Topic 1: Numerical Homogenization
Random homogenization, theoretical and numerical aspects.
Frederic Legoll, France
The aim of this course is to introduce the students to random homogenization and the associated numerical approaches that have been recently proposed in that context.
Consider a heterogeneous material modelled at the continuum scale with PDEs, e.g. by continuum mechanics. The properties of the materials are supposed to be varying on a small characteristic lengthscale. Think e.g. of composite materials (used e.g. in the aeronautics industry), of porous media in subsurface modelling (modeled by the Darcy law), etc. The aim of homogenization theory is to replace this heterogeneous material by a homogeneous, effective material, for which the computation of the material response is inexpensive.
We will first briefly review the case when the microstructure of the heterogeneous material is periodic. In that case, computing the homogenized coefficient is inexpensive.
The periodicity assumption may be restrictive. In a second stage, we will consider other cases, such as the random case. In that setting, computing the homogenized coefficient is very challenging. We will describe several practical approaches to address this question.
Students are expected to be familiar with simple linear elliptic PDEs (such as the Poisson problem). No background on multiscale problems is expected.
Numerical techniques for PDEs with rapidly varying coefficients.
Alexei Lozinski, France
Partial differential equations (PDEs) with with rapidly varying coefficients arising, for example, in modelling of heterogeneous materials as mentioned above, present a challenge for numerical simulations. Indeed, classical finite element (or other discretization) methods would require a computational mesh with a step sufficiently small with respect to the length scale of coefficient variation, which can make the simulation prohibitivly expensive. We shall present so called multi-scale numerical methods (MsFEM, MsFEM with oversampling, Crouzeix-Raviart-MsFEM, etc) which aim to overcome this issue by incorporating the typical behaviour of multi-scale solutions in the basis functions used in the approximation (unlike piecewise polynomial basis function used in standard finite elements). Although these methods can be implemented and used on a relatively general microstructure, their theoretical analysis can typically be performed only for the periodic microstructure and is based on the results from periodic homogenization. We shall finally discuss some methods (such as local orthogonal decomposition) with theoretically guaranteed convergence on a general microstructure.
Topic 2: Optimal Control and Navier Stokes Equations
Mythily Ramaswamy, India
First of all, divergence
free function spaces and
their properties,
required to develop the
existence theory for
Navier-Stokes equations,
will be introduced.
Using this, a basic
existence theory for
time dependent
NavierStokes equations
will be covered. The aim
of these lectures is to
pose an optimal control
problem and derive the
optimality conditions in
simple cases.
Topic 3: A posteriori error estimation and adaptivity
Goal oriented error estimation for multi-physics problems
Thomas Richter, Germany
1. Introduction to goal oriented error estimators for linear
problems, output functionals, adjoint solutions and first
examples.
2. The general concept of the Dual Weighted Residual
method for nonlinear problems with application to flow models.
3. Application to coupled multi-physics problem.
4. Advanced topics: nonstationary problems and space-time
adaptivity, relation to optimization problems, anisotropic meshes.
A posteriori error estimation based on equilibrated fluxes
Martin Vohralik, France
- A unified framework
for a posteriori error
estimation based on
equilibrated fluxes
(covers conforming,
nonconforming, and mixed
finite elements,
linear/nonlinear and
steady/unsteady partial
differential equations)
- Robustness in a
posteriori error
estimation (with respect
to the approximation
polynomial degree and to
problem data)
- Adaptive solvers
(adaptive stopping
criteria, adaptive
regularization, model
choice)
- Multiscale
approximations (multiscale
and mortar
discretizations,
multigrid solvers
- Applications to flows
in porous media
Topic 4: Finite volume methods for dissipative problems
Claire Chainais, France
We shall study finite
volume schemes for some
problems stemming from
physics or engineering
(semiconductors,
corrosion, porous
medium, etc). We shall
emphasize the importance
of preserving certain
physical proprieties
(positivity, Maximum
principle, energy or
entropy inequalities) to
prove the convergence of
the numerical schemes.
We shall demonstrate how
to apply the entropy
dissipation methods at
the discrete level to
study the long time
behavior of the schemes
or to ensure the correct
asymptotics in the limit
of certain parameters
going to 0.
Topic 5: Simulations of thin structures and fluid-structure interaction
Modeling and simulations of thin structures: application in biology
Olivier Pantz, France
- Non linear
elasticity: hyperelastic
solids.
- Asymptotic limits:
formal asymptotic
expansion. Nonlinear
membrane models,
isometric plates,
VonKarman model.
- Gamma convergence.
- Modeling of vesicles
and red blood cells:
strongly solids.
-
Numerical Simulations:
Finite elements, FreeFem++,
Gradient methods, Newton
method.
Monolithic Methods for Fluid-Structure Interactions.
Olivier Pironneau, France and Hiroshi Suito, Japan
- Fundamental equations of continuum mechanics for fluids and structures. Eulerian, Lagrangian and Arbitrary Lagrangian-Eulerian formulations (OP).
- Fluid-structure interaction with immersed boundary method and its applications for 3D problems (HS).
- Formulation with Lagrangian multipliers and error estimates (OP).
- Small displacements and/or thin structures: transpiration approximation (OP).
- Large displacement : Fully Eulerian formulation and discretizations. Incompressible materials. Generalisation to compressible materials (OP).
- Pre- and post-processing techniques for FEM applications (HS).
- Test cases for FSI (OP).
- Application of FEM and fluid structure interaction to medical problems (HS).
Topic 6: From a microscopic description of matter to a macroscopic one on a computer: computational statistical physics
Gabriel Stoltz, France
In this series of lectures, I will first introduce the fundamental principles of statistical physics from a mathematical viewpoint, by emphasizing how the computation of macroscopic properties of matter (transport coefficients such as the shear viscosity or the thermal conductivity, equation of states linking the pressure, density and temperature of a fluid, etc) reduces to the evaluation of high dimensional integrals with respect to appropriate probability measures. I will then present several strategies to approximate these integrals, relying on ergodic averages of deterministic or stochastic differential equations (with a pedagogical survey of stochastic differential equations, seen from a numerical perspective). I will also discuss the numerical analysis underpinning the error estimations of the computed quantities, by distinguishing the two main sources of errors: statistical errors, related to central limit theorems,
and systematic errors (biases).
