Hole Appearance Constraint Method in 2D Structural Topology Optimization

Abstract

A 2D topology optimization algorithm is proposed, which integrates the control of hole shape, hole number, and the minimum scale between holes through the utilization of an appearance target image. The distance between the structure and the appearance target image is defined as the hole appearance constraint. The appearance constraint is organized as inequality constraints to control the performance of the structure in an iterative optimization. Specifically, hole shapes are controlled by matching adaptable equivalent shape templates, the minimum scales between holes are controlled by a hole shrinkage strategy, and the hole number is controlled by a hole number calculation and filling method. Based on the SIMP interpolation topology optimization model, the effectiveness of the proposed method is verified through numerical examples.

1. Introduction

Topology optimization [1,2] is regarded as an effective conceptual design approach, enabling the identification of the optimal material distribution within a specified design domain. It has been successfully applied in various fields such as advanced structures and materials, the automotive industry, aerospace, and architecture in the past 30 years [3,4,5,6,7]. Among other factors, precise control of holes is essential for the designed structure, considering aspects such as manufacturing, functionality, and aesthetics [8]. The performance of a structure is determined by detailed information about the holes, such as their location, number, shape, and size. Therefore, precise control over the structural topology allows designers to create structures with well-defined features and exceptional performance. For example, avoiding complex structural topology can ensure that the optimization results are easily manufacturable [9,10]. In optimizing systems with embedded components, it becomes necessary to incorporate a certain number of holes with specific sizes and shapes to facilitate further assembly of functional devices [11]. The hole control in additive manufacturing can affect the manufacturability of structures, such as hole shape control in the design of self-supporting structures [12] and unclosed hole control in selective laser sintering [13].

In general, controlling the number of holes aims to limit the total number of holes within a structure. Current techniques for controlling the hole number in the structure can be categorized into implicit and explicit methods. Implicit methods do not directly quantify the hole number in the structure, for instance, sensitivity filtering methods [14], Heaviside projection-based methods [15,16], and structural skeleton-based methods [17,18]. By using explicit methods, designers can establish a direct relationship between hole quantities and design variables [19,20,21,22].

Hole shape control aims to achieve regularization of hole shapes in structures after topology optimization. A method utilizing shape templates to represent internal holes was proposed, aiming to simplify the structural model by using shape and size parameters [23,24]. Subsequently, a structure optimization system was proposed, which used image interpretation technology to describe complex hole shapes [25,26]. As these methods traditionally performed topology optimization and shape optimization separately, with shape optimization as a post-processing step, this approach could, in some cases, reduce the mechanical performance of the resulting structure. It was important to mitigate this effect by carrying out topology optimization and shape constraints in parallel. The integration of structural shape features and topology optimization has drawn significant attention [27]. The feature-based topology optimization method resulted in a structure composed of only simplified geometric primitives [28]. The flexible void regions method simultaneously optimized the structural performance, shapes, and positions of void regions using penalties based on inertia moments and centroid positions [11]. Various methods for explicit geometric topology optimization have been proposed, such as moving morphable components (MMC) [29] and moving morphable voids (MMV) [30]. Based on the level-set method, a comprehensive study was conducted on hole control using hole features as design primitives [8].

In general, while the aforementioned methods for controlling holes each have their merits, there are several significant issues that still require further exploration. The first challenge is the selection of characteristic shape templates and the measurement of complex hole shapes. The gap between the template and the hole shape cannot be too large; otherwise, the mechanical performance of the design may be greatly reduced. This is usually solved by selecting more templates, but this also increases the difficulty of the hole template matching problem. The second challenge is that the optimization process needs to consider both the mechanical performance of the structure and the hole number, shape, and spacing.

