Computational phase-field modeling

Phase-field modeling is emerging as a promising tool for the treatment of problems with interfaces. The classical description of interface problems requires the numerical solution of partial differential equations on moving domains in which the domain motions are also unknowns. The computational treatment of these problems requires moving meshes and is very difficult when the moving domains undergo topological changes. Phase-field modeling may be understood as a methodology to reformulate interface problems as equations posed on fixed domains. In some cases, the phase-field model may be shown to converge to the moving-boundary problem as a regularization parameter tends to zero, which shows the mathematical soundness of the approach. However, this is only part of the story because phase-field models do not need to have a moving-boundary problem associated and can be rigorously derived from classical thermomechanics. In this context, the distinguishing feature is that constitutive models depend on the variational derivative of the free energy. In all, phase-field models open the opportunity for the efficient treatment of outstanding problems in computational mechanics, such as, the interaction of a large number of cracks in three dimensions, cavitation, film and nucleate boiling, tumor growth or fully three-dimensional air-water flows with surface tension. In addition, phase-field models bring a new set of challenges for numerical discretization that will excite the computational mechanics community

Numerical methods for solving the Cahn-Hilliard equation and its applicability to related Energy-based models

In this paper, we review some numerical methods presented in the literature in the last years to approximate the Cahn-Hilliard equation. Our aim is to compare the main properties of each one of the approaches to try to determine which one we should choose depending on which are the crucial aspects when we approximate the equations. Among the properties that we consider desirable to control are the time accuracy order, energy-stability, unique solvability and the linearity or nonlinearity of the resulting systems. In particular, we concern about the iterative methods used to approximate the nonlinear schemes and the constraints that may arise on the physical and computational parameters. Furthermore, we present the connections of the Cahn-Hilliard equation with other physically motivated systems (not only phase field models) and we state how the ideas of efficient numerical schemes in one topic could be extended to other frameworks in a natural way.Ministry of Education, Youth and Sports of the Czech RepublicMinisterio de Economía y Competitivida