On a bisection algorithm that produces conforming locally refined simplicial meshes

Antti Hannukainen; Sergey Korotov; Michal Křížek

doi:10.1007/978-3-642-12535-5_68

On a bisection algorithm that produces conforming locally refined simplicial meshes

Antti Hannukainen^*, Sergey Korotov, Michal Křížek

^*Corresponding author for this work

Department of Mathematics and Systems Analysis

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Scientific › peer-review

3 Citations (Scopus)

Abstract

First we introduce a mesh density function that is used to define a criterion to decide, where a simplicial mesh should be fine (dense) and where it should be coarse. Further, we propose a new bisection algorithm that chooses for bisection an edge in a given mesh associated with the maximum value of the criterion function. Dividing this edge at its midpoint, we correspondingly bisect all simplices sharing this edge. Repeating this process, we construct a sequence of conforming nested simplicial meshes whose shape is determined by the mesh density function. We prove that the corresponding mesh size of the sequence tends to zero for d∈=∈2, 3 as the bisection algorithm proceeds. It is also demonstrated numerically that the algorithm seems to produce only a finite number of similarity-distinct triangles.

Original language	English
Title of host publication	Large-Scale Scientific Computing - 7th International Conference, LSSC 2009, Revised Papers
Pages	571-579
Number of pages	9
Volume	5910 LNCS
DOIs	https://doi.org/10.1007/978-3-642-12535-5_68
Publication status	Published - 2010
MoE publication type	A4 Article in a conference publication
Event	International Conference on Large-Scale Scientific Computations - Sozopol, Bulgaria Duration: 4 Jun 2009 → 8 Jun 2009 Conference number: 7

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	5910 LNCS
ISSN (Print)	03029743
ISSN (Electronic)	16113349

Conference

Conference	International Conference on Large-Scale Scientific Computations
Abbreviated title	LSSC
Country	Bulgaria
City	Sozopol
Period	04/06/2009 → 08/06/2009

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10.1007/978-3-642-12535-5_68

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Hannukainen, A., Korotov, S., & Křížek, M. (2010). On a bisection algorithm that produces conforming locally refined simplicial meshes. In Large-Scale Scientific Computing - 7th International Conference, LSSC 2009, Revised Papers (Vol. 5910 LNCS, pp. 571-579). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 5910 LNCS). https://doi.org/10.1007/978-3-642-12535-5_68

Hannukainen, Antti ; Korotov, Sergey ; Křížek, Michal. / On a bisection algorithm that produces conforming locally refined simplicial meshes. Large-Scale Scientific Computing - 7th International Conference, LSSC 2009, Revised Papers. Vol. 5910 LNCS 2010. pp. 571-579 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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abstract = "First we introduce a mesh density function that is used to define a criterion to decide, where a simplicial mesh should be fine (dense) and where it should be coarse. Further, we propose a new bisection algorithm that chooses for bisection an edge in a given mesh associated with the maximum value of the criterion function. Dividing this edge at its midpoint, we correspondingly bisect all simplices sharing this edge. Repeating this process, we construct a sequence of conforming nested simplicial meshes whose shape is determined by the mesh density function. We prove that the corresponding mesh size of the sequence tends to zero for d∈=∈2, 3 as the bisection algorithm proceeds. It is also demonstrated numerically that the algorithm seems to produce only a finite number of similarity-distinct triangles.",

author = "Antti Hannukainen and Sergey Korotov and Michal K{\v r}{\'i}{\v z}ek",

year = "2010",

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Hannukainen, A, Korotov, S & Křížek, M 2010, On a bisection algorithm that produces conforming locally refined simplicial meshes. in Large-Scale Scientific Computing - 7th International Conference, LSSC 2009, Revised Papers. vol. 5910 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 5910 LNCS, pp. 571-579, International Conference on Large-Scale Scientific Computations, Sozopol, Bulgaria, 04/06/2009. https://doi.org/10.1007/978-3-642-12535-5_68

On a bisection algorithm that produces conforming locally refined simplicial meshes. / Hannukainen, Antti; Korotov, Sergey; Křížek, Michal.

Large-Scale Scientific Computing - 7th International Conference, LSSC 2009, Revised Papers. Vol. 5910 LNCS 2010. p. 571-579 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 5910 LNCS).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Scientific › peer-review

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AU - Křížek, Michal

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N2 - First we introduce a mesh density function that is used to define a criterion to decide, where a simplicial mesh should be fine (dense) and where it should be coarse. Further, we propose a new bisection algorithm that chooses for bisection an edge in a given mesh associated with the maximum value of the criterion function. Dividing this edge at its midpoint, we correspondingly bisect all simplices sharing this edge. Repeating this process, we construct a sequence of conforming nested simplicial meshes whose shape is determined by the mesh density function. We prove that the corresponding mesh size of the sequence tends to zero for d∈=∈2, 3 as the bisection algorithm proceeds. It is also demonstrated numerically that the algorithm seems to produce only a finite number of similarity-distinct triangles.

AB - First we introduce a mesh density function that is used to define a criterion to decide, where a simplicial mesh should be fine (dense) and where it should be coarse. Further, we propose a new bisection algorithm that chooses for bisection an edge in a given mesh associated with the maximum value of the criterion function. Dividing this edge at its midpoint, we correspondingly bisect all simplices sharing this edge. Repeating this process, we construct a sequence of conforming nested simplicial meshes whose shape is determined by the mesh density function. We prove that the corresponding mesh size of the sequence tends to zero for d∈=∈2, 3 as the bisection algorithm proceeds. It is also demonstrated numerically that the algorithm seems to produce only a finite number of similarity-distinct triangles.

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Hannukainen A, Korotov S, Křížek M. On a bisection algorithm that produces conforming locally refined simplicial meshes. In Large-Scale Scientific Computing - 7th International Conference, LSSC 2009, Revised Papers. Vol. 5910 LNCS. 2010. p. 571-579. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-642-12535-5_68