We will host a 5-day introductory course to FFT-based homogenization methods. This workshop is open to research students (M2 onwards) as well as researchers from both academia and industrial R&D. Each of the nine sessions of this workshop is composed of a theoretical lecture followed by hands-on applications (mostly on computers).

This event is supported financially by “Université Franco–Allemande/Deutsche–Französische Hochschule” (NBV-61-20-III).

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As of 2022/04/29, the slides of the conference and the video recordings of (most of) the lectures are now available online on HAL

Long description

Analysis at the macroscopic scale of a structure that exhibits heterogeneities at the microscopic scale requires a first homogenization step that allows the heterogeneous constitutive material to be replaced with an equivalent, homogeneous material.

Approximate homogenization schemes (based on mean field/effective field approaches) as well as rigorous bounds have been around for several decades; they are extremely versatile and can address all kinds of material non-linearities. However, they rely on a rather crude description of the microstructure. For applications where a better account of the finest details of the microstructure is desirable, the solution to the so-called corrector problem (that delivers the homogenized properties) must be computed by means of full-field simulations. Such simulations are complex, and classical discretization strategies (e.g., interface-fitting finite elements) are ill-suited to the task.

During the 1990s, Hervé Moulinec and Pierre Suquet introduced a new numerical method for solving the corrector problem. This method is based on the discretization of an integral equation that is equivalent to the original boundary-value problem. Observing that the resulting linear system has a very simple structure (block-diagonal plus block-circulant), Moulinec and Suquet used the fast Fourier transform (FFT) to compute the matrix-vector products that are required to find the solution efficiently.

During the last decade, the resulting method has gained in popularity (the initial Moulinec–Suquet paper is cited 134 times over the 1998–2009 period and 619 times over the 2010–2020 period — source: Scopus). Significant advances have been made on various topics: theoretical analysis of the convergence, discretization strategies, innovative linear and non-linear solvers, etc.

Nowadays, FFT-based homogenization methods have become state-of-the-art techniques in materials science and are used for industry with increasing frequency. This workshop is intended to provide an accessible introduction to FFT-based computational homogenization methods and to provide the participants with a glimpse at the current research frontier.

About the organizers

  • Sébastien Brisard (Laboratoire Navier, Univ. Gustave Eiffel)
  • Felix Ernesti (Institute of Engineering Mechanics, Karlsruhe Institute of Technology (KIT))
  • Matti Schneider (Institute of Engineering Mechanics, Karlsruhe Institute of Technology (KIT))
  • François Willot (Centre for Mathematical Morphology, Mines ParisTech, PSL Research University)

Practical details

  • The workshop will be hosted at Université Gustave Eiffel, Champs sur Marne, France, 14th-18th march, 2022
  • More precisely, lectures will take place in the Georges Perec building and tutorials in the Copernic building
  • Classes will be held in english
  • Attendants are requested to bring their own laptop for the tutorial sessions. They will be required to install Python + Numpy + Scipy
  • It is recommended that attendants are familiar with the fundamental equations of continuum mechanics (linear elasticity in particular) and basic linear algebra. Some notions of optimization, weak formulations as well as programming in Python might be useful, too.
  • The nearest hotel is Ibis Marne-la-Vallée Champs
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