Topology Optimization for Quasi-Periodic Cellular Structures Using Hybrid Moving Morphable Components and the Density Approach

Abstract

Porous hierarchical structures are extensively utilized in engineering for their high specific strength, enhanced corrosion resistance, and multifunctionality. Over the past two decades, multiscale topology optimization for these structures has garnered significant attention. This paper introduces a novel hybrid MMCs (Moving Morphable Components)–density topology optimization method for quasi-periodic cellular structures. The term ‘quasi-periodic’ refers to microstructures whose different macroscopic points exhibit similar topologies with varying parameters. The primary concept involves using the MMC method to describe microstructural topology, while employing variable density to depict macro layouts. This approach leverages the advantage of MMCs in explicitly describing structural topology alongside the variable density of arbitrary microstructures. Sensitivity analyses of the optimization functions concerning design variables are shown, and a gradient optimization solver is employed to solve the optimization model. The examples effectively show the efficacy of the proposed method, illustrating that quasi-periodic cellular structures outperform single-scale solid structures.

1. Introduction

Cellular structures are abundant in natural biological entities, and are renowned for their exceptional structural performance [1]. The biomimetic emulation of these cellular structures has garnered significant interest across various industries including the aerospace, automotive, and biomedical fields. In recent years, propelled by the rapid advancements in advanced manufacturing technologies, particularly additive manufacturing, it has become feasible to fabricate cellular structures with increasingly intricate geometries [2,3,4]. Consequently, the systematic design of cellular structures tailored to meet specific engineering requirements has emerged as a pivotal focus.

The topology optimization (TO) method is an automated design approach that generates novel and often unexpected design solutions by determining the optimal material distribution to meet specified design objectives and constraints [5,6,7,8,9]. It is of great significance to search for a cellular structure with excellent performance and establish a design theory based on the TO method to promote the engineering application of hierarchical structures. Cellular structures, unlike single-scale structures, encompass two or more interconnected scales, posing challenges for topology optimization. A straightforward yet computationally costly approach involves discretizing the domain using fine meshes that span all scales. Subsequently, procedures such as those used in traditional TO can be applied [10]. However, this method typically necessitates a dense mesh, leading to significant computational overheads.

Another approach to mitigate computational demands is to separate the two scales using the multiscale finite element method [11] or homogenization method [12,13]. Combining topological optimization methods with homogenization theory, some scholars have carried out research on microstructure topology optimization methods with specific/specific properties, and obtained a large number of microstructure configurations with excellent properties, such as materials with a negative Poisson ratio, materials with zero expansion and materials with a high permeability. Notable studies include those by Rodrigues et al. [14] and Xia et al. [15,16], where both macro-scale material distribution and corresponding local microstructures were optimized concurrently. However, these methods still entail significant computational costs due to the necessity of inverse-homogenization for all macro elements, and they often lack connectivity between neighboring microstructures. To address these challenges, Yan et al. [17,18] proposed a concurrent design for periodic multiscale structures that provides simpler topological optimization formulas with fewer computational resources being required. Based on the homogenization method, the performance transfer relationship between the two scales is established, and the optimization variables of two levels are unified in a simple optimization model, and there is only one macro finite element calculation and one micro homogenization calculation for each optimization step. Therefore, this method shows advantages in terms of its simple optimization formulation, low calculation and easy implementation, and has been successfully applied to many physical problems, such as the thermodynamic coupling problem, dynamic problem, uncertainty problem and multifunctional problem. However, since such structures contain the same microscopic structure, the ability to change the material properties in the macro domain is limited, thus reducing the room for improvement in structural properties. In view of this, some scholars have proposed a TO method of periodic multiscale structure according to partitions. In this kind of algorithm, the domain can be divided into a series of regions either artificially or according to specific criteria, different regions have different microstructure configurations, and the same region is periodically filled by a single microstructure. The core problem of this method is how to find a suitable and efficient macro-structure partitioning strategy to reduce the computation while ensuring the performance.

