Topology Optimization – a Variational Formulation of the Problem and Example Application

A variational formulation of the topology optimization problem is presented. A strain energy functional, being an equivalent of compliance, was minimized while constraints were imposed on the body mass. A global mass constraint and a local constraint on the amount of mass accumulated in a single material point of the body were adopted. A penalization procedure was defined and implemented in the optimization process to speed up the latter. The procedure in the successive optimization process steps translocates mass within the design domain, from the less strained areas to the more strained ones. The optimization process was described as a series of sequences of topologies determined using various control parameters, including different threshold functions. This means that the optimization process is characterized by a sequence of objective functional values approaching a minimal value. Various functions updating Young’s modulus were considered. Primarily the updating method referred to as SIMP was adopted. Three ways of using the discrete strain energy value to update Young's modulus in the considered material point were taken into account. These were: the amount of energy accumulated in the preceding step, the sum of the amounts of energy from all the preceding steps and the average amount of energy from the last two steps. In order to ensure the global limiting condition a mass constancy satisfaction procedure was incorporated into the algorithm. The algorithm procedures are described in detail. Finally, the algorithm was used to analyze selected problem relating to the pavement structure and the structure of tall buildings.