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Lösung des fixen geodätischen Randwertproblems mit Hilfe der Randelementmethode / R. Klees (1992)

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Titre :	Lösung des fixen geodätischen Randwertproblems mit Hilfe der Randelementmethode
Titre original :	[Solution du problème géodésique de valeurs aux limites à l'aide de la méthode des éléments résiduels]
Type de document :	Thèse/HDR
Auteurs :	R. Klees, Auteur
Editeur :	Munich : Bayerische Akademie der Wissenschaften
Année de publication :	1992
Collection :	DGK - C
Sous-collection :	Dissertationen num. 382
Importance :	314 p.
Format :	21 x 30 cm
ISBN/ISSN/EAN :	978-3-7696-9428-4
Note générale :	Bibliographie
Langues :	Allemand (ger)
Descripteur :	[Vedettes matières IGN] Géodésie physique [Termes IGN] discrétisation [Termes IGN] équation intégrale [Termes IGN] linéarisation [Termes IGN] problème des valeurs limites
Index. décimale :	30.40 Géodésie physique
Résumé :	(Auteur)The présent study deals with thé numerical solution of thé fixed geodetic boundary value problem (bvp) using thé boundary élément method. Assuming a continuous covering of thé Earth's surface with gravity values thé mathematical formulation leads to a nonlinear outer bvp for thé Laplace-operator in three dimensional euclidean space. The theoretical part starts in section 1 with some of thé most important mathematical foundations we need such as properties of boundary surfaces embedded in R3, strongly singular surface intégrais, some aspects of thé theory of pseudodifferential operators and singular intégral operators forming thé theoretical frame for thé whole convergence analysis and finally some détails from thé theory of (bivariate) spline functions. Section 2 contains thé mathematical formulation of thé fixed geodetic bvp and gives a short overview over known geodetic literatures concerning this problem. Local existence and uniqueness assertions both in Hôlder and Sobolev spaces are given based on thé implicit function theorem of Hildebrandt and Graves. Besides, thé convergence of a perturbation expansion is shown based on an idea of Jorge (1987) leading to a constructive method for thé solution of thé nonlinear problem in Hôlder spaces. In both cases thé nonlinear problem is transformed into a séries of linear ones which can be classified as oblique bvps. Under thé assumption of their regularity thé Fredholm alternative is valid not only in thé case of Hôlder spaces but also in Sobolev spaces so that uniqueness implies existence. The uniqueness can be proved in both cases using thé generalized maximum principle of Hopf. The very restrictive requirement F e C3 in thé case of Sobolev spaces can be weakened under some suppositions so that F e G1*" is sufficient to apply thé maximum principle. Therefore thé nonlinear fixed bvp can be solved iteratively where in each step a classi-cal oblique bvp arises being thé subject of thé next sections. Both direct and indirect methods are discussed in section 4 to transform thé oblique boundary value prob-lem into an intégral équation leading to équations of différent types like strongly singular or hypersingular équations of thé second kind depending on thé methods used. It is shown that thé strongly singular intégral équation resulting from thé représentation of thé unknown disturbing potential as single layer potential is thé most appropriate one especially from thé numerical point of view. The principal symbol of this intégral operator is determined and it is proved that our intégral operator is strongly elliptic if and only if thé oblique direction is never tangential to thé Earth's surface, i.e. thé strong ellipticity is not automatically fulfilled by our intégral operator but has to be demanded explicitly. For thé transformation of thé intégral équation into a finite dimensional lirfear System différent discreti-zation methods are disussed in section 5 like allocation, Galerkin and least squares. It is shown that GalerkinBubnov discretization is thé best compromise between thé numerical effort, thé properties of thé boundary intégral operator to assure thé convergence of thé discretization procédure and thé order of convergence. For thé convergence of thé GalerkinBubnov method we need thé strong ellipticity of thé boundary intégral operator bringing on thé coerciveness in thé form of a Garding's inequality being thé starting point for thé whole convergence analysis. Section 6 contains thé définition of thé boundary élément method including thé pros and cons in relation to finite élément methods, thé définition of thé test and trial functions consisting of polynomial finite élément functions on thé boundary. Besides, thé coefficients of thé linear System for GalerkinBubnov discretization using piecewise constant test and trial functions are derived. In section 7 a nearly complète error analysis in Sobolev spaces is given, considering both discreti-zation and numerical cubature errors and giving thé state of thé art in estimating thé influence of surface approximation errors. Under thé assumption of strong