Topology Optimization of Distributed Parameter Systems

This chapter gives a brief introduction to formulations and solution techniques for topology optimization of elastic structures. As a starting point we formulate the problem of optimizing stiffness of a sheet by finding an optimal thickness distribution, which is basically a special case of the general stiffness optimization problem of the previous chapter and which relates closely to the truss problem of Chap. 5. The classical optimality criteria method has shown to be very efficient and is widely used for problems of this type. We show that this method can be seen as a special case of the sequential convex approximation method of Chap. 4. Formulations and solution techniques for topology optimization are next introduced as a modification of the variable thickness sheet problem where penalization is introduced to favor discrete-valued thickness distributions. We discuss the occurrence of ill-posedness of formulations and numerical instabilities, and possible cures of these difficulties based on restriction or relaxation. As a standard reference for structural topology optimization we mention Bendsøe and Sigmund (Topology Optimization: Theory, Methods and Applications, 2003, Springer).