Structural shape and topology optimization using an implicit free boundary parametrization method

In this paper, an implicit free boundary parametrization method is presented as an effective approach for simultaneous shape and topology optimization of structures. The moving free boundary of a structure is embedded as a zero level set of a higher dimensional implicit level set function. The radial basis functions (RBFs) are introduced to parametrize the implicit function with a high level of accuracy and smoothness. The motion of the free boundary is thus governed by a mathematically more convenient ordinary differential equation (ODE). Eigenvalue stability can be guaranteed due to the use of inverse multiquadric RBF splines. To perform both shape and topology optimization, the steepest gradient method is used to determine a velocity function. To guarantee that the optimal solution be in the feasible domain, a bi-sectioning algorithm is proposed to obtain the Lagrange multiplier. The velocity function is extended in a physically meaningful way and its dis-continuity at the free boundary is eliminated by using a smoothing filter. The usual periodic reinitialization process is avoided to allow for the nucleation of new holes. It is shown that simultaneous shape and topology optimization can be obtained and a mass-conservative stable evolution guaranteed due to the present extension velocities. The proposed method is implemented in the framework of classical minimum compliance design and its efficiency and accuracy over the existing methods are high-lighted. Numerical examples can demonstrate its excellence in accuracy, convergence speed and insensitivity to initial designs in structural shape and topology optimization of two dimensional (2D) problems.