Different to the above method, recently, some researchers focused on the design of quasi-periodic structures. The so-called quasi-periodic means that the microstructures share the same topology but different parameters. The key idea of these works is changing the parameters of a unit cell, in order to transfer the variable material property across the macro domain. Due to the invariance in topological form on the macro scale, the quasi-periodic structure avoids the discontinuity phenomenon at the boundary of the microstructure, and the spatial variation in material properties can be realized by adjusting the macro distribution of variable parameters, so it has a broader design space than the periodic cellular structure. Recently, the authors introduced a TO method for a quasi-periodic cellular structure. Previous studies achieved this by manipulating the microstructural topology using erode–dilate operators. Similar concepts have been implemented within the level set TO method [19,20], where quasi-periodic cells are described by cutting the signed distance function using different thresholds. Building upon this foundation, Zong et al. [21] introduced shape-function-based thresholds that interpolate height variables at nodes to ensure seamless geometric connections between adjacent cells. Through this formulation, the macro and micro concurrent design of a quasi-periodic double-level structure is realized. For this kind of algorithm, micro-structures at different locations have the same topological configuration but different parameters, which ensures different functional requirements at different locations, and creates a double-layer porous structure with better performance.

Compared to density or level set methods, the Moving Morphable Components (MMCs) method [22,23,24] offers a clear geometric representation of the design space, which simplifies the interpretation and manipulation of design variables. The above advantages make it convenient to incorporate variable parameters for quasi-periodic cellular structures. Therefore, we applied the MMCs method to describe microstructural topology. For the macro domain, the one to zero density is also used to give the optimal distribution for microstructures with different volume fractions. The study focuses on minimizing compliance under a volume constraint, and employs the Moving Asymptotes Method (MMA) [25] to solve the optimization models.

2. Problem Formulation

This section introduces a novel topological description for quasi-periodic cellular structures within the MMCs framework. First, a brief overview of the MMCs method is provided to ensure the paper is self-contained. Next, the design variables used to describe the quasi-periodic cellular structures are defined. Finally, the formulation for topology optimization is presented.

2.1. Moving Morphable Component Method

Different from the level set and density methods, the MMCs-based approach adopts a set of moving morphable components as basic building blocks for topology optimization. The implementation of the MMCs method involves optimizing the positions (center coordinates), sizes and orientations of a series of structural components to determine the final global topology description equation. This, in turn, defines the precise boundaries of the structure for its final representation. The basic forms and component descriptions are shown in Figure 1. This topological description method enables the final designed structure to have clear boundary descriptions and geometric features (such as the length and width of the components).

Based on the above component description, the final configuration of the structure can be obtained by designing the control parameter vectors $D=\left\{{\left(D^1\right)}^{\mathrm{T}},\dots ,{\left(D^i\right)}^{\mathrm{T}},\dots \right\}$ of multiple components. Here, $D^i={\left(x_{0i},y_{0i},L_i,{\theta }_i,d_i^{\mathrm{T}}\right)}^{\mathrm{T}}$ represents the topological control parameters of the ith component. $d_i$ represents the parameters in the component control equation (such as $t_i^1$, $t_i^2$ and $t_i^3$ in Figure 1), and L is the length of a component. The relationship between the background element density and the global Heaviside equation determined by

$${\rho }_j^{mi}={\displaystyle \sum _{k=1}^{4}H({\varphi }_k^j(D^i))}/4$$

(1)

where ${\varphi }_k^j$ represents the TD function value of the k-th node with the j-th element node. Many functions have been used to describe components, such as super-ellipses and closed B-splines (CBS). In this paper, the following TDF [26] is applied for its simplicity:

$${\varphi }_i\left(x,y\right)={\left({\displaystyle \frac{x^{\prime }}{L_i}}\right)}^p+{\left({\displaystyle \frac{y^{\prime }}{f\left(x^{\prime }\right)}}\right)}^p-1$$

(2)

with

$$\left\{\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right\}=\left[\begin{array}{cc}\cos {\theta }_i& \sin {\theta }_i\\ -\sin {\theta }_i& -\cos {\theta }_i\end{array}\right]\left\{\begin{array}{c}x-x_{0i}\\ y-y_{0i}\end{array}\right\}$$

