<figure><img src="image_1.png" /></figure>

HAL
open science

# Efficiency of boundary conditions on the computation of local fields in a Representative Volume Element

L Belgrand, I. Ramière, R. Largenton, F Lebon

► To cite this version:

L Belgrand, I. Ramière, R. Largenton, F Lebon. Efficiency of boundary conditions on the computation of local fields in a Representative Volume Element. ECCOMAS Congress 2022 - The 8th European Congress on Computational Methods in Applied Sciences and Engineering, Jun 2022, Oslo, Norway. cea-03777686

HAL Id: cea-03777686

https://cea.hal.science/cea-03777686

Submitted on 15 Sep 2022

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
# Efficiency of boundary conditions on the computation of local fields in a Representative Volume Element

L. Belgrand<fnref n="1" />,<fnref n="3" />,*, I. Ramière<fnref n="1" />, R. Largenton<fnref n="2" />, F. Lebon<fnref n="3" />

**Keywords:** *Numerical homogenization, Heterogeneous materials, Local behaviour, Boundary conditions*

Within the framework of numerical homogenization approaches, we focus on the effect of boundary conditions (BCs) on local mechanical fields computed by the Finite Element method. The influence of classical BCs (affine displacements, periodic conditions) imposed on the Representative Volume Element (RVE) has been largely studied with respect to the effective macroscopic behaviour [1]. When a periodic microstructure can be generated at the RVE scale (periodic or model materials typically), periodic conditions produce more accurate results. However, these conditions come with technical difficulties linked to the generation of the periodic mesh and additional costs in terms of computation time.

In a multiscale use of numerical homogenization [2], local fields are of great importance to detect phenomena arising at the local scale. Moreover these fields must be computed in reasonable calculation times to make these numerical coupling approaches efficient. Very few studies focus on the effects of the BCs on the local behaviour. Affine displacement conditions, which are the computationally most efficient technique, are subject to local boundary effects, located on cut inclusions in case of matrix-inclusion composites [3]. Different ways are followed in order to improve the ratio precision over cost of such approaches : truncation or filtering [4], homogenization-based Dirichlet values, RVE without cut inclusions.

## REFERENCES

[1] T. Kanit, S. Forest, I. Galliet, V. Mounoury, and D. Jeulin. Determination of the size of the representative volume element for random composites: statistical and numerical approach. *International Journal of Solids and Structures*, 40(13):3647-3679, 2003.

[2] S. Saeb, P. Steinmann, and A. Javili. Aspects of computational homogenization at finite deformations: A unifying review from Reuss' to Voigt's bound. *Applied Mechanics Reviews*, 68, 06 2016.

[3] M. Salmi, F. Auslender, M. Bornert, and M. Fogli. Apparent and effective mechanical properties of linear matrix-inclusion random composites: Improved bounds for the effective behavior. *International Journal of Solids and Structures*, 49(10):1195-1211, 2012.

[4] X. Yue and W. E. The local microscale problem in the multiscale modeling of strongly heterogeneous media: Effects of boundary conditions and cell size. *Journal of Computational Physics*, 222:556-572, 03 2007.

<footnote n="1">¹ CEA, DES, IRESNE, DEC, SESC, LSC ; Cadarache, F-13108 Saint-Paul-Lez-Durance
{louis.belgrand,isabelle.ramiere}@cea.fr</footnote>

<footnote n="2">² EDF R&D, Dpt MMC ; Cadarache, F-13108 Saint-Paul-Lez-Durance
rodrigue.largenton@edf.fr</footnote>

<footnote n="3">³ Aix-Marseille Université, CNRS, Centrale Marseille, LMA ; 13453 Marseille cedex 13
{belgrand,lebon}@lma.cnrs-mrs.fr</footnote>