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The study and exploration of the subsurface requires the construction of geological models. This task can be difficult, especially in complex geological settings, with various unconformities. These models are constructed from seismic or well data, which can be sparse and noisy. In this paper, we propose a new method to compute a stratigraphic function that represents geological layers in arbitrary settings. This function interpolates the data using piecewise quadratic C1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$C^1$$\end{document} Powell–Sabin splines and is regularized via a self-adaptive diffusion scheme. For the discretization, we use Powell–Sabin splines on triangular meshes. Compared to classical interpolation methods, the use of piecewise quadratic splines has two major advantages. First, they have the ability to produce surfaces of higher smoothness and regularity. Second, it is straightforward to discretize high-order smoothness energies like the squared Hessian energy. The regularization is considered as the most challenging part of any implicit modeling approach. Often, existing regularization methods produce inconsistent geological models, in particular for data with high thickness variations. To handle this kind of data, we propose a new scheme in which a diffusion term is introduced and iteratively adapted to the shapes and variations in the data while minimizing the interpolation error. [ABSTRACT FROM AUTHOR]

Abstract: A granular structure is usually modeled by a parallelepiped containing spherical balls in three dimensions or by a rectangle filled with disks in two dimensions. These grains (spherical balls or disks) are disjoint and their size correspond to a size distribution determined by experiments. In this paper, we consider the geometrical modeling and the meshing of these structures. To define the repartition of disjoint grains, we propose a new constructive algorithm based on an advancing-front approach. Often, the use of an advancing-front algorithm leads to a heterogeneity of the local density in the generated structure. In order to homogenize this density, we propose an optimization method based on local grain relocations. Furthermore, we introduce a method to transform spherical balls into polyhedral cells similar to realistic grain shapes. To generate quality meshes of granular models with, either spherical balls (disks) or polyhedral (polygonal) cells, an adaptive scheme is proposed. The mesh generation method is a combined advancing front-Delaunay approach governed by a metric field. The metric specification is based on the geometry and the proximity of grains. [Copyright &y& Elsevier]

Abstract: A granular structure is an aggregation of grains whose sizes are characterized by a given distribution. A sample of this kind of structure is often modeled by a rectangle filled with disks in two dimensions or a parallelepiped containing spherical balls in three dimensions. Therefore, the geometrical modeling of such a structure consists in determining a “dense” repartition of disjoint disks (or spherical balls) with respect to the specified size distribution. In this paper, a new sphere packing algorithm based on an advancing-front approach, commonly used in meshing methods, is proposed. Significant improvements in computing time and density have been obtained comparing to existing methods. In addition, we show that the generated circular grains can be transformed into polygonal (polyhedral) cells generally observed in reality. An example of a metallic nanostructure model is presented. To cite this article: P. Laug et al., C. R. Mecanique 336 (2008). [Copyright &y& Elsevier]