Explicit Isogeometric Topology Optimization Method with Suitably Graded Truncated Hierarchical B-Spline

1  Introduction

TO is an engineering optimization method that seeks the optimal material distribution in a prescribed design domain with specified conditions. In the last three decades, a series of TO methods have been proposed and evolved, including solid isotropic material with penalization [1,2], evolutionary structural optimization [3,4], and level set method (LSM) [5–7]. Most of these traditional TO methods use the finite element method (FEM) as the solver for the unknown physical field over the design domain, which suffers from numerical instability and geometric discretization errors, due to the low discontinuity between adjacent elements and the disconnection between computer aided design (CAD) and computer aided engineering (CAE).

Hughes et al. [8] proposed isogeometric analysis (IGA) to substitute the FEM method, which uses the CAD mathematical primitive to represent the field unknowns of the partial differential equations (PDEs). Since IGA has the advantages of high boundary continuity between adjacent elements and elimination of geometric discretization errors, Qian [9] used the relative density of control points as design variables and embedded the physical design domain into a rectangular parametric design domain parametrized by the tensor product B-spline. Gao et al. [10] used ITO for the continuum structure using the enhanced density distribution function. Wang et al. [11] designed a new high-efficiency ITO, which improves the efficiency in three aspects: mesh scale reduction, acceleration strategy for the solver, and design variables reduction. Yu et al. [12] presented a multiscale ITO method, where the configuration and layout of microstructures are optimized synchronously. Zhao et al. [13] developed an ITO method for the design problems with an arbitrarily shaped design domain in the framework of T-spline based IGA. Chen et al. [14,15] applied the isogeometric boundary element based TO method to the design problems of acoustic structures for enhancing the sound-absorption performance. Due to the element-wise discreteness inherited from the variable density method, the ITO methods mentioned above easily get into trouble with the curse of dimensionality for generating accurate structural boundaries.

To resolve the issues that occurred in the variable density-based ITO methods, one alternative is the explicit TO method by Guo et al. [16] using MMC, which originated from the fact that the TO results can be viewed as the optimal distribution of a limited number of components. In the explicit TO method, the geometric characteristic parameters of a limited number of discrete components, e.g., the coordinates of the centroid and the inclination angle of a component, are treated as the design variables, by which the explicit geometric information of structural boundaries can be generated directly from the converged design variables. Zhang et al. [17] devised the finite-circle method to avoid the overlapping of components for the explicit TO method. Hou et al. [18] adopted the NURBS-based IGA method to obtain the structural response analysis and sensitivity analysis for the MMC-based explicit TO method. Then, an explicit ITO method is proposed with the geometric parameters of moving morphable voids (MMV) [19] treated as the design variables, which utilizes two mesh resolution schemes at different discretization levels. Gai et al. [20] proposed an explicit ITO method that uses the closed B-splines curves to describe the MMV. Zhang et al. [21] integrated the MMV method and the NURBS-based IGA to optimize the 3D shell structure. Since the sensitivity analysis of the explicit ITO depends on the structural boundaries, the accuracy of sensitivities around the structural boundaries determines the optimization accuracy of the explicit ITO method. However, due to the tensor product structure of B-splines, the analytical mesh of explicit ITO cannot perform local refinement of the boundary through the adaptive refinement technique, which leads to a severe contradiction between the analytical efficiency and optimization accuracy.

To resolve the contradiction between the efficiency and accuracy of the explicit ITO method, the problem of enforcement of global refinement for IGA mesh must be solved first. Currently, the main solutions for this issue are listed as: hierarchical B-splines [22], T-splines [14], LR-splines [23] and hierarchical box-splines [24]. Among these methods, the hierarchical B-spline is the most widely used one because of its provable linear independence and subdivision property [25]. Noël et al. [26] used truncated hierarchical B-splines to describe the parametrization of the level set function for LSM, which improved the accuracy of the structural boundaries with limited increased computational effort for the updating of design variables. Xie et al. [27] proposed an adaptive explicit ITO method based on hierarchical B-splines, which refines the isogeometric mesh associated with the regions near the optimized structural boundaries locally and achieves a significant balance between the computational efficiency and optimization accuracy. It has been stated that the THB is superior to the hierarchical B-splines in adaptive IGA because it improves the conditionality of the stiffness matrix and reduces the bandwidth of the stiffness matrix [28]. Subsequently, Xie et al. [29] put forward the fully adaptive explicit ITO method with more advanced computing efficiency in terms of truncated hierarchical B-splines, where the isogeometric mesh is locally refined and coarsened simultaneously and the corresponding basis function space recovers the property of partition of unity. However, in the current adaptive explicit ITO method, tiny holes are easily observed in the optimized structures, which should be eliminated for the benefits of both the numerical robustness of the optimization process and the manufacturing of optimized structures. The underlying reason for the aforementioned weaknesses of the adaptive explicit ITO method is that the analytical accuracy of the THB-based IGA method is deteriorated in the regions of structural boundaries, resulting from the non-constrained hierarchical differences between the non-vanishing THB basis functions.

