Adaptive level set topology optimization using hierarchical B-splines

Abstract

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.

Appendix Hierarchical B-splines versus truncated hierarchical B-splines

The influence of the truncation operation on the B-splines is investigated by solving the 2D beam problem, presented in Section 8.1, with HB-splines and THB-splines on an adaptive mesh with an initial refinement level $l_{\text {ref}}^{0} = 2$ and applying the refinement operation every 25 iterations first for the initial hole seeding approach and then for the level set/density scheme.

It should be noted that extra treatment is required to impose bounds on the design variables when working with HB-splines, as they do not constitute a PU. Bounds are enforced by clipping the variable values with the upper or lower allowed values. The clipping operation is only applied to the combined level set/density scheme as the density values should remain between 0 and 1.

1.1 A.1 2D beam with initial hole seeding

The problem setting is identical to the one presented in Section 8.1.1, but with a comparison of HB-splines and THB-splines. The obtained designs and corresponding strain energy values are given in Fig. 25. The linear, quadratic, and cubic designs only differ slightly. The remaining differences between the designs can be explained by the support size of the HB- and THB-splines that differs, as shown in Fig. 7. This is further supported by the maximum stencil size recorded for each design variables, i.e., the maximum number of design coefficients affected by the change in a specific design coefficient. The stencil sizes for the HB-splines are 8, 16, and 33 for the linear, quadratic, and cubic orders, against 2, 9, and 16 for the THB-splines. The runtime ratios and the efficiency factors are given in Table 11 and match closely for HB- and THB-splines.

Fig. 25

2D beam using adaptively refined meshes with initial hole seeding. Initial refinement level of $l_{\text {ref}}^{0} = 2$, maximum refinement level of $l_{\text {ref},\max \limits } = 4$. Comparison of HB- and THB-splines

Table 11 Performance in terms of computational cost for designs in Fig. 25

1.2 A.2 2D beam with simultaneous hole seeding

The problem setting is identical to the one presented in Section 8.1.2 but with a comparison of HB-splines and THB-splines. The obtained designs and corresponding final strain energy values are given in Fig. 26. The designs differ more significantly for HB- and THB-splines than when working with level set only (see Fig. 25). These differences can be partly explained by the maximum stencil size difference between HB- and THB-splines, i.e., 7, 16, and 37 for linear, quadratic, and cubic HB-splines against 2, 9, and 16 for the THB-splines. On top of the stencil size mismatch, these differences in the final designs can be explained by the clipping operation applied to the density values to keep them between 0 and 1 when working with HB-splines. Clipping yields non-differentiability with respect to the clipped values which can influence the optimization process. The runtime ratios and the computational gain factors are given in Table 12 and match closely for both HB- and THB-splines.

Fig. 26

2D beam using adaptively refined meshes with simultaneous hole seeding. Initial refinement level of $l_{\text {ref}}^{0} = 2$ and maximum refinement level of $l_{\text {ref},\max \limits } = 4$. Comparison of HB- and THB-splines

Table 12 Performance in terms of computational cost for designs in Fig. 26