Multidimensional Random Fields Research Articles

Metamodel-assisted distributed genetic algorithms applied to structural shape optimization problems†

This article is concerned with the development of a general optimization tool based on distributed real genetic algorithms (DRGAs) assisted by metamodel evaluation and applied to structural shape optimization problems of general boundary-element models (BEMs). The evaluation fitness function is performed by a surrogate function based on multidimensional Gaussian random field models (MGRFMs) in order to minimize the computational cost of the evolutionary algorithms. The model boundary of a structural system or a mechanical tool is discretized using the BEM, and selected parts of the boundary are modelled using β-spline curves or surfaces in order to facilitate re-meshing and adaptation of the boundary to the external actions. Then a hypercube topology of populations of these models follows a genetic evolution process to determine the optimum shape of the system. The optimum models have minimum weight and satisfy all imposed constraints. A numerical example is presented and discussed in order to show the efficiency and robustness of the developed computational tool. The number of function evaluations is substantially reduced compared with previous versions of the optimization algorithm without the metamodel evaluation technique. †This is an extended and enhanced version of work presented at the mini-symposium on Evolutionary Algorithms: Recent Applications in Engineering and Science organized by Dr William Annicchiarico at the 7th World Congress on Computational Mechanics, Los Angeles, July 2006.

Adaptive Random Field Mesh Refinements in Stochastic Finite Element Reliability Analysis of Structures

A technique for adaptive random field refinement for stochastic finite element reliability analysis of structures is presented in this paper. Refinement indicator based on global importance measures are proposed and used for carrying out adaptive random field mesh refinements. Reliability index based error indicator is proposed and used for assessing the percentage error in the estimation of notional failure probability. Adaptive mesh refinement is carried out using hierarchical graded mesh obtained through bisection of elements. Spatially varying stochastic system parameters (such as Young's modulus and mass density) and load parameters are modeled in general as non-Gaussian random fields with prescribed marginal distributions and covariance functions in conjunction with Nataf's models. Expansion optimum linear estimation method is used for random field discretisation. A framework is developed for spatial discretisation of random fields for system/load parameters considering Gaussian/non-Gaussian nature of random fields, multidimensional random fields, and multiple-random fields. Structural reliability analysis is carried out using first order reliability method with a few refined features, such as, treatment of multiple design points and/or multiple regions of comparable importance. The gradients of the performance function are computed using direct differentiation method. Problems of multiple performance functions, either in series, or in parallel, are handled using the method based on product of conditional marginals. The efficacy of the proposed adaptive technique is illustrated by carrying out numerical studies on a set of examples covering linear static, free vibration and forced vibration problems.

3D effects in seismic liquefaction of stochastically variable soil deposits

The natural variability of soil properties within geologically distinct and uniform layers has been proven to greatly affect soil behaviour and to induce significant variability in the predicted response. Previous studies concluded that small-scale heterogeneity greatly affects the liquefaction potential of saturated soil deposits, and provided geotechnical design guidelines to account for the effects of various characteristics of spatial variability. Those studies were based on two-dimensional analyses of soil liquefaction (in a vertical plane) assuming plane strain behaviour. Therefore the correlation distance of soil variability in a direction normal to the plane of analysis was implicitly taken as infinite (i.e. no variability in the third direction). In this study, a Monte Carlo simulation approach involving generation of sample functions of non-Gaussian, multivariate, multidimensional random fields and non-linear finite element analyses is used to investigate the effects of soil heterogeneity on the liquefaction potential of a ‘stochastically heterogeneous’ soil deposit subjected to seismic loading. To assess the 3D effects, Monte Carlo simulation results obtained for a 3D soil deposit are compared with corresponding results from 2D plane strain analyses. The calculations are performed for a range of seismic acceleration intensities, and the results are presented in terms of fragility curves expressing the probability of exceeding various thresholds in the response as a function of earthquake intensity.

3D effects in seismic liquefaction of stochastically variable soil deposits

The natural variability of soil properties within geologically distinct and uniform layers has been proven to greatly affect soil behaviour and to induce significant variability in the predicted response. Previous studies concluded that small-scale heterogeneity greatly affects the liquefaction potential of saturated soil deposits, and provided geotechnical design guidelines to account for the effects of various characteristics of spatial variability. Those studies were based on two-dimensional analyses of soil liquefaction (in a vertical plane) assuming plane strain behaviour. Therefore the correlation distance of soil variability in a direction normal to the plane of analysis was implicitly taken as infinite (i.e. no variability in the third direction). In this study, a Monte Carlo simulation approach involving generation of sample functions of non-Gaussian, multivariate, multidimensional random fields and non-linear finite element analyses is used to investigate the effects of soil heterogeneity on the liquefaction potential of a ‘stochastically heterogeneous’ soil deposit subjected to seismic loading. To assess the 3D effects, Monte Carlo simulation results obtained for a 3D soil deposit are compared with corresponding results from 2D plane strain analyses. The calculations are performed for a range of seismic acceleration intensities, and the results are presented in terms of fragility curves expressing the probability of exceeding various thresholds in the response as a function of earthquake intensity.

