Two Methods for Analyzing Waves in Composites with Random Microstructure

1991 ◽  
Vol 253 ◽  
Author(s):  
John R. Willis
Keyword(s):  
Variational Principle ◽  
Variational Principles ◽  
Statistical Information ◽  
Random Microstructure ◽  
Self Consistent ◽  
The Mean ◽  
Stochastic Variational Principles

ABSTRACTThe problem of calculating the mean wave in a composite with random microstructure is addressed. Exact characterizations of the problem can be given, in the form of stochastic variational principles. Substitution of simple configuration-dependent trial fields into these generates approximations which are, in a sense, ‘optimal’. It is necessary in practice to employ only trial fields which will generate, in the variational principle, no more statistical information than is actually available. Trial fields that require knowledge of two-point statistics generate equations that can also be obtained directly, through use of the QCA. The same fields can be substituted into an alternative variational principle to yield an approximation that makes use of three-point statistics – this approximation is less easy to obtain by direct reasoning. When not even two-point information is available, some more elementary approximation is needed. One such approximation, which is simple and direct in its application, is an extension to dynamics of a “self-consistent embedding” scheme which is widely used in static problems. This is also discussed, together with some illustrative results for a matrix containing inclusions and for a polycrystal.

scholarly journals Simulating Solar Global Magnetism in 3-D

2012 ◽  
Vol 10 (H16) ◽  
pp. 101-103
Author(s):  
A. S. Brun ◽  
A. Strugarek
Keyword(s):  
Magnetic Field ◽  
Recent Progress ◽  
Convection Zone ◽  
Thermal Structure ◽  
Solar Convection ◽  
Self Consistent ◽  
The Mean ◽  
The Magnetic Field

AbstractWe briefly present recent progress using the ASH code to model in 3-D the solar convection, dynamo and its coupling to the deep radiative interior. We show how the presence of a self-consistent tachocline influences greatly the organization of the magnetic field and modifies the thermal structure of the convection zone leading to realistic profiles of the mean flows as deduced by helioseismology.

An explicit algebraic Reynolds-stress and scalar-flux model for stably stratified flows

2013 ◽  
Vol 723 ◽  
pp. 91-125 ◽  
Author(s):  
W. M. J. Lazeroms ◽  
G. Brethouwer ◽  
S. Wallin ◽  
A. V. Johansson
Keyword(s):  
Heat Flux ◽  
Kinetic Energy ◽  
Temperature Gradient ◽  
Reynolds Stress ◽  
Turbulent Heat Flux ◽  
Turbulent Heat ◽  
Homogeneous Shear ◽  
Self Consistent ◽  
The Mean ◽  
Homogeneous Shear Flow

AbstractThis work describes the derivation of an algebraic model for the Reynolds stresses and turbulent heat flux in stably stratified turbulent flows, which are mutually coupled for this type of flow. For general two-dimensional mean flows, we present a correct way of expressing the Reynolds-stress anisotropy and the (normalized) turbulent heat flux as tensorial combinations of the mean strain rate, the mean rotation rate, the mean temperature gradient and gravity. A system of linear equations is derived for the coefficients in these expansions, which can easily be solved with computer algebra software for a specific choice of the model constants. The general model is simplified in the case of parallel mean shear flows where the temperature gradient is aligned with gravity. For this case, fully explicit and coupled expressions for the Reynolds-stress tensor and heat-flux vector are given. A self-consistent derivation of this model would, however, require finding a root of a polynomial equation of sixth-order, for which no simple analytical expression exists. Therefore, the nonlinear part of the algebraic equations is modelled through an approximation that is close to the consistent formulation. By using the framework of a$K\text{{\ndash}} \omega $model (where$K$is turbulent kinetic energy and$\omega $an inverse time scale) and, where needed, near-wall corrections, the model is applied to homogeneous shear flow and turbulent channel flow, both with stable stratification. For the case of homogeneous shear flow, the model predicts a critical Richardson number of 0.25 above which the turbulent kinetic energy decays to zero. The channel-flow results agree well with DNS data. Furthermore, the model is shown to be robust and approximately self-consistent. It also fulfils the requirements of realizability.

