Adaptive level set topology optimization using hierarchical B-splines

This paper presents an adaptive discretization strategy for level set topology optimization of structures based on hierarchical B-splines. This work focuses on the influence of the discretization approach and the adaptation strategy on the optimization results and computational cost. The geometry of the design is represented implicitly by the iso-contour of a level set function. The extended finite element method is used to predict the structural response. The level set function and the state variable fields are discretized by hierarchical B-splines. While first-order B-splines are used for the state variable fields, up to third-order B-splines are considered for discretizing the level set function. The discretizations of the design and the state variable fields are locally refined along the material interfaces and selectively coarsened within the bulk phases. For locally refined meshes, truncated B-splines are considered. The properties of the proposed mesh adaptation strategy are studied for level set topology optimization where either the initial design is comprised of a uniform array of inclusions or inclusions are generated during the optimization process. Numerical studies employing static linear elastic material/void problems in 2D and 3D demonstrate the ability of the proposed method to start from a coarse mesh and converge to designs with complex geometries, reducing the overall computational cost. Comparing optimization results for different B-spline orders suggests that higher interpolation order promote the development of smooth designs and suppress the emergence of small features, without providing an explicit feature size control. A distinct advantage of cubic over quadratic B-splines is not observed.