CONCLUSION

This stage enriched me a lot. First I learned a lot of things.

• The first enrichment for me was the discovery of a foreign country: its way of life, the language. I discovered a lot of different cultures (Duke was crowded with foreign students): a lot of European people, but also from China and Turkey.
• I learned a lot about the media of the research in the United States. A lot of things are different from what happenses in France. For Example, the students are initialized very early for the research. Just after the Graduation (which is equivalent to the License in France), the students have to do research during their Master's Degree. The other main difference with what I knew from France was the financing fundings of the research, which is more a competition than in France.
• The others things I learned were the scientific knowledge: the Finite Element Method sure, Continuous Mechanics, Physical Behavior of the phase transformations, but also a lot concerning the programming (C++, MATLAB).

Concerning the presented method, the application of the X-FEM for problem including ``difficult locations'' (discontinuities, high variations, cracks, ...), in other words for problems invloving non-linear constitutive laws. proved to be a very interesting technique. In several phase transformations cases, the presence of terms of interface speed in the equation of temperature on the interface can make avoid the Latin in the presented algorithm, which could be the algorithm faster.