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# A filter-based computational homogenization method for handling non-separated scales problems

Julien Yvonnet, Amen Tognevi, Guy Bonnet, Mohamed Guerich

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Julien Yvonnet, Amen Tognevi, Guy Bonnet, Mohamed Guerich. A filter-based computational homogenization method for handling non-separated scales problems. 12e Colloque national en calcul des structures, CSMA, May 2015, Giens, France. hal-01515064

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# A filter-based computational homogenization method for handling non-separated scales problems

J. Yvonnet<fnref n="1" />, A. Tognevi<fnref n="1" />,<fnref n="2" />, G. Bonnet<fnref n="1" />, M. Guerich<fnref n="2" />

**Résumé** — In this work, a computational homogenization framework based on filters is proposed to handle homogenization with arbitrary scale separation. Classical homogenization is extended by using low-pass filters instead of averaging operators, leading to a naturally nonlocal elastic framework, where the kernel function is fully constructed by means of computations on a unit cell describing the micro-structure. An associated Finite Element Framework is proposed to deal with heterogeneous structures containing inclusions with characteristic length of the same order than loading fluctuation wavelength.

**Mots clés** — Homogenization, Non-separated scales, Filter-based homogenization.

## 1 Introduction

Classical homogenization assumes separation of scales. This assumption is only valid when the characteristic fluctuation wavelength of an applied strain field over a given microstructure is supposed to be large compared with the scale of the microstructure. Otherwise, the scales are considered as non-separated and the overall effective behavior at a given point depends on the strain states at other points in the neighborhood, defining a nonlocal effective behavior. Several families of methodologies have been proposed to handle lack of scale separation [1, 2, 3, 4]. The objective of this work is to provide a new computational homogenization method able to handle arbitrary scale separation.

## 2 Mesoscopic description of mechanical fields by numerical filters

In [5, 6], a framework for homogenization able to handle arbitrary scale separation has been proposed by replacing averaging operators by numerical filters (Gaussian, or least-square-based).

First, we define two scales, one scale associated to fine scale strain and stress fields $\varepsilon(\mathbf{x})$ and $\sigma(\mathbf{x})$, and another (upper) scale called "Mesoscale", associated to strain and stress fields denoted by $\hat{\varepsilon}(\mathbf{x})$ and $\hat{\sigma}(\mathbf{x})$, respectively, and called "Mesoscopic" strain and stress fields. These fields are related to each other through the following relations :

$$ \hat{\varepsilon}(\mathbf{x}) = \mathcal{F}(\varepsilon(\mathbf{x})), \quad \hat{\sigma}(\mathbf{x}) = \mathcal{F}(\sigma(\mathbf{x})), \qquad (1) $$

where $\mathcal{F}(\cdot)$ is a linear operator, acting as a low-pass filter on the fine scale fluctuations. Associating $\mathcal{F}$ with a characteristic length $h$ related to the fluctuations observation at the mesoscale, the following properties are assumed for $\mathcal{F}$ :

$$ \lim_{h \to 0} \mathcal{F}^h(\varepsilon(\mathbf{x})) = \varepsilon(\mathbf{x}), \quad \lim_{h \to \infty} \mathcal{F}^h(\sigma(\mathbf{x})) = \langle \varepsilon(\mathbf{x}) \rangle. \qquad (2) $$

In [6], we introduced a least-square-based filter in the form :

$$ \mathcal{F}(\varepsilon_{ij}(\mathbf{x})) = \sum_{p=1}^{P} M^p(\mathbf{x}) \hat{\varepsilon}_{ij}^p, \qquad (3) $$

where $M^p(\mathbf{x})$ are piece-wise polynomial basis functions (e.g. finite element shape functions) and $\hat{\varepsilon}_{ij}^p$ are coefficients, interpreted as the nodal values of the mesoscopic strain field at some nodes of a coarse grid

<footnote n="1">¹ Université Paris-Est, MSME UMR 8208 CNRS, 5 Bd Descartes 77454 Marne-la-Vallée Cedex 2, France, julien.yvonnet@univ-paris-est.fr</footnote>

<footnote n="2">² Département de Mécanique des Systèmes, Ecole Supérieure d'Ingénieurs Léonard de Vinci (ESILV), 92916 Paris la Défense Cedex, France</footnote>
associated with the mesoscopic mesh (see [6] for more details), for each component $ij$. These values are assumed associated with a compatible mesoscopic strain field.

