Relaxation Through Homogenization for Optimal Design Problems with Gradient Constraints

Abstract

Abstract. The problem of relaxation of optimal design problems for multiphase composite structures in the presence of constraints on the gradient of the state variable is addressed. A relaxed formulation for the problem is given in the presence of a finite or infinite number of constraints. The relaxed formulation is used to identify minimizing sequences of configurations of phases.

Design of functionally graded composite structures in the presence of stress constraints

“…(see Lipton, 2000Lipton, , 2001. The tensors S e and r are functions of the fiber radii r or equivalently functions of the area fraction h ¼ pr 2 .…”

“…However the approach presented here extends to fully three dimensional problems (see Lipton, 2001). The concept of a functionally graded material is introduced by considering a fiber reinforced shaft.…”

“…This is due to the fact that that the coupling between the macroscopic stress and local effects due to microstructure do not decouple in this limit (see Lipton, 2001). It has recently been shown that…”

“…Thus the covariance can be easily computed from the effective compliance if an explicit formula for the effective compliance is available. For the general case it is shown that r can be computed from the solutions of well defined microscopic equilibrium problems (see Lipton (2001); see also Lipton (2000) for the special case of periodic microstructures). We note here that r is a positive tensor so the coupling term in (6) is nonnegative.…”

“…It is evident from (4)-(6) that the optimal design of FGMs for maximum torsional rigidity with d ) e is approximated by the problem max S e in Ad 2 Z X/ / dx 1 dx 2 ; subject to : F ðs sÞ 6 C ð8Þ (see Lipton, 2001).…”