As pointed out in [30], the admissible mesh can effectively improve the accuracy of the THB-based IGA method. To overcome the shortcoming of the current adaptive explicit ITO method, the present work proposes an advanced adaptive explicit ITO method using MMC in terms of suitably graded truncated hierarchical B-spline (SGTHB-ITO-MMC). Based on the marking strategy proposed in [31], this work develops a constrained marking strategy for imposing the suitably graded constraint on the hierarchical mesh, which enlarges the set of active elements marked to be refined and shrinks the set of deactivated elements marked to be coarsened. With the imposed suitably graded constraint, the accuracy and robustness of the adaptive explicit ITO method are improved. Besides, to further improve the efficiency in computing the TDF values for the hierarchical mesh, we designed an improved TDF calculation strategy for the deletion of the micro components and the local TDF computing strategy for the Gaussian quadrature points.

The rest of this paper is organized as follows: Section 2 “Suitably graded THB based ITO using MMC” introduces the basic theory of the SGTHB-ITO-MMC method, including the definitions corresponding to THB and admissible hierarchical meshes. The optimization model and marking strategy as well as the improved TDF calculation strategy for SGTHB-ITO-MMC are introduced in detail in Section 3 “Optimization model of SGTHB-ITO-MMC”. Section 4 “Overview scheme for SGTHB-ITO-MMC” presents an overall procedure for SGTHB-ITO-MMC. In Section 5 “Numerical examples”, the effectiveness of SGTHB-ITO-MMC is validated by several 2D and 3D benchmarks. Finally, this work is concluded in Section 6 “Conclusions”.

2  Suitably Graded THB Based ITO Using MMC

This section provides the theoretical foundation for the proposed SGTHB-ITO-MMC, which mainly includes the following two aspects: the construction of THB and the related concepts of admissible hierarchical meshes.

2.1 Truncated Hierarchical B-Splines

2.1.1 Hierarchical B-Spline Spaces

For a given knot vector Ξ={ξ1,ξ2,⋯,ξn+p+1}, n and p represent the number of control points and the order of basis function, respectively. Supposing that several function spaces of B-spline are nested: V0⊂V1⊂…⊂Vn−1, these nested spaces are obtained by a sequence of knot vectors {Ξ0,Ξ1,…,ΞN−1} and defined on an initial domain Ω0. For multivariate case, Vl(l=1,2,…,N−1) is constructed by the tensor product structure of the univariate basis function space with the knot vector Ξl bisected from Ξ0 l-times. Then, the knot intervals constituting a mesh Ql and an arbitrary knot interval are referred to as an element of level l. Define Ω0⊇Ω1⊇…⊇ΩN−1 as a nested sequence of domains, where Ωl is the domain of level l and represents the union of the elements on l−1 level marked to be subdivided. Subsequently, a hierarchical mesh HE is made up of active elements on different levels and is defined as:

HE={Q∈Γl,l=0,⋯,N−1}(1)

where Γl represents the union of active cells of level l with Q⊂Ωl∧Q⊄Ωl+1.

Once the hierarchical mesh HE is determined [32], the corresponding hierarchical B-spline function space can be constructed by the following steps:

•   Initialization: H(HE)0={β∈B0: supp β≠∅};

•   A recursive formula: H(HE)l+1=HAl+1∪HBl+1,l=0,1,…,N−2,

where HAl+1={β∈H(HE)l: supp β⊄Ωl+1} and HBl+1={β∈Bl+1: supp β⊆Ωl+1},

with supp β={x:β(x)≠0∧x∈Ω0};

•   H(HE)=H(HE)N−1.

2.1.2 Truncated Operation

The hierarchical B-splines basis functions are inevitably spanning the active elements belonging to different levels, which results in the lacking of the essential property of partition of unity for numerical analysis. To recover this important property, a truncated operation should be applied to the two-scale relationship existing in the basis functions of hierarchical B-splines on two consecutive levels, which is formulated as:

truncl(s)=∑β∈Bl+1,supp β∉Ωl+1cβl+1(s)β,wheres=∑β∈Bl+1cβl+1(s)β,s∈Vl(2)

where truncl(s) is a truncated active basis function of s of level l, cβl+1(s) is termed as the coefficient of β belonging to level l+1. Therefore, the THB function space ℋ