The Use of Dual B-Spline Representations for the Double de Rham Complex of Discrete Differential Forms

Yi Zhang; Varun Jain; Artur Palha; Marc Gerritsma

doi:10.1007/978-3-030-49836-8_11

The Use of Dual B-Spline Representations for the Double de Rham Complex of Discrete Differential Forms

Yi Zhang^*, Varun Jain, Artur Palha, Marc Gerritsma

^*Corresponding author for this work

Aerodynamics

Research output: Chapter in Book/Conference proceedings/Edited volume › Conference contribution › Scientific › peer-review

1 Citation (Scopus)

Abstract

In ℝⁿ, let Λ^k(Ω) represent the space of smooth differential k-forms in Ω. The de Rham complex consists of a sequence of spaces, Λ^k(Ω), k = 0, 1…, n, connected by the exterior derivative, d: Λ^k(Ω) → Λ^k+1(Ω). Appropriately chosen B-spline spaces together with their associated dual B-spline spaces form a discrete double de Rham complex. In practical applications, this discrete double de Rham complex leads to very sparse systems. In this paper, this construction will be explained and illustrated by means of a non-trivial three-dimensional example.

Original language	English
Title of host publication	Isogeometric Analysis and Applications 2018
Editors	Harald van Brummelen, Cornelis Vuik, Matthias Möller, Clemens Verhoosel, Bernd Simeon, Bert Jüttler
Publisher	Springer
Pages	227-242
Number of pages	16
ISBN (Print)	9783030498351
DOIs	https://doi.org/10.1007/978-3-030-49836-8_11
Publication status	Published - 2021
Event	3rd Conference on Isogeometric Analysis and Applications, 2018 - Delft, Netherlands Duration: 23 Apr 2018 → 26 Apr 2018

Publication series

Name	Lecture Notes in Computational Science and Engineering
Volume	133
ISSN (Print)	1439-7358
ISSN (Electronic)	2197-7100

Conference

Conference	3rd Conference on Isogeometric Analysis and Applications, 2018
Country/Territory	Netherlands
City	Delft
Period	23/04/18 → 26/04/18

Keywords

B-splines
Differential forms
Discrete double de Rham complex
Dual representations

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10.1007/978-3-030-49836-8_11

Cite this

Zhang, Y., Jain, V., Palha, A., & Gerritsma, M. (2021). The Use of Dual B-Spline Representations for the Double de Rham Complex of Discrete Differential Forms. In H. van Brummelen, C. Vuik, M. Möller, C. Verhoosel, B. Simeon, & B. Jüttler (Eds.), Isogeometric Analysis and Applications 2018 (pp. 227-242). (Lecture Notes in Computational Science and Engineering; Vol. 133). Springer. https://doi.org/10.1007/978-3-030-49836-8_11

Zhang, Yi ; Jain, Varun ; Palha, Artur et al. / The Use of Dual B-Spline Representations for the Double de Rham Complex of Discrete Differential Forms. Isogeometric Analysis and Applications 2018. editor / Harald van Brummelen ; Cornelis Vuik ; Matthias Möller ; Clemens Verhoosel ; Bernd Simeon ; Bert Jüttler. Springer, 2021. pp. 227-242 (Lecture Notes in Computational Science and Engineering).

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title = "The Use of Dual B-Spline Representations for the Double de Rham Complex of Discrete Differential Forms",

abstract = "In ℝn, let Λk(Ω) represent the space of smooth differential k-forms in Ω. The de Rham complex consists of a sequence of spaces, Λk(Ω), k = 0, 1…, n, connected by the exterior derivative, d: Λk(Ω) → Λk+1(Ω). Appropriately chosen B-spline spaces together with their associated dual B-spline spaces form a discrete double de Rham complex. In practical applications, this discrete double de Rham complex leads to very sparse systems. In this paper, this construction will be explained and illustrated by means of a non-trivial three-dimensional example.",

keywords = "B-splines, Differential forms, Discrete double de Rham complex, Dual representations",

author = "Yi Zhang and Varun Jain and Artur Palha and Marc Gerritsma",

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doi = "10.1007/978-3-030-49836-8_11",

language = "English",

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series = "Lecture Notes in Computational Science and Engineering",

