PhD in Pure and Applied Mathematics

Multiscale mathematical modeling in engineering, biology and medicine

Multiscale mathematical modeling in engineering, biology and medicine

Academic year 2018/2019

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Course objectives

The course is based on the courses given for the master and Ph.D. students in 2015‐2018 at Skoltech (Moscow), University of Chile (Santiago), University of Lyon (University Jean Monnet). The course introduces main mathematical models describing mechanical behavior at microscopic level of heterogeneous media and for blood flow in network of vessels. The homogenization technique is applied for multiscale analysis of heterogeneous media. For the network of vessels the asymptotic methods (matching, boundary layers) is presented. The method of asymptotic partial decomposition of the domain defines hybrid dimension models combining one‐dimensional description obtained by the dimension reduction with three‐dimensional zooms. It justifies the special exponentially precise junction conditions at the interface of 1D and 3D parts. It can be applied to model the blood flow in vessels with trombs or stents.

Course delivery

• Mon 18 room 7D, 10‐12 + 14‐16
• Tue 19 room 1D, 10‐12 + 14‐16
• Wed 20 room Buzano (DISMA) 10‐13
• Thu 21 room Buzano, 10‐13 14‐16
• Fri 22 room Buzano, 10‐12 + 14‐16
• Mon 25 room Buzano, 10‐12 + 14‐16

Program

The course introduces main mathematical models for heterogeneous media and homgenization techniques for multiscale analysis of heterogeneous media. The special case of blood flow in network of vessels is studied with the method of asymptotic partial decomposition of the domain that defines hybrid dimension models combining one-dimensional description obtained by the dimension reduction with three-dimensional zooms.

1) Introduction to the main equations of mathematical physics used in the mathematical modeling and boundary and initial conditions.

• Diffusion‐convection equation
• Viscous flows equations (Navier‐Stokes equations, Stokes equations, non‐newtonian flows)
• Elasticity equations, visco‐elasticity equations
• Dirichlet's, Neumann's, Robin's and periodic boundary conditions; number of initial conditions; periodic in time problems
• Derivation from physic laws (ideas) and notion of mathematical analysis (weak formulation, existence, uniqueness and stability of the solution, i.e. well‐posedness).

2) Modeling of composite materials and meta‐materials. Homogenization technique in mechanics of solids: passage from microscopic scale to the macroscopic scale.

3) Models of flows. Thin tube structures and multi‐structures. Asymptotic analysis. Method of partial asymptotic decomposition of the domain for flows in a tube structure with rigid walls. Elastic and viscoelastic walls of the flows: special boundary conditions for the fluid.

More info: valeriachiadopiat@polito.it

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