Density and Level Set-XFEM Schemes for Topology Optimization of 3-D   Structures
  As the capabilities of additive manufacturing techniques increase, topology optimization provides a promising approach to design geometrically sophisticated structures which can be directly manufactured. Traditional topology optimization methods aim at finding the conceptual design but often lack a sufficient resolution of the geometry and structural response, needed to directly use the optimized design for manufacturing. To overcome these limitations, this paper studies the viability and characteristics of the eXtended Finite Element Method (XFEM) in combination with the Level-Set Method (LSM) for topology optimization of three dimensional structural design problems. The LSM describes the geometry by defining the nodal level set values via explicit functions of the optimization variables. The structural response is predicted by a generalized version of the XFEM. The LSM-XFEM approach is compared against results from a traditional Solid Isotropic Material with Penalization (SIMP) method for two-phase ``solid-void'' and ``solid-solid'' problems. The numerical results demonstrate that the LSM-XFEM approach can describe crisply the geometry and predict the structural response of complex three-dimensional structures with acceptable accuracy even on coarse meshes. However, the LSM-XFEM studied here lacks a robust and intuitive formulation to control the minimum feature size, and the optimization results may depend on the initial design. 
Carlos H. Villanueva and Kurt Maute
math.OC