Isogeometric analysis of phase-field models: Application to the cahn-hilliard equation

H. Gomez; V. M. Calo; T. J.R. Hughes

doi:10.1007/978-1-4020-9231-2_1

Isogeometric analysis of phase-field models: Application to the cahn-hilliard equation

H. Gomez, V. M. Calo, T. J.R. Hughes

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

10 Scopus citations

Abstract

The Cahn-Hilliard equation involves fourth-order spatial derivatives. Finite element solutions to the Cahn-Hilliard equation are not common because primal variational formulations of fourth-order operators are only well defined and inte-grable if the finite element basis functions are piecewise smooth and globally l¹ -continuous. There are a very limited number of two-dimensional finite elements possessing l¹ -continuity applicable to complex geometries, but none in three-dimensions. We propose isogeometric analysis as a technology that possesses a unique combination of attributes for complex problems involving higher-order differential operators, namely, higher-order accuracy, robustness, two- and three-dimensional geometric flexibility, compact support, and, most importantly, the possibility of l¹ and higher-order continuity. A NURBS-based variational formulation for the Cahn-Hilliard equation was tested on two- and three-dimensional problems. We present steady state solutions in two-dimensions and, for the first time, in three-dimensions. To achieve these results an adaptive time-stepping method is introduced. We also present a technique for desensitizing calculations to dependence on mesh refinement. This enables the calculation of topologically correct solutions on coarse meshes, opening the way to practical engineering applications of phase-field methodology.

Original language	English (US)
Title of host publication	ECCOMAS Multidisciplinary Jubilee Symposium
Subtitle of host publication	New Computational Challenges in Materials, Structures, and Fluids
Editors	Josef Eberhardsteiner, Christian Hellmich, Herbert A. Mang, Jacques Périaux
Publisher	Springer
Pages	1-16
Number of pages	16
ISBN (Print)	9781402092305
DOIs	https://doi.org/10.1007/978-1-4020-9231-2_1
State	Published - 2009
Event	International ECCOMAS Multidisciplinary Jubilee Symposium - New Computational Challenges in Materials, Structures, and Fluids, EMJS 2008 - Vienna, Austria Duration: Feb 18 2008 → Feb 20 2008

Publication series

Name	Computational Methods in Applied Sciences
Volume	14
ISSN (Print)	1871-3033

Other

Other	International ECCOMAS Multidisciplinary Jubilee Symposium - New Computational Challenges in Materials, Structures, and Fluids, EMJS 2008
Country/Territory	Austria
City	Vienna
Period	02/18/08 → 02/20/08

Bibliographical note

Publisher Copyright:
© Springer Science + Business Media B.V. 2009.

Keywords

Cahn-Hilliard
Isogeometric analysis
NURBS
Phase-field

ASJC Scopus subject areas

Civil and Structural Engineering
Modeling and Simulation
Biomedical Engineering
Computer Science Applications
Fluid Flow and Transfer Processes
Computational Mathematics
Electrical and Electronic Engineering

Access to Document

10.1007/978-1-4020-9231-2_1

Cite this

Gomez, H., Calo, V. M., & Hughes, T. J. R. (2009). Isogeometric analysis of phase-field models: Application to the cahn-hilliard equation. In J. Eberhardsteiner, C. Hellmich, H. A. Mang, & J. Périaux (Eds.), ECCOMAS Multidisciplinary Jubilee Symposium: New Computational Challenges in Materials, Structures, and Fluids (pp. 1-16). (Computational Methods in Applied Sciences; Vol. 14). Springer. https://doi.org/10.1007/978-1-4020-9231-2_1

Gomez, H. ; Calo, V. M. ; Hughes, T. J.R. / Isogeometric analysis of phase-field models : Application to the cahn-hilliard equation. ECCOMAS Multidisciplinary Jubilee Symposium: New Computational Challenges in Materials, Structures, and Fluids. editor / Josef Eberhardsteiner ; Christian Hellmich ; Herbert A. Mang ; Jacques Périaux. Springer, 2009. pp. 1-16 (Computational Methods in Applied Sciences).

