Removal and Contraction Operations in nnD Generalized Maps for Efficient Homology Computation

In this paper, we show that contraction operations preserve the homology of $n$D generalized maps, under some conditions. Removal and contraction operations are used to propose an efficient algorithm that compute homology generators of $n$D generalized maps. Its principle consists in simplifying a generalized map as much as possible by using removal and contraction operations. We obtain a generalized map having the same homology than the initial one, while the number of cells decreased significantly. Keywords: $n$D Generalized Maps; Cellular Homology; Homology Generators; Contraction and Removal Operations.Comment: Research repor

Computing Homology Generators for Volumes Using Minimal Generalized Maps

International audienceIn this paper, we present an algorithm for computing efficiently homology generators of 3D subdivided orientable objects which can contain tunnels and cavities. Starting with an initial subdivision, represented with a generalized map where every cell is a topological ball, the number of cells is reduced using simplification operations (removal of cells), while preserving homology. We obtain a minimal representation which is homologous to the initial object. A set of homology generators is then directly deduced on the simplified 3D object

Conversion between chains of maps and chains of surfaces; application to the computation of incidence graphs homology

Many combinatorial cellular structures have been defined in order to represent the topology of subdivided geometric objects. Two main classes can be distinguished. According to the terminology of [8], one is related to incidence graphs and the other to ordered models. Both classes have their own specificities and their use is relevant in different contexts. It is thus important to create bridges between them. So we define here chains of surfaces (a subclass of incidence graphs) and chains of maps without multi-incidence (a subclass of ordered models), which are able to represent the topology of subdivided objects, whose cells have " manifold-like " properties. We show their equivalence by providing conversion operations. As a consequence, it is hence possible to directly apply on each model results obtained on the other. We extend here classical results related to homology computation obtained for incidence graphs corresponding to regular CW −complexes and recent results about combinatorial cell complexes where cells are not necessarily homeomorphic to balls

A Boundary Operator for Computing the Homology of Cellular Structures

71 pagesThe paper focuses on homology computation over cellular structures through the computation of incidence numbers. Roughly speaking, if two cells are incident, then their incidence number characterizes how they are attached. Having these numbers naturally leads to the definition of a boundary operator, which induces a cellular homology. More precisely, the two main families of cellular structures (incidence graphs and ordered models) are studied through various models. A boundary operator is then proposed for the most general structure, and is optimized for the other structures. It is proved that, under specific conditions, the cellular boundary operator proposed in this paper defines a cellular homology equivalent to the simplicial one

Incremental Computation of the Homology of Generalized Maps: An Application of Effective Homology Results

This paper deals with the incremental computation of the homology of " cellular " combinatorial structures, namely combinatorial maps and incidence graphs. " Incremental " is related to the operations which are applied to construct such structures: basic operations, i.e. the creation of cells and the identification of cells, are considered in the paper. Such incremental computation is done by applying results of effective homology [RS06]: a correspondence between the chain complex associated with a given combinatorial structure is maintained with a " smaller " chain complex , from which the homology groups and homology generators can be more efficiently computed

Homology of Cellular Structures Allowing Multi-incidence

International audienceThis paper focuses on homology computation over ‘cellular’ structures that allow multi-incidence between cells. We deal here with combinatorial maps, more precisely chains of maps and subclasses such as maps and generalized maps. Homology computation on such structures is usually achieved by computing simplicial homology on a simplicial analog. But such an approach is computationally expensive because it requires computing this simplicial analog and performing the homology computation on a structure containing many more cells (simplices) than the initial one. Our work aims at providing a way to compute homologies directly on a cellular structure. This is done through the computation of incidence numbers. Roughly speaking, if two cells are incident, then their incidence number characterizes how they are attached. Having these numbers naturally leads to the definition of a boundary operator, which induces a homology. Hence, we propose a boundary operator for chains of maps and provide optimization for maps and generalized maps. It is proved that, under specific conditions, the homology of a combinatorial map as defined in the paper is equivalent to the homology of its simplicial analogue

