scholarly journals On the meaning of the chosen set-averaging method within Eshelby-Kröner self-consistent scale transition model: the geometric mean versus the classical arithmetic average

2011 ◽  
Vol 91 (9) ◽  
pp. 689-698 ◽  
Author(s):  
S. Fréour ◽  
E. Lacoste ◽  
J. Fajoui ◽  
F. Jacquemin
Keyword(s):  
Averaging Method ◽  
Geometric Mean ◽  
Transition Model ◽  
Arithmetic Average ◽  
Scale Transition ◽  
Self Consistent ◽  
Classical Arithmetic

Arithmetic-geometric means of positive matrices

1987 ◽  
Vol 101 (2) ◽  
pp. 209-219 ◽  
Author(s):  
Joel E. Cohen ◽  
Roger D. Nussbaum
Keyword(s):  
Geometric Mean ◽  
Real Numbers ◽  
Positive Real ◽  
Geometric Means ◽  
Arithmetic Average ◽  
Positive Elements ◽  
Positive Matrices

AbstractWe prove the existence of unique limits and establish inequalities for matrix generalizations of the arithmetic–geometric mean of Lagrange and Gauss. For example, for a matrix A = (aij) with positive elements aij, define (contrary to custom) A½ elementwise by [A½]ij = (aij)½. Let A(0) and B(0) be d × d matrices (1 < d < ∞) with all elements positive real numbers. Let A(n + 1) = (A(n) + B(n))/2 and B(n + 1 ) = (d−1A(n)B(n))½. Then all elements of A(n) and B(n) approach a common positive limit L. When A(0) and B(0) are both row-stochastic or both column-stochastic, dL is less than or equal to the arithmetic average of the spectral radii of A(0) and B(0).

Leak Width of a Cusp-Confined Plasma

1982 ◽  
Vol 37 (8) ◽  
pp. 780-784
Author(s):  
G. Knorr ◽  
D. Willis
Keyword(s):  
Steady State ◽  
Charge Distribution ◽  
Geometric Mean ◽  
The Self ◽  
Magnetic Mirror ◽  
Plasma Stream ◽  
Experimental Measurements ◽  
State Calculation ◽  
Self Consistent

The theoretical and numerical steady-state calculation of the width of an escaping plasma stream through a magnetic mirror solves the self-consistent potential and charge distribution for a low beta plasma. The leak width is obtained as the geometric mean of the ion and electron Lar-mor radius, also called hybrid width, in agreement with some experimental measurements. If the out-streaming plasma is unstable or collisional, the leak width can only larger, contrary to earlier results in the literature

Self-consistent scale transition with imperfect interfaces: Application to nanocrystalline materials

2008 ◽  
Vol 56 (7) ◽  
pp. 1546-1554 ◽  
Author(s):  
L. Capolungo ◽  
S. Benkassem ◽  
M. Cherkaoui ◽  
J. Qu
Keyword(s):  
Nanocrystalline Materials ◽  
Imperfect Interfaces ◽  
Scale Transition ◽  
Self Consistent

scholarly journals Examples of numerical simulations of two-dimensional unsaturated flow with VS2DI code using different interblock conductivity averaging schemes

Geologos ◽  
2015 ◽  
Vol 21 (3) ◽  
pp. 161-167 ◽  
Author(s):  
Adam Szymkiewicz ◽  
Witold Tisler ◽  
Kazimierz Burzyński
Keyword(s):  
Hydraulic Conductivity ◽  
Unsaturated Flow ◽  
Water Pressure ◽  
Geometric Mean ◽  
Pressure Head ◽  
Richards Equation ◽  
Test Problems ◽  
Simulation Code ◽  
Arithmetic Average ◽  
The Difference

AbstractFlow in unsaturated porous media is commonly described by the Richards equation. This equation is strongly nonlinear due to interrelationships between water pressure head (negative in unsaturated conditions), water content and hydraulic conductivity. The accuracy of numerical solution of the Richards equation often depends on the method used to estimate average hydraulic conductivity between neighbouring nodes or cells of the numerical grid. The present paper discusses application of the computer simulation code VS2DI to three test problems concerning infiltration into an initially dry medium, using various methods for inter-cell conductivity calculation (arithmetic mean, geometric mean and upstream weighting). It is shown that the influence of the averaging method can be very large for coarse grid, but that it diminishes as cell size decreases. Overall, the arithmetic average produced the most reliable results for coarse grids. Moreover, the difference between results obtained with various methods is a convenient indicator of the adequacy of grid refinement.

scholarly journals The metric geometric mean transference and the problem of the average eye

2008 ◽  
Vol 67 (2) ◽  
Author(s):  
W. F. Harris
Keyword(s):  
Geometric Mean ◽  
Optical Systems ◽  
Symmetric Matrices ◽  
Refractive Errors ◽  
Arithmetic Average ◽  
First Order ◽  
Optical Character ◽  
All Optical ◽  
The Mean ◽  
Definition Of

An average refractive error is readily obtained as an arithmetic average of refractive errors.  But how does one characterize the first-order optical character of an average eye?  Solutions have been offered including via the exponential-mean-log transference.  The exponential-mean-log transference ap-pears to work well in practice but there is the niggling problem that the method does not work with all optical systems.  Ideally one would like to be able to calculate an average for eyes in exactly the same way for all optical systems. This paper examines the potential of a relatively newly described mean, the metric geometric mean of positive definite (and, therefore, symmetric) matrices.  We extend the definition of the metric geometric mean to matrices that are not symmetric and then apply it to ray transferences of optical systems.  The metric geometric mean of two transferences is shown to satisfy the requirement that symplecticity be pre-served.  Numerical examples show that the mean seems to give a reasonable average for two eyes.  Unfortunately, however, what seem reasonable generalizations to the mean of more than two eyes turn out not to be satisfactory in general.  These generalizations do work well for thin systems.  One concludes that, unless other generalizations can be found, the metric geometric mean suffers from more disadvantages than the exponential-mean-logarithm and has no advantages over it.

scholarly journals A Multi-Scale Study of Residual Stresses Created during the Cure Process of a Composite Tooling Material

2011 ◽  
Vol 681 ◽  
pp. 309-314 ◽  
Author(s):  
Emmanuel Lacoste ◽  
Sylvain Fréour ◽  
Frédéric Jacquemin
Keyword(s):  
Effective Properties ◽  
Local Stress ◽  
Transition Model ◽  
Cure Process ◽  
Multi Scale ◽  
Scale Transition ◽  
Composite Tooling ◽  
Stress States ◽  
Isotropic Carbon ◽  
Transition Procedure

This paper deals with the cure of an in-plane isotropic carbon-polymer tooling material, with a complex microstructure [1]. The Mori-Tanaka (MT) and Eshelby-Kröner self-consistent (EKSC) models are used in order to achieve a two-steps scale transition procedure, relating the microscopic properties of the material to their macroscopic counterparts. This procedure enables estimating the multi-scale mechanical states experienced by the material, i.e. the local (microscopic) stresses due to thermal and chemical shrinkage of the resin, along a typical, macroscopic stress-free, cure process. The influence of the chosen scale transition model on both the calculated effective properties of the material and its local stress states, is investigated. These results are a first step for investigating the service life fatigue of the material, as well as its failure behaviour.

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