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Body-fitting Meshes for the Discontinuous Galerkin Method

Cui, Jian (2013):
Body-fitting Meshes for the Discontinuous Galerkin Method.
TU Darmstadt, [Online-Edition: http://tuprints.ulb.tu-darmstadt.de/3541],
[Ph.D. Thesis]

Abstract

Abstract

In this dissertation, a scheme capable of generating highly accurate, body fitting meshes and its application with the Discontinuous Galerkin Method is introduced. Unlike most other mesh generators, this scheme generates meshes consisting of quadrilateral or hexahedral elements exclusively. The high approximation quality is achieved by means of high order curved elements. The resulting meshes are highly suited for the application with high order methods, specifically the Discontinuous Galerkin Method, as large portions of the mesh are fully structured while matching the high order of the numerical method with a high order geometry representation at object and domain boundaries.

The mesh scheme works in two steps. It chooses an interior volume that can be represented using a fully structured Cartesian mesh and connects it to the embedded objects and domain boundaries with a so called buffer-layer in a second step. Inside the layer, high order curved elements are applied for yielding high geometric representation accuracy. The resulting meshes are ideal for the application of high order methods. In the interior part the Cartesian structure can be exploited for obtaining high efficiency of the numerical method while the accuracy potential can be realized also in the presence of curved objects and boundaries.

After introducing the mesh scheme, the Discontinuous Galerkin Method is described and applied for solving Maxwell’s equations. As a high order method it achieves exponential convergence under p-refinement. It is shown that using meshes produced by the novel scheme this property is achieved for curved domains as well. As an example, optimal convergence rates are demonstrated in a cylindrical cavity problem. In another example, the abilities of the method to produce correct spectral properties of closed resonator problems are investigated. To this end, a time-domain signal is recorded during the transient analysis. After applying the Fourier transform accurate frequency spectra are observed, which are free of spurious modes.

Item Type: Ph.D. Thesis
Erschienen: 2013
Creators: Cui, Jian
Title: Body-fitting Meshes for the Discontinuous Galerkin Method
Language: English
Abstract:

Abstract

In this dissertation, a scheme capable of generating highly accurate, body fitting meshes and its application with the Discontinuous Galerkin Method is introduced. Unlike most other mesh generators, this scheme generates meshes consisting of quadrilateral or hexahedral elements exclusively. The high approximation quality is achieved by means of high order curved elements. The resulting meshes are highly suited for the application with high order methods, specifically the Discontinuous Galerkin Method, as large portions of the mesh are fully structured while matching the high order of the numerical method with a high order geometry representation at object and domain boundaries.

The mesh scheme works in two steps. It chooses an interior volume that can be represented using a fully structured Cartesian mesh and connects it to the embedded objects and domain boundaries with a so called buffer-layer in a second step. Inside the layer, high order curved elements are applied for yielding high geometric representation accuracy. The resulting meshes are ideal for the application of high order methods. In the interior part the Cartesian structure can be exploited for obtaining high efficiency of the numerical method while the accuracy potential can be realized also in the presence of curved objects and boundaries.

After introducing the mesh scheme, the Discontinuous Galerkin Method is described and applied for solving Maxwell’s equations. As a high order method it achieves exponential convergence under p-refinement. It is shown that using meshes produced by the novel scheme this property is achieved for curved domains as well. As an example, optimal convergence rates are demonstrated in a cylindrical cavity problem. In another example, the abilities of the method to produce correct spectral properties of closed resonator problems are investigated. To this end, a time-domain signal is recorded during the transient analysis. After applying the Fourier transform accurate frequency spectra are observed, which are free of spurious modes.
Uncontrolled Keywords: Discontinuous Galerkin Method, Finite Element, Body-fitting Meshes, Curved Element, High Order Convergence.
Divisions: 18 Department of Electrical Engineering and Information Technology > Institute of Electromagnetic Field Theory (from 01.01.2019 renamed Institute for Accelerator Science and Electromagnetic Fields)
Study Areas > Study area Computational Engineering
18 Department of Electrical Engineering and Information Technology
Zentrale Einrichtungen
Study Areas
Exzellenzinitiative
Exzellenzinitiative > Graduate Schools > Graduate School of Computational Engineering (CE)
Exzellenzinitiative > Graduate Schools
Date Deposited: 04 Aug 2013 19:55
Official URL: http://tuprints.ulb.tu-darmstadt.de/3541
URN: urn:nbn:de:tuda-tuprints-35419
Referees: Weiland, Prof. Thomas and Zhang, Prof. Min
Refereed / Verteidigung / mdl. Prüfung: 17 June 2013
Alternative Abstract:
Alternative abstract Language
Kurzfassung der Arbeit In der vorliegenden Dissertation wird ein Verfahren zur Erzeugung hochpräziser, randangepasster Gitter vorgestellt und im Rahmen der Diskontinuierlichen Galerkin (DG) Methode angewendet. Im Gegensatz zu den meisten anderen Gittergeneratoren erzeugt das vorgestellte Verfahren ein Gitter, welches ausschließlich aus vierseitigen oder hexaedrischenElementen besteht. Die hohe Qualität der Randapproximation wird mittels gekrümmter Elemente hoher Ordnung erreicht. Die erzeugten Gitter sind bestens für den Einsatz von numerischen Methoden höherer Ordnung geeignet. Dies gilt insbesondere für die DG Methode, da große Teile des Gitters vollständig strukturiert sind und zugleich die Ordnung der Geometrierepräsentation der hohen Ordnung der numerischen Methode entsprechen kann. Die Gittererzeugung erfolgt in zwei Schritten. Zunächst wird ein inneres Volumen ausgewählt, welches mittels eines vollständig strukturierten kartesischen Gitters abgebildet wird. Im Anschluss werden alle eingebetteten Objekte sowie die Randflächen mittels einer Pufferschicht mit dem inneren Gitter verbunden. Innerhalb dieser Schicht werden gekrümmte Elemente hoher Ordnung verwendet, wodurch eine hohe Approximationsgenauigkeit nicht ebener Objekte und Ränder erzielt wird. Die resultierenden Gitter sind optimal für die Anwendung numerischer Methoden hoher Ordnung geeignet, da einerseits eine hohe Effizienz durch die kartesische Struktur im Inneren und andererseits eine hohe Genauigkeit, selbst in Anwesenheit gekrümmten Objekte und Ränder, gegeben ist. Nach der Beschreibung des Gittergenerierungsalgorithmus, wird die Diskontinuierliche Galerkin Methode für die Maxwellschen Gleichungen beschrieben. Die DG Methode erlaubt die Verwendung von Basisfunktionen beliebig hoher Ordnung. Damit kann für hinreichend glatte Lösungen exponentielle Konvergenz des Fehlers bezüglich der Anzahl der Freiheitsgrade erreicht werden. Es wird anhand von Beispielen gezeigt, dass mit der DG Methode und dem in dieser Arbeit vorgestellten Gittergenerierungsverfahren exponentielle Konvergenz für nicht-geradlinig berandete Rechengebiete erreicht werden kann. Weiterhin wird exemplarisch für einen Resonator gezeigt, dass die mittels Fouriertransformation eines Zeitsignals erhaltenen Spektren frei von nicht-physikalischen Eigenwerten sind.German
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