| 講演要旨： |
A level set based topological shape optimization method is developed incorporating topological derivatives for nucleation in a generic domain. In the level set method, the initial domain is kept fixed and its boundary is represented by an implicit moving boundary (IMB) embedded in the level set function, which facilitates to handle complicated topological shape changes. The sensitivity of objective function and constraint with respect to the boundary variations are efficiently determined using an adjoint design sensitivity analysis method. The optimization process is to move the IMB, according to the obtained sensitivity with respect to the boundary variations, under the condition of allowable material volume. The “Hamilton-Jacobi (H-J) equation”and computationally robust numerical technique of “up-wind scheme” lead the initial implicit boundary to an optimal one according to the normal velocity field, while minimizing the objective function of compliance as well as satisfying the required constraint of the allowable volume. For the optimization process, the required velocity field to update the H-J equation is determined from the descent direction of Lagrangian derived from Kuhn-Tucker optimality conditions. The optimization process would terminate when the normal velocity vanishes throughout the whole domain. The major drawback of existing level set approaches is that it has no nucleation mechanism. To overcome this difficulty, a topological derivative is introduced to embed the nucleation capability in the level set framework. The topological derivative is defined as the gradient of performance functional when an infinitesimal hole is introduced in the generic domain. Based on the asymptotic regularization concept, the topological derivative is considered as the limit of shape derivative as the radius of a hole approaches to zero. It turns out that the developed method is very efficient and able to create the sufficient number of holes using indicators obtained from the topological derivatives whenever and wherever necessary during the optimization process. In numerical examples, together with linear elasticity problems, some extensive works are demonstrated such as heat conduction problems, energy flow problems, and geometrically nonlinear structures. In the existing level set based methods, either implicit function or ersatz material is introduced to represent the IMB and domain in initial reference domain. However, these approaches lead to a convergence difficulty in geometrically nonlinear problems. In this extensive research, we demonstrate a topological shape optimization method using actual boundary and domain converted from the IMB. Unstructured mesh is utilized for more accurate response and sensitivity analyses in each optimization step. The necessary design sensitivity or the velocity field is computed efficiently in the actual domain. The velocity field outside the domain is obtained using an extension method based on a fast marching algorithm. Since homogeneous material property and actual boundary are utilized, the well known convergence difficulty is significantly relieved. The necessity of geometrically nonlinear topology optimization and the applicability to large deformation problems are demonstrated. |