(3)

where $f\left(x^{\prime }\right)$ describes the thickness of the components along the x’ direction. p = 6 is applied here. By using these parameters, the boundary and geometry features of a component can be described explicitly:

$$f\left(x^{\prime }\right)={\displaystyle \frac{t^1+t^2-2t^3}{2L^2}}{\left(x^{\prime }\right)}^2+{\displaystyle \frac{t^2-t^1}{2L^2}}{x}^{\prime }+t^3$$

(4)

where $t^1$, $t^2$ and $t^3$ denote the thickness of a component. The specific expression of Heaviside function is as follows:

$$H_{\epsilon }(x)=\left\{\begin{array}{cc}1,\hfill & if x>\epsilon ,\\ {\displaystyle \frac{3(1-\alpha )}{4}}({\displaystyle \frac{x}{\epsilon }}-{\displaystyle \frac{x^3}{3{\epsilon }^3}})+{\displaystyle \frac{1+\alpha }{2}},\hfill & if-\epsilon \le x\le \epsilon ,\\ 0,\hfill & otherwise.\end{array}\right.$$

(5)

Here, $\epsilon$ represents the transition region as well as degree of nonlinearity in the filtering process. $\alpha$ is a small value for avoiding singularities for the linear equations solver.

2.2. Topological Description Formulation

Compared to the traditional density or level set-based method, the design variable in the MMC method changes from the unit density ${\rho }_i^{mi}$ to the component variable $D^i$ of the structure. Thus, we can easily apply the varied thickness of the components based on MMC algorithm to map the microscopic base unit cell (BUC) into a series of quasi-periodic microstructures.

Two procedures are involved in describing a quasi-periodic cellular structure. One procedure involves describing the base unit cell, while the other involves choosing a variable method to generate quasi-microstructures from BUC. Since the MMC method provides explicit parameters for describing components, it offers a more convenient approach to defining variable parameters compared to other methods. Therefore, the MMCs method is applied here to describe the topology of the BUC. The topology of the BUC can be determined by a design vector $D^{mi}$. To obtain a series of quasi-periodic microstructures using a simple alterable parameter from the BUC, we define a parameter R which can scale the thickness of all components, as shown in Figure 2, this means:

$${\left[{\mathrm{t}}_i^1,{\mathrm{t}}_i^2,{\mathrm{t}}_i^3\right]}_Q=R\times {\left[{\mathrm{t}}_i^1,{\mathrm{t}}_i^2,{\mathrm{t}}_i^3\right]}_B$$

(6)

where the subscripts Q and B represent the quasi-periodic microstructures and base unit cell (BUC), respectively. When R < 1, the thickness of the components decreases, resulting in microstructures with a smaller volume fraction. Conversely, when R > 1, the thickness of the components increases, leading to microstructures with a larger volume fraction. By gradually increasing R from zero in small increments until the cell is completely filled with solid material, different microstructures can be obtained.

After obtaining quasi-periodic microstructures, the next step is to determine the optimal macro distribution of these microstructures. The most direct approach is to set vector $R_{ma}$ as the design variable. By optimizing R for the desired objective and constraint functions, the optimal quasi-periodic cellular structure can be achieved. However, determining the bounds of R in this way can be challenging. To address this problem, the element density $0\le {\rho }^{ma}\le 1$ is defined as the design variable instead of R in the macro design. Here, ${\rho }_e^{ma}=0$ and ${\rho }_e^{ma}=1$ represent the void and solid, respectively. For $0<{\rho }_e^{ma}<1$, the corresponding microstructure with the same volume fraction is placed. The subscript e denotes index of the design variable.

In a conclusion, the component parameter is the Micro Topology description variable and R is the Macro Topology description variable. By optimizing the distribution of the quasi-periodic microstructures, in other words, by optimizing the distribution of the Macro Topology description variable R, a multi-scale structure with quasi-periodic microstructures can be obtained.

