Level set based topological shape optimization of geometrically nonlinear structures using unstructured mesh

Using hyperelastic materials and unstructured mesh, a level set based topological shape optimization method is developed for geometrically nonlinear structures in total Lagrangian framework. In the level set method, the initial domain is kept fixed and its boundary is represented by an implicit moving boundary embedded in a level set function, which facilitates to handle complicated topological shape changes and eventually leads the initial implicit boundary to an optimal one according to the normal velocity field while both minimizing the objective function of instantaneous structural compliance and satisfying an allowable material volume constraint. In existing level set based methods, an initial reference domain or an ersatz material is employed for the penalization of whole domain to represent the current domain. However, these approaches end up with a convergence difficulty in nonlinear response analysis due to the inaccurate tangent stiffness. To overcome this difficulty, taking advantage of the obtained level set function, the current structural boundary is actually represented using a Delaunay triangulation scheme and a hyperelastic material law is employed to handle the large strain problem. The required velocity field in the actual domain to update the level set equation is determined from the descent direction of Lagrangian derived from optimality conditions. The velocity field outside the actual domain is determined through a velocity extension scheme based on a fast marching method. Since homogeneous material property and actual boundary are utilized, the convergence difficulty is significantly relieved.