Homogeneous Random Field Research Articles

Effective thermoelastic moduli and stress concentrator factors in nanocomposites

Nanocomposites are modeled as a linearly elastic composite medium, which consists of a homogeneous matrix containing a statistically homogeneous random field of spheroid nanofibers with prescribed random orientation. An estimation of the effective thermoelastic properties of NC was performed by the effective field method (see Buryachenko, [10]) taking into account the random orientation of nanofibers as well as justified selection of spatial correlations of fiber location. The independent justified choice of shapes of inclusions and correlation holes provides the matrix of effective moduli which is symmetric (in contrast to the Mori-Tanaka approach). One estimates also the effective tensor of thermal expansion and stress concentrator factors depending on the orientation of the fiber being considered as well as on the justified choice of the shape of correlation holes, concentration and orientation distribution functions of nanofibers.

Effective elastic moduli of nanocomposites with prescribed random orientation of nanofibers

Nanocomposite is modeled as a linearly elastic composite medium, which consists of a homogeneous matrix containing a statistically homogeneous random field of homogeneous prolate spheroidal nanofibers with prescribed random orientation. Estimation of effective elastic moduli of nanocomposites was performed by the version of effective field method (see for references Buryachenko VA. Multiparticle effective field and related methods in micromechanics of composite materials. Appl Mech Rev 2001;54:1–47) developed in the framework of quasi-crystalline approximation when the spatial correlations of inclusion location take particular ellipsoidal forms. The independent justified choice of shapes of inclusions and correlation holes provide the formulae of effective moduli which are symmetric, completely explicit and easily to use. The parametric numerical analyses revealed the most sensitive parameters influencing the effective moduli which are defined by the axial elastic moduli of nanofibers rather than their transversal moduli as well as by the justified choice of correlation holes, concentration and prescribed random orientation of nanofibers.

A Meshless Method for Computational Stochastic Mechanics

This paper presents a stochastic meshless method for probabilistic analysis of linear-elastic structures with spatially varying random material properties. Using Karhunen-Loève (K-L) expansion, the homogeneous random field representing material properties was discretized by a set of orthonormal eigenfunctions and uncorrelated random variables. Two numerical methods were developed for solving the integral eigenvalue problem associated with K-L expansion. In the first method, the eigenfunctions were approximated as linear sums of wavelets and the integral eigenvalue problem was converted to a finite-dimensional matrix eigenvalue problem that can be easily solved. In the second method, a Galerkin-based approach in conjunction with meshless discretization was developed in which the integral eigenvalue problem was also converted to a matrix eigenvalue problem. The second method is more general than the first, and can solve problems involving a multi-dimensional random field with arbitrary covariance functions. In conjunction with meshless discretization, the classical Neumann expansion method was applied to predict second-moment characteristics of the structural response. Several numerical examples are presented to examine the accuracy and convergence of the stochastic meshless method. A good agreement is obtained between the results of the proposed method and the Monte Carlo simulation. Since mesh generation of complex structures can be far more time-consuming and costly than the solution of a discrete set of equations, the meshless method provides an attractive alternative to the finite element method for solving stochastic-mechanics problems.

Stochastic spline Ritz method based on stochastic variational principle

In the previous versions of stochastic variational principles, the virtual displacements are often assumed to vary arbitrarily and independent of any uncertainties. This paper investigates the stochasticity of the virtual displacements via the relationship among virtual displacements, admissible displacements and real displacements. A stochastic variational principle is proposed so that the stochastic properties of the random parameters involved can be incorporated into the functional of the total potential energy. Cubic B-spline functions are employed as the trial functions to reduce the number of degrees of freedom. Then, the stochastic spline Ritz method is developed based on the stochastic variational principle via the second-order perturbation techniques. The spline Ritz method is applied to the structures consisting of elastic beams and plates by a local average of the homogeneous random field. Numerical examples are given to illustrate the efficiency and accuracy of the presented method.

Markov approximation of homogeneous lattice random fields

We refine some well-known approximation theorems in the theory of homogeneous lattice random fields. In particular, we prove that every translation invariant Borel probability measure μ on the space X of finite-alphabet configurations on ℤd, d≥1, can be weakly approximated by Markov measures μn with supp(μn)=X and with the entropies h(μn)→h(μ). The proof is based on some facts of Thermodynamic Formalism; we also present an elementary constructive proof of a weaker version of this theorem.

