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 (US)|
|Number of pages||36|
|Journal||Computer Methods in Applied Mechanics and Engineering|
|State||Published - Dec 27 2016|
Bibliographical noteKAUST Repository Item: Exported on 2020-10-01
Acknowledgements: We would like to acknowledge the open source software packages that made this work possible: PETSc  and , NumPy , matplotlib , ParaView . We would also like to thank Adel Sarmiento for proofreading the manuscript and providing useful comments. This work was supported by 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. 644602. 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).