We study a level set method for numerical shape optimization of elastic structures. Our approach combines the level set algorithm of Osher and Sethian with the classical shape gradient. Although this method is not specifically designed for topology optimization, it can easily handle topology changes for a very large class of objective functions. Its cost is moderate since the shape is captured on a fixed Eulerian mesh.

 2d cantilever (100K) 2d bridge. Multi-load optimization (104K) 3d cantilever (1.4M) 3d electric mast (276K) Starfish under hydrostatic pressure (2.6M) Chair (2.2M) 3d grip - shape evolution (588K) 3d grip - deformation of the optimal shape (772K) Another 3d grip (616K) Car suspension triangle (multiple loads on an unstructured mesh) (942K) 2d cantilever in nonlinear elasticity (517K)

 The level set method, based on the classical shape derivative, is seen to easily handle boundary propagation with topological changes. However, in practice it does not allow for the nucleation of new holes (at least in 2-d). For this reason, we couple the level set method with another approach, known as the bubble or topological gradient method, which is precisely designed for introducing new holes in the optimization process. The coupling of these two method yields an efficient algorithm which can escape from local minima in a given topological class of shapes. Both methods relies on a notion of gradient computed through an adjoint analysis, and have a low CPU cost since they capture a shape on a fixed Eulerian mesh. The main advantage of our coupled algorithm is to make the resulting optimal design largely independent of the initial guess.

 2d cantilever starting from the full domain, with topological gradient computations (424K) 2d electric mast starting from the full domain, with topological gradient computations (580K) 2d bridge, with topological gradient computations (572K) 3d cantilever with topological gradient computations (1.7M) Negative Poisson's ratio (evolution of the shape using level set + topological gradient) (5.0M) Negative Poisson's ratio (animation of the final design) (496K)

(Last updated: 8 Jun 2005, 16:20)