Mixed variational potentials and inherent symmetries of the Cahn-Hilliard theory of diffusive phase separation.
This work shows that the Cahn-Hilliard theory of diffusive phase separation is related to an intrinsicmixed variational principlethat 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 aninherent symmetryin the two-field representation of the Cahn-Hilliard theory. This can be exploited in the numerical implementation by the construction of time- and space-discreteincremental 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 onlow-orderbasis functions, leading to monolithicsymmetric algebraic systemsof iterative update procedures based on a linearization of the nonlinear problem. They induce in a natural format the choice ofsymmetric solversfor 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.
Institute of Applied Mechanics (CE), Chair I, University of Stuttgart, Pfaffenwaldring 7, 70550 Stuttgart, Germany.
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Miehe
Institute of Applied Mechanics (CE), Chair I, University of Stuttgart, Pfaffenwaldring 7, 70550 Stuttgart, Germany.
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Hildebrand
Institute of Applied Mechanics (CE), Chair I, University of Stuttgart, Pfaffenwaldring 7, 70550 Stuttgart, Germany.
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Böger
coupled problems; diffusion; finite-element method; phase separation; variational principles