2.3. Optimization Formulation

In this study, the typical optimization problem of minimizing structural compliance is considered to make an easy comparison with traditional methods. By minimizing structural compliance, maximum stiffness for a prescribed force can be achieved. This function, being convex, is well-suited for finding the optimal solution and is widely applied in topology optimization to test new methods. The formulations are expressed as follows:

$$\begin{array}{ll}\mathrm{find}& {\rho }^{ma}, D^{\mathrm{mi}}\\ \min & c=F^{\mathrm{T}}U=U^{\mathrm{T}}KU\\ \mathrm{s}.\mathrm{t}. & K({\rho }^{ma}, D^{\mathrm{mi}})U=F\\ & {\displaystyle {\int }_{D^{mi}}H\left({\varphi }^s\left(x\right)\right)dV}-{\overline{V}}^{mi}=0\\ & {\displaystyle \sum _{e=1}^{N^{ma}}{\rho }_e^{ma}{v}_e^{ma}}-{\overline{V}}^{ma}\le 0\\ & 0\le {\rho }_e^{ma}\le 1, (e=1,\dots ,N^{ma})\\ & D^{mi}\in {\Xi }_D\end{array}$$

(7)

where ${\rho }^{ma}={\left({\rho }_1^{ma},\dots ,{\rho }_e^{ma},\dots ,{\rho }_{N^{ma}}^{ma}\right)}^{\mathrm{T}}$ and $D^{mi}={\left({\left(D^1\right)}^{\mathrm{T}},\dots ,{\left(D^i\right)}^{\mathrm{T}},\dots {\left(D^{N^{mi}}\right)}^{\mathrm{T}}\right)}^{\mathrm{T}}$ denote the vectors of design variables. ${\overline{V}}^{mi}$ and ${\overline{V}}^{ma}$ are the upper limits for the volume constraint. $N^{mi}$ and $N^{ma}$ the number of design variables within the macro and micro domains, respectively. Here, the limit of the micro volume is added for a stable convergence. Here, ${\overline{V}}^{mi}=0.2$ is applied in the following examples. ${\Xi }_D$ is the admissible set for the design variables. c is the structural compliance which can be computed by:

$$c=U^{T}KU={\displaystyle \sum _{e=1}^{N^{ma}}{U}_e^T{K}_e{U}_e}$$

(8)

where $K$, $U$ and $F$ denote the global stiffness matrix, displacement vector and load vector, respectively. The subscript e represents the element form:

$$K_e={\displaystyle {\int }_{{\Omega }_e}{B}^T{D}_e^{ma}Bd\Omega }$$

(9)

where ${\Omega }_e$ denotes the element domain. $D_e^{ma}\left({\rho }_{\mathrm{e}}^{ma},D^{\mathrm{mi}}\right)$ is the elastic matrix which relates to the microstructural topology and macro density. B is the element strain–displacement matrix.

3. Numerical Implementations

In this section, we initially outline interpolation strategies that correlate material properties with design variables. Subsequently, a sensitivity analysis of the functions is carried out using gradient-based optimization solvers.

3.1. Interpolation Scheme

Before computing the element stiffness matrix, the elastic matrices of microstructures should be given. In this study, the widely applied asymptotic homogenization (AH) method [27] is used:

$$D^{\mathrm{H}}={\displaystyle \frac{1}{\left|Y\right|}}{\displaystyle \underset{Y}{\int }\left[D\left(y\right)-D\left(y\right){\epsilon }_y\left(\varphi ,y\right)\right]dy}$$

(10)

where Y is the local coordinate in the unit cell, and y denotes the coordinate vector. $D\left(y\right)$ is the elastic matrix for the elements in the microstructure. To obtain a clear 0–1 topology, the Solid Isotropic Material with Penalty method [28] is applied here as:

$$D_i=\left(\underset{¯}{\rho }+(\overline{\rho }-\underset{¯}{\rho })\left({\overline{\tilde{\rho }}}_i^{mi}\right)\right)D_0$$

