Constitutive Mixing Rules
In the proposed design framework, each finite element comprising the spatial domain Ds of the structure will generally contain a spatially varying mixture of the structural and void materials. It is necessary to prescribe the stiffness (or elastic moduli) of such mixtures in terms of the stiffness characteristics of the solid material Csolid, those of the fictitious void material C void, and the local volumetric density of the
structural material 9(X). Here, the well-known powerlaw formula (Bendsoe and Sigmund 1999) is used to accomplish this task, providing the local effective stiffness of the mixture C* as:
C* = vpCso„d + (1 – Vp)CVoitl (3)
where typically the mixing rule parameter p є I1,4]. With p=1, the Voigt rule of mixtures is obtained which does not penalize mixtures, but which does yield a convex formulation for compliance minimization problems (Swan and Kosaka, 1997) so that only one solution exists for the design problem. With p=4, mixtures are penalized in the final design, so that regions of Ds tend to be either solid or void, but the optimization problem is not convex, and will admit a number of solutions that satisfy the first order optimality conditions.