$$ \newcommand{\C}{{\mathbb{{C}}}} \newcommand{\R}{{\mathbb{{R}}}} \newcommand{\Q}{{\mathbb{{Q}}}} \newcommand{\Z}{{\mathbb{{Z}}}} \newcommand{\N}{{\mathbb{{N}}}} \newcommand{\uu}[1]{{\boldsymbol{{#1}}}} \newcommand{\uuuu}[1]{{\symbb{{#1}}}} \newcommand{\uv}[1]{{\underline{{#1}}}} \newcommand{\ve}[1]{{\uv{{e}}_{{#1}}}} \newcommand{\x}{{\uv{{x}}}} \newcommand{\n}{{\uv{{n}}}} \newcommand{\eps}{{\uu{{\varepsilon}}}} \newcommand{\E}{{\uu{{E}}}} \newcommand{\sig}{{\uu{{\sigma}}}} \newcommand{\Sig}{{\uu{{\Sigma}}}} \newcommand{\cod}{{\uv{{\symscr{b}}}}} % \newcommand{\trans}[1]{{{}^{t}{#1}}} \newcommand{\trans}[1]{{{#1}{}^\intercal}} \newcommand{\sotimes}{{\stackrel{s}{\otimes}}} \newcommand{\sboxtimes}{\stackrel{s}{\boxtimes}} \newcommand{\norm}[1]{{\lVert{{#1}}\rVert}} \newcommand{\ud}{{\,\mathrm{d}}} \newcommand{\mat}{\mathsf} \DeclareMathOperator{\arcosh}{arcosh} \DeclareMathOperator{\divz}{div} \DeclareMathOperator{\divu}{\uv{div}} \DeclareMathOperator{\hess}{hess} \DeclareMathOperator{\gradu}{\uv{grad}} \DeclareMathOperator{\graduu}{\uu{grad}} \DeclareMathOperator{\Mat}{Mat} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\ISO}{ISO} \newcommand{\volt}[1]{{#1}^{-1\circ}} \newcommand{\dcirc}{\overset{\circ}{:}} \newcommand{\jump}[1]{\mathopen{[\![}\,#1\,\mathclose{]\!]}} $$

Echoes

Extended Calculator of HOmogEnization Schemes

Author

Jean-François Barthélémy

Published

July 1, 2026

Welcome

The library echoes allows to implement various mean-field homogenization schemes of random media involving different types of heterogeneities in the framework of elasticity, conductivity, viscoelasticity as well as nonlinear homogenization.

This book gathers tutorials presenting the main features of the library:

elements of tensor calculus,
Hill and Eshelby tensors and their derivatives with respect to reference medium moduli,
concentration problems,
RVEs and schemes in linear homogenization,
extension to nonlinear homogenization,
extension to linear time-dependent behaviors.

Download

The core of echoes has been developed in C++ and wrapped by a Python interface. Hence its use requires first the installation of a Python environment including pip executable (for instance Anaconda).

Wheel packages can be downloaded for various versions of Python under Windows or Linux by choosing the appropriate file for your configuration under the link

Once in possession of the relevant .whl file, the package can be installed in a console (Anaconda console or any console allowing to run pip) by

pip install -U echoes-XYZ.whl
# replacing echoes-XYZ.whl by the correct path to the whl file

Citation

If you use echoes, please cite it as (Barthélémy, 2022) or in bibtex style

@software{echoes,
  title = {Echoes: {{Extended Calculator}} of {{HOmogEnization Schemes}}},
  shorttitle = {Echoes},
  author = {Barthélémy, Jean-François},
  date = {2022-11-22},
  doi = {10.5281/ZENODO.19679057},
  url = {https://echoes.barthelemy.xyz},
  organization = {Zenodo},
  version = {v1.0.0},
}

This documentation is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License license .

About the author

Jean-François Barthélémy is a researcher at Cerema in the research team UMR MCD.

The scientific content of this documentation — models, algorithms, and numerical examples — was developed by the author of the library or the authors of mentioned papers over the course of their research. The editorial work (chapter structure, code formatting, and notebook generation pipeline) was assisted by Claude Code (Anthropic).

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