K. Deb et al. (Eds.): GECCO 2004, LNCS 3103, pp. 981–992, 2004. © Springer-Verlag Berlin Heidelberg 2004 Optimization of Constructive Solid Geometry Via a Tree-Based Multi-objective Genetic Algorithm Karim Hamza 1 and Kazuhiro Saitou 2* 1 Ph.D. Candidate, 2 Associate Professor, Mechanical Engineering Department University of Michigan, Ann Arbor, MI 48109-2102, USA {khamza, kazu}@.umich.edu Abstract. This paper presents the multi-objective evolutionary optimization of three-dimensional geometry represented via constructive solid geometry (CSG), a binary tree of boolean operations of solid primitives. NSGA-II is ex- tended for binary tree chromosomes with customized crossover and mutation operators tailored for the evolution of CSG trees and applied for two-objective shape optimization of indoor modular space truss joints. The results show suc- cess in generating a variety of shapes over the Pareto front. A selection of Pa- reto-optimal shapes are manufactured using a solid freeform fabrication proc- ess. 1 Introduction Shape optimization of three-dimensional solid bodies is a challenging task that finds many applications across a multitude of disciplines such as machine design, struc- tures, micro electro-mechanical systems, fluid mechanics and aerospace. Automated shape optimization of three-dimensional geometry often requires the computer repre- sentation of the solid geometry [1,2], such as i) voxel representation, ii) octree repre- sentation, iii) constructive solid geometry (CSG) and iv) boundary representation (B- rep). In voxel representations [1,2], a solid is represented as a collection of small volumetric cells in uniform sizes. The octree representation [1] is a structured variant of voxel representation based on a hierarchical subdivision, which allows the shape representation with locally varying details. By far, the family of shape optimization methods, which appear in the literature most often, is known as topology optimization [3,4]. Topology optimization relies mostly on voxel representation and sometimes on the octree. A possible reason for the popularity could be the ease of encoding and interpreting the design variables in terms of material/no material decisions. Many algorithms have been applied for to- pology optimization, including gradient based, evolutionary, stochastic and hybrid techniques. However, the fact remains that the final output of topology optimization * Corresponding Author