J C Michel

P Suquet

Computational micromechanics: a variational derivation of FFT-based schemes

It is a well-established result in the theory of linear composites that the local fields in such materials satisfy an integral equation which can be written symbolically as [START_REF] Kröner | Statistical Continuum Mechanics[END_REF][START_REF] Willis | Variational and related methods for the overall properties of composites[END_REF][START_REF] Milton | The Theory of Composites[END_REF][START_REF] Jeulin | Random structures in physics[END_REF])

I + Γ 0 δL ε = E, ( 1
)
where ε is the local field under investigation, E is its average, Γ 0 is the Green's operator associated with a homogeneous reference medium L 0 and δL = L -L 0 is the deviation of the actual material properties of the composite from the homogeneous reference medium.

As recognized by Kröner, this integral equation has the same form as the Lippmann-Schwinger (LS) equation in scattering theory. The resolution of (1) requires the inverse of I + Γ 0 δL. One of the earlier method to perform this inversion iteratively is based on a fixed point method:

ε (k+1) = -Γ 0 δLε (k) + E, i.e. ε (k) = k j=0 -Γ 0 δL j E. (2)
This scheme with one of the phases as reference medium has been used by [START_REF] Brown | Solid mixture permittivities[END_REF] to study the first few terms of the series and [START_REF] Moulinec | A fast numerical method for computing the linear and nonlinear properties of composites[END_REF] used an optimized reference medium, together with the discrete Fourier transform to turn the formal expansion (2) into an iterative computational scheme. Although the simple fixed point technique is quite efficient for composites with moderate contrast, it is sensitive to the choice of the reference medium and its convergence can be slow for materials with high contrast. Accelerated iterative methods have been proposed, and some of them have been used by D. Jeulin and his coworkers [START_REF] Escoda | Estimation of local stresses and elastic properties of a mortar sample by FFT computation of fields on a 3D image[END_REF]. According to [START_REF] Mishra | A comparative study on low-memory iterative solvers for FFT-based homogenization of periodic media[END_REF] conjugate gradient methods (see in particular [START_REF] Zeman | Accelerating a FFT-based solver for numerical homogenization of periodic media by conjugate gradients[END_REF][START_REF] Gélébart | Non linear extension of FFT-based methods accelerated by conjugate gradients to evaluate the mechanical behavior of composite materials[END_REF] are among the most efficient existing techniques. However, as noted by these authors, the operator I + Γ 0 δL is not a self-adjoint operator (the corresponding matrix for the discrete problem is not symmetric) since the operators Γ 0 and δL do not commute. So the use of biconjugate gradient methods is in principle required to solve (1).

The starting point of this study is the following question: can the Lippmann-Schwinger operator be made self-adjoint by some magic? The answer is "yes" [START_REF] Schneider | An FFT-based fast gradient method for elastic and inelastic unit cell homogenization problems[END_REF], the proof is elementary and relies on a simple change of scalar product (similar, but different from that used in Jeulin, 2005 p189).

1

The variational property of the LS equation opens new avenues for computational schemes. Its extension to constituents governed by one non-quadratic potential is straightforward. The extension to materials governed by two potentials relies on incremental variational principles [START_REF] Mialon | Eléments d'analyse et de résolution numérique des relations de l'élastoplasticité[END_REF][START_REF] Ortiz | The variational formulation of viscoplastic constitutive updates[END_REF][START_REF] Bourdin | The variational approach to fracture[END_REF][START_REF] Michel | A model-reduction approach in micromechanics of materials preserving the variational structure of constitutive relations[END_REF]. Assume that the constitutive laws for each constituents can be written as:

σ = ∂w ∂ε (ε, α), ∂w ∂α (ε, α) + ∂ϕ ∂ α ( α) = 0, (3)
where α denote a set of internal variables. Then, after time-discretization with an implicit scheme (backward Euler), the fields (strain and internal variables) solution of the local problem (expressing equilibrium of the stress field, compatibility of the strain field with appropriate boundary) at time t n+1 has the variational property :

(ε n+1 , α n+1 ) = Argmin ε =ε Argmin α w(ε, α) + ∆tϕ α -α n ∆t . ( 4
)
This variational property is used to derive computational schemes for the class of composites described by (3) which are significantly faster than the usual schemes.