Given the fine scale strain field $\varepsilon(\mathbf{x})$ on a discrete fine mesh composed of $N$ nodes $\mathbf{x}^m$, $m = 1, \dots, N$, the unknown coefficients $\hat{\varepsilon}_{ij}^p$ are found by minimizing the following functional :

$$ U = \sum_{m=1}^{N} \left( \sum_{p=1}^{P} M^p(\mathbf{x}^m) \hat{\varepsilon}_{ij}^p - \varepsilon_{ij}(\mathbf{x}^m) \right)^2 . \quad (4) $$

Following [5], we then introduce the following split of microscopic strain into a filtered (mesoscopic) part and a remaining fluctuation $\tilde{\varepsilon}(\mathbf{x})$ :

$$ \varepsilon(\mathbf{x}) = \hat{\varepsilon}(\mathbf{x}) + \tilde{\varepsilon}(\mathbf{x}). \quad (5) $$

Let us consider a unit cell characterizing the microstructure defined in a domain $\Omega \in \mathbb{R}^D$, $D$ being the dimension of the space, with boundary $\partial\Omega$. We define the following localization problem on the unit cell for non-separated scales : assuming known an applied (non-constant) mesoscopic strain field $\hat{\varepsilon}(\mathbf{x})$, find $\varepsilon(\mathbf{x})$ satisfying :

$$ \nabla \cdot (\sigma(\mathbf{x})) = 0 \quad \text{in } \Omega \quad (6) $$

and

$$ \mathcal{F}(\varepsilon(\mathbf{x})) = \hat{\varepsilon}(\mathbf{x}) \quad \text{in } \Omega \quad (7) $$

with

$$ \sigma(\mathbf{x}) = \mathbb{C}(\mathbf{x}) : \varepsilon(\mathbf{x}) \quad (8) $$

where $\mathbb{C}(\mathbf{x})$ is a fourth-order elasticity tensor. Condition (7) is an extension of the classical averaging condition of homogenization in the case of separated scales. It can be interpreted as follows : the filtered part of the compatible strain field satisfying (6)-(8) must match $\hat{\varepsilon}(\mathbf{x}) \quad \forall \mathbf{x} \in \Omega$. An iterative scheme has been proposed in [6] to enforce this condition.

### 3 Mesoscopic homogenized model

In this framework, we have shown in [5] that the constitutive law at the meso scale is given by :

$$ \hat{\sigma}(\mathbf{x}) = \int_{\Omega} \hat{\mathbb{C}}(\mathbf{x}, \mathbf{y}) : \hat{\varepsilon}(\mathbf{y}) d\mathbf{y}, \quad \hat{\mathbb{C}}(\mathbf{x}, \mathbf{y}) = \mathcal{F}\{\mathbb{C}(\mathbf{x}) : \hat{\mathbb{A}}(\mathbf{x}, \mathbf{y})\}, \quad (9) $$

where $\hat{\mathbb{C}}(\mathbf{x}, \mathbf{y})$ is a nonlocal elastic tensor defined in $\Omega$. Note that the constitutive law (9) is general for linear filter operators satisfying conditions (2), and then remains valid when $\mathcal{F}$ is chosen e.g. as a Gaussian filter. Another remark is that in contrast to classical nonlocal elasticity [1], the present nonlocal elasticity operator does not have translational invariance, i.e. $\hat{\mathbb{C}}(\mathbf{x}, \mathbf{y}) \neq \hat{\mathbb{C}}(\mathbf{x} - \mathbf{y})$. This constitutes a key feature of the the model to continuously define mesoscopic constitutive laws for arbitrary scale separation.

A discrete framework can be formulated as follows. Using (3), we define $\hat{\varepsilon}(\mathbf{x})$ in the form

$$ \hat{\varepsilon}_{ij}(\mathbf{x}) \approx \sum_{p} M^p(\mathbf{x}) \hat{\varepsilon}_{ij}^p \quad (10) $$

where the nodal values $\hat{\varepsilon}_{ij}^p$ are defined on a coarse grid discretizing the structure at the mesoscopic scale and $M^p$ are interpolations functions. Using the filter operator $\mathcal{F}$ defined by (3), we obtain the following relationships :

