Inverse homogenization and design of microstructure for pointwise stress control

A Computational Method for an Optimal Design Problem in Uniform Torsion Under Local Stress Constraints

“…The problem is that in approximating the optimal solutions by classical designs (meaning admissible for problem (VP)) we could have small areas (with vanishing measure in the limit) in which the stress constraint is violated. In this sense, the results in the very interesting paper [18] are quite remarkable. In this work the authors proposed a method for designing optimal graded microstructure for a problem similar to the one we consider here.…”

“…We refer to the very interesting recent work [16], where the authors beyond introducing a new effective technique for the numerical simulation of black and white optimal designs (ruling out the possibility of micro-structures among the components), report intensively on previous work on the stress-constrained problem in the framework of topology optimization. Other works dealing with the numerical simulation of the problem are [11,13,14,17,18], where the authors, based on partial relaxation results, develop a method for the simulation of optimal designs allowing composites or functionally graded materials.…”

Representation formulas forL∞norms of weakly convergent sequences of gradient fields in homogenization

“…We pick an open subset S ⊂ Ω of interest and the gradient constraint for the multi-scale problem is written in terms of the modulation function. We set When the constraint M is chosen such that there exists a control β ∈ Ad γ for which C i (β) ≤ M then an optimal design β * exists for the design problem (6.8), this is established in [28], [22]. The optimal design β * specifies characteristic functions χ i * (x, y) = χ i (β * (x), y) from which we recover continuously graded microgeometries χ i * kj (x, n j x) and coefficient matrices A * ,kj (x, n j x) of the form (5.5).…”

“…The feature common to all of these problems is that they involve weakly convergent sequences of gradients and their composition with L ∞ norms of the type given by (1.3) and (1.4). Motivated by the applications we develop an explicit local representation formula for the lower bound on (1.5) for continuously graded periodic microstructures introduced for optimal design problems in [25], [22], [23], see section 5. A similar set of lower bounds have appeared earlier within the context of two-scale homogenization [27].…”

Multi-scale quasistatic damage evolution for polycrystalline materials

“…The average stress over each grain is obtained through the solution of a global or homogenized problem posed in terms of an effective elastic tensor depending locally on the damage volume fraction and the orientation of the principle slip plane within each crystal grain. The recent analysis of Lipton and Stuebner (2006) shows that the global-local modeling used here captures the asymptotic behavior of the actual stress fields inside the polycrystal when the layered damage microstructure is sufficiently fine with respect to the grain size. The numerical implementation of the quasistatic damage evolution is illustrated for long shafts subjected to plane strain loading.…”