Concept# Théorème de la divergence

Résumé

En analyse vectorielle, le théorème de la divergence (également appelé théorème de Green-Ostrogradski ou théorème de flux-divergence), affirme l'égalité entre l'intégrale de la divergence d'un champ vectoriel sur un volume dans \R^3 et le flux de ce champ à travers la frontière du volume (qui est une intégrale de surface).
L'égalité est la suivante :
\int!!!!!\int!!!!!\int_{\mathcal{V}} \overrightarrow\nabla \cdot\overrightarrow F , {\rm d}V =
\int!!!!!!!\subset!!!\supset!!!!!!!\int_{\partial \mathcal{V}} \overrightarrow F \cdot \mathrm d \overrightarrow{S}
où :

- \mathcal{V}, est le volume ;
- \partial\mathcal{V}, est la frontière de \mathcal{V}
- {\rm d} \overrightarrow S est le vecteur normal à la surface, dirigé vers l'extérieur et de norme égale à l'élément de surface qu'il représente *\overrightarrow F est continûment dérivable en tout point de \mat

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Water vapor is a fundamental constituent of the atmosphere and is the most abundant green house gas thus having an important influence on climate. It is as well a key prognostic variable for numerical weather prediction models (NWP). Currently, the vertical profiles of tropospheric water vapor are provided by twice a day radiosondes. The routine observations have rather low temporal resolution that is insufficient to resolve fast-running meteorological phenomena. The aim of the thesis work was to design and build a Raman lidar instrument capable of continuous vertical profiling of tropospheric water vapor field with high temporal and vertical resolution. The provided observations will improve the database available for direct meteorological applications and could increase the accuracy of numerical weather prediction. RALMO – RAman Lidar for Meteorological Observations is developed as fully automated, eye-safe instrument for operational use by the Swiss meteorological service – MeteoSwiss. The lidar is able to provide vertical profiles of water vapor mixing ratio with time resolution from 5 to 30 min and vertical resolution from 15 m in boundary layer and 75 – 500 m in free troposphere. The daytime vertical operational range of the lidar extends from about 50 m to mid-troposphere and the detection limit is 0.5 g/kg. In night-time conditions the vertical operational range extends up to the tropopause with 0.01 g/kg detection limit. To allow daytime operation with extended vertical range and required detection limit the lidar is designed with narrow field-of-view receiver, narrow band detection, and it uses high pulse power laser with wavelength in the UV but out of solar blind region. The lidar transmitter uses flash-lamps pumped frequency tripled Nd: YAG laser generating 8 ns pulses with 0.3 J energy and 355 nm wavelength. To reduce the beam irradiance, required for eye-safe operation, and to reduce the divergence required for narrow field of view receiver, the laser beam is expanded by 15x refractive type expander. The backscattered laser light is collected by four 30 cm in diameter telescopes with focal length of 1 m. For better long term alignment stability the telescopes are tightly arranged around the beam expander in compact assembly. The field-of-view of the telescopes is reduced to 0.2 mrad to decrease the collected sky-scattered sunlight thus allowing daytime measurements up to mid troposphere. Fibers transmit the light collected by the telescopes to the lidar polychromator. Fiber coupling was preferred against free space connection because it separates the units mechanically and increases the overall lidar stability. In addition the fibers perform aperture scrambling which prevents range dependence of the receiver parameters. An additional "near range" fiber is installed in one of the telescopes to enhance the near range signal and to allow daytime measurements starting from approximately 50 m above the lidar. A high throughput diffraction grating polychromator is designed for narrow band isolation of water vapor, nitrogen, and oxygen Q branches of ro-vibrational Raman spectra. Water vapor mixing ratio is derived from the ratio of water vapor to nitrogen Raman signals and the oxygen signal is used to correct for aerosol differential extinction at water and nitrogen wavelengths. The water vapor detection channel passband is 0.3 nm. The narrow band detection increases the lidar sensitivity and operational range in daytime conditions. For maximum throughput of the polychromator, the exit and entrance slits are matched and the polychromator entrance accepts the divergent beam from the fibers without losses. Photomultipliers at the exit of the polychromator detect the Raman lidar signals, which are then acquired by transient recorders (Licel). The signals are simultaneously recorded in analog and photon-counting modes. The analog signals are used in daytime conditions with sky background signal that saturates completely the photon counter, whereas in night-time conditions, photon counting signals are used. Two computers provide for the automated operation of the lidar instrument. The first one controls all lidar units relevant to the lidar operation, including the laser and the data acquisition. For this purpose, Lidar Automat software has been developed under LabView and requires only activation by an operator. Automated data treatment software, developed under Matlab, is run on the second lidar computer. It reads the initial lidar data, treats them, and stores the final result in files ready for upload to the meteorological service. The files contain vertical profiles of water vapor mixing ratio and relative error. The lidar was completed in July 2007 and installed at the aerological station of MeteoSwiss in Payerne. Since October 2007 the lidar has been in experimental operation and the software for automated data treatment was completed. Since September 2008 the instrument has been fully operational, providing continuous vertical water vapor mixing ratio profiles uploaded to MeteoSwiss every 30 min. Regular comparisons with Vaisala RS-92 and Snow White® radiosondes were performed during the experimental lidar operation at Payerne. Long-term stability study of the calibration coefficient was performed as well.

