Residual Minimization for Isogeometric Analysis in Reduced and Mixed Forms

Victor M. Calo; Quanling Deng; Sergio Rojas; Albert Romkes

doi:10.1007/978-3-030-22741-8_33

Residual Minimization for Isogeometric Analysis in Reduced and Mixed Forms

Victor M. Calo, Quanling Deng, Sergio Rojas, Albert Romkes

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

1 Citation (Scopus)

Abstract

Most variational forms of isogeometric analysis use highly-continuous basis functions for both trial and test spaces. Isogeometric analysis results in excellent discrete approximations for differential equations with regular solutions. However, we observe that high continuity for test spaces is not necessary. In this work, we present a framework which uses highly-continuous B-splines for the trial spaces and basis functions with minimal regularity and possibly lower order polynomials for the test spaces. To realize this goal, we adopt the residual minimization methodology. We pose the problem in a mixed formulation, which results in a system governing both the solution and a Riesz representation of the residual. We present various variational formulations which are variationally-stable and verify their equivalence numerically via numerical tests.

Original language	English
Title of host publication	Computational Science – ICCS 2019
Subtitle of host publication	19th International Conference Faro, Portugal, June 12–14, 2019 Proceedings, Part II
Editors	João M.F. Rodrigues, Pedro J.S. Cardoso, Jânio Monteiro, Roberto Lam, Valeria V. Krzhizhanovskaya, Michael H. Lees, Peter M.A. Sloot, Jack J. Dongarra
Place of Publication	Switzerland
Publisher	Springer
Pages	463-476
Number of pages	14
ISBN (Print)	9783030227401
DOIs	https://doi.org/10.1007/978-3-030-22741-8_33
Publication status	Published - 2019
Externally published	Yes
Event	International Conference on Computational Science 2019 - Faro, Portugal Duration: 12 Jun 2019 → 14 Jun 2019 Conference number: 19th https://link.springer.com/book/10.1007/978-3-030-22741-8

Publication series

Name	Lecture Notes in Computer Science
Volume	11537 LNCS
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	International Conference on Computational Science 2019
Abbreviated title	ICCS 2019
Country/Territory	Portugal
City	Faro
Period	12/06/19 → 14/06/19
Internet address	https://link.springer.com/book/10.1007/978-3-030-22741-8

Keywords

Discontinuous Petrov-Galerkin
Finite elements
Isogeometric analysis
Mixed formulation

Access to Document

10.1007/978-3-030-22741-8_33

Cite this

Calo, V. M., Deng, Q., Rojas, S., & Romkes, A. (2019). Residual Minimization for Isogeometric Analysis in Reduced and Mixed Forms. In J. M. F. Rodrigues, P. J. S. Cardoso, J. Monteiro, R. Lam, V. V. Krzhizhanovskaya, M. H. Lees, P. M. A. Sloot, & J. J. Dongarra (Eds.), Computational Science – ICCS 2019: 19th International Conference Faro, Portugal, June 12–14, 2019 Proceedings, Part II (pp. 463-476). (Lecture Notes in Computer Science; Vol. 11537 LNCS). Springer. https://doi.org/10.1007/978-3-030-22741-8_33

Calo, Victor M. ; Deng, Quanling ; Rojas, Sergio et al. / Residual Minimization for Isogeometric Analysis in Reduced and Mixed Forms. Computational Science – ICCS 2019: 19th International Conference Faro, Portugal, June 12–14, 2019 Proceedings, Part II. editor / João M.F. Rodrigues ; Pedro J.S. Cardoso ; Jânio Monteiro ; Roberto Lam ; Valeria V. Krzhizhanovskaya ; Michael H. Lees ; Peter M.A. Sloot ; Jack J. Dongarra. Switzerland : Springer, 2019. pp. 463-476 (Lecture Notes in Computer Science).

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title = "Residual Minimization for Isogeometric Analysis in Reduced and Mixed Forms",

abstract = "Most variational forms of isogeometric analysis use highly-continuous basis functions for both trial and test spaces. Isogeometric analysis results in excellent discrete approximations for differential equations with regular solutions. However, we observe that high continuity for test spaces is not necessary. In this work, we present a framework which uses highly-continuous B-splines for the trial spaces and basis functions with minimal regularity and possibly lower order polynomials for the test spaces. To realize this goal, we adopt the residual minimization methodology. We pose the problem in a mixed formulation, which results in a system governing both the solution and a Riesz representation of the residual. We present various variational formulations which are variationally-stable and verify their equivalence numerically via numerical tests.",

keywords = "Discontinuous Petrov-Galerkin, Finite elements, Isogeometric analysis, Mixed formulation",

author = "Calo, {Victor M.} and Quanling Deng and Sergio Rojas and Albert Romkes",

note = "Publisher Copyright: {\textcopyright} Springer Nature Switzerland AG 2019.; International Conference on Computational Science 2019, ICCS 2019 ; Conference date: 12-06-2019 Through 14-06-2019",

year = "2019",

doi = "10.1007/978-3-030-22741-8_33",

language = "English",

isbn = "9783030227401",

series = "Lecture Notes in Computer Science",

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pages = "463--476",

editor = "Rodrigues, {Jo{\~a}o M.F.} and Cardoso, {Pedro J.S.} and J{\^a}nio Monteiro and Roberto Lam and Krzhizhanovskaya, {Valeria V.} and Lees, {Michael H.} and Sloot, {Peter M.A.} and Dongarra, {Jack J.}",

