Abstract：The PDE-based local level set method (LLSM) has higher computational efficiency than the conventional level set methods (LSMs) with global models. Nevertheless, the LLSM possesses no mechanism to nucleate new holes in the material domain for two-dimensional structures. A local-level-set-based algorithm is needed to be developed, with the ability of hole nucleation and bi-directional evolutionary features during topology optimization. A novel algorithm is proposed by combining the LLSM with the bi-directional evolutionary structural optimization (BESO) method. Two kinds of local level set models are constructed in this algorithm: one adopts the proposed discrete level set functions (DLSFs); the other chooses the local level set function (LLSF) of the LLSM. Firstly, a proposed bi-directional evolutionary algorithm using the DLSFs is implemented according to the optimization criteria of the BESO method until a stable topological solution is found. Then the LLSM is applied to further evolve the local details of topology and the shape of structure. The DLSFs are treated as nodal design variables in the bi-directional evolutionary algorithm. As nodal sensitivities, topological derivatives are taken instead of elemental sensitivities of the BESO method. The Shepard interpolation is selected in the sensitivity filtering to fit for the node-based variables and sensitivities. This algorithm transforms the final DLSFs into the initial LLSF by iteratively solving a distance-regularized equation (DRE). To increase computational efficiency of the LLSM, the DRE is included in the LLSM instead of the reinitialization equation. To eliminate the unnecessary diffusion effect, a new and balanced formulation of the diffusion term is introduced into the DRE. Despite that parts of the diffusion rates in the DRE are negative, a conditionally stable difference scheme under reverse diffusion constraints is formulated to ensure the numerical stability of DRE. Typical examples are used to demonstrate the effectiveness of the proposed algorithm, and the numerical results show higher convergence. The bi-directional evolutionary algorithm can not only nucleate new holes inside the design domain but also prevent multiple local minima of topology optimization. The LLSM is able to further improve the convergence to obtain at least one local optimal solution.
2018, Vol. 39 
