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In Figures 3a and 3b the original and rebinned spectral density of a typical microstructure is shown, where the x- and yaxis are the frequency axis and greyscale colors show the normalized values of the SD. Please note that the origin of the frequency domain is located in the center of the illustrations. In order to improve the efficiency of the method a threshold value FE2 -Simulation of Microheterogeneous Steels 23 A (a) (b) Fig. 4 Simple test example: (a) generated target structure and (b) initial configuration in the optimization process for finding the SSRVE.

MARK – The marking was done selecting a minimal subset M of T with the bulk criterion 1/α ≤ ηT η T ∈M for some global parameter 0 < < 1. 42 Advancements in the Computational Calculus of Variations REFINE – The triangles in the set M are refined via the red-refinement procedure. Further refinement then occurs in order to avoid hanging nodes. This computes new triangulation with red-green-blue refinement. Explanations on the red-green-blue refinement procedure here applied can be found in [23]. We would like to emphasise that thanks to the adaptive method described above with any of the three indicators the numerical method recovers its optimal convergence rate in the sense that σ −σ L4/3 ( ) ∝ N −1/2 ≈ h .

36 Advancements in the Computational Calculus of Variations Rather than showing analogous convergence results for the convex and quasiconvex cases, that anyway include relevant problems such as those of Manià type [5] and those proposed by Foss et al. [17] and Ball [2], we would like to state more precisely the relation between this new discrete formulation and the original continuous problem. To this end, let us define the functionals ⎧ if v ∈ V , η = det Dv, ε = 0, ⎨ E(v) F (v, η, ε) = (v, η) + ε−1 (det Dv, η) if v ∈ V , 0 < ε < ∞, ⎩ +∞ otherwise; F (v, η, ε) = (v, η) + ε−1 (det Dv, η) if v ∈ V , η ∈ Y , 0 < ε < ∞, +∞ otherwise.