Analysis of Optimality Conditions for Some Topology Optimization Problems in Elasticity

The subject of topology optimization has undergone an enormous practical development since the appearance of the paper by Bends\o e and Kikuchi (1988), where some ideas from homogenization theory were put into practice. Since then, several engineering applications as well as different approaches have been developed successfully. However, it is difficult to find in the literature some analytical examples that might be used as a test in order to assess the validity of the solutions obtained with different algorithms. As a matter of fact, one is often faced with numerical instabilities requiring a fine tuning of the algorithm for each specific case. In this work, we develop a family of analytical solutions for very simple topology optimization problems, in the framework of elasticity theory, including bending and extension of rods, torsion problems as well as plane stress and plane strain elasticity problems. All of these problems are formulated in a simplified theoretical framework. A key issue in this type of problems is to be able to evaluate the sensitivity of the homogenized elastic coefficients with respect to the microstructure parameter(s). Since we are looking for analytical solutions, we use laminates for which an explicit dependence of the homogenized coefficients on the microstructure is known.