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.

Strip foundation on 2-D and 3-D random subsoil

The paper affords a stochastic description of a three-dimensional (3-D) soil medium and its modelling under plane strain conditions. The main aim of the paper is to work out a computational model enabling incorporation of 3-D variability of soil properties into the plane strain state analysis. First, the 2-dimensional (2-D) problem and its reduction to the uni-axial strain state is considered. Then this is extended for the 2-D schematisation of the 3-D state. The resulting standard deviations for the 3-D state of stress and strain are compared with those for the 2-D state. It is assumed that the soil medium is statistically homogeneous and its mechanical behaviour governed by the linear elasticity theory. It is also assumed that elastic parameters can be modelled as the multidimensional random fields. The strip foundation on a soil layer in the 3-D and the 2-D strain states is analysed. Stochastic 2-D and 3-D finite element methods, based on the Monte Carlo technique were used. The analysis performed enables determination of the standard deviations of components of the stress tensor and the displacement vector for the 3-D state, based on the solution for the 2-D plane strain state.

On the asymptotic distributions of mean, autocovariance, autocorrelation, crossgovariancb and impulse response estimators of a stationary multidimensional random field

Asymptotic properties of mean, autocovariance, autocorrelation, crosscovariance and impulse response estimators of a stationary M-dimensionai (M-D) random field are studied. It is shown that only unbiased-type estimators of autocovariances, autocorrelations, crosscovariances and impulse responses have the asymptotic distributions when M≧ 2. Moreover, the asymptotic distributions of mean, autocovariance, autocorrelation, crosscovariance and impulse response estimators are presented.

Simulation of Homogeneous and Partially Isotropic Random Fields

A rigorous methodology for the simulation of homogeneous and partially isotropic multidimensional random fields is introduced. The property of partial isotropy of the random field is explicitly incorporated in the derivation of the algorithm. This consideration reduces significantly the computational effort associated with the generation of sample functions, as compared with the case when only the homogeneity in the field is taken into account. The approach is based on the spectral representation method, utilizes the fast Fourier transform, and generates simulations with random variability in both their amplitudes and phases, or in their phases only. Spatially variable seismic ground motions experiencing loss of coherence are generated as an example application of the developed approach.

Multidimensional Nonstationary Maximum Entropy Spectral Analysis by Using Neural Net

A multidimensional version of the nonstationary maximum entropy spectral estimation method has been developed. By applying the neural net technique, the estimated maximum entropy spectral density function has more resolution power than the conventional nonstationary periodogram spectral density function. The application of spectral density functions to simulate multidimensional nonstationary random fields is also investigated.

An efficient multivariate random field generator using the fast Fourier transform

It is often convenient to use synthetically generated random fields to study the hydrologic effects of spatial heterogeneity. Although there are many ways to produce such fields, spectral techniques are particularly attractive because they are fast and conceptually straightforward. This paper describes a spectral algorithm for generating sets of random fields which are correlated with one another. The algorithm is based on a discrete version of the Fourier-Stieltjes representation for multidimensional random fields. The Fourier increment used in this representation depends on a random phase angle process and a complex-valued spectral factor matrix which can be readily derived from a specified set of cross-spectral densities (or cross-covariances). The inverse Fourier transform of the Fourier increment is a complex random field with real and imaginary parts which each have the desired coveriance structure. Our complex-valued spectral formulation provides an especially convenient way to generate a set of random fields which all depend on a single underlying (independent) field, provided that the fields in question can be related by space-invariant linear transformations. We illustrate this by generating multi-dimensional mass conservative groundwater velocity fields which can be used to simulate solute transport through heterogeneous anisotropic porous media.

Efficient Simulation of Multidimensional Random Fields

An efficient stochastic simulation method is presented for generating sample time histories of multidimensional, spatially correlated, stationary Gaussian random fields. A key feature of the method is the use of filtering functions in directly expressing the conditional mean value of any component of the random field. The general formulation and an updating algorithm for determining the required filtering functions for simulation of any general multidimensional random field are established. Also, emphasis is placed on the efficiencies, which can be achieved when relatively simple restrictions are placed on the cross-spectral densities and the incoherence structure of the random field. These restrictions, which are commonly used in modeling seismic ground motion, are shown to allow the computational effort to be reduced by a factor of more than 100 for a three-dimensional random field. Numerical results are presented to illustrate the application of the direct simulation method to the generation of time histories of seismic ground motion. The method is shown to be both efficient and accurate.

Error of digitization of multidimensional random hydrophysical fields with power spectra

Exact and approximate formulae are proposed to evaluate the errors of spatial/temporal digitization (aliasing) of then-dimensional random fields withm-power spectra atn=1, …, 4 and 1≤m≤5, which satisfactorily approximate temperature, velocity, density, and other fields in the sea and atmosphere.

