Parametric Shape and Topology Optimization via Radial Basis Functions, Partition of Unity and Level Set Method

This paper presents a parametric shape and topology optimization approach via radial basis functions (RBF), partition of unity (POU) and the level set method. This approach is a further development of the RBF-based level set method for shape and topology optimization. Level set method allows for easy treatment of shape and topological changes of dynamic interfaces and moreover, the RBF based parameterized representation makes the implementation of the level set method even simpler and more flexible. In the proposed approach, the curve embedding function (level set function) is represented via RBF and the evolution of the implicit boundary is determined by the parameters of RBF. In other words, the optimal boundary of a structure is achieved by updating the parameters. Through this approach, the shape and topology optimization problem is converted to a parametric optimization problem. By combining the results of shape sensitivity analysis and the Hamilton-Jacobi equation, we obtain the sensitivities of the objective function and the constraints with respect to parameters. Thereafter, the parameters are updated via a proper optimization algorithm. In existing methods, RBF with global support leads to dense coefficient matrices, and as the number of RBFs becomes large, the coefficient matrices will become ill-conditioned. We propose to utilize the POU method to partition the design domain into several small domains in which the curve embedding function can be more efficiently interpolated and the global solution can be obtained by combination of these local solutions. This method is more flexible and efficient tha