CURRENTS, SPIN DENSITIES AND MEAN-FIELD FORM FACTORS IN ROTATING NUCLEI: A SEMI-CLASSICAL APPROACH

2004 ◽  
Vol 13 (01) ◽  
pp. 225-233 ◽  
Author(s):  
J. BARTEL ◽  
K. BENCHEIKH ◽  
P. QUENTIN
Keyword(s):  
Mean Field ◽  
Form Factors ◽  
Spin Vector ◽  
Self Consistent ◽  
Field Form ◽  
Local Densities ◽  
The Mean ◽  
Self Consistency ◽  
Spin Densities ◽  
Rotating Nuclei

We present self-consistent semi-classical local densities characterising the structure of rotating nuclei. A particular emphasis is put on those densities which are generated by the breaking of time-reversal symmetry through the cranking piece of the Routhian, namely the current density and the spin vector density. Our approach which is based on the Extended-Thomas-Fermi method goes beyond the Inglis cranking approach and contains naturally the Thouless-Valatin self-consistency terms expressing the response of the mean field to the time-odd part of the density matrix.

Criterion for DNA melting in the mean-field modified self-consistent phonon theory

1991 ◽  
Vol 43 (11) ◽  
pp. 9284-9286
Author(s):  
Y. Feng ◽  
E. W. Prohofsky
Keyword(s):  
Mean Field ◽  
Dna Melting ◽  
Self Consistent ◽  
The Mean ◽  
Phonon Theory

Variational Principles

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Variational Principle ◽  
Lower Semicontinuity ◽  
Variational Principles ◽  
Perfect Information ◽  
Perturbation Functions ◽  
Abstract Version ◽  
Recursive Constructions ◽  
Natural Description

This chapter describes smooth variational principles (of Ekeland type) as infinite two-player games. These variational principles are based on a simple but careful recursive choice of points where certain functions that change during the process have values close to their infima. Like many other recursive constructions, the choice has a natural description using the language of infinite two-player games with perfect information. The chapter first considers the perturbation game used in Theorem 7.2.1 to formulate an abstract version of the variational principle before showing how to specialize it to more standard formulations. It then examines the bimetric variant of the smooth variational principle, along with the perturbation functions that are relatively simple. It concludes with an assessment of cases when completeness and lower semicontinuity hold only in a bimetric sense.

scholarly journals Self-consistent triple decomposition of the turbulent flow over a backward-facing step under finite amplitude harmonic forcing

2019 ◽  
Vol 475 (2225) ◽  
pp. 20190018
Author(s):  
E. Yim ◽  
P. Meliga ◽  
F. Gallaire
Keyword(s):  
Turbulent Flow ◽  
Finite Amplitude ◽  
Navier Stokes ◽  
Reynolds Stresses ◽  
Mean Flow ◽  
Backward Facing Step ◽  
Eddy Viscosity Model ◽  
Coherent Interaction ◽  
Self Consistent ◽  
The Mean

We investigate the saturation of harmonically forced disturbances in the turbulent flow over a backward-facing step subjected to a finite amplitude forcing. The analysis relies on a triple decomposition of the unsteady flow into mean, coherent and incoherent components. The coherent–incoherent interaction is lumped into a Reynolds averaged Navier–Stokes (RANS) eddy viscosity model, and the mean–coherent interaction is analysed via a semi-linear resolvent analysis building on the laminar approach by Mantič-Lugo & Gallaire (2016 J. Fluid Mech. 793 , 777–797. ( doi:10.1017/jfm.2016.109 )). This provides a self-consistent modelling of the interaction between all three components, in the sense that the coherent perturbation structures selected by the resolvent analysis are those whose Reynolds stresses force the mean flow in such a way that the mean flow generates exactly the aforementioned perturbations, while also accounting for the effect of the incoherent scale. The model does not require any input from numerical or experimental data, and accurately predicts the saturation of the forced coherent disturbances, as established from comparison to time-averages of unsteady RANS simulation data.

scholarly journals Mixed variational potentials and inherent symmetries of the Cahn–Hilliard theory of diffusive phase separation

