ABSTRACT
The topology optimization problem in its basic form is a discrete programming problem. Binary material variables, x(s) ∈ [0, 1], describe if material is present or void at spatial position s in the design domain. To implement efficient continuous non-linear programming algorithms the binary requirements on the material variables are relaxed. Variables at intermediate values are then penalised to encourage purely binary or ‘black-white’ designs with the ‘simple isotropic material with penalization’ (SIMP) approach. Suggested amongst others by Bendsøe (1989).The general topology optimization problem with local stress constraints is stated as (Duysinx & Bendsøe 1998)
where x ∈Rn, q and x ∈Ru. The objective function f0 : Rn+u →R and constraints fj : Rn+u →R, j = 1, 2, . . ., m. Respectively, m and n are the number of constraints and the number of finite elements in the mesh. The vector q denotes the vector of nodal displacements and w the vector of nodal forces (assumed to be design independent), the dimension of which, u, depend on the method of discretization. The elemental stiffness matrix, K , is derived according to the finite element method with SIMP penalization.
