This work puts forward an explicit isogeometric topology optimization (ITO) method using moving morphable components (MMC), which takes the suitably graded truncated hierarchical B-Spline based isogeometric analysis as the solver of physical unknown (SGTHB-ITO-MMC). By applying properly basis graded constraints to the hierarchical mesh of truncated hierarchical B-splines (THB), the convergence and robustness of the SGTHB-ITO-MMC are simultaneously improved and the tiny holes occurred in optimized structure are eliminated, due to the improved accuracy around the explicit structural boundaries. Moreover, an efficient computational method is developed for the topological description functions (TDF) of MMC under the admissible hierarchical mesh, which consists of reducing the dimensionality strategy for design space and the locally computing strategy for hierarchical mesh. We apply the above SGTHB-ITO-MMC with improved efficiency to a series of 2D and 3D compliance design problems. The numerical results show that the proposed SGTHB-ITO-MMC method outperforms the traditional THB-ITO-MMC method in terms of convergence rate and efficiency. Therefore, the proposed SGTHB-ITO-MMC is an effective way of solving topology optimization (TO) problems.

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 [

Hughes et al. [

To resolve the issues that occurred in the variable density-based ITO methods, one alternative is the explicit TO method by Guo et al. [

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 [

As pointed out in [

The rest of this paper is organized as follows:

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.

For a given knot vector

Once the hierarchical mesh

Initialization:

A recursive formula:

where

with

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:

An illustration of hierarchical B-splines basis function space and its corresponding THB basis function space are shown in

To ensure that the hierarchical computational mesh of the proposed SGTHB-ITO-MMC model satisfies the specified suitably graded constraints, this section reviews the key definitions related to the admissible hierarchical mesh for THB [

The mathematical optimization model of the SGTHB-ITO-MMC method is firstly proposed in this section. Then, the constrained marking strategy is devised by integrating the fully adaptive marking strategy with the suitable constraint imposed on the hierarchical mesh, which controls the local refinement and coarsening of the hierarchical mesh for SGTHB-ITO-MMC. Finally, the improved TDF calculation strategy is described to improve the efficiency of SGTHB-ITO-MMC.

According to the optimization model presented in [

In the above formula,

It should be noted that the ersatz material model and the sensitivity analysis are rather straightforward for SGTHB-ITO-MMC, which can be referred to [

To implement the adaptivity of hierarchical mesh under the specified suitably graded constraint, a constrained marking strategy is developed. The proposed marking strategy in this work consists of three aspects: triggering strategy, local refining, and local coarsening strategies under suitably graded constraints, where the triggering strategy is calculated in the same way as the one proposed in [

Once the marking strategy is triggered, the initial local refining of the marking strategy formulated as

If the initial refining element set marked to be refined is obtained, the updating procedures for the initial marked set are described in Algorithms 1 and 2 to fulfill the specified suitably graded constraint during the local refinement of the hierarchical mesh. Algorithm 1 takes the original hierarchical mesh

When the local refinement of the hierarchical mesh is accomplished, the elements away from the structural boundaries should be marked to be coarsened for SGTHB-ITO-MMC. Similar to the refining strategy, the local coarsening strategy is also divided into two stages: (1) obtaining the initial marking set to be coarsened; (2) updating the coarsened element set. According to the TDF values of the Gaussian quadrature points of the children for each deactivated element, the status of that deactivated element to be refined or not is determined as follows:

Then, the marked coarsened set is updated by the procedures presented in Algorithm 3 to guarantee the admissible requirement resulting from the suitably graded constraint. In Algorithm 3, the deactivated element should be reactivated and its children should remain unchanged in the hierarchical mesh, if its coarsening neighborhood with admissibility

To improve the computational efficiency of SGTHB-ITO-MMC method, this work proposes the following strategy for the benefit of accelerating the calculation of TDF values for the hierarchical mesh. On the one hand, the computing complexity of TDF is reduced by removing the unnecessary TDF computing associated with micro components, with the micro components removed from the design space. On the other hand, the TDF generation is calculated locally inconsistent with the hierarchy feature of the hierarchical mesh.

