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

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

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.

Citation

Vignal, P., Collier, N., Dalcin, L., Brown, D., & Calo, V. (2017). An energy-stable time-integrator for phase-field models. Computer Methods in Applied Mechanics and Engineering, 316, https://doi.org/10.1016/j.cma.2016.12.017

• Journal Article Type: Article
• Acceptance Date: Dec 27, 2016
• Online Publication Date: Dec 27, 2016
• Publication Date: Apr 1, 2017
• Deposit Date: Mar 23, 2017
• Publicly Available Date: Mar 23, 2017
• Journal: Computer Methods in Applied Mechanics and Engineering
• Print ISSN: 0045-7825
• Electronic ISSN: 1879-2138
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 316
• DOI: https://doi.org/10.1016/j.cma.2016.12.017
• Keywords: Phase-field models; PetIGA; High-order partial differential equation; Mixed finite elements; Isogeometric analysis; Time integration
• Public URL: https://nottingham-repository.worktribe.com/output/854175
• Publisher URL: http://www.sciencedirect.com/science/article/pii/S0045782516317972