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This course gives an elementary introduction to the phase-field method and to its applications for the modeling of crystal growth. Two different interpretations of the phase-field variable are given and discussed. It can be seen as a physical order parameter that characterizes a phase transition, or as a smoothed indicator function that tracks domain(More)
Bacillus subtilis swarms rapidly over the surface of a synthetic medium creating remarkable hyperbranched dendritic communities. Models to reproduce such effects have been proposed under the form of parabolic Partial Differential Equations representing the dynamics of the active cells (both motile and multiplying), the passive cells (non-motile and(More)
The bacterium Bacillus subtilis frequently forms biofilms at the interface between the culture medium and the air. We present a mathematical model that couples a description of bacteria as individual discrete objects to the standard advection-diffusion equations for the environment. The model takes into account two different bacterial phenotypes. In the(More)
The formation of macropores in silicon during electrochemical etching processes has attracted much interest. Experimental evidences indicate that charge transport in silicon and in the electrolyte should realistically be taken into account in order to be able to describe the macropore morphology. However, up to now, none of the existing models has the(More)
Phase-field models have become popular in recent years to describe a host of free-boundary problems in various areas of research. The key point of the phase-field approach is that surfaces and interfaces are implicitly described by continuous scalar fields that take constant values in the bulk phases and vary continuously but steeply across a diffuse front.(More)
The phase-field method has become the method of choice for simulating microstructure formation during solidification. Recent progress, both on the formulation of the model and on the numerical implementation, makes it now possible to simulate quantitatively the evolution of complex microstructures in three dimensions. This is illustrated by simulations of(More)
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