Asymptotics for the boundary parabolic Anderson problem in a half space

We study the large time behavior of the solutions of the Cauchy problem for the Anderson model restricted to the upper half space and/or when the potential is a homogeneous random field concentrated on the boundary . In other words we consider the problem: with an appropriate initial condition. We determine the large time asymptotics of the moments of the solutions as well as their almost sure asymptotic behavior when t ↠ ∞ and when the distance from the boundary, i.e. y = y (t) goes simultaneously to infinity as a function of the time t . We identify the rates of escape of y (t) which correspond to specific behaviors of the solutions and different types of dependence upon the diffusivity constant κ . We also show that the case of the lattice differs drastically from the continuous case when it comes to the existence of the moments and the influence of κ . Intermittency is proved as a consequence of the large time behavior of the solutions.

Phase correlations in cosmic microwave background temperature maps

We study the statistical properties of spherical harmonic modes of temperature maps of the cosmic microwave background. Unlike other studies, which focus mainly on properties of the amplitudes of these modes, we look instead at their phases. In particular, we present a simple measure of phase correlation that can be diagnostic of departures from the standard assumption that primordial density fluctuations constitute a statistically homogeneous and isotropic Gaussian random field, which should possess phases that are uniformly random on the unit circle. The method we discuss checks for the uniformity of the distribution of phase angles using a non-parametric descriptor based on the use order statistics, which is known as Kuiper's statistic. The particular advantage of the method we present is that, when coupled to the judicious use of Monte Carlo simulations, it can deliver very interesting results from small data samples. In particular, it is useful for studying the properties of spherical harmonics at low l for which there are only small number of independent values of m and which therefore furnish only a small number of phases for analysis. We apply the method to the COBE-DMR and WMAP sky maps, and find departures from uniformity in both. In the case of WMAP, our results probably reflect Galactic contamination or the known variation of signal-to-noise across the sky rather than primordial non-Gaussianity.

モンテカルロシミュレーション法による物性値に不確定性をもつ傾斜機能平板の確率論的熱弾性問題の解析

The statistics of the temperature and thermal stress are quantitatively obtained by Monte Carlo simulation method for functionally graded material plates (FGM plates) with uncertainties in the thermal conductivity and coefficient of linear thermal expanssion and with arbitrary nonhomogeneous thermal and mechanical properties through the thickness of plate. The stochastic temperature and thermal stress fields are analyzed by approximating the FGM plates as multilayered plates with random thermal conductivity and coefficient of linear thermal expanssion in each layer. Autocorrelation coefficients of the random properties and crosscorrelation coefficients between the random properties are expressed in exponential function forms as homogeneous Markov random field of discrete coordinate. Numerical calculations are carried out for FGM plates composed of PSZ and SUS304, which have two different variability of random properties. The effects of the step-wise change in material composition, boundary condition and correlation coefficients on the standard deviations of temperature and thermal stress are discussed.

Fourier Transforms of Distributions of Homogeneous Random Fields with Independent Increments and Complex Markov--Maslov Chains

We prove formulas that describe, under certain sufficiently general assumptions, the Fourier transforms of distributions of processes and fields mentioned in the title. A similar method applied to complex Markov chains on a finite segment can be used to compute the Fourier transform of the complex Poisson measure from [1, Chap. IX] (where it is found by series expansion of the measure) given its finite-dimensional distributions. We assume that the value space of a random field is a Polish (i.e., complete, separable, and metrizable) locally convex space; however, the results seem to be new for real processes as well. An additional assumption on the processes is that their trajectories have no discontinuities of the second kind in the (weakened) topology of the value space; in particular, such are the Poisson jump processes. Earlier [2] such a formula was proved only for processes with strongly continuous trajectories. Also discussed in [2] are applications of a similar construction for continuous fields to the particular case of Wiener–Levy–Chentsov fields [3].

Asymptotics for the boundary parabolic Anderson problem in a half space

We study the large time behavior of the solutions of the Cauchy problem for the Anderson model restricted to the upper half space and/or when the potential is a homogeneous random field concentrated on the boundary . In other words we consider the problem:

About approximation of 3-D random fields and statistical simulation

The estimator of the mean-square approximation of 3-D homogeneous and isotropic random field is investigated. The problem of statistical simulation of realizations of random fields in threedimensional space is considered. The algorithm for the receiving of this realization has been formulated, which has been constructed on the base the mean-square approximation of random fields estimator. It has been constructed the statistical model for the Gaussian random fields in three-dimensional space, which has been given by its statistical characteristics.

