A static formulation for topology optimization with control action

Resumen

 In this work we consider the classical minimum compliance topology optimization in linear elasticity. In our formulation we consider a control functional that represents control forces acting on a structural piece intending to minimize a specific cost functional. The cost functional in question comprises the work of applied body and traction forces. The optimization procedure is performed under suitable constraints for external forces, control energy, volume restriction and a set of project variables. We highlight the variational problem addressed may be summarized as the search for the optimal density distributions of material and control forces which minimize the inner structural work. The topology optimization in this work uses Solid Isotropic Material with Penalization approach (SIMP) [3, 4], based on the concept of optimizing the material distribution, through a density distribution, while the control force is inserted in steady-state form. In order to model the structure, namely a cantilever beam, a bilinear iso-parametric element was considered relating the Finite Element Method (FEM). At this point we start to describe the primal variational formulation. Let Ω be a Lipschitzian domain with a boundary Γ. The boundary Γ is divided in two parts, Γt and Γu, such thatmeasureΩ (Γt  Γu) 0 and ΓtΓu  Γ. We define pseudo-density functions ρ1 (x) and ρ2 (x) on Ω such that the fourth order elasticity tensor E (ρ1) depends, through a penalty parameter, non-linearly on ρ1(x), whereas h(ρ2) depends non-linearly on ρ2. The compliance optimization can be formulated as […]