Journal
PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES
Volume 470, Issue 2164, Pages -Publisher
ROYAL SOC
DOI: 10.1098/rspa.2013.0641
Keywords
variational principles; phase separation; diffusion; coupled problems; finite-element method
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Funding
- German Research Foundation (DFG) within the Cluster of Excellence Exc 310 Simulation Technology at the University of Stuttgart
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This work shows that the Cahn-Hilliard theory of diffusive phase separation is related to an intrinsic mixed variational principle that determines the rate of concentration and the chemical potential. The principle characterizes a canonically compact model structure, where the two balances involved for the species content and microforce appear as the Euler equations of a variational statement. The existence of the variational principle underlines an inherent symmetry in the two-field representation of the Cahn-Hilliard theory. This can be exploited in the numerical implementation by the construction of time-and space-discrete incremental potentials, which fully determine the update problems of typical time-stepping procedures. The mixed variational principles provide the most fundamental approach to the finite-element solution of the Cahn-Hilliard equation based on low-order basis functions, leading to monolithic symmetric algebraic systems of iterative update procedures based on a linearization of the nonlinear problem. They induce in a natural format the choice of symmetric solvers for Newton-type iterative updates, providing a speed-up and reduction of data storage when compared with non-symmetric implementations. In this sense, the potentials developed are believed to be fundamental ingredients to a deeper understanding of the Cahn-Hilliard theory.
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