Spectral representation and asymptotic properties of certain deterministic fields with innovation components

In this paper we derive the spectral and ergodic properties of a special class of homogeneous random fields, which includes an important family of evanescent random fields. Based on a derivation of the resolution of the identity for the operators generating the homogeneous field, and using the properties of measurable transformations, the spectral representation of both the field and its covariance sequence are derived. A necessary and sufficient condition for the existence of such representation is introduced. Using an analysis approach that employs the solution to the linear Diophantine equations, further characterization and modeling of the spectral properties of evanescent fields are provided by considering their spectral pseudo-density function, defined in this paper. The geometric properties of the spectral pseudo-density of the evanescent field are investigated. Finally, necessary and sufficient conditions for mean and strong ergodicity of the first and second order moments of these fields are derived. The analysis, initially carried out for complex valued random fields, is later extended to include the case of real valued fields.

Relationship between juxtaposed, overlapping, and fractal representations of multimodal spatial variability

It has been shown by Neuman [1990], Di Federico and Neuman [1997], and Di Federico et al. [1999] that the power variogram of a statistically isotropic or anisotropic fractal field can be constructed as a weighted integral from zero to infinity of exponential or Gaussian variograms of overlapping, homogeneous random fields (modes) having mutually uncorrelated increments and variance proportional to a power 2H of the integral (spatial correlation) scale where H is the Hurst coefficient. Low‐ and high‐frequency cutoffs are related to length scales of the sampling window (domain) and data support (sample volume), respectively. Intermediate cutoffs account for lacunarity due to gaps in the multiscale hierarchy, created by a hiatus of modes associated with discrete ranges of scales. Alternative mathematical representations of multimodal spatial variability were formulated by various authors in which space is filled by a discrete number of juxtaposed (mutually exclusive) materials or categories, each having its own architecture and attributes. The spatial distribution of categories is characterized by indicator random variables and their attributes by random fields. This paper focuses on expressions developed by Lu and Zhang [2002] in which the indicator variables and their attributes are mutually uncorrelated while each is autocorrelated and cross correlated within and between categories. Upon rewriting their expressions for statistically homogeneous and anisotropic media, it is demonstrated mathematically that in the limit as the categories stretch to occupy each point in space (overlap) in a way that preserves their local architecture, their attributes become mutually uncorrelated. Categories are said to overlap if they are found in fixed proportions within a representative sampling volume centered about any mathematical point throughout space; the idea is analogous to the well‐known dual continuum concept in which two categories, most commonly fractures and porous blocks, are considered to overlap. In reality, the categories do not overlap but are considered to preserve their relative architectural arrangement throughout space. The variogram of an attribute, sampled jointly over overlapping statistically homogeneous and anisotropic categories, is the sum of the variograms of statistically homogeneous and anisotropic components of this attribute sampled over individual categories, weighted by their volumetric proportions. It is further demonstrated that when the overlapping categories represent an infinite or truncated hierarchy of modes whose attributes have exponential or Gaussian variograms, such that the product of category probability density function (pdf) and attribute variance is proportional to a power 2H + 1 of the attribute integral scale, they represent a fractal field characterized by a power or truncated power variogram as in the work of Di Federico and Neuman. Any category pdf that satisfies this condition, and any attribute pdf having a variance that satisfies the same condition, are admissible; neither of them need to be Gaussian (to form fractional Brownian motion) or Levy‐like (to approximate fractional Levy motion) for the attribute to form a random fractal. Intermediate cutoffs render the latter model discrete, as is that of Lu and Zhang. Hence the two multimodal models are mutually consistent and complementary.

Asymptotic inference for LSE in multivariate continuous regression models with long-memory random fields

This paper establishes the asymptotic behaviour for the covariance matrix and the limit distributions of the least squares estimators for a regression coefficients in a multivariate continuous regression models with long-memory Gaussian errors. The used method is based on the asymptotic analysis of orthogonal expansion of non-linear functionals of homogeneous and isotropic Gaussian random fields.

Asymptotic properties of LSE of regression coefficients on singular random fields observed on a sphere

We present some upper bounds on the rate of convergence in the central limit theorem for normalized least square estimates (LSE) in a spherical regression model with long range dependence (LRD) stationary errors. The used method is based on the asymptotic analysis of orthogonal expansion of non-linear functionals of homogeneous isotropic Gaussian random fields and on the Kolmogorov distance. The theory have many applications in science for instance in evaluating the COBE data.