Electroencephalography (EEG) is a key modality to monitor brain activity with high temporal resolution. EEG makes use of an array of electrodes to measure the electrical potential on the scalp. While most traditional EEG analyses have looked at EEG rhythms in different frequency bands, another important application of EEG is source imaging; i.e., map back the measured scalp potential to the underlying source distribution. However, source localization is an ill-posed problem. Most approaches consider a grid of neurobiologically relevant dipoles with fixed positions and directions of their moments, which makes for many more unknown intensities than the number of measures. To make the solution unique, one needs to add regularization strategies such as smoothness or sparsity. The alternative, which is the approach that we will follow in this work, is to impose a sparse and parametric source model with few unknown parameters, but which include both dipole positions and moments thus rendering the problem solution highly non-linear. The proposed method, for which we coin the term "analytic sensing", is based on two main working principles. First, based on the divergence theorem, the sensing principle relates the boundary potential to a volumetric information about the sources. This principle makes use of a mathematical test function ("analytic sensor") that needs to be a homogeneous solution of the Poisson equation, which is the governing equation of the quasi-static electromagnetic setting. The analytic sensor can be defined for different geometries and conductivity profiles of the domain of interest. We derive closed-form expressions for 3D multi-layer spherical models that are often used as a head model in EEG. Recent advances in signal processing known as "annihilation filter" or "finite rate of innovation", have extended Prony's method to recover sparse sources in a robust way, typically a stream of Diracs. We propose to apply the annihilation principle by imposing a particular choice of multiple analytic sensors; i.e., the virtual measurements obtained with these functions can be annihilated by a filter that allows us to recover the dipoles' positions in a non-iterative way. The dipoles' moments can subsequently be retrieved by solving a linear system of equations. While the application of the sensing principle is intrinsically 2D or 3D, the annihilation principle gives only access to the orthographic projection of the source distribution. Two approaches have been developed. First, using coordinate transformations, one can obtain multiple projections of the sources and recombine them to reconstruct the full 3D information. Second, using a second set of analytic sensors, it is possible to retrieve the missing coordinate by solving another linear system of equations. Successful application of the method requires careful implementation of both principles. For the sensing principle, we project the boundary measurements on a set of spherical harmonics with proper regularization to cover parts of the boundary that are not measured. We also show how the annihilation step can be implemented to be robust to noise and numerically stable. We demonstrate the precision and robustness of the method by both experimental results and theoretical Cramèr-Rao bounds. Finally, we show effective source localization for real-world experimental EEG data; i.e., we identify the underlying sources for several time instants of a visual evoked potential.

Pablo Antolin Sanchez, Thibaut Hirschler

This paper presents a novel method for solving partial differential equations on three-dimensional CAD geometries by means of immersed isogeometric discretizations that do not require quadrature schemes. It relies on a newly developed technique for the evaluation of polynomial integrals over spline boundary representations that is exclusively based on analytical computations. First, through a consistent polynomial approximation step, the finite element operators of the Galerkin method are transformed into integrals involving only polynomial integrands. Then, by successive applications of the divergence theorem, those integrals over B-Reps are transformed into the first surface and then line integrals with polynomials integrands. Eventually, these line integrals are evaluated analytically with machine precision accuracy. The performance of the proposed method is demonstrated by means of numerical experiments in the context of 2D and 3D elliptic problems, retrieving optimal error convergence order in all cases. Finally, the methodology is illustrated for 3D CAD models with an industrial level of complexity.