M1 - Modeling Biological and Environmental Systems - Mention MECHANICS - IDIL

The aim of this course is to introduce students to the finite element method as applied to one-, two- and three-dimensional problems in engineering and applied science. This presentation is made within the framework of linear elasticity and small perturbations in statics. Starting with prerequisites in mathematics and solid mechanics, the principle of discretization is first addressed through the approaches of Ritz and Gallerkine for one-dimensional media. Next, the problem of numerical integration is approached using the Gauss method. Meshing and validation of computational models is then addressed in the study of surface modeling with 2D elements. Finally, these notions will be used to set up the complete formalism of the finite element method within the framework of bar and beam elements, then triangle-type elements. A practical application of these important theoretical notions is carried out on an industrial calculation code (ANSYS) during practical work and a project.

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The first part of this course concerns complements of probability theory: conditional expectation, Gaussian vectors. The second part presents one of the main families of stochastic processes in discrete time : Markov chains. These are sequences of dependent random variables, whose dependence relation is relatively simple since each variable depends only on the previous one. It is also a very powerful modeling tool. We will study the main properties of these processes, as well as their behavior in long time and the estimation of their parameters.

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The first part of this course concerns complements of probability theory: conditional expectation, Gaussian vectors. The second part presents one of the main families of stochastic processes in discrete time : Markov chains. These are sequences of dependent random variables, whose dependence relation is relatively simple since each variable depends only on the previous one. It is also a very powerful modeling tool. We will study the main properties of these processes, as well as their behavior in long time and the estimation of their parameters.

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This 42h course is divided in two parts (1/3, 2/3) in order to give the basic elements in heat transfer and fluid mechanics (3D). Fluids will be considered as continuous media. We will call a particle an element of infinitely small volume for a mathematical description but large enough compared to the molecules to be described by continuous functions. This course is an extension of the course on modeling elastic media in L3 and the course on fluid mechanics (1D).

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The course aims to provide a general introduction to cellular and molecular biology and to put into context the use of modern physics, through its quantitative methods and approaches, to describe biological systems and their complexity from the molecular to the cellular and tissue scales.

A fundamental point addressed is also the quantification of phenomena, their physical interpretation and their physico-mathematical modeling. The course opens to the philosophy and to the set of themes of this master's course centered on the study of the physical principles of the organization and the dynamics of the living and complex matter.

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