Conformal Mapping on Rough Boundaries I: Applications to harmonic problems

The aim of this study is to analyze the properties of harmonic fields in the vicinity of rough boundaries where either a constant potential or a zero flux is imposed, while a constant field is prescribed at an infinite distance from this boundary. We introduce a conformal mapping technique that is tailored to this problem in two dimensions. An efficient algorithm is introduced to compute the conformal map for arbitrarily chosen boundaries. Harmonic fields can then simply be read from the conformal map. We discuss applications to "equivalent" smooth interfaces. We study the correlations between the topography and the field at the surface. Finally we apply the conformal map to the computation of inhomogeneous harmonic fields such as the derivation of Green function for localized flux on the surface of a rough boundary

Signature Sequence of Intersection Curve of Two Quadrics for Exact Morphological Classification

We present an efficient method for classifying the morphology of the intersection curve of two quadrics (QSIC) in PR3, 3D real projective space; here, the term morphology is used in a broad sense to mean the shape, topological, and algebraic properties of a QSIC, including singularity, reducibility, the number of connected components, and the degree of each irreducible component, etc. There are in total 35 different QSIC morphologies with non-degenerate quadric pencils. For each of these 35 QSIC morphologies, through a detailed study of the eigenvalue curve and the index function jump we establish a characterizing algebraic condition expressed in terms of the Segre characteristics and the signature sequence of a quadric pencil. We show how to compute a signature sequence with rational arithmetic so as to determine the morphology of the intersection curve of any two given quadrics. Two immediate applications of our results are the robust topological classification of QSIC in computing B-rep surface representation in solid modeling and the derivation of algebraic conditions for collision detection of quadric primitives

On Weingarten transformations of hyperbolic nets

Weingarten transformations which, by definition, preserve the asymptotic lines on smooth surfaces have been studied extensively in classical differential geometry and also play an important role in connection with the modern geometric theory of integrable systems. Their natural discrete analogues have been investigated in great detail in the area of (integrable) discrete differential geometry and can be traced back at least to the early 1950s. Here, we propose a canonical analogue of (discrete) Weingarten transformations for hyperbolic nets, that is, C^1-surfaces which constitute hybrids of smooth and discrete surfaces "parametrized" in terms of asymptotic coordinates. We prove the existence of Weingarten pairs and analyse their geometric and algebraic properties.Comment: 41 pages, 30 figure

Fast directional continuous spherical wavelet transform algorithms

We describe the construction of a spherical wavelet analysis through the inverse stereographic projection of the Euclidean planar wavelet framework, introduced originally by Antoine and Vandergheynst and developed further by Wiaux et al. Fast algorithms for performing the directional continuous wavelet analysis on the unit sphere are presented. The fast directional algorithm, based on the fast spherical convolution algorithm developed by Wandelt and Gorski, provides a saving of O(sqrt(Npix)) over a direct quadrature implementation for Npix pixels on the sphere, and allows one to perform a directional spherical wavelet analysis of a 10^6 pixel map on a personal computer.Comment: 10 pages, 3 figures, replaced to match version accepted by IEEE Trans. Sig. Pro

Morphological analysis of stylolites for paleostress estimation in limestones surrounding the Andra Underground Research Laboratory site

We develop and test a methodology to infer paleostress from the morphology of stylolites within borehole cores. This non-destructive method is based on the analysis of the stylolite trace along the outer cylindrical surface of the cores. It relies on an automatic digitization of high-resolution photographs and on the spatial Fourier spectrum analysis of the stylolite traces. We test and show, on both synthetic and natural examples, that the information from this outer cylindrical surface is equivalent to the one obtained from the destructive planar sections traditionally used. The assessment of paleostress from the stylolite morphology analysis is made using a recent theoretical model, which links the morphological properties to the physical processes acting during stylolite evolution. This model shows that two scaling regimes are to be expected for the stylolite height power spectrum, separated by a cross-over length that depends on the magnitude of the paleostress during formation. We develop a non linear fit method to automatically extract the cross-over lengths from the digitized stylolite profiles. Results on cores from boreholes drilled in the surroundings of the Andra Underground Research Laboratory located at Bure, France, show that different groups of sedimentary stylolites can be distinguished, and correspond to different estimated vertical paleostress values. For the Oxfordian formation, one group of stylolites indicate a paleostress of around 10 MPa, while another group yields 15 MPa. For the Dogger formation, two stylolites indicate a paleostress of around 10 MPa, while others appear to have stopped growing at paleostresses between 30 and 22 MPa, starting at an erosion phase that initiated in the late Cretaceous and continues today. This method has a high potential for further applications on reservoirs or other geological contexts where stylolites are present.Comment: International Journal of Rock Mechanics and Mining Sciences (2013) online firs