Trois approches cohérentes pour modéliser la multifissuration des composites 1D

International audienceIl existe de nombreux modèles pour décrire la fissuration des composites 1D, avec, pour la plupart, les points communs suivants : la mécanique est simplifiée en 1D, le comportement des matériaux (fibres et matrice) est élastique, l'interphase est souvent modélisée par une contrainte de cisaillement. Le point délicat de ces modèles consiste à gérer le caractère aléatoire de la fissuration, souvent décrit par un modèle deWeibull. Pour ce faire, plusieurs stratégies ont été déployées qui peuvent se classer en trois approches : la première approche (CL) discrétise le composite, et tire aléatoirement des contraintes limites suivant une statistique de Weibull ; la seconde approche (PF) détermine de manière aléatoire les différentes fissures de manière séquentielle ; la troisième approche (AC) modélise un composite infini, et propose des formules analytiques pour décrire la statistique des fragments de composites. Nous montrons, dans le cas simplifié du micro-composite, que ces trois approches conduisent à des résultats équivalents

Testing the independence of maxima: from bivariate vectors to spatial extreme fields

International audienceCharacterizing the behaviour of multivariate or spatial extreme values is of fundamental interest to understand how extreme events tend to occur. In this paper we propose to test for the asymptotic independence of bivariate maxima vectors. Our test statistic is derived from a madogram, a notion classically used in geostatistics to capture spatial structures. The test can be applied to bivariate vectors, and a generalization to the spatial context is proposed. For bivariate vectors, a comparison to the test by Falk and Michel (2006) is conducted through a simulation study. In the spatial case, special attention is paid to pairwise dependence. A multiple test procedure is designed to determine at which lag asymptotic independence takes place. This new procedure is based on the bootstrap distribution of the number of times the null hypothesis is rejected. It is then tested on maxima of three classical spatial models and finally applied to two climate datasets

Conditional simulation of a positive random vector subject to max-linear constraints. A geometric perspective

Full text available for free at http://geostats2012.nr.no/pdfs/1748421.pdfInternational audiencePredicting natural phenomena modeled by max-stable random fields with Fréchet margins is not simple because these models do not possess finite first and second order moments. In such situations, a Monte Carlo approach based on conditional simulations can be considered. In this paper we examine a recent algorithm set up by Wang and Stoev to conditionally simulate a max-stable random field with discrete spectrum. Besides presenting this algorithm, we provide it with a geometric interpretation and put emphasis on several implementation details to obviate its combinatorial complexity. Along the way, a number of other critical issues are mentioned that are not often addressed in the current practice of conditional simulations. An illustrative example is given

Conditional Sampling for Max-Stable Processes with a Mixed Moving Maxima Representation

This paper deals with the question of conditional sampling and prediction for the class of stationary max-stable processes which allow for a mixed moving maxima representation. We develop an exact procedure for conditional sampling using the Poisson point process structure of such processes. For explicit calculations we restrict ourselves to the one-dimensional case and use a finite number of shape functions satisfying some regularity conditions. For more general shape functions approximation techniques are presented. Our algorithm is applied to the Smith process and the Brown-Resnick process. Finally, we compare our computational results to other approaches. Here, the algorithm for Gaussian processes with transformed marginals turns out to be surprisingly competitive.Comment: 35 pages; version accepted for publication in Extremes. The final publication is available at http://link.springer.co

Level sets estimation and Vorob'ev expectation of random compact sets

The issue of a "mean shape" of a random set $X$ often arises, in particular in image analysis and pattern detection. There is no canonical definition but one possible approach is the so-called Vorob'ev expectation \E_V(X), which is closely linked to quantile sets. In this paper, we propose a consistent and ready to use estimator of \E_V(X) built from independent copies of $X$ with spatial discretization. The control of discretization errors is handled with a mild regularity assumption on the boundary of $X$: a not too large 'box counting' dimension. Some examples are developed and an application to cosmological data is presented

Modelling aggregation on the large scale and regularity on the small scale in spatial point pattern datasets

We consider a dependent thinning of a regular point process with the aim of obtaining aggregation on the large scale and regularity on the small scale in the resulting target point process of retained points. Various parametric models for the underlying processes are suggested and the properties of the target point process are studied. Simulation and inference procedures are discussed when a realization of the target point process is observed, depending on whether the thinned points are observed or not. The paper extends previous work by Dietrich Stoyan on interrupted point processes

Attribute Controlled Reconstruction and Adaptive Mathematical Morphology

ISBN : 978-3-642-38293-2International audienceIn this paper we present a reconstruction method controlled by the evolution of attributes. The process begins from a marker, propagated over increasing quasi-flat zones. The evolution of several increasing and non-increasing attributes is studied in order to select the appropriate region. Additionally, the combination of attributes can be used in a straightforward way. To demonstrate the performance of our method, three applications are presented. Firstly, our method successfully segments connected objects in range images. Secondly, input-adaptive structuring elements (SE) are defined computing the controlled propagation for each pixel on a pilot image. Finally, input-adaptive SE are used to assess shape features on the image. Our approach is multi-scale and auto-dual. Compared with other methods, it is based on a given attribute but does not require a size parameter in order to determine appropriate regions. It is useful to extract objects of a given shape. Additionally, our reconstruction is a connected operator since quasi-flat zones do not create new contours on the image

High-resolution truncated plurigaussian simulations for the characterization of heterogeneous formations

Integrating geological concepts, such as relative positions and proportions of the different lithofacies, is of highest importance in order to render realistic geological patterns. The truncated plurigaussian simulation method provides a way of using both local and conceptual geological information to infer the distributions of the facies and then those of hydraulic parameters. The method (Le Loc'h and Galli 1994) is based on the idea of truncating at least two underlying multi-Gaussian simulations in order to create maps of categorical variable. In this manuscript we show how this technique can be used to assess contaminant migration in highly heterogeneous media. We illustrate its application on the biggest contaminated site of Switzerland. It consists of a contaminant plume located in the lower fresh water Molasse on the western Swiss Plateau. The highly heterogeneous character of this formation calls for efficient stochastic methods in order to characterize transport processes.Comment: 12 pages, 9 figure