Numerical instabilities in level set topology optimization with the extended finite element method

This paper studies level set topology optimization of structures predicting the structural response by the eXtended Finite Element Method (XFEM). In contrast to Ersatz material approaches, the XFEM represents the geometry in the mechanical model by crisp boundaries. The traditional XFEM approach augments the approximation of the state variable fields with a fixed set of enrichment functions. For complex material layouts with small geometric features, this strategy may result in interpolation errors and non-physical coupling between disconnected material domains. These defects can lead to numerical instabilities in the optimized material layout, similar to checker-board patterns found in density methods. In this paper, a generalized Heaviside enrichment strategy is presented that adapts the set of enrichment functions to the material layout and consistently interpolates the state variable fields, bypassing the limitations of the traditional approach. This XFEM formulation is embedded into a level set topology optimization framework and studied with material-void and material-material design problems, optimizing the compliance via a mathematical programming method. The numerical results suggest that the generalized formulation of the XFEM resolves numerical instabilities, but regularization techniques are still required to control the optimized geometry. It is observed that constraining the perimeter effectively eliminates the emergence of small geometric features. In contrast, smoothing the level set field does not provide a reliable geometry control but mainly improves the convergence rate of the optimization process.