Study of some optimal XFEM type methods

The XFEM method in fracture mechanics is revisited. A first improvement is considered using an enlarged fixed enrichment subdomain aroud the crack tip and a bonding condition for the corresponding degree of freedom. An efficient numerical integration rule is introduced for the nonsmooth enrichment functions. The lack of accuracy due to the transition layer between the enrichment aera and the rest of the domain leads to consider a pointwise matching condition at the boundary of the subdomain. An optimal numerical rate of convergence is then obtained using such a nonconformal method

Extrapolating the Fractal Characteristics of an Image Using Scale-Invariant Multiple-Point Statistics

The resolution of measurement devices can be insufficient for certain purposes. We propose to stochastically simulate spatial features at scales smaller than the measurement resolution. This is accomplished using multiple-point geostatistical simulation (direct sampling in the present case) to interpolate values at the target scale. These structures are inferred using hypothesis of scale invariance and stationarity on the spatial patterns found at the coarse scale. The proposed multiple-point super-resolution mapping method is able to deal with "both continuous and categorical variables,” and can be extended to multivariate problems. The advantages and limitations of the approach are illustrated with examples from satellite imagin

Parallel Multiple-Point Statistics Algorithm Based on List and Tree Structures

Multiple-point statistics are widely used for the simulation of categorical variables because the method allows for integrating a conceptual model via a training image and then simulating complex heterogeneous fields. The multiple-point statistics inferred from the training image can be stored in several ways. The tree structure used in classical implementations has the advantage of being efficient in terms of CPU time, but is very RAM demanding and then implies limitations on the size of the template, which serves to make a proper reproduction of complex structures difficult. Another technique consists in storing the multiple-point statistics in lists. This alternative requires much less memory and allows for a straightforward parallel algorithm. Nevertheless, the list structure does not benefit from the shortcuts given by the branches of the tree for retrieving the multiple-point statistics. Hence, a serial algorithm based on list structure is generally slower than a tree-based algorithm. In this paper, a new approach using both list and tree structures is proposed. The idea is to index the lists by trees of reduced size: the leaves of the tree correspond to distinct sublists that constitute a partition of the entire list. The size of the indexing tree can be controlled, and then the resulting algorithm keeps memory requirements low while efficiency in terms of CPU time is significantly improved. Moreover, this new method benefits from the parallelization of the list approac

Hybrid discretization of the Signorini problem with Coulomb friction. Theoretical aspects and comparison of some numerical solvers

International audienceThe purpose of this work is to present in a general framework the hybrid discretization of unilateral contact and friction conditions in elastostatics. A projection formulation is developed and used. An existence and uniqueness results for the solutions to the discretized problem is given in the general framework. Several numerical methods to solve the discretized problem are presented (Newton, SOR, fixed points, Uzawa) and compared in terms of the number of iterations and the robustness with respect to the value of the friction coefficient

Conditioning Facies Simulations with Connectivity Data

When characterizing and simulating underground reservoirs for flow simulations, one of the key characteristics that needs to be reproduced accurately is its connectivity. More precisely, field observations frequently allow the identification of specific points in space that are connected. For example, in hydrogeology, tracer tests are frequently conducted that show which springs are connected to which sink-hole. Similarly well tests often allow connectivity information in a petroleum reservoir to be provided. To account for this type of information, we propose a new algorithm to condition stochastic simulations of lithofacies to connectivity information. The algorithm is based on the multiple-point philosophy but does not imply necessarily the use of multiple-point simulation. However, the challenge lies in generating realizations, for example of a binary medium, such that the connectivity information is honored as well as any prior structural information (e.g. as modeled through a training image). The algorithm consists of using a training image to build a set of replicates of connected paths that are consistent with the prior model. This is done by scanning the training image to find point locations that satisfy the constraints. Any path (a string of connected cells) between these points is therefore consistent with the prior model. For each simulation, one sample from this set of connected paths is sampled to generate hard conditioning data prior to running the simulation algorithm. The paper presents in detail the algorithm and some examples of two-dimensional and three-dimensional applications with multiple-point simulation

High order extended finite element method for cracked domains

The aim of the paper is to study the capabilities of the Extended Finite Element Method (XFEM) to achieve accurate computations in non smooth situations such as crack problems. Although the XFEM method ensures a weaker error than classical finite element methods, the rate of convergence is not improved when the mesh parameter h is going to zero because of the presence of a singularity. The difficulty can be overcome by modifying the enrichment of the finite element basis with the asymptotic crack tip displacement solutions as well as with the Heaviside function. Numerical simulations show that the modified XFEM method achieves an optimal rate of convergence (i.e. like in a standard finite element method for a smooth problem