Velocity field level-set method for topological shape optimization using freely distributed design variables

The velocity field level-set topological shape optimization method combines the implicit representation in the standard level-set method and the capabilities of general mathematical programming algorithms in handling multiple constraints and additional design variables. The key concept is to construct the normal velocity field using basis functions and the velocity design variables at specified points (referred to as velocity knots) in the entire design domain. In this study, the velocity design variables are decoupled from the level-set grid points. Making use of this property, we can adaptively change the arrangement of the velocity knots as the structural boundary evolves. This provides more design freedom in the optimization and allows for a significant reduction in the number of design variables. Several numerical examples in two- and three-dimensional design domains are presented to demonstrate the robustness and efficiency of the proposed method. We also show that changing the number of velocity knots may implicitly exert certain control on topological complexity and length scale.