Homogenisation error

Homogenisation is the process of  representing a multimaterial domain by a single material that captures the behaviour that we are interested in modelling. The use of homogenisation can be justified  by the lack of an accurate description of the domain. It is not inusual to have only a general description of the composition of domain (volume fraction of the different components, average diameter of the particles …) in contrast to a complete description, where the position and shape of each of the particles is known.  And even when a complete description is available, the homogenised model might still be useful, due to its simplicity (easier to discretise).

The objective of this work is to estimate the error due to homogenisation.  We are interested in determining when the simplified model describes well the heterogenous material. This work is structured around 2 ideas. Firstly, the construction of a reference problem. This reference problem must capture the heterogeneous domain and since the most of time, we lack an accurate description, the reference problem is a stochastic problem where the position of the particles is a random variable. In this setting, all the possible particle layouts can be represented.

The second idea is to use the constitutive relation error(CRE) to compare the solution of both problems without having to solve the reference problem. This comparison results in error bounds that can be applied to the error in energy-norm and also to quantities of interests, scalar outputs of the solution, such as the average flux exiting through a surface or the average temperature in the domain. It is worth remarking that the quantities of interest in the homogenised problem are scalars while they are a probability density function in the reference problem. Therefore, we compare their expectations. The authors have also proposed a bound on the variance.

Mesh of the homogeneous model (~ 2 000 elements).
Mesh of the homogeneous model (~ 2 000 elements).
Mesh of the heterogeneous model (~60 000 elements).
Mesh of the heterogeneous model (~60 000 elements).
Temperature of the homogenised model.
Temperature of the homogenised model.
Temperature of the heterogeneous model (particles of radio 0.05, volume fraction of 20%).
Temperature of the heterogeneous model (particles of radio 0.05, volume fraction of 20%).
Temperature gradient of the homogenised model.
Temperature gradient of the homogenised model.
Temperature gradient of the heterogenous model(particles of radio 0.05, volume fraction of 20%).
Temperature gradient of the heterogenous model(particles of radio 0.05, volume fraction of 20%).

The method here presented guarantees that the average temperature on the upper face is in the interval [444.30, 445.80].
The method here presented guarantees that the average temperature on the upper face is in the interval [444.30, 445.80].
Slides presentation given in the ITN Insist workshop in Strobl (January, 2015)

Conference papers

Paladim, Daniel, Pierre Kerfriden, and Stéphane Bordas. “Efficient modeling of random heterogeneous materials with an uniform probability density function.” 11th World Congress on Computational Mechanics. 2014.
Kerfriden, Pierre, Daniel Alves Paladim, and Stephane Pierre Alain Bordas. “Homogenisation methods with guaranteed accuracy: quantifying the scale separability.” (2014).