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

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...

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Bibliographic Details
Published in:Computer methods in applied mechanics and engineering 2017-04, Vol.316, p.1179-1214
Main Authors: Vignal, P., Collier, N., Dalcin, L., Brown, D.L., Calo, V.M.
Format: Article
Language:English
Subjects:
Approximation
Discretization
Errors
Finite element analysis
Finite element method
High-order partial differential equation
Isogeometric analysis
Mathematical models
Mixed finite elements
Partial differential equations
PetIGA
Phase-field models
Robustness (mathematics)
Stability
Studies
Taylor series
Time integration
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Summary: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.
ISSN:0045-7825
1879-2138
DOI:10.1016/j.cma.2016.12.017