(11)

where $D_0$ is the elastic material matrix and subscript i represents the ith element in unit cell. $\underset{¯}{\rho }=0.001$ is a small value to avoid a singular global stiffness matrix, $\overline{\rho }=1$. ${\epsilon }_y\left(\cdot \right)$ represents the strain calculator. $\varphi =\left[{\varphi }^{11},{\varphi }^{22},{\varphi }^{12}\right]$ denotes the characteristic displacements which are obtained by solving:

$${\displaystyle {\int }_Y{\epsilon }_y^{\mathrm{T}}\left(v\right)\left[D\left(y\right)-D\left(y\right){\epsilon }_y\left(\varphi \right)\right]dy}=0,\forall \varphi \in V_y$$

(12)

where $V_y$ denotes the function space of periodic functions. Due to computational constraints, calculating the elastic matrices $D^{\mathrm{H}}$ for all microstructures generated by the BUC is challenging. As a result, only a subset of samples is selected for computation. Subsequently, cubic B-splines are employed as the basis functions, and an explicit formulation is derived using the least squares method:

$$D^{\mathrm{H}}\left({\rho }_e^{ma},{\rho }_i^{mi}\right)=f_{spline}\left({\rho }_e^{ma},{\rho }_i^{mi}\right)$$

(13)

To prevent the occurrence of microstructures with low volume fractions, which are challenging to manufacture, a penalization scheme is implemented:

$$D_e^{ma}\left({\rho }_e^{ma},{\rho }_i^{mi}\right)={\left({\rho }_e^{ma}\right)}^q{f}_{spline}\left({\rho }_e^{ma},{\rho }_i^{mi}\right)$$

(14)

where q is the penalization power. The following values of q are suggested:

$$q=\left\{\begin{array}{ll}3& {\rho }_e^{ma}<0.1\\ 0& {\rho }_e^{ma}\ge 0.1\end{array}\right.$$

(15)

3.2. Sensitivity Analysis

Since topology optimization has a large number of design variables, gradient-based optimization solver is often applied. Here, the adjoint method is used to derive the sensitivities of the functions with respect to the design variables in micro and macro domain. Frist, the general formulation of the structural compliance with respect to $x_j$ (=${\rho }_i^{mi},{\rho }_e^{ma}$) can be written as:

$${\displaystyle \frac{\partial c}{\partial x_j}}=-{\displaystyle \sum _e^{N^{ma}}{U}_e^T\left({\displaystyle {\int }_{{\Omega }_e}{B}^T\frac{\partial D_e^{ma}}{\partial x_j}BdV}\right)U_e}$$

(16)

where $x_j$ (=${\rho }_i^{mi},{\rho }_e^{ma}$) includes the two types of the design variable. For the two design variables, the primary difference lies in the derivation of the elastic matrix $D_e^{ma}$. Therefore, we will give the detail formulations of the design variables ${\rho }_e^{ma}$ and ${\rho }_i^{mi}$ separately.

(1) Sensitivities with respect to. ${\rho }_e^{ma}$

Differentiating Equation (14) to ${\rho }_e^{ma}$ yields:

$${\displaystyle \frac{\partial D_e^{ma}}{\partial {\rho }_j^{ma}}}=\left\{\begin{array}{cc}0& e=j\\ q{\left({\rho }_j^{ma}\right)}^{q-1}{f}_{spline}\left({\rho }_e^{ma},{\rho }_i^{mi}\right)+{\left({\rho }_j^{ma}\right)}^q{\displaystyle \frac{\partial f_{spline}\left({\rho }_e^{ma},{\rho }_i^{mi}\right)}{\partial {\rho }_j^{ma}}}& e\ne j\end{array}\right.$$

(17)

Then, substituting it into Equation (16), and we can obtain:

$$\begin{array}{l}{\displaystyle \frac{\partial c}{\partial {\rho }_e^{ma}}}=\\ -p{\left({\rho }_e^{ma}\right)}^{p-1}{U}_e^{\mathrm{T}}\left({\displaystyle {\int }_{{\Omega }_e}{B}^{\mathrm{T}}{f}_{spline}\left({\rho }_e^{ma},{\rho }_i^{mi}\right)BdV}\right)U_e-{\left({\rho }_e^{ma}\right)}^p{U}_e^{\mathrm{T}}\left({\displaystyle {\int }_{{\Omega }_e}{B}^{\mathrm{T}}\frac{\partial f_{spline}\left({\rho }_e^{ma},{\rho }_i^{mi}\right)}{\partial {\rho }_e^{ma}}BdV}\right)U_e\end{array}$$

(18)

where subscript $e=1,2,\dots ,N_{ma}$.