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booktitle = "Isogeometric Analysis and Applications 2018",

note = "3rd Conference on Isogeometric Analysis and Applications, 2018 ; Conference date: 23-04-2018 Through 26-04-2018",

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Zhang, Y , Jain, V , Palha, A & Gerritsma, M 2021, The Use of Dual B-Spline Representations for the Double de Rham Complex of Discrete Differential Forms. in H van Brummelen, C Vuik, M Möller, C Verhoosel, B Simeon & B Jüttler (eds), Isogeometric Analysis and Applications 2018. Lecture Notes in Computational Science and Engineering, vol. 133, Springer, pp. 227-242, 3rd Conference on Isogeometric Analysis and Applications, 2018, Delft, Netherlands, 23/04/18. https://doi.org/10.1007/978-3-030-49836-8_11

The Use of Dual B-Spline Representations for the Double de Rham Complex of Discrete Differential Forms. / Zhang, Yi ; Jain, Varun ; Palha, Artur et al.
Isogeometric Analysis and Applications 2018. ed. / Harald van Brummelen; Cornelis Vuik; Matthias Möller; Clemens Verhoosel; Bernd Simeon; Bert Jüttler. Springer, 2021. p. 227-242 (Lecture Notes in Computational Science and Engineering; Vol. 133).

Research output: Chapter in Book/Conference proceedings/Edited volume › Conference contribution › Scientific › peer-review

TY - GEN

T1 - The Use of Dual B-Spline Representations for the Double de Rham Complex of Discrete Differential Forms

AU - Zhang, Yi

AU - Jain, Varun

AU - Palha, Artur

AU - Gerritsma, Marc

PY - 2021

Y1 - 2021

N2 - In ℝn, let Λk(Ω) represent the space of smooth differential k-forms in Ω. The de Rham complex consists of a sequence of spaces, Λk(Ω), k = 0, 1…, n, connected by the exterior derivative, d: Λk(Ω) → Λk+1(Ω). Appropriately chosen B-spline spaces together with their associated dual B-spline spaces form a discrete double de Rham complex. In practical applications, this discrete double de Rham complex leads to very sparse systems. In this paper, this construction will be explained and illustrated by means of a non-trivial three-dimensional example.

AB - In ℝn, let Λk(Ω) represent the space of smooth differential k-forms in Ω. The de Rham complex consists of a sequence of spaces, Λk(Ω), k = 0, 1…, n, connected by the exterior derivative, d: Λk(Ω) → Λk+1(Ω). Appropriately chosen B-spline spaces together with their associated dual B-spline spaces form a discrete double de Rham complex. In practical applications, this discrete double de Rham complex leads to very sparse systems. In this paper, this construction will be explained and illustrated by means of a non-trivial three-dimensional example.

KW - B-splines

KW - Differential forms

KW - Discrete double de Rham complex

KW - Dual representations

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DO - 10.1007/978-3-030-49836-8_11

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AN - SCOPUS:85101408728

SN - 9783030498351

T3 - Lecture Notes in Computational Science and Engineering

SP - 227

EP - 242

BT - Isogeometric Analysis and Applications 2018

A2 - van Brummelen, Harald

A2 - Vuik, Cornelis

A2 - Möller, Matthias

A2 - Verhoosel, Clemens

A2 - Simeon, Bernd

A2 - Jüttler, Bert

PB - Springer

T2 - 3rd Conference on Isogeometric Analysis and Applications, 2018

Y2 - 23 April 2018 through 26 April 2018

ER -

Zhang Y , Jain V , Palha A , Gerritsma M. The Use of Dual B-Spline Representations for the Double de Rham Complex of Discrete Differential Forms. In van Brummelen H, Vuik C, Möller M, Verhoosel C, Simeon B, Jüttler B, editors, Isogeometric Analysis and Applications 2018. Springer. 2021. p. 227-242. (Lecture Notes in Computational Science and Engineering). doi: 10.1007/978-3-030-49836-8_11

The Use of Dual B-Spline Representations for the Double de Rham Complex of Discrete Differential Forms

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