# A FFT-based variational procedure for homogenization of maxwellian viscoelastic composites

Antoine Martin

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Antoine Martin. A FFT-based variational procedure for homogenization of maxwellian viscoelastic composites. WCCM ECCOMA 2026 - 17th World Congress on Computational Mechanics - 10th European Congress on Computational Methods in Applied Sciences and Engineering, Jul 2026, Munich, Germany. cea-05506211

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Submitted on 11 Feb 2026

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# A FFT-BASED VARIATIONAL PROCEDURE FOR HOMOGENIZATION OF MAXWELLIAN VISCOELASTIC COMPOSITES

A. Martin*

* Commissariat à l'Energie Atomique et aux Energies Alternatives (CEA)
CEA, DES, IRESNE, DEC, SESC
Cadarache, F 13108 St Paul Lez Durance, France
e-mail: antoine.martin@cea.fr, web page: https://cadarache.cea.fr/cad

## ABSTRACT

Homogenization of non-linear composites is a difficult task, generally addressed with two types of approaches at the microstructure scale: mean-field schemes (MFS), or full-field computation. At the structure scale, MFS are sometimes preferred due to their low computational cost, although their accuracy need to be improved. In our approach, we propose a homogenized behaviour as fast as MFS for the integration, and whose accuracy is close to the exact solution.

Variational approaches are powerful tools often used in homogenization. In a purely linear elastic framework, the homogenized stiffness minimizes the Helmholtz potential over the kinematic strain fields. Also, it minimizes the Hashin-Shtrikman functional over unconstrained polarization fields. In the framework of generalized standard materials, the solution minimizes at each time step the incremental potential [1]. This incremental potential was recently rewritten by Idiart [2] within the framework of maxwellian viscoelasticity by means of kinematic and divergence-free fields as:

$$ \mathbf{J}(\bar{\varepsilon}, \alpha_n, \Delta t) = \min_{\bar{\alpha}} \min_{\alpha_s \in S(C_0)} \left\{ \min_{e \in \mathcal{K}(\bar{\varepsilon}-\bar{\alpha})} \langle w(\mathbf{e} - \alpha_s) \rangle + \Delta t \min_{\dot{\alpha}_c \in \mathcal{K}(\dot{\bar{\alpha}})} \langle \varphi(\dot{\alpha}_c + \dot{\alpha}_s) \rangle \right\} \quad (1) $$

where $\dot{\alpha} = \frac{\alpha - \alpha_n}{\Delta t}$, and $S(C_0)$ is the set of fields $\alpha_s$ such that $\mathrm{div}(C_0 : \alpha_s) = 0$, with $\langle \alpha_s \rangle = 0$; and $\mathcal{K}(\bar{\varepsilon})$ the set of fields $\varepsilon$ such that $\mathbf{e} = \bar{\varepsilon} + \mathrm{grad}^s \xi (\xi)$ being a displacement field which checks the boundary conditions). Besides, the potentials are given by $w(\varepsilon) = \frac{1}{2}\varepsilon : \mathbf{C} : \varepsilon$, and $\varphi(\dot{\alpha}) = \frac{1}{2}\dot{\alpha} : \mathbf{M} : \dot{\alpha}$.

The set of fields in $\mathcal{K}(\bar{\varepsilon})$ can be generated by means of the Green operator:

$$ \mathbf{e} = \bar{\mathbf{e}} + \Gamma_0(\tau) \quad (2) $$

where $\Gamma_0$ is the Green operator related to a reference medium whose elasticity is $C_0$, and $\tau$ is an unconstrained polarization field. Similarly, the set $S(C_0)$ can be generated by means of the stress Green operator:

$$ \alpha_s = S_0 : \Delta_0(\mathbf{a}) - \langle \mathbf{a} \rangle \quad (3) $$

where $\Delta_0 = C_0 - C_0\Gamma_0C_0$ and $\mathbf{a}$ is an unconstrained eigenstrain field.

The first set of compatible fields was already explored analytically by Torquato [3] for spherical particles within the framework of conductivity. However, for a general morphology, this set is rather difficult to describe, analytically.

Here, we propose to explore the sets $S$ and $\mathcal{K}$ within a numerical framework. Indeed, Green operators are easily computed by means of Fast Fourier Transform. These operators generate kinematically/statically admissible fields: one example of such a field belonging to $\mathcal{K}(\bar{\varepsilon})$ is:

$$ \mathbf{e} = \bar{\mathbf{e}} + \Gamma_0(\chi \mathbf{e}_i \otimes \mathbf{e}_j)\bar{\tau}_{ij} \quad (4) $$

where $\chi$ is the characteristic function of one phase, and $\bar{\tau}_{ij}$ is the unknown to be optimized by the variational procedure. These fields, called modes, can be computed off-line, and they do not depend
on the material parameters. Hence, comparing to NTFA approach [4], computation of the modes is
immediate, and do not change if the material changes. After this computation, the integration of the
behaviour is very fast, allowing to go to the structure scale easily.

The author acknowledges the ANR – France (French National Research Agency) for its financial support of the AnoHonA project number AAPG2023.

REFERENCES

[1] Mialon, P. *Eléments d'analyse et de résolution numérique des relations de l'élasto-plasticité*, Bul-letin de la Direction des études et recherches-Electricité de France, Série C, Mathématiques, Infor-matique, (3), pp. 57-88, 1986.

[2] Idiart, M. *Thermodynamic potentials for viscoelastic composites*, Journal of the Mechanics and Physics of Solids, Vol. 194, p. 105936, 2025.

[3] Torquato, S. *Bulk properties of two-phase disordered media. III. New bounds on the effective conductivity of dispersions of penetrable spheres*, The Journal of chemical physics, Vol. 84, pp. 6345-6359, 1986.

[4] Michel, J.C. and Suquet, P. *A model-reduction approach in micromechanics of materials preserving the variational structure of constitutive relations*, Journal of the Mechanics and Physics of Solids, Vol. 90, pp. 254-285, 2016.