Structural shape and topology optimization in a level-set-based framework of region representation

In this paper we present a new framework to approach the problem of structural shape and topology optimization. We use a level-set method as a region representation with a moving boundary model. As a boundary optimization problem, the structural boundary description is implicitly embedded in a scalar function as its "iso-surfaces." Such level-set models are flexible in handling complex topological changes and are concise in describing the material regions of the structure. Furthermore, by using a simple Hamilton-Jacobi convection equation, the movement of the implicit moving boundaries of the structure is driven by a transformation of the objective and the constraints into a speed function that defines the level-set propagation. The result is a 3D structural optimization technique that demonstrates outstanding flexibility in handling topological changes, the fidelity of boundary representation, and the degree of automation, comparing favorably with other methods in the literature based on explicit boundary variation or homogenization. We present two numerical techniques of conjugate mapping and variational regularization for further enhancement of the level-set computation, in addition to the use of efficient up-wind schemes. The method is tested with several examples of a linear elastic structure that are widely reported in the topology optimization literature.