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abstract = "The Cahn-Hilliard equation involves fourth-order spatial derivatives. Finite element solutions to the Cahn-Hilliard equation are not common because primal variational formulations of fourth-order operators are only well defined and inte-grable if the finite element basis functions are piecewise smooth and globally l1 -continuous. There are a very limited number of two-dimensional finite elements possessing l1 -continuity applicable to complex geometries, but none in three-dimensions. We propose isogeometric analysis as a technology that possesses a unique combination of attributes for complex problems involving higher-order differential operators, namely, higher-order accuracy, robustness, two- and three-dimensional geometric flexibility, compact support, and, most importantly, the possibility of l1 and higher-order continuity. A NURBS-based variational formulation for the Cahn-Hilliard equation was tested on two- and three-dimensional problems. We present steady state solutions in two-dimensions and, for the first time, in three-dimensions. To achieve these results an adaptive time-stepping method is introduced. We also present a technique for desensitizing calculations to dependence on mesh refinement. This enables the calculation of topologically correct solutions on coarse meshes, opening the way to practical engineering applications of phase-field methodology.",

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Gomez, H, Calo, VM & Hughes, TJR 2009, Isogeometric analysis of phase-field models: Application to the cahn-hilliard equation. in J Eberhardsteiner, C Hellmich, HA Mang & J Périaux (eds), ECCOMAS Multidisciplinary Jubilee Symposium: New Computational Challenges in Materials, Structures, and Fluids. Computational Methods in Applied Sciences, vol. 14, Springer, pp. 1-16, International ECCOMAS Multidisciplinary Jubilee Symposium - New Computational Challenges in Materials, Structures, and Fluids, EMJS 2008, Vienna, Austria, 02/18/08. https://doi.org/10.1007/978-1-4020-9231-2_1

Isogeometric analysis of phase-field models: Application to the cahn-hilliard equation. / Gomez, H.; Calo, V. M.; Hughes, T. J.R.
ECCOMAS Multidisciplinary Jubilee Symposium: New Computational Challenges in Materials, Structures, and Fluids. ed. / Josef Eberhardsteiner; Christian Hellmich; Herbert A. Mang; Jacques Périaux. Springer, 2009. p. 1-16 (Computational Methods in Applied Sciences; Vol. 14).

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

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N2 - The Cahn-Hilliard equation involves fourth-order spatial derivatives. Finite element solutions to the Cahn-Hilliard equation are not common because primal variational formulations of fourth-order operators are only well defined and inte-grable if the finite element basis functions are piecewise smooth and globally l1 -continuous. There are a very limited number of two-dimensional finite elements possessing l1 -continuity applicable to complex geometries, but none in three-dimensions. We propose isogeometric analysis as a technology that possesses a unique combination of attributes for complex problems involving higher-order differential operators, namely, higher-order accuracy, robustness, two- and three-dimensional geometric flexibility, compact support, and, most importantly, the possibility of l1 and higher-order continuity. A NURBS-based variational formulation for the Cahn-Hilliard equation was tested on two- and three-dimensional problems. We present steady state solutions in two-dimensions and, for the first time, in three-dimensions. To achieve these results an adaptive time-stepping method is introduced. We also present a technique for desensitizing calculations to dependence on mesh refinement. This enables the calculation of topologically correct solutions on coarse meshes, opening the way to practical engineering applications of phase-field methodology.

AB - The Cahn-Hilliard equation involves fourth-order spatial derivatives. Finite element solutions to the Cahn-Hilliard equation are not common because primal variational formulations of fourth-order operators are only well defined and inte-grable if the finite element basis functions are piecewise smooth and globally l1 -continuous. There are a very limited number of two-dimensional finite elements possessing l1 -continuity applicable to complex geometries, but none in three-dimensions. We propose isogeometric analysis as a technology that possesses a unique combination of attributes for complex problems involving higher-order differential operators, namely, higher-order accuracy, robustness, two- and three-dimensional geometric flexibility, compact support, and, most importantly, the possibility of l1 and higher-order continuity. A NURBS-based variational formulation for the Cahn-Hilliard equation was tested on two- and three-dimensional problems. We present steady state solutions in two-dimensions and, for the first time, in three-dimensions. To achieve these results an adaptive time-stepping method is introduced. We also present a technique for desensitizing calculations to dependence on mesh refinement. This enables the calculation of topologically correct solutions on coarse meshes, opening the way to practical engineering applications of phase-field methodology.

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Gomez H, Calo VM, Hughes TJR. Isogeometric analysis of phase-field models: Application to the cahn-hilliard equation. In Eberhardsteiner J, Hellmich C, Mang HA, Périaux J, editors, ECCOMAS Multidisciplinary Jubilee Symposium: New Computational Challenges in Materials, Structures, and Fluids. Springer. 2009. p. 1-16. (Computational Methods in Applied Sciences). doi: 10.1007/978-1-4020-9231-2_1

Isogeometric analysis of phase-field models: Application to the cahn-hilliard equation

Abstract

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Keywords

ASJC Scopus subject areas

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