Error Evaluation of Three Random‐Field Generators

The use of multidimensional random fields to model real engineering systems is gaining acceptance as personal computer systems become increasingly powerful. The accuracy of such models depends dire...

Multiple window based minimum variance spectrum estimation for multidimensional random fields

Spectrum analysis is viewed as the problem of estimating the power of a process contained within narrow bands. This view leads naturally to consideration of filter-based methods for estimating spectra. Multiple window-based estimators where the power in a band is estimated as the average of the powers estimated at the outputs of multiple filters are considered. The filters are designed using a linearly constrained minimum variance criterion commonly employed in adaptive beamforming. This results in filters that automatically adjust their sidelobes to minimize leakage of energy from outside the band of interest. Expressions for the bias and variance of the power estimates are derived assuming the sample covariance matrix estimate is used to estimate the data covariance matrix and that the data are independent and identically Gaussian distributed. These expressions lead to the definition of a performance factor that indicates the degree of variance reduction obtained via multiple window processing. A technique for obtaining increased suppression of energy leaking through the filter sidelobes at the expense of the response fidelity to energy in the band is presented. Simulations are presented to illustrate the effectiveness of the technique. >

Space transformation methods in the representation of geophysical random fields

Various aspects of multidimensional random fields are studied by means of space transformations. The latter are elegant and comprehensive Radon operations which can solve complex multidimensional problems by transforming them to a suitable unidimensional setting, where analysis is considerably simpler. It is shown that spatial correlation functions in R/sup n/ are uniquely determined by means of their space transformations in R/sup 1/. Necessary and sufficient conditions are established in order that a spatial random field (in R/sup n/) be represented as the linear combination of pairwise uncorrelated random processes (in R/sup 1/). Space transformations provide analytically tractable criteria for testing the permissibility of correlation functions and constitute an attractive instrument for spatial and spatiotemporal random field simulation and for studying stochastic partial differential equations. Several examples and a case study are discussed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Using occupancy grids for mobile robot perception and navigation

An approach to robot perception and world modeling that uses a probabilistic tesselated representation of spatial information called the occupancy grid is reviewed. The occupancy grid is a multidimensional random field that maintains stochastic estimates of the occupancy state of the cells in a spatial lattice. To construct a sensor-derived map of the robot's world, the cell state estimates are obtained by interpreting the incoming range readings using probabilistic sensor models. Bayesian estimation procedures allow the incremental updating of the occupancy grid, using readings taken from several sensors over multiple points of view. The use of occupancy grids from mapping and for navigation is examined. Operations on occupancy grids and extensions of the occupancy grid framework are briefly considered.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Statistical opportunities for analyzing spatial and temporal heterogeneity of field soils

Statistical techniques for analyzing data in the agricultural sciences have traditionally followed the pioneering efforts of R.A. Fisher who assumed that observations in the field were independent and identically distributed. Such techniques, proven useful in the past and still being used today for comparing the merits of different management practices or different treatments, are presently giving way to additional methods that are based upon observations being spatially or temporally correlated. It is physically more sensible to expect soil attributes to be correlated when they are measured at adjacent points in space or time. Spatially repetitious patterns of soil attributes for physical and biological processes occurring at distances of a few molecules to those of kilometers are also expected. Opportunities to use geostatistics, time series analyses, state-space models, spectral analyses of variance, lagged regression models and other alternative techniques for analyzing multidimensional random fields are available to enhance the understanding of agro-ecosystems. Approaches to modeling and fitting data using stochastic partial differential equations and scaling techniques also help reveal the underlying processes occurring in field soils. Inclusion of these alternatives in the development of an agro-ecological technology leads to improved sampling designs to better entire management units, rather than ascertaining the impact of particular, sometimes arbitrary treatments applied to a set of small plots using analysis of variance methods.

The space transformation in the simulation of multidimensional random fields

Space transformations are proposed as a mathematically meaningful and practically comprehensive approach to simulate multidimensional random fields. Within this context the turning bands method of simulation is reconsidered and improved in both the space and frequency domains.

Convergence theorems of sampler processes with applications to image processing

We consider sampling methods for multidimensional random fields. We study the convergence of a sampler process constructed by the methods stated in Introduction. We can apply convergence theorems to the restoration of degraded images in image processing.

Control, Accuracy and Reliability of a Statistical Simulation of Non-Stationary Systems

Abstract The problem of control and estimating the reliability of a complex stochastic systems modelling is examined. In order to solve this problem the estimates of statistical characteristics of a random processes and fields are used. New method of computing the distribution function of spectral density and correlation function of random processes and fields is suggested. We discuss the new results in application of a curve estimates by Parzen and Rozenblatt to estimation the distribution density of random processes and fields. The effective and quasieffective estimates for variance and mathematical expectation of non-stationary processes are observed. New method of statistical simulation of multidimensional random fields, defined by spectral decomposition, is suggested. The numerical algorithms of a parametric modelling of some classes of random functions are considered.