Optimal Topological Design of Stiffened Plates Using the SRV Constraint for 0/1 Solutions

1. Abstract This paper applies optimum structural topological design methodologies to the stiffening of plates in bending. The purpose of the design is to use a given amount of material as orthogonal stiffeners to minimize the compliance of plates in bending. Cheng and Olhoff had shown that if left unchecked, this formulation includes dense sets of infinitely thin stiffeners – a solution though optimal, obviously impractical. In order to circumvent the obstacle, the present formulation assumes a plate meshed by an orthogonal lattice with rectangular elements allowing the stiffeners be positioned only along the lines of the mesh. Assuming that the mesh has l longitudinal and t transversal lines, the basic problem is to add, in an optimal fashion, uniform beams along these lines, subjected to a constraint on the amount of material. The stiffening beams are assumed of rectangular cross-sections of constant height, with their widths or densities taken as the design variables xi, i=1,.., l+t of the problem. In the present form, we have stated a standard topological design problem that calls for the minimization of the compliance, subjected to a constraint on the amount of material, which can be solved by classical optimization techniques, not unlike the erstwhile topology optimization for 2D membrane problems, using density design variables. We, then, limit the stiffening grid to stiffeners of a constant cross-section and ask to locate a given length of stiffeners material along its longitudinal and/or transversal lines, in order to minimize the compliance. The design variables are still the continuous widths xi of the beams, with the understanding that eventually a 0/1 solution is to be reached. In standard topological design, a SIMP material is usually employed for that purpose. In the present work we use the SRV constraint, which was recently introduced in topology design of structures to obtain 0/1 designs. This is a constraint on the Sum of the Reciprocals of the Variables (sum of the inverse values of the widths of the beams in our case), which, in conjunction with a constant material constraint, enforces a discrete design. The problem is formulated as a classical minimization problem with two constraints (constant volume of material and constant SRV) and is solved via MMA, the Method of Moving Asymptotes of Svanberg, using a Matlab version provided by the author. The method is implemented on a Matlab platform and is illustrated by several examples of the optimum stiffening of plates under various loading conditions.