(2) Sensitivities with respect to. $D_i^{mi}$

Substituting Equation (14) to Equation (16), the sensitivity of ${\rho }_i^{mi}$ can be written as:

$${\displaystyle \frac{\partial c}{\partial D_i^{mi}}}={\displaystyle \frac{\partial c}{\partial {\rho }_j^{mi}}}\cdot {\displaystyle \frac{\partial {\rho }_j^{mi}}{\partial D_i^{mi}}}=-{\displaystyle \sum _{r=1}^{N^{ma}}{u}_r^T\cdot (\mathrm{f}({\rho }_r^{ma})\cdot {\displaystyle \underset{{\Omega }_e}{\int }{B}^T}\frac{\partial D_r({\rho }_j^{mi},{\rho }_e^{ma})}{\partial {\rho }_j^{mi}}\cdot \frac{\partial {\rho }_j^{mi}}{\partial D_i^{mi}}B\cdot d{\Omega }_e)\cdot u_r}$$

(19)

Note that $\partial c/\partial {\rho }_i^{mi}$ is a summation of the elements in the macro domain:

$${\displaystyle \frac{\partial {\rho }_j^{mi}}{\partial D_i^{mi}}}={\displaystyle \sum _{k=1}^4(q\cdot H{({\varphi }_k^j)}^{q-1}\cdot \frac{\partial H({\varphi }_k^j)}{\partial D_i^{mi}}})/4$$

(20)

4. Results and Discussions

To validate the effectiveness of the method, two design problems including the short and long cantilever beam problems are shown here. The cantilever beam is a common design problem in structural optimization. These problems usually involve optimizing the material distribution of a structure to meet specific performance indicators under given constraints. The detailed numerical implantations and parameter settings are provided in the subsequent subsections, including an optimization model, parameter setting, optimization results, and so on. For the finite element analysis, four-node bilinear rectangular element grids are applied to discretize the domains. In two examples, the micro-design domain is discretized into 50 × 50 rectangular elements. The material properties of the two examples are set as Young’s modulus E = 206 MPa and Poisson’s ratio $\mu =0.28$. To obtain a design with a smooth gradient for the density in the macro domain and prevent the phenomenon of a numerical instability such as the checkerboard scheme that occurs in the topology optimization design of a continuum structure, the density filtering technique is applied. Here, the filtering radius is set to $r^{ma}=1.5$. The objective function of the example is the maximization of structural stiffness; that is, the minimization of structural compliance. It is the most commonly used objective function in topology optimization design because it has the characteristics of a simple form and reflects structural stiffness properties. By updating the design variables based on MMA after the sensitivity analysis, the optimal micro-structure layout of the cantilever beam can be obtained to meet the design requirements. The convergence criterion of the MMA algorithm was selected as $\max ‖x^{i+1}-x^i‖\le {10}^{-3}$, and the maximum number of iteration steps was set to 300.

4.1. Short Cantilever Beam Problem

The first test case involves minimizing compliance for a short cantilever beam. Figure 3 illustrates the design domain for the short cantilever beam, including the force, boundary conditions and sizes. In this case, the size of the design domain is 20 mm by 15 mm. The degree of freedom at the left end of the short beam is completely fixed. A vertical point load of F = 1 KN is applied at the bottom-right corner. The short cantilever beam is discretized using a 20 × 15 grid of elements. The upper volume fraction of the material is 0.4.

