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Numerical solution to the oblique derivative boundary value problem on non-uniform grids above the Earth topography / Matej Medl’a in Journal of geodesy, vol 92 n° 1 (January 2018)  

Public [article]  inJournal of geodesy > vol 92 n° 1 (January 2018) . - pp 1 - 19 Titre : Numerical solution to the oblique derivative boundary value problem on non-uniform grids above the Earth topography Type de document : Article/Communication Auteurs : Matej Medl’a, Auteur ; Karol Mikula, Auteur ; Robert Cunderlik, Auteur ; Marek Macák, Auteur Année de publication : 2018 Article en page(s) : pp 1 - 19 Note générale : Bibliographie Langues : Anglais (eng) Descripteur : [Vedettes matières IGN] Géodésie physique [Termes IGN] champ de pesanteur local [Termes IGN] discrétisation [Termes IGN] équation de Laplace [Termes IGN] méthode des éléments finis [Termes IGN] problème des valeurs limites Résumé : (Auteur) The paper presents a numerical solution of the oblique derivative boundary value problem on and above the Earth’s topography using the finite volume method (FVM). It introduces a novel method for constructing non-uniform hexahedron 3D grids above the Earth’s surface. It is based on an evolution of a surface, which approximates the Earth’s topography, by mean curvature. To obtain optimal shapes of non-uniform 3D grid, the proposed evolution is accompanied by a tangential redistribution of grid nodes. Afterwards, the Laplace equation is discretized using FVM developed for such a non-uniform grid. The oblique derivative boundary condition is treated as a stationary advection equation, and we derive a new upwind type discretization suitable for non-uniform 3D grids. The discretization of the Laplace equation together with the discretization of the oblique derivative boundary condition leads to a linear system of equations. The solution of this system gives the disturbing potential in the whole computational domain including the Earth’s surface. Numerical experiments aim to show properties and demonstrate efficiency of the developed FVM approach. The first experiments study an experimental order of convergence of the method. Then, a reconstruction of the harmonic function on the Earth’s topography, which is generated from the EGM2008 or EIGEN-6C4 global geopotential model, is presented. The obtained FVM solutions show that refining of the computational grid leads to more precise results. The last experiment deals with local gravity field modelling in Slovakia using terrestrial gravity data. The GNSS-levelling test shows accuracy of the obtained local quasigeoid model. Numéro de notice : A2018-011 Affiliation des auteurs : non IGN Thématique : POSITIONNEMENT Nature : Article nature-HAL : ArtAvecCL-RevueIntern DOI : 10.1007/s00190-017-1040-z Date de publication en ligne : 30/05/2017 En ligne : https://doi.org/10.1007/s00190-017-1040-z Format de la ressource électronique : URL Article Permalink : https://documentation.ensg.eu/index.php?lvl=notice_display&id=89054 [article]

Finite element method for solving geodetic boundary value problems / Z. Faskova in Journal of geodesy, vol 84 n° 2 (February 2010)  

Public [article]  inJournal of geodesy > vol 84 n° 2 (February 2010) . - pp 135 - 144 Titre : Finite element method for solving geodetic boundary value problems Type de document : Article/Communication Auteurs : Z. Faskova, Auteur ; Robert Cunderlik, Auteur ; Karol Mikula, Auteur Année de publication : 2010 Article en page(s) : pp 135 - 144 Note générale : Bibliographie Langues : Anglais (eng) Descripteur : [Vedettes matières IGN] Géodésie physique [Termes IGN] champ de pesanteur terrestre [Termes IGN] Earth Gravity Model 2008 [Termes IGN] géoïde [Termes IGN] méthode des éléments finis [Termes IGN] problème de Dirichlet [Termes IGN] problème des valeurs limites Résumé : (Auteur) The goal of this paper is to present the finite element scheme for solving the Earth potential problems in 3D domains above the Earth surface. To that goal we formulate the boundary-value problem (BVP) consisting of the Laplace equation outside the Earth accompanied by the Neumann as well as the Dirichlet boundary conditions (BC). The 3D computational domain consists of the bottom boundary in the form of a spherical approximation or real triangulation of the Earth’s surface on which surface gravity disturbances are given. We introduce additional upper (spherical) and side (planar and conical) boundaries where the Dirichlet BC is given. Solution of such elliptic BVP is understood in a weak sense, it always exists and is unique and can be efficiently found by the finite element method (FEM). We briefly present derivation of FEM for such type of problems including main discretization ideas. This method leads to a solution of the sparse symmetric linear systems which give the Earth’s potential solution in every discrete node of the 3D computational domain. In this point our method differs from other numerical approaches, e.g. boundary element method (BEM) where the potential is sought on a hypersurface only. We apply and test FEM in various situations. First, we compare the FEM solution with the known exact solution in case of homogeneous sphere. Then, we solve the geodetic BVP in continental scale using the DNSC08 data. We compare the results with the EGM2008 geopotential model. Finally, we study the precision of our solution by the GPS/levelling test in Slovakia where we use terrestrial gravimetric measurements as input data. All tests show qualitative and quantitative agreement with the given solutions. Copyright Springer Numéro de notice : A2010-108 Affiliation des auteurs : non IGN Thématique : POSITIONNEMENT Nature : Article nature-HAL : ArtAvecCL-RevueIntern DOI : 10.1007/s00190-009-0349-7 Date de publication en ligne : 13/10/2009 En ligne : https://doi.org/10.1007/s00190-009-0349-7 Format de la ressource électronique : URL article Permalink : https://documentation.ensg.eu/index.php?lvl=notice_display&id=30304 [article]

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