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Calo, VM, Deng, Q, Rojas, S & Romkes, A 2019, Residual Minimization for Isogeometric Analysis in Reduced and Mixed Forms. in JMF Rodrigues, PJS Cardoso, J Monteiro, R Lam, VV Krzhizhanovskaya, MH Lees, PMA Sloot & JJ Dongarra (eds), Computational Science – ICCS 2019: 19th International Conference Faro, Portugal, June 12–14, 2019 Proceedings, Part II. Lecture Notes in Computer Science, vol. 11537 LNCS, Springer, Switzerland, pp. 463-476, International Conference on Computational Science 2019, Faro, Portugal, 12/06/19. https://doi.org/10.1007/978-3-030-22741-8_33

Residual Minimization for Isogeometric Analysis in Reduced and Mixed Forms. / Calo, Victor M.; Deng, Quanling; Rojas, Sergio et al.
Computational Science – ICCS 2019: 19th International Conference Faro, Portugal, June 12–14, 2019 Proceedings, Part II. ed. / João M.F. Rodrigues; Pedro J.S. Cardoso; Jânio Monteiro; Roberto Lam; Valeria V. Krzhizhanovskaya; Michael H. Lees; Peter M.A. Sloot; Jack J. Dongarra. Switzerland: Springer, 2019. p. 463-476 (Lecture Notes in Computer Science; Vol. 11537 LNCS).

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

TY - GEN

T1 - Residual Minimization for Isogeometric Analysis in Reduced and Mixed Forms

AU - Calo, Victor M.

AU - Deng, Quanling

AU - Rojas, Sergio

AU - Romkes, Albert

N1 - Conference code: 19th

PY - 2019

Y1 - 2019

N2 - Most variational forms of isogeometric analysis use highly-continuous basis functions for both trial and test spaces. Isogeometric analysis results in excellent discrete approximations for differential equations with regular solutions. However, we observe that high continuity for test spaces is not necessary. In this work, we present a framework which uses highly-continuous B-splines for the trial spaces and basis functions with minimal regularity and possibly lower order polynomials for the test spaces. To realize this goal, we adopt the residual minimization methodology. We pose the problem in a mixed formulation, which results in a system governing both the solution and a Riesz representation of the residual. We present various variational formulations which are variationally-stable and verify their equivalence numerically via numerical tests.

AB - Most variational forms of isogeometric analysis use highly-continuous basis functions for both trial and test spaces. Isogeometric analysis results in excellent discrete approximations for differential equations with regular solutions. However, we observe that high continuity for test spaces is not necessary. In this work, we present a framework which uses highly-continuous B-splines for the trial spaces and basis functions with minimal regularity and possibly lower order polynomials for the test spaces. To realize this goal, we adopt the residual minimization methodology. We pose the problem in a mixed formulation, which results in a system governing both the solution and a Riesz representation of the residual. We present various variational formulations which are variationally-stable and verify their equivalence numerically via numerical tests.

KW - Discontinuous Petrov-Galerkin

KW - Finite elements

KW - Isogeometric analysis

KW - Mixed formulation

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SN - 9783030227401

T3 - Lecture Notes in Computer Science

SP - 463

EP - 476

BT - Computational Science – ICCS 2019

A2 - Rodrigues, João M.F.

A2 - Cardoso, Pedro J.S.

A2 - Monteiro, Jânio

A2 - Lam, Roberto

A2 - Krzhizhanovskaya, Valeria V.

A2 - Lees, Michael H.

A2 - Sloot, Peter M.A.

A2 - Dongarra, Jack J.

PB - Springer

CY - Switzerland

T2 - International Conference on Computational Science 2019

Y2 - 12 June 2019 through 14 June 2019

ER -

Calo VM, Deng Q, Rojas S, Romkes A. Residual Minimization for Isogeometric Analysis in Reduced and Mixed Forms. In Rodrigues JMF, Cardoso PJS, Monteiro J, Lam R, Krzhizhanovskaya VV, Lees MH, Sloot PMA, Dongarra JJ, editors, Computational Science – ICCS 2019: 19th International Conference Faro, Portugal, June 12–14, 2019 Proceedings, Part II. Switzerland: Springer. 2019. p. 463-476. (Lecture Notes in Computer Science). doi: 10.1007/978-3-030-22741-8_33

Residual Minimization for Isogeometric Analysis in Reduced and Mixed Forms

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