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

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1 A filter-based comutational homogenization method for handling non-searated scales roblems Julien Yvonnet, Amen Tognevi, Guy Bonnet, Mohamed Guerich To cite this version: Julien Yvonnet, Amen Tognevi, Guy Bonnet, Mohamed Guerich. A filter-based comutational homogenization method for handling non-searated scales roblems. 12e Colloque national en calcul des structures, May 2015, Giens, France. CSMA 2015, 2015, <htts://csma2015.ec-nantes.fr/>. <hal > HAL Id: hal htts://hal.archives-ouvertes.fr/hal Submitted on 27 Ar 2017 HAL is a multi-discilinary oen access archive for the deosit and dissemination of scientific research documents, whether they are ublished or not. The documents may come from teaching and research institutions in France or abroad, or from ublic or rivate research centers. Public Domain L archive ouverte luridiscilinaire HAL, est destinée au déôt et à la diffusion de documents scientifiques de niveau recherche, ubliés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires ublics ou rivés.

2 CSMA e Colloque National en Calcul des Structures Mai 2015, Presqu île de Giens (Var) A filter-based comutational homogenization method for handling non-searated scales roblems J. Yvonnet 1, A. Tognevi 1,2, G. Bonnet 1, M. Guerich 2 1 Université Paris-Est, MSME UMR 8208 CNRS, 5 Bd Descartes Marne-la-Vallée Cedex 2, France, julien.yvonnet@univ-aris-est.fr 2 Déartement de Mécanique des Systèmes, Ecole Suérieure d Ingénieurs Léonard de Vinci (ESILV), Paris la Défense Cedex, France Résumé In this work, a comutational homogenization framework based on filters is roosed to handle homogenization with arbitrary scale searation. Classical homogenization is extended by using low-ass filters instead of averaging oerators, leading to a naturally nonlocal elastic framework, where the kernel function is fully constructed by means of comutations on a unit cell describing the microstructure. An associated Finite Element Framework is roosed to deal with heterogeneous structures containing inclusions with characteristic length of the same order than loading fluctuation wavelength. Mots clés Homogenization, Non-searated scales, Filter-based homogenization. 1 Introduction Classical homogenization assumes searation of scales. This assumtion is only valid when the characteristic fluctuation wavelength of an alied strain field over a given microstructure is suosed to be large comared with the scale of the microstructure. Otherwise, the scales are considered as nonsearated and the overall effective behavior at a given oint deends on the strain states at other oints in the neighborhood, defining a nonlocal effective behavior. Several families of methodologies have been roosed to handle lack of scale searation [1, 2, 3, 4]. The objective of this work is to rovide a new comutational homogenization method able to handle arbitrary scale searation. 2 Mesoscoic descrition of mechanical fields by numerical filters In [5, 6], a framework for homogenization able to handle arbitrary scale searation has been roosed by relacing averaging oerators by numerical filters (Gaussian, or least-square-based). First, we define two scales, one scale associated to fine scale strain and stress fields ε(x) and σ(x), and another (uer) scale called "Mesoscale", associated to strain and stress fields denoted by ˆε(x) and ˆσ(x), resectively, and called "Mesoscoic" strain and stress fields. These fields are related to each other through the following relations : ˆε(x) = F (ε(x)), ˆσ(x) = F (σ(x)), (1) where F (.) is a linear oerator, acting as a low-ass filter on the fine scale fluctuations. Associating F with a characteristic length h related to the fluctuations observation at the mesoscale, the following roerties are assumed for F : lim F h (ε(x)) = ε(x), lim F h (ε(x)) = ε(x). (2) h 0 h In [6], we introduced a least-square-based filter in the form : F (ε i j (x)) = P M (x)ˆε i j, (3) =1 where M (x) are iece-wise olynomial basis functions (e.g. finite element shae functions) and ˆε i j are coefficients, interreted as the nodal values of the mesoscoic strain field at some nodes of a coarse grid 1

