symbolic dependencies can be readily extended for advection by another, Newtonian or non-Newtonian, flow profile in any-shape open-tabular conduits. It is established that the truncation errors in the three transport coefficients k_{T}, Sk, and Ku decay with the second-order accuracy. While the physical values of the three transport coefficients are set by Péclet number, their truncation corrections additionally depend on the two adjustable relaxation rates and the two adjustable equilibrium weight families which independently determine the convective and diffusion discretization stencils. We identify flow- and dimension-independent optimal strategies for adjustable parameters and confront them to stability requirements. Through specific choices of two relaxation rates and weights, we expect our results be directly applicable to forward-time central differences and leap-frog central-convective Du Fort-Frankel-diffusion schemes. In straight channel, a quasi-exact validation of the truncation predictions through the numerical moments becomes possible thanks to the specular-forward no-flux boundary rule. In the staircase description of a cylindrical capillary, we account for the spurious boundary-layer diffusion and dispersion because of the tangential constraint of the bounce-back no-flux boundary rule.

Moment method for unsteady flows with heterogeneous condensation

Abstract The classic moment method for homogeneous condensation is extended to unsteady flows with heterogeneous condensation. The moments are defined on the size distribution function of the droplet including the particle core and the liquid layer. The simple instantaneous-wetting model is used for the heterogeneous nucleation and the Gyarmathy model is employed to evaluate the droplet growth. Combining the first three lowest moments with the two-dimensional Euler equations, the governing equations of unsteady flows with heterogeneous condensation are obtained. As a demonstration, the heterogeneous condensation in a closed shock tube problem is simulated with a top-hat radius distribution of insolvable particles for different mean particle radius and particle concentration. It is shown that the extended moment method can not only give the flow details including pressure, density and wave pattern caused by condensation, but also provide the important physical properties of the aerosol.

Numerical Study on Homogeneous Condensation in a Vortex

The homogeneous water condensation in a vortex is numerically studied in the present work by using an extended method of moments (EMOM). The EMOM is based on the liquid phase so that the gas phase and the liquid phase can be treated separately, having their own governing equations respectively. The EMOM is integrated into a well validated numerical method-VAS2D (two-dimensional & axisymmetric vectorized adaptive solver), which is a MUSCL-type finite-volume method developed for the unstructured quadrilateral grids. The interactions between the two phases are realized through the source terms including the exchange of momentum and heat. The new numerical method is then applied to a vortex flow with a mixture of nitrogen and water vapor. Several cases with different initial situations are investigated. The evolution of the flow field is discussed. The case with the undersaturated gas apart from the nucleation region has a smooth diffusion of the droplets. But for the case with the supersaturated gas, the droplets continuously grow when they move to the new supersaturated region under the effect of centrifugal force. It leads to a more complex interaction between the vortex and the condensation.

Taylor dispersion in heterogeneous porous media: Extended method of moments, theory, and modelling with two-relaxation-times lattice Boltzmann scheme

This article describes a generalization of the method of moments, called extended method of moments (EMM), for dispersion in periodic structures composed of impermeable or permeable porous inclusions. Prescribing pre-computed steady state velocity field in a single periodic cell, the EMM sequentially solves specific linear stationary advection-diffusion equations and restores any-order moments of the resident time distribution or the averaged concentration distribution. Like the pioneering Brenner's method, the EMM recovers mean seepage velocity and Taylor dispersion coefficient as the first two terms of the perturbative expansion. We consider two types of dispersion: spatial dispersion, i.e., spread of initially narrow pulse of concentration, and temporal dispersion, where different portions of the solute have different residence times inside the system. While the first (mean velocity) and the second (Taylor dispersion coefficient) moments coincide for both problems, the higher moments are different. Our perturbative approach allows to link them through simple analytical expressions. Although the relative importance of the higher moments decays downstream, they manifest the non-Gaussian behaviour of the breakthrough curves, especially if the solute can diffuse into less porous phase. The EMM quantifies two principal effects of bi-modality, as the appearance of sharp peaks and elongated tails of the distributions. In addition, the moments can be used for the numerical reconstruction of the corresponding distribution, avoiding time-consuming computations of solute transition through heterogeneous media. As illustration, solutions for Taylor dispersion, skewness, and kurtosis in Poiseuille flow and open/impermeable stratified systems, both in rectangular and cylindrical channels, power-law duct flows, shallow channels, and Darcy flow in parallel porous layers are obtained in closed analytical form for the entire range of Péclet numbers. The high-order moments and reconstructed profiles are compared to their predictions from the advection-diffusion equation for averaged concentration, based on the same averaged seepage velocity and Taylor dispersion coefficient. In parallel, we construct Lattice-Boltzmann equation (LBE) two-relaxation-times scheme to simulate transport of a passive scalar directly in heterogeneous media specified by discontinuous porosity distribution. We focus our numerical analysis and assessment on (i) truncation corrections, because of their impact on the moments, (ii) stability, since we show that stable Darcy velocity amplitude reduces with the porosity, and (iii) interface accuracy which is found to play the crucial role. The task is twofold: the LBE supports the EMM predictions, while the EMM provides non-trivial benchmarks for the numerical schemes.

