The mechanical response of a heterogeneous medium results from the interactions of mechanisms spa n- ning several length scales. The computational homogenization method captures direct influence of underlying constituents and morphology with a numerically efficient framework. This stud y reviews the performance of first order computational homogenization technique with a flat punch indentation problem. Results obtained are benchmarked against those using direct numerical simulations (DNS) with full microstructural details. It is shown th at the computational homogenization method is able to capture structural response adequately , even for constituent materials with nonlinear behavior. However, the first order computational homogeniz a- tion method becomes problematic when localized macroscopi c deformation occurs. In this context, some r e- cent trends addressing the issues are discussed. Figure 1. First -order computational homogenization scheme 2.1 Identification of RVE The representative volume element or RVE can be defined as a sample that reflects the material micr o- structure, and from which effective material pr operties can be estimated with desired level of precision. The size of the RVE should be large enough to contain sufficient number of inclusion s so that the extracted overall properties are independent of the applied boundary conditions and at the same time sufficiently smaller than the macroscopic characteristic lengt h scale for the continuum assumption s to be valid at the macro level. This condition is known as the principle of separation of scales. In case of a periodic medium, the unit cell can be taken as the representative of the whole domain. 2.2 Macro to micro tra nsition The objective of this step is to formulate a boundary value problem on the RVE based on macroscopi c field values . The l ocal displa cement field (u) inside the RVE can be decomposed as: w.yEuM (1) where w is a fluctuation fiel d due to presence of the heterogeneities, y represents the local coordinate s (see Fig 1) and EM is the macroscopic strain tensor. The kinematic coupling between the se two scales is achieved by constraining the volume average of local strain field within th e RVE to be equal to EM, i.e. the volume average of fluctuation field w has to vanish. In literature, this r equirement is commonly achieved by having periodic boundary conditions for the RVE, which have been shown to provide better estimates for the appare nt properties, with smaller RVE size s. Furthe r- more, t he periodic boundary conditions ensure (i) identical shapes of the opposite edges by imposing equal fluctuation values on analogous points of the RVE; and (ii) stress continuity across the boundaries by satisf y- ing anti-periodic traction s on opposite edges . 2.3 Micro to m acro t ransition The Hill -Mandel condition, which requires an equivalence of power density between the two scales, is im- posed to effect the micro to macro transition. F ollowing this condition , the m acroscopic stress tensor (σM) is defined as the volume average of the local stress field (σ):  RVEdVV1σ σM (2) In this study, t he tangent stiffness matrix (Ct), defined as ΔσM = Ct ΔEM, is computed by performing static condensat ions on structu ral stiffness matrix of the RVE to improve computational efficiency over the conve n- tional perturbation method (Kouznetsova 2002 ). 2.4 Boundary value problem at macro -scale The macro level problem is solved in the commercial software ABAQUS with the following sequence : (i) discretize the macrostructure and assign RVE at each integration points; (ii) compute element stiffness matr i- ces and residual nodal force vectors by using the tangent stiffness, and macroscopic stress tensor provided by RVE ana lyses, and (iii ) check for convergence. 3 EXAMPLE 3.1 Flat punch indentation This section considers the plane strain flat punch indentation on a perforated plate as illustrated in Figure 2a. The plate comprises of 200 by 40 periodic unit cells. Two different morphologies with the same volume fraction (30%) of voids are considered , as shown in Figure 2 b. Displacement boundary condition is applied on the plate surface co rresponding to the location of indenters . The matrix material follows an isotropi c von Mi s- es plasticity model : n P o y )(εh σσ (3) with initial yield strength (σo) = 400 MPa , hardening modulus (h) = 2 GPa , and n = 0.5. The effective plastic strain is denoted as εp. Other material p roperties of the matrix include: elastic modulus (E) =200 GPa and Poisson ratio (υ) = 0.3 . (a) (b) Figure 2. (a) Flat punch indentati on of perforated plate . (b) Two different RVE s with same void volume fraction considered 3.2 Results and discussions The problem is solved using : (i) direct numerical simulation (DNS), and (ii) first-order computational h o- mogenization . In Figure 3, the indent ation load (P) vs indentation depth (δ) graphs are presented for the two morphologies considered . From this figure, it can be observed that the mechanical behaviour of the plate is well approximated by the first -order computational homogenization technique. Furthermore, the comput a- tional cost is reduced significantly. Figure 3. P - δ response of the plate composed of RVE s with one void (left), and three voids (right) Figure 4 depicts contour plots of the S22 stress component in the deformed RVE at three locations of the macrostructure (see Fig 2 ). As observed , the deformed shape strongly depend s on the macroscopic strain components at the respective locations. Also , maximum stress values inside the RVEs are observed to be much higher than that of homogenized macrostructure due to the stress concentration effect around the hole . For the same reason , macro stress -strain fields obtained from direct numerical simulations and the comput a- tional homogenization approach are not directly comparable . However, if fields within the indiv idual cells of the plate are represented by a constant value i.e. the volume average, then S22 contour plots resulting from both approaches exhibit good match, as illustrated in Figure 5. Figur e 4. Contour plots for S22 ( MPa) inside the defor med RVE at locations 1,2, and 3 in Fig 2a . Figure 5. Contour plots for S22 (MPa) obtained from multi -scale computational homogenization (top), and direct numerical simul a- tion (bottom ), at the indentation depth of 0.014 m . 4 REMARKS Multi -scale computational homogenization methods have become increasingly popular with advances in computational resources. These methods do not make any constitutive assum ptions at the macro scale. I n- stead , numerical stress -strain relationships are extracted on the fly at various materials points by solving the RVE boundary value problem . This approach can easily be extended to nonlinear cases; incorporating the e f- fects of large displacements, large rotations, and non linear history depende nt micro -constituent material be- havior. However, because of the inherent quadratic nature of the micro -macro con current coupling strategy, it is still computationally expensive. Nevertheless, this concern can be mitigated by making use of parallel computation (Feyel & Chaboche 2000). Despite overall success, this technique suffers from a few limitations e.g. macroscopic localization problems and structural size effects cannot be modeled properly . In this context, extensions of classical homogenization methods have emerged e.g. second -order computational homogeniz a- tion (Kouznetsova 2002) . These methods rely on introduction of higher order gradients to capture the rapid fluctuations within RVE.