Regular grid

A regular grid is a tessellation of n-dimensional Euclidean space by congruent parallelotopes (e.g. bricks). Its opposite is irregular grid. Grids of this type appear on graph paper and may be used in finite element analysis, finite volume methods, finite difference methods, and in general for discretization of parameter spaces. Since the derivatives of field variables can be conveniently expressed as finite differences, structured grids mainly appear in finite difference methods. Unstructured grids offer more flexibility than structured grids and hence are very useful in finite element and finite volume methods. Each cell in the grid can be addressed by index (i, j) in two dimensions or (i, j, k) in three dimensions, and each vertex has coordinates (i\cdot dx, j\cdot dy) in 2D or (i\cdot dx, j\cdot dy, k\cdot dz) in 3D for some real numbers dx, dy, and dz representing the grid spacing. Related grids A Cartesian grid is a special case where the e

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Geometric, variational discretization of continuum theories

Mathieu Desbrun

This study derives geometric, variational discretization of continuum theories arising in fluid dynamics, magnetohydrodynamics (MHD), and the dynamics of complex fluids. A central role in these discretizations is played by the geometric formulation of fluid dynamics, which views solutions to the governing equations for perfect fluid flow as geodesics on the group of volume-preserving diffeomorphisms of the fluid domain. Inspired by this framework, we construct a finite-dimensional approximation to the diffeomorphism group and its Lie algebra, thereby permitting a variational temporal discretization of geodesics on the spatially discretized diffeomorphism group. The extension to MHD and complex fluid flow is then made through an appeal to the theory of Euler-Poincare systems with advection, which provides a generalization of the variational formulation of ideal fluid flow to fluids with one or more advected parameters. Upon deriving a family of structured integrators for these systems, we test their performance via a numerical implementation of the update schemes on a cartesian grid. Among the hallmarks of these new numerical methods are exact preservation of momenta arising from symmetries, automatic satisfaction of solenoidal constraints on vector fields, good long-term energy behavior, robustness with respect to the spatial and temporal resolution of the discretization, and applicability to irregular meshes. (C) 2011 Elsevier B.V. All rights reserved.

2011

Automated Assessment of Digital Terrain Models Derived From Airborne Laser Scanning

Philipp Schär, Jan Skaloud

This paper presents the derivation of surface-related quality indicators describing the confidence-metrics of the final geo-products de- rived from airborne laser scanning (ALS). We first discuss the number of factors influencing the qual- ity of the digital terrain models (DTM) and review the rigorous derivation of quality metric per each laser target when considering all the elements of direct-georeferencing as well as the scanning ge- ometry. As in the context of DTM creation, how- ever, the laser measurements are rarely used as sin- gle values; we extend this approach by considering other factors as classification, sampling density and interpolation. Further, we propose a novel proce- dure that enables an automated generation of a DTM quality map encapsulating all these factors assuming that the following conditions are ful- filled: i) the accuracy of each ground point involved in DTM generation is known or derived; ii) the DTM is represented as a regular grid where the el- evation values are calculated by projecting the grid-cell centre coordinates on the corresponding facet of the TIN-model whose nodes are the irregu- larly sampled laser-points reflected from the ground. The derived DTM-quality map is thus in- fluenced by the choice of grid resolution with re- spect to the actual density of the laser point-cloud, as well as the accuracy of individual laser returns. Finally, we present an example that demonstrates surface-quality map computed for an ALS point- cloud where the distribution of automatically clas- sified ground points is very disparate and contains important gaps due to dense vegetation or insuffi- cient surface-reflectance. We conclude with sug- gestions on possible applications of such quality- maps that can be associated as metadata to the DTM.

2012

ME-372: Finite element method

L'étudiant acquiert une initiation théorique à la méthode des éléments finis qui constitue la technique la plus courante pour la résolution de problèmes elliptiques en mécanique. Il apprend à appliquer cette méthode à des cas simples et à l'exploiter pour résoudre les problèmes de la pratique.

ENV-140: Fundamentals of geomatics

Bases de la géomatique pour les ingénieurs civil et en environnement. Présentation des méthodes d'acquisition, de gestion et de représentation des géodonnées. Apprentissage pratique avec des méthodes topométriques et d'imagerie du territoire.

COM-500: Statistical signal and data processing through applications

Building up on the basic concepts of sampling, filtering and Fourier transforms, we address stochastic modeling, spectral analysis, estimation and prediction, classification, and adaptive filtering, with an application oriented approach and hands-on numerical exercises.