$$ \hat{\sigma}(\mathbf{x}) = \sum_{p} \hat{\mathbb{C}}^p(\mathbf{x}) : \hat{\varepsilon}^p, \quad \hat{\mathbb{C}}^p(\mathbf{x}) = \mathcal{F}\{\mathbb{C}(\mathbf{x}) : \hat{\mathbb{A}}^p(\mathbf{x})\}, \quad (11) $$

and

$$ \hat{\mathbb{A}}^p(\mathbf{x}) = M^p(\mathbf{x})\mathbb{I} + \Psi^p(\mathbf{x}) - \mathcal{F}(\Psi^p(\mathbf{x})), \quad (12) $$
where $\Psi^p(\mathbf{x})$ are transformation tensors associated to each node $p$ of the coarse mesh on the unit cell, defined by:

$$ \Psi_{ijkl}^{p}(\mathbf{x}) = e_{ij}^{p,(kl)}(\mathbf{x}), \qquad (13) $$

where $e^{p,(kl)}$ is the strain solution field obtained by solving on the unit cell an elementary problem (see details in [6]) and where $\chi^{p,(kl)}(\mathbf{x})$ is a non-uniform eigenstrain prescribed over the support of $M^p(\mathbf{x})$ and defined as

$$ \chi^{p,(kl)}(\mathbf{x}) = \frac{M^p(\mathbf{x})}{2} (\mathbf{v}_k \otimes \mathbf{v}_l + \mathbf{v}_l \otimes \mathbf{v}_k) \qquad (14) $$

where $\mathbf{v}_i$ are the base vectors of a Cartesian coordinate system. Then the local strain field can be re-localized by the relationship:

$$ \varepsilon(\mathbf{x}) = \sum_{p} \hat{A}^{p}(\mathbf{x}) \varepsilon^{p}. \qquad (15) $$

# 4 Numerical examples

The methodology is illustrated in the following example. A heterogeneous structure, as depicted in Figure 1 (b), is subjected to 4 points bending. The unit cell described in Fig. 1 (a) is used to compute the nonlocal constitutive operators described in section 3. Results in Figures 2 and 3 show that the proposed method allows reproducing both the simplified model (mesoscopic model) computed on a coarse grid, but also provides a good approximation for local fields, which can be obtained by post-treatment through localization rule (15). We also show the limits of classical homogenization methods and extended versions, in such situations when scales cannot be separated.

<figure><img src="image_2.png" /><figcaption>FIGURE 1 – (a) Unit cell; (b) heterogeneous structure subjected to 4 points bending.</figcaption></figure>

# Références

[1] A.C. Eringen, D.G.B. Edelen. *On nonlocal elasticity*, International Journal of Engineering Science, 10:233-243, 1972.

[2] V. Kouznetsova, M.G.D. Geers, W.A.M. Brekelmans. *Multi-scale second-order computational homogenization of multi-phase materials: a nested finite element solution strategy*, Computer Methods in Applied Mechanics and Engineering, 193:5525-5550, 2004.

[3] S. Forest, K. Sab. *Cosserat overall modelling of heterogeneous materials*, Mechanics Research Communications, 25(4):449-454, 1998.

[4] T.-H. Tran, V. Monchiet, G. Bonnet. *A micromechanics-based approach for the derivation of constitutive elastic coefficients of strain-gradient media*, International Journal of Solids and Structures, 49:783-792, 2012.

[5] J. Yvonnet, G. Bonnet. *A consistent nonlocal scheme based on filters for the homogenization of heterogeneous linear materials with non-separated scales*, International Journal of Solids and Structures, 51:196-209, 2014.

[6] J. Yvonnet, G. Bonnet. *Nonlocal/coarse graining homogenization of linear elastic media with non-separated scales using least-square polynomial filters*, International Journal for Multiscale Computational Engineering, 12(5):375-395, 2014.
<figure><img src="image_3.png" /><figcaption>(a) Reference</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>(c) Standard homogenization</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>(b) Filter-based homogenization</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>(d) Standard homogenization with enhanced re-localization</figcaption></figure>

FIGURE 2 – Stress fields obtained at the mesoscopic scale (on the coarse grid) : comparisons between several models.

<figure><img src="image_7.png" /><figcaption>(a) Reference</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>(c) Standard homogenization</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>(b) Filter-based homogenization</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>(d) Standard homogenization with enhanced re-localization</figcaption></figure>

FIGURE 3 – Stress fields obtained at the microscopic scale (on the finest grid) : comparisons between several models.