Efficient Derivative Pricing by the Extended Method of Moments

In this paper, we introduce the extended method of moments (XMM) estimator. This estimator accommodates a more general set of moment restrictions than the standard generalized method of moments (GMM) estimator. More specifically, the XMM differs from the GMM in that it can handle not only uniform conditional moment restrictions (i.e., valid for any value of the conditioning variable), but also local conditional moment restrictions valid for a given fixed value of the conditioning variable. The local conditional moment restrictions are of special relevance in derivative pricing to reconstruct the pricing operator on a given day by using the information in a few cross sections of observed traded derivative prices and a time series of underlying asset returns. The estimated derivative prices are consistent for a large time series dimension, but a fixed number of cross sectionally observed derivative prices. The asymptotic properties of the XMM estimator are nonstandard, since the combination of uniform and local conditional moment restrictions induces different rates of convergence (parametric and nonparametric) for the parameters.

Extended method of moments for deterministic analysis of stochastic multistable neurodynamical systems

The analysis of transitions in stochastic neurodynamical systems is essential to understand the computational principles that underlie those perceptual and cognitive processes involving multistable phenomena, like decision making and bistable perception. To investigate the role of noise in a multistable neurodynamical system described by coupled differential equations, one usually considers numerical simulations, which are time consuming because of the need for sufficiently many trials to capture the statistics of the influence of the fluctuations on that system. An alternative analytical approach involves the derivation of deterministic differential equations for the moments of the distribution of the activity of the neuronal populations. However, the application of the method of moments is restricted by the assumption that the distribution of the state variables of the system takes on a unimodal Gaussian shape. We extend in this paper the classical moments method to the case of bimodal distribution of the state variables, such that a reduced system of deterministic coupled differential equations can be derived for the desired regime of multistability.

Extended moment method for electrons in semiconductors

The semiclassical Boltzmann equation for electrons in semiconductors with the Kane dispersion law or the parabolic band approximation is considered and systems of moment equations with an arbitrary number of moments are derived. First, the paper deals with spherical harmonics in the formalism of symmetric trace-free tensors. The collision frequencies are carefully studied for the physical properties of silicon. Then, for the parabolic band approximation, the hierarchy of equations for full moments of the phase density and the corresponding closure problem is discussed. In particular, a set of 2 R scalar and vectorial moments is considered. To answer the question which number R one has to chose in order to retain the physical contents of the Boltzmann equation, the moment equations are examined in the drift-diffusion limit and in an infinite crystal in a homogeneous electric field (transient and stationary cases) for increasing number of moments R. The number R must be considered to be sufficient, if its further increase does not change the result considerably and the appropriate numbers for the processes are given.

Ray analysis of low-grazing scattering from a breaking water wave

Low-grazing-angle backscattering from a modeled breaking-wave surface profile has been calculated using a ray-optical approach and compared with reference scattering found using an extended moment method. The calculations show that interference between the direct backscatter from the breaking plume and multipath scattering between the plume and wave face can lead to the HH-to-VV polarization-backscattering ratios of greater than 9 dB that characterize sea-spike events. The multipath effects can be accurately predicted from simple reflection from the front face at the smallest grazing angles. At higher angles, diffraction from rapid changes in the surface curvature must also be considered.

On the Number of Moments in Radiative Transfer Problems

In a recent paper (Struchtrup, Annals of Physics,257, 1997) we set up an extended moment method for radiative transfer problems, which involves matrices of mean absorption and scattering coefficients. In the present paper, we examine the resulting moment equations for one-dimensional radiative transfer problems. In particular we are interested in the number of moments which one has to choose in order to have satisfactory agreement between solutions of the moment equations and solutions of the radiative transfer equation. We show that the moment theory will describe a one-dimensional beam properly, if moments with a tensorial rank of about 30 are taken into account.