Extracting and Unfolding a Stratigraphic Unit to Update Property Population

International audienceWith the wide usage of geo-modelling tools, users could have the need to enhance their previous geostatiscal population without rebuilding an entire stratigraphic model In this paper we explain how we can extract non explicit information from a stratigraphic model (reference iso-chronological surfaces, faults used to constraint the model), and then, use this information to realise 3D flattening on iso-chronological surfaces prior to geostatiscal population. Three methods were presented here: traditional by topological correspondence, vertical shear and an original isometric unfolding process based on the minimization of the elastic tensor deformation. These methods could be applied for every type of deposit: Horizontal, Parallel to Top, parallel to Bottom, Proportional. Then, we compare the application of these methods on several case studies and develop the advantages to reengineer a stratigraphic model and repopulate it after flattening. Even if the "traditional" and vertical shear methods could be applied on certain situations, following multiple test bed as the ones presented in this paper, we are thinking that the isometric unfolding presented here is much reliable. As a consequence, we will exploit more and more this isometric unfolding method in the next future and process each lithostratigraphic unit independently than the others

Architectural effect on 3D elastic properties and anisotropy of cubic lattice structures

This article investigates the elastic properties of a large panel of lattice architectures using a continuous description of geometry. The elastic constants of the orthotropic material are determined, and discussed in terms of specific stiffness and of its density dependence. Different kind of topology families are emerging depending on their specific deformation behavior. For some of them, interesting properties in term of traction-compression were measured, while some other families are predominantly adapted to shear loading. Homogenization technique also allows to quantify the anisotropy of the structures and to compare them. Specific structures having quasi-isotropic properties even at low relative densities were detected. Experimental works demonstrated the validity of the numerical models, and highlighted the necessity to consider carefully the effect of defects on the specific strength, which are of the second-order however not negligible. Finally, this article provides user-friendly maps for selection of optimal architectures for a large variety of specific needs, like a target stiffness or anisotropy

The Sumatra subduction zone: A case for a locked fault zone extending into the mantle

A current view is that the portion of the subduction interface that remains locked in the time interval between large interplate earthquakes, hereinafter referred to as the locked fault zone (LFZ), does not extend into the mantle because serpentinization of the mantle wedge would favor stable aseismic sliding. Here, we test this view in the case of the Sumatra subduction zone where the downdip end of the LFZ can be well constrained from the pattern and rate of uplift deduced from coral growth and from GPS measurements of horizontal deformation. These geodetic data are modeled from a creeping dislocation embedded in an elastic half-space and indicate that the LFZ extends 132 ± 10/7 km from the trench, to a depth between 35 and 57 km. By combining this information with the geometry of the plate interface as constrained from two-dimensional gravimetric modeling and seismicity, we show that the LFZ extends below the forearc Moho, which is estimated to lie at a depth of ~30 km, at a horizontal distance of 110 km from the trench. So, in this particular island arc setting, the LFZ most probably extends into the mantle, implying that either the mantle is not serpentinized, or that the presence of serpentine does not necessarily imply stable sliding. From thermal modeling, the temperature at the downdip end of the LFZ is estimated to be 260 ± 100°C. This temperature seems too low for thermally activated ductile flow, so that aseismic slip is most probably due to pressure and/or temperature induced steady state brittle sliding, possibly favored by fluids released from the subducting slab

Persistent elastic behavior above a megathrust rupture patch: Nias island, West Sumatra

We quantify fore-arc deformation using fossil reefs to test the assumption commonly made in seismic cycle models that anelastic deformation of the fore arc is negligible. Elevated coral microatolls, paleoreef flats, and chenier plains show that the Sumatran outer arc island of Nias has experienced a complex pattern of relatively slow long-term uplift and subsidence during the Holocene epoch. This same island rose up to 2.9 m during the Mw 8.7 Sunda megathrust rupture in 2005. The mismatch between the 2005 and Holocene uplift patterns, along with the overall low rates of Holocene deformation, reflects the dominance of elastic strain accumulation and release along this section of the Sunda outer arc high and the relatively subordinate role of upper plate deformation in accommodating long-term plate convergence. The fraction of 2005 uplift that will be retained permanently is generally <4% for sites that experienced more than 0.25 m of coseismic uplift. Average uplift rates since the mid-Holocene range from 1.5 to −0.2 mm/a and are highest on the eastern coast of Nias, where coseismic uplift was nearly zero in 2005. The pattern of long-term uplift and subsidence is consistent with slow deformation of Nias along closely spaced folds in the north and trenchward dipping back thrusts in the southeast. Low Holocene tectonic uplift rates provide for excellent geomorphic and stratigraphic preservation of the mid-Holocene relative sea level high, which was under way by ∼7.3 ka and persisted until ∼2 ka