Material Science
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Physico-chemical Homogenization
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Total Sintering and Dense Ceramics
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Partial Sintering and Porous Ceramics
Mathematics
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Mathematical Homogenization
Numerous physical phenomenon happen within materials with a highly complex microstructure. For example composite materials with a high degree of mixing, materials with a ruguous surface or porous materials exhibiting a significant amount of microporosities.
In these materials, physical properties can vary widely in space. If one wants to model these phenomenons, they will obtain partial differential equations whose coefficients can vary a lot at small scale. Describing these highly varying coefficients can be an harduous or even impossible task.
The goal of homogenization theory is to understand the global behavior of these systems as the variations becomes small. The main idea relies on the fact that, despite the material being heterogeneous, it is much easier to model its global behavior using a simpler model.
In this limit model, called homogenized model, the material is replaced by an homogeneous one with equivalent macroscopic properties. This method aims at linking the microscopic properties to its macroscopic behavior and plays an important in the study of composite materials as well as porous media and many other physical phenomenon.
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Random Tessellations
We consider uniformely distributed random points in the plane (also known as a Poisson Point Process).
For each of these points, called germ, we define its Voronoi set as the points of the plane closest to it. The family of Voronoi sets obtained from the given germs constitutes a random tessellation of the plane called the Poisson-Voronoi tessellation.
Statistical properties of these random tessellations, such as the expected surface area of the cells, the expected number of vertices per cell, extreme values, etc. have extensively studied since the 20th century in the case of the euclidean plane but also in R^n or even in more general spaces.
Poisson-Voronoi tessellations provide a wide range of applications from telecommunication (antennas positioning) to biology (studies of cells) or physics (crystallography, microstructure modeling).
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Gaussian Random Fields