An Extended Moment Method in Radiative Transfer: The Matrices of Mean Absorption and Scattering Coefficients

In extension of the ideas of Anderson and Spiegel (1972) the radiative transfer equation is replaced by moment equations for the momentsAA1A2…ANr=∫:(p0RL)r+1−NpA1pA2…pANfdP,r=0,1,…,R. HerepAis the photon 4-momentum,cp0RLis the photon energy in the rest Lorentz frame andfis the photon phase density. From these follow moment equations for the projected symmetric trace free moments introduced by Thorne (1981). The required closure of the equations is achieved by use of a series expansion of the phase density which is motivated by the entropy maximum principle. This procedure provides a coupling of the moment equations by means ofmatricesof mean absorption and scattering coefficients. It is shown that the extension fromr=1 (Anderson and Spiegel, 1972; Thorne, 1981; Schweizer, 1988) tor=0,1,…,Rgives reasonable results: In the limit of local radiative equilibrium (LRE) the well-known Rosseland mean of the absorption coefficient is recovered. For a simple non-LRE experiment, the homogeneous compression and relaxation of radiation, the radiative transfer equation, and the moment equations are solved. The comparison of the results in the case of pure bremsstrahlung (free–free) absorption shows an excellent agreement forR⩾6.

Computation of electromagnetic wave scattering from an arbitrary three-dimensional inhomogeneous dielectric object

A scattering matrix formulation for scattering of electromagnetic fields by an arbitrary three-dimensional dielectric objects is presented. It is based on an extended moment method that digitizes the scattering equation by dividing the scatterer as well as the measurement region into small cells. The Green's function within each cell is derived analytically by integrating over the cell. The accuracy of the method is demonstrated by computing vector scattering fields from a uniform dielectric sphere and comparing the results with those calculated using the Mie scattering formula. The formulation also provides an inverse solution, namely, determination of the dielectric profile of the scatterer from the scattering field measured in a finite region outside the scatterer.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Broad-band characteristics of rotationally symmetric antennas and thin wire constructs

The rotationally symmetric monopole antenna is given by a set of simultaneous equations on the rotationally symmetric surface current of a conductor, separated with respect to the variable of the envelope length on the surface of the rotationally symmetric antenna and the variable in the direction of revolution. The structure is analyzed in detail by means of an extended moment method based on the present theory. As an application, this analysis is applied to a conical monopole antenna. Studies are performed to find body shapes providing broadband characteristics. It is found that the antennas with lengthwise cross sections approximated by straight lines, ellipses, and circles have an impedance of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">50 \Omega</tex> over a wide frequency band. When these shapes are replaced with wire elements for analysis, 16 elements are found sufficient to approximate the characteristics of the rotationally symmetric antenna. Theoretical calculations and experimental results are compared. Agreement is excellent and the method is found useful.

Broadband characteristics of bodies of revolution

AbstractA monopole antenna consisting of a body of revolution is given by a set of simultaneous equations on the rotationally symmetric surface current of a conductor separated with respect to the variable of the envelope length on the surface of the body of revolution and the variable in the rotating direction. The structure is analyzed in detail by means of an extended moment method based on the present theory.As an application, this analysis is applied to a conical monopole antenna. Studies are performed to find body shapes providing broadband characteristics. It is found that antennas with cross sections approximated by straight lines, ellipses and circles have an impedance of 50 ω over a wide range of frequency. When these shapes are replaced with wire elements for analysis, 16 elements are found sufficient to provide the characteristics for the bodies of revolution. Theoretical calculations and experimental results are compared. Agreement is excellent and the method is found useful.

Extended moment analysis for binomial parameters of transmitter release

The quantum hypothesis proposes that a binomial distribution should fit the amplitude distribution for synaptic potentials. Since importance is now being attached to significant changes in the n and p parameters of the binomial model during various treatments of synaptic preparations, this paper describes an important extension of the method of moments which can be used to extract binomial parameters in difficult experimental circumstances. Essentially, the skewness (third moment) of the observed amplitude distribution of synaptic responses is used to provide the additional information needed in cases where spontaneous miniature responses are absent. Computer simulations are used to assess the reliability of the proposed new estimators. The estimator bias due to non-uniform unit responses is also evaluated. Other applications of the extended method of moments, including a new test of the binomial hypothesis, are also described.

On non-Newtonian flow past a sphere

The approximate solutions for flow of and Ostwald-de Waele fluid past a sphere at R e·0 = 60 and 1 ⩾ n ⩾ 0·8 are obtained by the use of an extended method of moments. As n decreases, (1) friction drag decreases, (2) pressure drag increases for flow past a blunt body, (3) total drag increases for flow past a sphere, (4) wake length increases for flow past a sphere, (5) separation point moves forward for flow past a sphere.