Adaptive Refinement for Unstructured T-Splines with Linear Complexity

Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review

Autoren

  • Roland Maier
  • Philipp Morgenstern
  • Thomas Takacs

Organisationseinheiten

Externe Organisationen

  • Friedrich-Schiller-Universität Jena
  • Johann Radon Institute for Computational and Applied Mathematics (RICAM)
  • Johannes Kepler Universität Linz (JKU)
Forschungs-netzwerk anzeigen

Details

Titel in ÜbersetzungAdaptive Verfeinerung für unstrukturierte T-Splines mit linearer Komplexität
OriginalspracheEnglisch
Aufsatznummer102117
FachzeitschriftCAD Computer Aided Design
Jahrgang96
Frühes Online-Datum3 Juni 2022
PublikationsstatusVeröffentlicht - Juni 2022

Abstract

Wir stellen einen adaptiven Verfeinerungsalgorithmus für T-Splines auf unstrukturierten 2D-Netzen vor. Während man bei strukturierten 2D-Netzen Elemente abwechselnd in horizontaler und vertikaler Richtung verfeinern kann, lässt sich ein solcher Ansatz nicht direkt auf unstrukturierte Netze verallgemeinern, bei denen keine zwei eindeutigen globalen Netzrichtungen zugewiesen werden können. Um dieses Problem zu lösen, führen wir das Konzept der Richtungsindizes ein, d.h. ganzzahlige Werte, die jeder Kante zugeordnet sind und die von der Theorie der höherdimensionalen strukturierten T-Splines inspiriert sind. Zusammen mit den Verfeinerungsstufen der Kanten steuern diese Indizes im Wesentlichen das Verfeinerungsschema. Wir kombinieren diese Ideen mit einem Verfahren zur Kantenunterteilung, das I-Knoten zulässt, und erhalten so ein sehr flexibles Verfeinerungsschema, das die T-Knoten gut verteilt und dabei die globale lineare Unabhängigkeit, die lokale lineare Unabhängigkeit außer in der Nähe "außerordentlicher Knoten", die Dünnbesetztheit der Systemmatrix und die Formregularität der Netzelemente bewahrt. Darüber hinaus zeigen wir, dass die Verfeinerungsprozedur lineare Komplexität hat im Sinne von garantierten oberen Schranken für a) den Abstand zwischen markierten und zusätzlich verfeinerten Elementen und für b) das Verhältnis der Anzahl der erzeugten und markierten Netzelemente.

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ASJC Scopus Sachgebiete

Zitieren

Adaptive Refinement for Unstructured T-Splines with Linear Complexity. / Maier, Roland; Morgenstern, Philipp; Takacs, Thomas.
in: CAD Computer Aided Design, Jahrgang 96, 102117, 06.2022.

Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review

Maier R, Morgenstern P, Takacs T. Adaptive Refinement for Unstructured T-Splines with Linear Complexity. CAD Computer Aided Design. 2022 Jun;96:102117. Epub 2022 Jun 3. doi: 10.48550/arXiv.2109.00448, 10.1016/j.cagd.2022.102117
Maier, Roland ; Morgenstern, Philipp ; Takacs, Thomas. / Adaptive Refinement for Unstructured T-Splines with Linear Complexity. in: CAD Computer Aided Design. 2022 ; Jahrgang 96.
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N2 - We present an adaptive refinement algorithm for T-splines on unstructured 2D meshes. While for structured 2D meshes, one can refine elements alternatingly in horizontal and vertical direction, such an approach cannot be generalized directly to unstructured meshes, where no two unique global mesh directions can be assigned. To resolve this issue, we introduce the concept of direction indices, i.e., integers associated to each edge, which are inspired by theory on higher-dimensional structured T-splines. Together with refinement levels of edges, these indices essentially drive the refinement scheme. We combine these ideas with an edge subdivision routine that allows for I-nodes, yielding a very flexible refinement scheme that nicely distributes the T-nodes, preserving global linear independence, analysis-suitability (local linear independence) except in the vicinity of extraordinary nodes, sparsity of the system matrix, and shape regularity of the mesh elements. Further, we show that the refinement procedure has linear complexity in the sense of guaranteed upper bounds on a) the distance between marked and additionally refined elements, and on b) the ratio of the numbers of generated and marked mesh elements.

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