  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. An immersed boundary technique, here the extended finite element
method, is used to predict the structural response. Both the level set function
and the state variable fields are discretized by hierarchical B-splines. While
only 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 level set function and the state variable fields are
locally refined along the material interfaces and selectively coarsened within
the bulk phases. For locally refined/coarsened 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 and fine features, reducing the overall computational cost.
Comparing optimization results for different B-spline orders suggests that
higher order basis functions 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.
