| This thesis discusses two topics pertaining to structural topology optimization: reliability-based topology optimization and the topological derivative.We first perform reliability-based topology optimization by combining reliability analysis and material distribution topology design methods to design linear elastic structures subject to random loadings. Both component reliability and system reliability are considered. In component reliability, we satisfy numerous probabilistic constraints which quantify the failure of different events. In system reliability, we satisfy a single probabilistic constraint which encompasses the component events. To solve the probabilistic optimization problem, we use a variant of the single loop method, which eliminates the need for an inner reliability analysis loop. The proposed method is amenable to implementation with existing deterministic topology optimization software, and hence suitable for practical applications.The topological derivative provides the variation of a functional when an infinitesimal hole is introduced in the domain. It was first introduced in the context of topology optimization as means to nucleate holes within a structure. Here we use the topological derivative to approximate the energy release rate field corresponding to a small crack at any boundary location and at any orientation. Our proposed method offers significant computational advantages over current finite element based methods since it requires a single analysis whereas the others require a distinct analysis for each crack size-location-orientation combination. Moreover, the proposed method evaluates the topological derivative in the non-cracked domain which eliminates the need for tailored meshes in the crack region.To improve our fracture mechanics analyses we next propose an algorithm to obtain higher order terms in the topological derivative expansion corresponding to the introduction of a circular hole, not a crack, in this preliminary study. In this way, we are able to obtain better estimates for the response functional when larger circular holes, and eventually cracks, are introduced into the domain. The primary element of our algorithm involves the asymptotic expansion for the stress on the hole boundary as the hole size approaches zero. |
