Level set based shape optimization using trimmed hexahedral meshes

This paper presents a new approach for level set based shape optimization using trimmed hexahedral meshes that geometrically conform to shape and topology changes during the optimization process. The marching cubes algorithm is employed to generate trimmed hexahedral meshes at each optimization iteration by splitting a regular background hexahedral mesh with the zero-isosurface of a level set function. A simple method is proposed to handle nonconforming polyhedral elements at the interface due to the face ambiguity arising in the marching cubes algorithm. The motion of the zero-isosurface of a level set function is obtained from the solution of the Hamilton–Jacobi transport equation using a normal velocity field defined on the regular background hexahedral mesh. Numerical results show that the present approach provides a natural and effective tool to solve shape optimization problems relying on explicitly discretized meshes of the shapes throughout the optimization process.