An energy-stable time-integrator for phase-field models

P. Vignal; N. Collier; L. Dalcin; D. L. Brown; V. M. Calo

doi:10.1016/j.cma.2016.12.017

An energy-stable time-integrator for phase-field models

P. Vignal, N. Collier, L. Dalcin, D. L. Brown, V. M. Calo

Computational Earth Sciences Group

Research output: Contribution to journal › Article › peer-review

24 Scopus citations

Abstract

We introduce a provably energy-stable time-integration method for general classes of phase-field models with polynomial potentials. We demonstrate how Taylor series expansions of the nonlinear terms present in the partial differential equations of these models can lead to expressions that guarantee energy-stability implicitly, which are second-order accurate in time. The spatial discretization relies on a mixed finite element formulation and isogeometric analysis. We also propose an adaptive time-stepping discretization that relies on a first-order backward approximation to give an error-estimator. This error estimator is accurate, robust, and does not require the computation of extra solutions to estimate the error. This methodology can be applied to any second-order accurate time-integration scheme. We present numerical examples in two and three spatial dimensions, which confirm the stability and robustness of the method. The implementation of the numerical schemes is done in PetIGA, a high-performance isogeometric analysis framework.

Original language	English
Pages (from-to)	1179-1214
Number of pages	36
Journal	Computer Methods in Applied Mechanics and Engineering
Volume	316
DOIs	https://doi.org/10.1016/j.cma.2016.12.017
State	Published - Apr 1 2017

Funding

We would like to acknowledge the open source software packages that made this work possible: PETSc[59,60], NumPy [61], matplotlib [62], ParaView [63]. We would also like to thank Adel Sarmiento for proofreading the manuscript and providing useful comments. This work was partially supported by the CSIRO Professorial Chair in Computational Geoscience at Curtin University and the Center for Numerical Porous Media (NumPor) at King Abdullah University of Science and Technology (KAUST) This work is part of the European Union's Horizon 2020 research and innovation programme of the Marie Skłodowska-Curie Grant Agreement No. 644202. This work was also supported by the Agencia Nacional de Promoción Científica y Tecnológicagrants PICT 2014–2660 and PICT-E 2014–0191, and the National Priorities Research Programgrant 7-1482-1-278 from the Qatar National Research Fund (a member of The Qatar Foundation).

Funders	Funder number
Qatar National Research Fund
Horizon 2020 Framework Programme
H2020 Marie Skłodowska-Curie Actions	644202
European Commission
Commonwealth Scientific and Industrial Research Organisation
Agencia Nacional de Promoción Científica y Tecnológica	7-1482-1-278, PICT 2014–2660, PICT-E 2014–0191
King Abdullah University of Science and Technology

Keywords

High-order partial differential equation
Isogeometric analysis
Mixed finite elements
PetIGA
Phase-field models
Time integration

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10.1016/j.cma.2016.12.017

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title = "An energy-stable time-integrator for phase-field models",

abstract = "We introduce a provably energy-stable time-integration method for general classes of phase-field models with polynomial potentials. We demonstrate how Taylor series expansions of the nonlinear terms present in the partial differential equations of these models can lead to expressions that guarantee energy-stability implicitly, which are second-order accurate in time. The spatial discretization relies on a mixed finite element formulation and isogeometric analysis. We also propose an adaptive time-stepping discretization that relies on a first-order backward approximation to give an error-estimator. This error estimator is accurate, robust, and does not require the computation of extra solutions to estimate the error. This methodology can be applied to any second-order accurate time-integration scheme. We present numerical examples in two and three spatial dimensions, which confirm the stability and robustness of the method. The implementation of the numerical schemes is done in PetIGA, a high-performance isogeometric analysis framework.",

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AU - Collier, N.

AU - Dalcin, L.

AU - Brown, D. L.

AU - Calo, V. M.

PY - 2017/4/1

Y1 - 2017/4/1

N2 - We introduce a provably energy-stable time-integration method for general classes of phase-field models with polynomial potentials. We demonstrate how Taylor series expansions of the nonlinear terms present in the partial differential equations of these models can lead to expressions that guarantee energy-stability implicitly, which are second-order accurate in time. The spatial discretization relies on a mixed finite element formulation and isogeometric analysis. We also propose an adaptive time-stepping discretization that relies on a first-order backward approximation to give an error-estimator. This error estimator is accurate, robust, and does not require the computation of extra solutions to estimate the error. This methodology can be applied to any second-order accurate time-integration scheme. We present numerical examples in two and three spatial dimensions, which confirm the stability and robustness of the method. The implementation of the numerical schemes is done in PetIGA, a high-performance isogeometric analysis framework.

AB - We introduce a provably energy-stable time-integration method for general classes of phase-field models with polynomial potentials. We demonstrate how Taylor series expansions of the nonlinear terms present in the partial differential equations of these models can lead to expressions that guarantee energy-stability implicitly, which are second-order accurate in time. The spatial discretization relies on a mixed finite element formulation and isogeometric analysis. We also propose an adaptive time-stepping discretization that relies on a first-order backward approximation to give an error-estimator. This error estimator is accurate, robust, and does not require the computation of extra solutions to estimate the error. This methodology can be applied to any second-order accurate time-integration scheme. We present numerical examples in two and three spatial dimensions, which confirm the stability and robustness of the method. The implementation of the numerical schemes is done in PetIGA, a high-performance isogeometric analysis framework.

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KW - Isogeometric analysis

KW - Mixed finite elements

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An energy-stable time-integrator for phase-field models

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