Level Set Based Shape Optimization of Geometrically Nonlinear Structures

Abstract

Using the level set method and topological derivatives, a topological shape optimization method that is independent of initial topology is developed for geometrically nonlinear structures in Total Lagrangian framework. In nonlinear topology optimization, response analysis may not converge due to relatively sparse material distribution driven by the conventional topology optimization such as homogenization and density methods. In the level set method, the initial domain is kept fixed and its boundary is represented by an implicit moving boundary embedded in the level set function, which facilitates to handle complicated topological shape changes. 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 both minimizing the objective function of instantaneous structural compliance and satisfying the required constraint of allowable material volume. In this paper, based on the obtained level set function, structural boundaries are actually represented in the response analysis. The developed method is able to create holes whenever and wherever necessary during the optimization and minimize the compliance through both shape and topological variations at the same time. The required velocity field in the initial domain to update the H-J equation is determined from the descent direction of Lagrangian derived from optimality conditions. The rest of velocity field is determined through a velocity extension method. Since the homogeneous material property and explicit boundary are utilized, the convergence difficulty is effectively prevented.