Mean Value Point Research Articles

Influence of Spatial Variability on Slope Reliability Using 2-D Random Fields

The paper investigates the probability of failure of slopes using both traditional and more advanced probabilistic analysis tools. The advanced method, called the random finite-element method, uses elastoplasticity in a finite-element model combined with random field theory in a Monte-Carlo framework. The traditional method, called the first-order reliability method, computes a reliability index which is the shortest distance (in units of directional equivalent standard deviations) from the equivalent mean-value point to the limit state surface and estimates the probability of failure from the reliability index. Numerical results show that simplified probabilistic analyses in which spatial variability of soil properties is not properly accounted for, can lead to unconservative estimates of the probability of failure if the coefficient of variation of the shear strength parameters exceeds a critical value. The influences of slope inclination, factor of safety (based on mean strength values), and cross correlation between strength parameters on this critical value have been investigated by parametric studies in this paper. The results indicate when probabilistic approaches, which do not model spatial variation, may lead to unconservative estimates of slope failure probability and when more advanced probabilistic methods are warranted.

A Different Version of Flett's Mean Value Theorem and an Associated Functional Equation

In this paper we present a mean value theorem derived from Flett's mean value theorem. It turns out that cubic polynomials have the midpoint of the interval as their mean value point. To answer what class of functions have this property, we consider a functional equation associated with this mean value theorem.

Structural Reliability Assessment Method Based on Adaptive Directional Simulation

This study outlines a structural failure probability estimation method based on an improved directional simulation, where directional vector samples are generated adaptively with a situation of the failure domain of the structural systems. The conditional failure probability on a certain sample direction is formulated as a piecewise integration with respect to the radial variable. To improve the sampling efficiency, the dominant failure modes effective for structural failure probability are selected and the directional samples are generated from the sampling density uniformly distributed over the surface of the unit hypersphere centered at the mean value point of the failure domain, which is composed of the selected dominant failure modes. The dominant failure modes and the mean value point are estimated through a preliminary simulation. Numerical examples are presented to illustrate that the computational efficiency of the structural failure probability estimation based on the proposed method is improved.

Mean-value second-order uncertainty analysis method: application to water quality modelling

Uncertainty analysis in hydrology and water quality modelling is an important issue. Various methods have been proposed to estimate uncertainties on model results based on given uncertainties on model parameters. Among these methods, the mean-value first-order second-moment (MFOSM) method and the advanced mean-value first-order second-moment (AFOSM) method are the most common ones. This paper presents a method based on a second-order approximation of a model output function. The application of this method requires the estimation of first- and second-order derivatives at a mean-value point in the parameter space. Application to a Streeter–Phelps prototype model is presented. Uncertainties on two and six parameters are considered. Exceedance probabilities (EP) of dissolved oxygen concentrations are obtained and compared with EP computed using Monte Carlo, AFOSM and MFOSM methods. These results show that the mean-value second-order method leads to better estimates of EP.

On the Asymptotics of Mean Value Points for Some Finite-Difference Operators

We prove some asymptotic estimates for mean value points of finite-difference differentiation operators of order n.

Elementary calculation of the branch-point energy in the continuum of interface-induced gap states

The band alignment at ideal semiconductor interfaces may be described by the continuum of interface-induced gap states. These intrinsic interface states derive from the virtual gap states (ViGS) of the complex semiconductor band structure. The energy at which their character changes from predominantly donorlike to acceptor-like is defined as their branch point. This branch point is located slightly below the middle of its band gap at the mean-value point in the Brillouin zone. By comparison with GW band-structure data it is demonstrated that, first, the width of the band gap at the mean-value point equals the dielectric band gap and that, second, the empirical tight-binding approximation reproduces the dispersion of the GW valence bands from the middle to the mean-value point of the Brillouin zone. Branch points computed by Tersoff for 15 semiconductors are found slightly below the middle of the respective dielectric band gaps at the mean-value point of the Brillouin zone. These results make possible simple calculations of branch points for other zincblende-structure semiconductors.

