Estimating and Choosing

Matheron, G.

doi:10.1007/978-3-642-48817-7

Cited by 153 publications

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“…Since the theoretical covariance of the fields (σ or ε) is not available, the integral range can be estimated according to the procedure proposed by Georges Matheron for any random function (30): working with realizations of Z(x) on domains B with an increasing volume V (or in the present case considering subdomains of large simulations, with a wide range of sizes), we can estimate the parameter A 3 by fitting the obtained variance according to the expression (9.2). The point variance D 2 Z of the corresponding field is directly estimated from the experimental variance of the field.…”

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confidence: 99%

Elastic and Electrical Behavior of Some Randommultiscale Highly-Contrasted Composites

Willot¹,

Jeulin²

2011

Int J Mult Comp Eng

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Abstract. The role of a non-uniform distribution of heterogeneities on the elastic as well as electrical properties of composites is studied numerically and compared with available theoretical results. Specifically, a random model made of embedded Boolean sets of spherical inclusions (see e.g. Jean et al, 2007) serves as the basis for building simple two-scales microstructures of "granular"-type. Materials with "infinitely-contrasted" properties are considered, i.e. inclusions elastically behave as rigid particles or pores, or as perfectly-insulating or highly-conducting heterogeneities. The inclusion spatial dispersion is controlled by the ratio between the two characteristic lengths of the microstructure. The macroscopic behavior as well as the local response of composites are computed using full-field computations, carried out with the "Fast Fourier Transform" method (Moulinec and Suquet, 1994). The entire range of inclusion concentration, and dispersion ratios up to the separation of length scales are investigated.As expected, the non-uniform dispersion of inhomogeneities in multi-scale microstructures leads to increased reinforcing or softening effects compared to the corresponding one-scale model (Willot and Jeulin, 2009); these effects are however still significantly far apart from Hashin-Shtrikman bounds. Similar conclusions are drawn regarding the electrical conductivity.

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Elastic and Electrical Behavior of Some Randommultiscale Highly-Contrasted Composites

Willot¹,

Jeulin²

2011

Int J Mult Comp Eng

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“…Since the theoretical covariance of the fields (σ or ε) is not available, the integral range can be estimated according to the procedure proposed by Georges Matheron for any random function (19): working with realizations of Z(x) on domains B with an increasing volume V (or in the present case considering subdomains of large simulations, with a wide range of sizes), we can estimate the parameter A 3 by fitting the obtained variance according to the expression (12). The point variance D 2 Z of the corresponding field is directly estimated from the experimental variance of the field.…”

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Elastic behavior of composites containing Boolean random sets of inhomogeneities

Willot

Jeulin

2009

International Journal of Engineering Science

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The overall mechanical response as well as strain and stress field statistics of an heterogeneous material made of two randomly distributed, linear-elastic phases, are investigated numerically. The Boolean model of spheres is used to generate microstructures consisting of either porous or rigid inclusions, at any volume fraction of the phases. Stress and strain field integral ranges, or equivalently the representative volume element, are computed and linked to features of the field statistics, and to the microstructure geometry.

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“…Wider classes of problems can be formed, e.g. by non-linear transformation of variables (as in generalized linear models, McCullagh and Nelder, 1989), or by defining linear relations locally (approximating more complex relations with local linear models, as in generalized additive models, Hastie and Tibshirani, 1990).Linear models are a means of expressing real-life problems in a mathematical form, and are therefore helpful in the abstraction of problems and in communication. Comprehensive treatments of linear models are found in Rao (1973), Searle (1971), Christensen (1987, and in Draper and Smith (1981).…”

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“…Furthermore local estimation only calls for variogram values at distances that allow a meaningful estimation of the variogram (i.e. values at distances less than approximately half the dimensions of the area used for variogram calculation, Journel and Huijbregts, 1978, III.B.7;Matheron, 1989) and that are less affected by biased estimation of the trend parameters ¥ (Cressie and Zimmerman, 1992). In practice V and v 0 are not known but estimated withV andv 0 from sample data, causinĝ (Harville, 1985, Christensen, 1991.…”

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Mapping groundwater quality in the Netherlands

Pebesma

Kwaadsteniet

1997

Journal of Hydrology

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Niets uit deze uitgave mag worden vermenigvuldigd en/of openbaar gemaakt worden door middel van druk, fotokopie of op welke andere wijze dan ook zonder voorafgaande schriftelijke toestemming van de uitgevers.All rights reserved. No part of this publication may be reproduced in any form, by print of photoprint, microfilm or any other means, without written permission by the publishers. Omslagontwerp: Mark Oonk MAPPING GROUNDWATER QUALITY IN THE NETHERLANDSHET KARTEREN VAN Problem definitionPollution from agricultural and industrial origin threatens the groundwater quality in the Netherlands (Van Duijvenbooden, 1989). Locally, this pollution is measured in the groundwater at tens of metres depth (CCRX, 1994(CCRX, , 1995. Since groundwater is the main source of fresh water, pollution causes a decrease in the long-term resources of water suitable for human consumption, and currently water companies have to increase their efforts for making groundwater suitable for drinking (Versteegh et al., 1995). Furthermore, because most groundwater finally flows to the surface, a deterioration of groundwater quality affects the ecohydrological conditions of the receiving seepage areas as well as the receiving seas. In order to get insight in the current situation of groundwater quality and the systematic changes of groundwater quality over time, the national groundwater quality monitoring network (NGM) was established from 1978 to 1984 (Van Duijvenbooden et al., 1985). This network consists of 370 permanent wells spread fairly evenly over the country (Fig. 3.2, page 27), with screens at 8-10 and 23-25 metre below the soil surface. The network is sampled yearly, at first for 19 variables, and more recently for 25 variables. In recent years all provinces commenced the installation of similar monitoring network sites, the provincial groundwater quality monitoring networks (PGMs). Currently, they provide a doubling of the number of wells in the NGM.Because the monitoring network is a major financial investment, questions like 'do we have sufficient measurements?'or 'can a reduced monitoring network be sufficient?'arise. Such questions can only be answered when we have answers to the questions 'sufficient for what?' and 'what exactly do we need to know about groundwater quality?' Clearly, this calls for the quantification of what can be inferred from available information about the quality of all the groundwater of interest. Therefore we need (a) to choose a suitable, effective and estimable measure of groundwater quality (what characteristic of groundwater quality do we need to know?), and (b) to handle the estimation uncertainty in a meaningful way (to what extent can we infer this characteristic from available information?). The uncertainty about the true groundwater quality that results from limited estimation accuracy is the key parameter in optimization studies, where a trade-off between physical cost and losses due to limited knowledge should be made: if we need to know everything about groundwater then we will have to mea...

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Estimating and Choosing

Cited by 153 publications

References 0 publications

Elastic and Electrical Behavior of Some Randommultiscale Highly-Contrasted Composites

Elastic and Electrical Behavior of Some Randommultiscale Highly-Contrasted Composites

Elastic behavior of composites containing Boolean random sets of inhomogeneities

Mapping groundwater quality in the Netherlands

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