Three‐dimensional steady state flow to a well in a randomly heterogeneous bounded aquifer

We consider flow in a confined aquifer of uniform thickness due to a well of zero radius that fully penetrates the aquifer and discharges at a constant rate. If the lateral extent of the aquifer is infinite, a steady state flow regime never develops. It is, however, well known that if the aquifer is additionally uniform, a quasi‐steady state region extends from the well out to a cylindrical surface whose radius expands as the square root of time. On the expanding surface, head is uniform and time invariant. Inside this surface, head at any time is described by a steady state solution. A rigorous analysis of the analogous situation in a randomly heterogeneous aquifer would require the solution of a three‐dimensional transient stochastic flow problem in an aquifer of infinite lateral extent. Here we take a different approach by developing a three‐dimensional steady solution for mean flow to a well in a randomly heterogeneous aquifer with a cylindrical prescribed head boundary. In analogy to the uniform case we expect our solution to approximate a quasi‐steady state region whose radius is initially small in comparison to the horizontal correlation scale of log conductivity but grows with time to become eventually much larger. We treat log conductivity as a statistically homogeneous random field characterized by a Gaussian spatial covariance function that may have different horizontal and vertical correlation scales. Our solution consists of analytical expressions for the ensemble mean and variance of head in the aquifer to second order in the standard deviation of log conductivity. It is based on recursive approximations of exact nonlocal moment equations that are free of distributional assumptions and so apply to both Gaussian and non‐Gaussian log conductivity fields. The analytical solution is supported by numerical Monte Carlo simulations. It clarifies the manner in which relationships between the horizontal and vertical scales of the quasi‐steady state region and those of statistical anisotropy impact the statistical moments of drawdown and the equivalent and apparent hydraulic conductivities of the aquifer. Both conductivities are shown to exhibit a scale effect by growing with distance from the well within a radius of one to two horizontal integral scales from it.

Comparing filters for the detection of point sources

This paper considers filters (the Mexican hat wavelet, the matched and the scale-adaptive filters) that optimize the detection/separation of point sources on a background. We make a one-dimensional treatment, we assume that the sources have a Gaussian profile, i. e. $\tau (x) = e^{- x^2/2R^2}$, and a background modelled by an homogeneous and isotropic Gaussian random field, characterised by a power spectrum $P(q)\propto q^{-\gamma}, \gamma \geq 0$. Local peak detection is used after filtering. Then, the Neyman-Pearson criterion is used to define the confidence level for detections and a comparison of filters is done based on the number of spurious and true detections. We have performed numerical simulations to test theoretical ideas and conclude that the results of the simulations agree with the analytical results.

About approximation of 3-D random fields and statistical simulation

The estimator of the mean-square approximation of 3-D homogeneous and isotropic random field is investigated. The problem of statistical simulation of realizations of random fields in threedimensional space is considered. The algorithm for the receiving of this realization has been formulated, which has been constructed on the base the mean-square approximation of random fields estimator. It has been constructed the statistical model for the Gaussian random fields in three-dimensional space, which has been given by its statistical characteristics.

Mean anisotropy of homogeneous Gaussian random fields and anisotropic norms of linear translation‐invariant operators on multidimensional integer lattices

Sensitivity of output of a linear operator to its input can be quantified in various ways. In Control Theory, the input is usually interpreted as disturbance and the output is to be minimized in some sense. In stochastic worst‐case design settings, the disturbance is considered random with imprecisely known probability distribution. The prior set of probability measures can be chosen so as to quantify how far the disturbance deviates from the white‐noise hypothesis of Linear Quadratic Gaussian control. Such deviation can be measured by the minimal Kullback‐Leibler informational divergence from the Gaussian distributions with zero mean and scalar covariance matrices. The resulting anisotropy functional is defined for finite power random vectors. Originally, anisotropy was introduced for directionally generic random vectors as the relative entropy of the normalized vector with respect to the uniform distribution on the unit sphere. The associated a-anisotropic norm of a matrix is then its maximum root mean square or average energy gain with respect to finite power or directionally generic inputs whose anisotropy is bounded above by a ≥ 0. We give a systematic comparison of the anisotropy functionals and the associated norms. These are considered for unboundedly growing fragments of homogeneous Gaussian random fields on multidimensional integer lattice to yield mean anisotropy. Correspondingly, the anisotropic norms of finite matrices are extended to bounded linear translation invariant operators over such fields.

Scale‐adaptive Filters for the Detection/Separation of Compact Sources

This paper presents scale-adaptive filters that optimize the detection/separation of compact sources on a background. We assume that the sources have a multiquadric profile, i. e. $\tau (x) = {[1 + {(x/r_c)}^2]}^{-\lambda}, \lambda \geq {1/2}, x\equiv |\vec{x}|$, and a background modeled by a homogeneous and isotropic random field, characterized by a power spectrum $P(q)\propto q^{-\gamma}, \gamma \geq 0, q\equiv |\vec{q}|$. We make an n-dimensional treatment but consider two interesting astrophysical applications related to clusters of galaxies (Sunyaev-Zel'dovich effect and X-ray emission).