Geometry projection methods for shape and topology optimization

The objective of this thesis is to develop fictitious domain methods for shape and topology optimization of continuum structures in which an unambiguous definition of the geometry is available. We use fictitious domain methods because they simplify the response analysis by eliminating the need for remeshing when the design changes and because they are naturally suitable for topology optimization. Here, the unambiguous geometry is projected onto the analysis space by means of a filtering technique. The filter is based on a bounded sample window whose diameter is proportional to the local grid spacing in the mesh used for response analysis. Thus, the errors associated with both the geometry projection and the response discretization vanish in the limit of mesh refinement. Accordingly, the numerical response solution converges to the continuum solution of the underlying boundary value problem. This projection algorithm is used in conjunction with (a) parameterized geometry models to develop a method for fixed topology shape optimization and (b) the topological derivative to develop a method for variable topology shape optimization.