ellipticity, being équivalent to thé positive definiteness of thé principal symbol of thé boundary intégral operator, it is shown that ail Galerkin methods are stable and converge. The asymptotic order of convergence is quasioptimal i.e. it has thé same order as thé best approximation of thé theoretical solution by our boundary élément trial functions. For piecewise constant functions a convergence of thé order 0(h2) for h-iO is proved if we measure thé error in thé Sobolev norm H''(F). This implies convergence of thé same order for thé potential in points away from thé boundary surface in any norm. The numerical part mainly deals with thé development of efficient methods for thé approximation and représentation of parts of thé Earth's surface and of cubature formulas for calculating thé coefficients of thé linear System and thé potential in thé Earth's outer space. It is shown in section 9 that Overhauser splines and bicubic BSplines are very suitable for thé approximation and représentation of thé boundary surface and superior to many other methods. For 1 1 km2 boundary éléments this leads to mean approximation errors of approximately 810 m where larger errors are possible depending on thé roughness of thé real topography. For thé représentation of thé surface normal field it is also shown that Overhauser splines are thé most appropriate approximation especially for finer grids. In Section 10 thé problem of efficient cubatures is considered for calculating thé coefficients of thé linear System defined by single and double intégrais over boundary éléments where both regular, weakly singular, strongly singular and quasisingular intégrais arise. Several cubature formulas for thé différent types of intégrais are considered. Extensive analytical and numerical developments show their efficiency compared with other cubature formulas used in engineering sciences. A detailed analysis shows that thé highly prob- lematic strongly singular intégrais can be calculated analytically and that thé resulting kernel has a loga-rithmic singularity over thé whole boundary of thé fini te élément. A local solution is calculated for a région of size 100-100 km2 using 10000 boundary éléments. The GalerkinBubnov discretization with piecewise constant trial functions leads to a dense unsymmetric linear System of thé order 10000-10000 with approximately 108 coefficients. In order to achieve a relative accuracy of at least 10'3 - 10'4 for each coefficient setting up thé linear System takes about 1 CPU-hour. In section 11.2 an efficient direct équation solver is developed based on a LUfactorization using a spécial Gaussian élimination well adapted to thé architecture of thé pipeline computer Siemens/Fujitsu VP 400-EX resp. S400/10. It is shown that thé time for thé I/O opérations can be reduced by minimizing thé amount of data to be transferred and by speeding up thé I/Oopérations using spécial runtime options of thé operational System and asynchronous I/O so that CPUusage and I/O opérations are done in parallel. The test compu-tations show that thé linear System mentioned before can be solved in approximately 10.5 resp. 7 minutes corresponding to a performance of 1.1 resp. 1.6 GFLOPS. In section 11.4 thé well-conditioning of thé linear System is proved leading to a very small sensitivity against errors of thé input data; thé influence of round off errors is proved to be below 10-". The local solution is based on thé use of a degree and order 360 spherical harmonie expansion of thé gravitational potential as référence potential and thé neglect of thé outer zone. As a detailed analysis shows, this leads to large errors only near by thé boundary of thé local région and a drastic decrease of thé layer density errors with increasing distance from thé boundary so that only relative différences of thé order 10'3 10'5 occur in most parts of our local région. In section 12 thé calculation of thé gravitational potential in points of thé Earth's outer space is considered. Efficient cubatures for thé numerical calculation of thé single layer potential are developed especially for points near by thé Earth's surface, and they are compared with other cubatures used in engineering sciences. The results show that such formulas can be found based on spécial parameter transformations smoothing out thé strong variation of thé integrand in points near by both thé Earth's surface and thé boundary of thé fmite élément. ' Section 13 contains some proposais for future work which has to be done in relation to thé numerical sol-ution of thé fixed geodetic bvp and thé boundary élément method.
Numéro de notice :	28089
Affiliation des auteurs :	non IGN
Thématique :	POSITIONNEMENT
Nature :	Thèse étrangère
Permalink :	https://documentation.ensg.eu/index.php?lvl=notice_display&id=63436

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Code-barres	Cote	Support	Localisation	Section	Disponibilité
28089-01	30.40	Livre	Centre de documentation	Géodésie	Disponible
28089-02	30.40	Livre	Centre de documentation	Géodésie	Disponible

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