A Physically Meaningful Level Set Method for Topology Optimization of Structures

This paper aims to present a physically meaningful level set method for shape and topology optimization of structures. Compared to the conventional level set method which represents the design boundary as the zero level set, in this study the boundary is embedded into non-zero constant level sets of the level set function, to implicitly implement shape fidelity and topology changes in time via the propagation of the discrete level set function. A point-wise nodal density field, non-negative and value-bounded, is used to parameterize the level set function via the compactly supported radial basis functions (CSRBFs) at a uniformly defined set of knots. The set of densities are used to interpolate practical material properties in finite element approximation via the standard Lagrangian shape function. CSRBFs knots are supposed to be consistent with finite element nodes only for the sake of numerical simplicity. By doing so, the discrete values of the level set function are assigned with practical material properties via the physically meaningful interpolation. The original more difficult shape and topology optimization of the Hamilton-Jacobi partial differential equations (PDEs) is transformed to a relatively easier size optimization of the nodal densities, to which more efficient optimization algorithms can be directly applied. In this way, the dynamic motion of the design boundary is just a question of transporting the discrete level set function until the optimal criteria of the structure is satisfied. Two widely studied examples are applied to demonstrate the effectiveness of the proposed method.