2014 ◽  
Vol 470 (2164) ◽  
pp. 20130641 ◽  
Author(s):  
C. Miehe ◽  
F. E. Hildebrand ◽  
L. Böger
Keyword(s):  
Phase Separation ◽  
Variational Principle ◽  
Data Storage ◽  
Variational Principles ◽  
Chemical Potential ◽  
Field Representation ◽  
Time Stepping ◽  
Speed Up ◽  
Cahn Hilliard Equation ◽  
Variational Statement

This work shows that the Cahn–Hilliard theory of diffusive phase separation is related to an intrinsic mixed variational principle that determines the rate of concentration and the chemical potential. The principle characterizes a canonically compact model structure, where the two balances involved for the species content and microforce appear as the Euler equations of a variational statement. The existence of the variational principle underlines an inherent symmetry in the two-field representation of the Cahn–Hilliard theory. This can be exploited in the numerical implementation by the construction of time- and space-discrete incremental potentials , which fully determine the update problems of typical time-stepping procedures. The mixed variational principles provide the most fundamental approach to the finite-element solution of the Cahn–Hilliard equation based on low-order basis functions, leading to monolithic symmetric algebraic systems of iterative update procedures based on a linearization of the nonlinear problem. They induce in a natural format the choice of symmetric solvers for Newton-type iterative updates, providing a speed-up and reduction of data storage when compared with non-symmetric implementations. In this sense, the potentials developed are believed to be fundamental ingredients to a deeper understanding of the Cahn–Hilliard theory.

A fully automated remote refraction system

2000 ◽  
Vol 6 (2_suppl) ◽  
pp. 16-18 ◽  
Author(s):  
Alan M Dyer ◽  
Angus H Kirk
Keyword(s):  
Statistical Information ◽  
Acceptable Level ◽  
Traditional Methods ◽  
Subjective Testing ◽  
The Subject ◽  
The Mean

Traditional methods of performing refractions depend on a trained refractionist being present with the subject and conducting an interactive form of subjective testing. A fully automated refraction system was installed in 13 optical dispensaries and after 15 months the patient and statistical information was gathered. The data from all operators were consistent and suggested a lack of operator effect on the refraction results. The mean of the SD of subjective sphere measurements was 0.2, or slightly less than a quarter dioptre, which would be an acceptable level of accuracy for ordering corrective lenses. The present study suggests an absence of operator influence on the results of the refractions and a degree of consistency and accuracy compatible with the prescription of lenses.

A self-consistent solution of the coupled Dirac-Einstein equations in the mean-field Hartree approximation

1978 ◽  
Vol 22 (10) ◽  
pp. 391-396
Author(s):  
M. Soffel ◽  
U. Heinz ◽  
B. Müller ◽  
W. Greiner
Keyword(s):  
Mean Field ◽  
Einstein Equations ◽  
Hartree Approximation ◽  
Self Consistent ◽  
Consistent Solution ◽  
The Mean

A note on the inviscid Orr-Sommerfeld equation

1962 ◽  
Vol 13 (3) ◽  
pp. 427-432 ◽  
Author(s):  
John W. Miles
Keyword(s):  
Differential Equation ◽  
Velocity Profile ◽  
Numerical Integration ◽  
Riccati Equation ◽  
Variational Principles ◽  
Wave Number ◽  
Null Condition ◽  
Variational Approximations ◽  
The Mean ◽  
Sommerfeld Equation

The inviscid Orr-Sommerfeld equation for ϕ(y) in y > 0 subject to a null condition as y → ∞ is attacked by considering separately the intervals (0, y1) and (y1, ∞), such that the solution in (0, y1) can be expanded in powers of the wave-number (following Heisenberg) and the solution of (y1, ∞) regarded as real and non-singular. Complementary variational principles for the latter solution are determined to bound an appropriate parameter from above and below. It also is shown how the original differential equation may be transformed to a Riccati equation in such a way as to facilitate both the Heisenberg expansion of the solution in (0, y1) and numerical integration in (y1, ∞). These methods are applied to a velocity profile that is linear in (0, y1) and asymptotically logarithmic as y → ∞, and it is found that the mean of the two variational approximations is in excellent agreement with the results of numerical integration of the Riccati equation.

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