Figure 4a shows the optimized topology of the base unit cell which is described by the MMCs method, while the optimized macrostructural density layout is shown in Figure 4b. It should be noted that, here, the black color means the design variable ${\rho }_e^{ma}$ = 1, the gray color means 0 < ${\rho }_e^{ma}$ < 1 and the white color means ${\rho }_e^{ma}$ = 0. For different parameters of the operator R, the corresponding microstructural is shown in Figure 5. Here, we just give microstructural topology for some ${\rho }_e^{ma}$, actually we can obtain an arbitrary microstructural topology for all of the ${\rho }_e^{ma}$ using the method shown in Section 2. According to the macrostructural density layout, the base unit cell with the closest volume fraction is placed at each macroscopic unit to assemble the quasi-periodic cellular structure, and the result is shown in Figure 6. The compliance of the optimized quasi-periodic cellular structure obtained by the proposed method is 21.73.

Firstly, based on the operator R described in the MMCs framework proposed in this paper, the microstructure form of the quasi-periodic cellular structure is described, and a microstructure optimization design is created.

Furthermore, in order to further verify the effectiveness of the algorithm, the structure with less compliance can be optimized under the same condition. Therefore, the problem was solved using the classical single-scale method and periodic cellular structural design, and the results are presented in Figure 7a and Figure 7b, respectively. The structural compliances obtained by these two methods were 27.85 and 52.81, respectively, both of which are higher than the compliance achieved by the proposed method. It also proves that the feasible domain of the structure can be greatly extended by a quasi-periodic design, and a cellular structure with a better performance can be obtained. Since the porous hierarchical structure has lager stiffness than traditional solid structures, it has been widely used in aerospace now. Often, the design results have been fabricated by additive manufacturing.

4.2. Long Cantilever Beam Problem

The second case addressed here is the minimum compliance for a long cantilever beam. The design domain, depicted in Figure 8, is a rectangle measuring 120 mm by 30 mm, with the left side fixed. The domain is discretized using a 120 × 30 grid of rectangle elements, and a vertical point load of F = 1 KN is applied at the bottom-right corner. The volume fraction of the total material is set at 0.4. Figure 9 shows the optimization iteration curve, highlighting the change in the base cell in the micro domain and the density distribution of the macro domain with the number of optimization steps. It can be seen from the curve that the convergence of the optimization objective function is good and the volume fraction is satisfied. The slight oscillations observed in the iteration curve are due to variations in the parameters. The final optimized quasi-periodic cellular structure, with a compliance of 269.73, is presented in Figure 10. At the top of the picture is the microstructure topology optimized at the microscopic level and the base unit cell library constructed by the erode–dilate operator. The volume fraction of base unit cells varies from zero to one, corresponding to the macroscopic density layout. It can be seen from the optimization results that the microstructure topology can ensure the connectivity between cellular structures, although the volume fraction of each base unit cell is variable. At the bottom of the image is the optimal density layout obtained by macroscopic optimization. As can be seen from the figure, the external outline of the structure is solid material, and the base unit cells are mainly distributed in the middle part of the beam. These numerical examples verify the effectiveness of the proposed erode-dilate algorithm under the MMCs framework, which can improve the design space, ensure the connectivity and manufacturability of the structure and obtain cellular structures with excellent performance.

5. Conclusions

In this study, we developed a novel topology optimization (TO) method for designing quasi-periodic cellular structures using a hybrid material-and-morphology-based (MMCs–density) approach. This method enables integrated optimization at both macro and micro scales. The MMCs method was employed to describe the quasi-periodic microstructures by varying the thickness of the bars. The topology of the base unit microstructure is optimized to construct a library of quasi-periodic structures, followed by the macrostructure optimization to determine the distribution of these quasi-microstructures. Interpolation functions are established to derive sensitivities for use in gradient-based optimization. This approach allows us to simultaneously optimize the topology and parameters of microstructures, which vary within the macro design domain, thereby enhancing the performance of graded structures. Moreover, the resulting neighboring microstructures are seamlessly connected, facilitating rapid prototyping via additive manufacturing. The numerical examples demonstrate that quasi-periodic structures offer significant performance improvements over periodic structures, with only a modest increase in the computational cost.