To avoid the numerical singularity, the lower limit of the geometric design variables of MMC is set to an extremely small positive value. It can lead to the numerical phenomenon that the islanding micro component exists in the optimized structure, as shown in

These islanding micro components result in the continuously refined mesh around them and that leads to the increasing deteriorative mesh quality. It also results from an excessive number of degrees of freedom, which decreases the computational efficiency of IGA. Therefore, it is necessary to reduce the dimensionality of the geometric design space by removing the design variables associated with micro components in the optimization process. If a component satisfies the requirements of

Apart from the micro components, the redundant TDF calculation also resulted from the way of calculating the TDF values for determining the ersatz material model. In this model, the Gaussian points of the hierarchical mesh inside the active elements are properly graded, and the Gaussian quadrature points do not span different levels of active elements, which enables the local updating of TDF for the hierarchical mesh. Besides, the active elements away from the structural boundaries do not require calculating the TDF values of the Gaussian quadrature points, since SGTHB-ITO-MMC is an essentially boundary-driven TO method. Therefore, the values of TDF of a structural component can be calculated only by Gaussian quadrature points near the boundary of the component, as shown in

In the proposed local TDF computing strategy, the set of active Gaussian quadrature points is defined in

This section mainly describes the execution process of the SGTHB-ITO-MMC method. The execution flow of the SGTHB-ITO-MMC method is shown in

Among them, the input module takes the mathematical model parameters in

Four numerical examples are used to demonstrate the effectiveness of the SGTHB-ITO-MMC framework with an improved TDF calculation strategy, which is running on MATLAB R2021a with Windows 10 as the software operating system, Intel (R) Core (TM) i5-10210U CPU @ 1.60 GHz–2.11 GHz and 16 GB of RAM as the hardware system. These examples adopted MMA as the optimizer. Young's modulus

Through the Messerschmidt-Bolkow-Blohm beam (MBB) problem,

This section chooses the MBB beam as the first example to verify the effectiveness of SGTHB-ITO-MMC.

The convergence histories and the variations of the number of DOFs are illustrated in

For validating the effectiveness of the improved TDF calculation strategy described in

This section discusses the effectiveness of the SGTHB-ITO-MMC for the short beam and validates the improved TDF calculation strategy under two different dimensionalities of the geometric design space.

The problem setting and initial design of MMC are described in

To verify the effectiveness of the proposed SGTHB-ITO-MMC in an enlarged geometric design space, the initial designs are shown in

Based on the optimized results illustrated in

In this section, the SGTHB-ITO-MMC method is extended to the 3D design problem by taking the 3D cantilever as the last numerical example. As shown in

The initial structural components layout for the 3D cantilever, and two intermediate optimized structural layouts, as well as the final convergent result, are shown in

This work proposes an explicit ITO method in the framework of suitably graded truncated hierarchical B-splines, which is established based on taking the suitably graded constraint into the consideration of the marking strategy. Furthermore, an improved TDF calculation strategy is put forward by reducing the dimensionality of the geometric design space and generating the TDF values locally for the active elements of the hierarchical mesh. By incorporating the suitably graded constraint into the explicit adaptive ITO method, the drawbacks existing in the optimal MMC designs, such as component discontinuities, tiny holes inside the structure, and zigzag boundaries, can be either eliminated or mitigated to a large extent. Besides, with the stricter suitably graded imposed, the convergence rate of SGTHB-ITO-MMC is increasingly accelerated. With the aid of the proposed improved TDF calculation strategy, the computational efficiency in generating the TDF values and convergence rate are simultaneously improved for SGTHB-ITO-MMC without affecting the optimal designs. Moreover, the proposed method can be applied to 3D design problems.

In the current explicit adaptive ITO method, the versatility is limited by the basic geometric configuration of MMC. To overcome the aforementioned issue, we will extend the current explicit adaptive ITO method to the MMV framework in the future. Furthermore, the stress-constrained problem solved by the adaptive explicit ITO method is also taken as one of our aims.