3 associated with the mesoscoic mesh (see [6] for more details), for each comonent i j. These values are assumed associated with a comatible mesoscoic strain field. Given the fine scale strain field ε(x) on a discrete fine mesh comosed of N nodes x m, m = 1,...,N, the unknown coefficients ˆε i j are found by minimizing the following functional : U = N P m=1( =1 M (x m )ˆε i j ε i j(x m )) 2. (4) Following [5], we then introduce the following slit of microscoic strain into a filtered (mesoscoic) art and a remaining fluctuation ε(x) : ε(x) = ˆε(x) + ε(x). (5) Let us consider a unit cell characterizing the microstructure defined in a domain Ω R D, D being the dimension of the sace, with boundary Ω. We define the following localization roblem on the unit cell for non-searated scales : assuming known an alied (non-constant) mesoscoic strain field ˆε(x), find ε(x) satisfying : and with (σ(x)) = 0 in Ω (6) F (ε(x)) = ˆε(x) in Ω (7) σ(x) = C(x) : ε(x) (8) where C(x) is a fourth-order elasticity tensor. Condition (7) is an extension of the classical averaging condition of homogenization in the case of searated scales. It can be interreted as follows : the filtered art of the comatible strain field satisfying (6)-(8) must match ˆε(x) x Ω. An iterative scheme has been roosed in [6] to enforce this condition. 3 Mesoscoic homogenized model In this framework, we have shown in [5] that the constitutive law at the meso scale is given by : ˆσ(x) = Ĉ(x,y) : ˆε(y)dy, Ĉ(x,y) = F { C(x) : Â(x,y) }, (9) Ω where Ĉ(x,y) is a nonlocal elastic tensor defined in Ω. Note that the constitutive law (9) is general for linear filter oerators satisfying conditions (2), and then remains valid when F is chosen e.g. as a Gaussian filter. Another remark is that in contrast to classical nonlocal elasticity [1], the resent nonlocal elasticity oerator does not have translational invariance, i.e. Ĉ(x, y) Ĉ(x y). This constitutes a key feature of the the model to continuously define mesoscoic constitutive laws for arbitrary scale searation. A discrete framework can be formulated as follows. Using (3), we define ˆε(x) in the form ˆε i j (x) M (x)ˆε i j (10) where the nodal values ˆε i j are defined on a coarse grid discretizing the structure at the mesoscoic scale and M are interolations functions. Using the filter oerator F defined by (3), we obtain the following relationshis : and ˆσ(x) = Ĉ (x) : ˆε, Ĉ (x) = F { C(x) : Â (x) }, (11) Â (x) = M (x)i + Ψ (x) F (Ψ (x)), (12) 2

4 where Ψ (x) are transformation tensors associated to each node of the coarse mesh on the unit cell, defined by : Ψ i jkl (x) = e,(kl) i j (x), (13) where e,(kl) is the strain solution field obtained by solving on the unit cell an elementary roblem (see details in [6]) and where χ,(kl) (x) is a non-uniform eigenstrain rescribed over the suort of M (x) and defined as χ,(kl) (x) = M (x) (v k v l + v l v k ) (14) 2 where v i are the base vectors of a Cartesian coordinate system. Then the local strain field can be relocalized by the relationshi : ε(x) = Â (x)ˆε. (15) 4 Numerical examles The methodology is illustrated in the following examle. A heterogeneous structure, as deicted in Figure 1 (b), is subjected to 4 oints bending. The unit cell described in Fig. 1 (a) is used to comute the nonlocal constitutive oerators described in section 3. Results in Figures 2 and 3 show that the roosed method allows reroducing both the simlified model (mesoscoic model) comuted on a coarse grid, but also rovides a good aroximation for local fields, which can be obtained by osttreatment through localization rule (15). We also show the limits of classical homogenization methods and extended versions, in such situations when scales cannot be searated. (a) (b) FIGURE 1 (a) Unit cell ; (b) heterogeneous structure subjected to 4 oints bending. Références [1] A.C. Eringen, D.G.B. Edelen. On nonlocal elasticity, International Journal of Engineering Science, 10 : , [2] V. Kouznetsova, M.G.D. Geers, W.A.M. Brekelmans. Multi-scale second-order comutational homogenization of multi-hase materials : a nested finite element solution strategy, Comuter Methods in Alied Mechancs and Engineering, 193 : , [3] S. Forest, K. Sab. Cosserat overall modelling of heterogeneous materials, Mechanics Research Communications, 25(4) : , [4] T.-H. Tran, V. Monchiet, G. Bonnet. A micromechanics-based aroach for the derivation of constitutive elastic coefficients of strain-gradient media, International Journal of Solids and Structures, 49 : , [5] J. Yvonnet, G. Bonnet. A consistent nonlocal scheme based on filters for the homogenization of heterogeneous linear materials with non-searated scales, International Journal of Solids and Structures, 51 : , [6] J. Yvonnet, G. Bonnet. Nonlocal/coarse graining homogenization of linear elastic media with non-searated scales using least-square olynomial filters, International Journal for Multiscale Comutational Engineering, 12(5) : ,

5 (a) Reference (b) Filter-based homogenization (c) Standard homogenization (d) Standard homogenization with enhanced re-localization FIGURE 2 Stress fields obtained at the mesoscoic scale (on the coarse grid) : comarisons between several models. (a) Reference (b) Filter-based homogenization (c) Standard homogenization (d) Standard homogenization with enhanced re-localization FIGURE 3 Stress fields obtained at the microscoic scale (on the finest grid) : comarisons between several models. 4