Topological Variations and Invariant-Based Optimal Design

The present Chapter discusses the topological optimization for elastic energy of elastic bodies. The topological genus of the optimized body is changed step by step by introduction of the new cavities, of bubbles. All newly introduced voids, cavities or hard inclusions will be referred to as “bubbles”. This means, that under “bubbles” we understand the introduced inhomogeneities with the elastic properties, which differs from the elastic properties of the initial material of the body. The initial material of the body will be traditionally referred to as “matrix”. This label is customarily in the theory of composite materials. The principal question is where to introduce the new bubble and what is the best possible form of the new bubble. In the utmost case, the inhomogeneities should be introduced in the most advantageous position, which guarantee the maximal effect for the assumed volume or mass. The position will be determined using the definite “characteristic function”. This function is equal to the absolute value of the configuration force, acting on an infinitesimally small introduced inclusion. The center of the void, which corresponds to the point of the extremum of the characteristic function, is the optimal position for the insertion of the new inhomogeneity. If the extremum of the characteristic function is on the boundary of the body, there is no benefit in the introduction of the new inhomogeneities and the conventional shape variation without the alternation of the topology must be performed. The idea is expanded to the optimization of the fundamental frequencies.