An adaptive X-FEM for shape optimization using h-adaptive scheme and posteriori error estimation

A procedure is proposed to generate optimal grid with minimal user intervention while keeping a prescribed level of accuracy, using an adaptive X-FEM and applied to shape optimization. In the finite element method (FEM), mesh generation is still a time consuming process, especially when subscale features such as small holes, crack, and inclusions are present, especially for an iterative optimization process or crack propagation analysis. One of the viable strategies for dealing with the problems is X-FEM proposed by Belyschhko. The key point of X-FEM is to represent arbitrary discontinuities by means of the discontinuous enrichment functions associated with nodes through the notion of partition of unity. In spite of various advantages of X-FEM, however, there are several obstacles for practical applications. Because of using a uniform background mesh and additional degree of freedoms for enrichment, an X-FEM is usually computationally more expensive than traditional finite element method. Furthermore, there are often accuracy problems. Although the performance of most elements is equivalent to or slightly superior to that of the FEM if they have the same shapes, the accuracy tends to be poor locally near a geometric boundary. An adaptive procedure can be very effective in the shape optimization area. The convergence rate of a gradient-based optimization process may be affected if the FE-mesh is adapted during the optimization process in order to restrict the discretization error. For an automatic procedure of optimal mesh generation, an h-adaptive scheme and a posteriori error estimation obtained by a post-processing process are utilized. The procedure is shown by 2-D shape optimization examples.