Empirical tight-binding calculation of the branch-point energy of the continuum of interface-induced gap states

The band lineup at metal–semiconductor contacts as well as at semiconductor heterostructures may be described by one and the same physical concept, the continuum of interface-induced gap states. These intrinsic interface states derive from the virtual gap states (ViGS) of the complex semiconductor band structure and their character varies from predominantly donorlike closer to the valence band to mostly acceptorlike nearer to the conduction band. Calculations are presented of the respective branch points for elemental and binary as well as ternary compound semiconductors which make use of Baldereschi’s concept of mean-value points in the Brillouin zone [Phys. Rev. B 7, 5212 (1973)], Penn’s idea of dielectric band gaps [Phys. Rev. 128, 2093 (1962)], and the empirical tight-binding approximation (ETB). The results are as follows. First, at the mean-value point the band gaps calculated in the GW approximation have the same widths as the dielectric band gaps. Second, the ETB approximation reproduces the GW valence-band energies at the mean-value point. Third, the branch points of the ViGS are slightly below midgap at the mean-value point. The ETB branch-point energies excellently reproduce the barrier heights of gold Schottky contacts on 19 semiconductors and the valence-band offsets of Al1−xGaxAs/GaAs heterostructures.

Band structure of silicon by the self-consistent variational cellular method

The self-consistent formulation of the variational cellular method has been developed in order to calculate the electronic structure of crystals with an arbitrary number of atoms per unit cell. Applications for silicon have been carried out. Silicon, chosen here as a test case, is treated as a face-centered cubic lattice with four “atoms” per unit cell by adding empty cells. The electronic charge density was taken muffin-tin, assuming a constant value in the interstitial region between the inscribed sphere and the Wigner–Seitz polyhedrum. The spherical symmetric electronic charge density in the inscribed sphere was obtained by adding a limited number of contributions of Brillouin zone states using the “mean value point theory” developed by Baldereschi and Chadi-Cohen. Our results are in good agreement with those obtained by other methods.

Self-consistent formulation of the variational cellular method applied to periodic structures—results for sodium and silicon

The theoretical formulation of the self-consistent variational cellular method (SCVCM) has been developed in order to calculate the electronic structure of crystals with an arbitrary number of atoms per unit cell. Calculations for metallic sodium and silicon have been carried out. The electronic charge density within the atomic polyhedron was calculated assuming two regions, one corresponding to the inscribed sphere and the other corresponding to the interstitial region. The radial electronic charge density in the inscribed sphere was obtained by adding a limited number of contributions of Brillouin zone states using the “mean value point theory”, developed by Baldereschi and Chadi-Cohen. In the interstitial region, the electronic density was taken as of a constant value. The model can be extended to open structures by adding empty cells. Our results for sodium and silicon are in good agreement with those obtained by other methods. The self-consistent scheme proposed is accurate and fast enough to be applied to more complex structures.

Mean-value point and dielectric properties of semiconductors and insulators

The mean-value-point technique for Brillouin-zone averages is shown to be applicable to the calculation of electronic dielectric screening matrices. The dielectric-matrix elements for which the technique is not accurate are only a few and for them a separate simple prescription is given. Detailed results are presented for the $\ensuremath{\omega}=0$, $\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}=0$, and $\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}\ensuremath{\rightarrow}0$ $\ensuremath{\epsilon}$ matrices of diamond, Si, Ge, $\ensuremath{\alpha}\ensuremath{-}\mathrm{Sn}$, MgO, and NaCl. Comparison with a nearly-free-electron model is also discussed.

Special points in the two-dimensional Brillouin zone

Using the method of Chadi and Cohen, we present, for each of the two-dimensional lattice types, the mean-value point, the set of generating wave vectors, and the sets of special points in the two-dimensional Brillouin zone which are the most efficient in finding accurate averages of a periodic function over the Brillouin zone.

Mean-Value Point in the Brillouin Zone

A new special point in the Brillouin zone is introduced. It is defined as the point such that the value which any given periodic function of wave vector assumes at this point is an excellent approximation to the average value of the same function throughout the Brillouin zone. This special point is termed the point, and is dictated by the crystal symmetry. The coordinates of the mean-value point for cubic lattices are explicitly given.

A local mean value theorem for the complex plane

The equationneed not have a solution z in the complex plane, even when ƒ is entire. For example, let ƒ(z) = ez, z1 = z0+2kπi. Thus the classical mean value theorem does not extend to the complex plane. McLeod has shown (2) that if ƒ is analytic on the segment joining z1 and z0, then there are points w1 and w2 on the segment such that where The purpose of this article is to give a local mean value theorem in the complex plane. We show that there is at least one point z satisfying (1), which we will call a mean value point, near z1 and z0 but not necessarily on the segment joining them, provided z1 and z0 are sufficiently close. The proof uses Rouché's Theorem (1).