$$ \newcommand{\C}{{\mathbb{{C}}}} \newcommand{\R}{{\mathbb{{R}}}} \newcommand{\Q}{{\mathbb{{Q}}}} \newcommand{\Z}{{\mathbb{{Z}}}} \newcommand{\N}{{\mathbb{{N}}}} \newcommand{\uu}[1]{{\boldsymbol{{#1}}}} \newcommand{\uuuu}[1]{{\symbb{{#1}}}} \newcommand{\uv}[1]{{\underline{{#1}}}} \newcommand{\ve}[1]{{\uv{{e}}_{{#1}}}} \newcommand{\x}{{\uv{{x}}}} \newcommand{\n}{{\uv{{n}}}} \newcommand{\eps}{{\uu{{\varepsilon}}}} \newcommand{\E}{{\uu{{E}}}} \newcommand{\sig}{{\uu{{\sigma}}}} \newcommand{\Sig}{{\uu{{\Sigma}}}} \newcommand{\cod}{{\uv{{\symscr{b}}}}} \newcommand{\trans}[1]{{{}^{t}{#1}}} \newcommand{\sotimes}{{\stackrel{s}{\otimes}}} \newcommand{\sboxtimes}{\stackrel{s}{\boxtimes}} \newcommand{\norm}[1]{{\lVert{{#1}}\rVert}} \newcommand{\ud}{{\,\mathrm{d}}} \DeclareMathOperator{\arcosh}{arcosh} \DeclareMathOperator{\divz}{div} \DeclareMathOperator{\divu}{\uv{div}} \DeclareMathOperator{\hess}{hess} \DeclareMathOperator{\gradu}{\uv{grad}} \DeclareMathOperator{\graduu}{\uu{grad}} \DeclareMathOperator{\Mat}{Mat} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\ISO}{ISO} \newcommand{\jump}[1]{[\hspace*{-.15em}[\hspace*{.1em}{#1}% \hspace*{.1em}]\hspace*{-.15em}]} $$

Echoes

Extended Calculator of HOmogEnization Schemes

Welcome

The library Echoes allows to implement various homogenization schemes involving different types of heterogeneities in the framework of elasticity, conductivity, viscoelasticity as well as tools to properly calculate the derivatives of macroscopic stiffness with respect to lower scale moduli (fundamental tool of the modified secant method in nonlinear homogenization).

This manual aims at recalling some fundamental aspects of the theory of homogenization of random media along with a presentation of the main features of the library Echoes as well as code examples.

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 with pip executable (for instance Anaconda).

Wheel packages can be downloaded for Python 3.7, 3.8, 3.9, 3.10 and 3.11 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, Jean-François, 2022)

Barthélémy, Jean-François, 2022. Echoes: Extended Calculator of HOmogEnization Schemes. https://doi.org/10.5281/ZENODO.10559657

or in bibtex style

@misc{Echoes,
 author  = {Barthélémy, Jean-François},
 title   = {Echoes library (Extended Calculator of HOmogEnization Schemes)},
 url     = {https://jfbarthelemy.github.io/echoes/},
 doi     = {10.5281/zenodo.10559657}
 version = {v1.0.0},
 year    = {2022},
 month   = {11}
}

\(\,\)

Barthélémy, Jean-François, 2022. Echoes: Extended Calculator of HOmogEnization Schemes. https://doi.org/10.5281/ZENODO.7348759