Appearance constraint is the distance between a 2D design and a 2D image composed of predefined pattern matching [31,32]. Inspired by this approach of measuring appearance constraint, this paper integrates the control of hole shapes, hole number, and the minimum scale between holes as an appearance constraint. By adopting three adaptable equivalent shape templates, including isosceles triangles, rectangles, and ellipses [33,34], more flexible hole shapes can be adapted through certain deformations while keeping the number of shape templates small. This paper introduces corresponding appearance target images based on the original 2D design, optimized by replacing hole shapes with matching equivalent shapes and filling redundant holes. Simultaneously, appearance constraint is defined as the distance between the 2D design and these appearance target images, enabling control over structural appearance features during the optimization process. In summary, a flexible implementation of the implicit constraint to control holes is proposed in this paper.

The structure of this paper is organized as follows: Section 2 primarily introduces the structural optimization framework. Section 3 describes the information extraction of holes in the structure and the best equivalent shape matching of hole shape. Section 4 gives the method of controlling the number of holes. Section 5 discusses the solving of hole interference problems and the calculation of corresponding appearance target image and appearance constraints. Section 6 uses multiple 2D examples to show the effect of the algorithm proposed. The summary of this study is in Section 7.

2. Implementation of Topology Optimization

2.1. The Topological Optimization Model

The purpose of topology optimization is to find the optimal distribution of materials in a given design domain according to a set of objectives and constraints. Our approach utilizes a comprehensive framework that includes SIMP interpolation and the method of moving asymptotes (MMA) optimization algorithm [35] to achieve structural optimization, as it effectively handles complex physical problems with multiple constraints. We take the minimization of compliance as the objective function and the appearance function and volume as design constraints. The design domain Ω is divided into n finite element meshes in 2D. For a fixed external load F and the resulting displacement U, the optimization problem can be defined as follows:

$$\begin{array}{c}Min:c(\rho )=F^{T}U={\displaystyle \sum _{e=1}^n{\rho }_e{ }^p}{u}_e{ }^T{k}_0{u}_e\\ s.t.:V({\rho }_e)\le V^{\ast }\\ KU=F\\ 0<{\rho }_{min}\le {\rho }_e\le 1\\ A\left({\rho }_e\right)\le {\mathrm{A}}^{\ast }\end{array}$$

(1)

For each element $e$, the effective Young’s modulus $Y$ is determined by the power-law functions $E_{\left({\rho }_e\right)}={\rho }_e^p{E}_0$ of the specified isotropic material $E_0$, where the penalty coefficient $p=3$. $K$ is the global stiffness matrix, ${\rho }_e$ is the density of the $e$-th element, ${\rho }_{\mathrm{m}\mathrm{i}\mathrm{n}}$ is the lower bound of the element design variables, and $V\left({\rho }_e\right)$ is subject to the constraint $V*$. The appearance function $A\left({\rho }_e\right)$ is defined by appearance target graph with the scalar constraint value of A*. The specific expression of $A\left({\rho }_e\right)$ can be found in Section 5.2.

2.2. Workflow

The proposed topological optimization algorithm with appearance function constraints integrates image processing techniques, equivalent shape measurement matching algorithms, hole interference handling methods, hole filling strategies, and appearance function constraints simultaneously. The flowchart of the hole appearance constraint method proposed in this paper is depicted in Figure 1. The proposed algorithm takes the classical topology optimization approach with the judgment of the hole number as the first stage of optimization and the optimization process with the appearance function constraint as the second stage, in which the gray image of the 2D design obtained in the first stage is the initial value of the second stage of optimization.

3. Hole Extraction and Feature Shape Measurement Algorithm

3.1. Extraction of Hole Shape Information

In a two-dimensional design domain, a hole is conceptually a region encircled by solid material, with its shape defined by the binary image of the structure. The classic SIMP method involves relaxing the intermediate density of design variables, resulting in the presence of grayscale elements in each iteration of the optimization process. Therefore, in order to extract the shape information of holes, we first need to define a binarization method to project the design variables to 0 or 1 to obtain a binary image of the structure.

There are many algorithms that can achieve projection operations, in order to convert grayscale images to binary images with equal volume. In this article, we choose the volume-preserving projection method [36], where the threshold $\overline{\mathsf{\rho }}$ is calculated by the classical optimality criteria (OC) updated equal division method under the assumption of volume equivalence. Then, the design variables ${\mathsf{\rho }}_{\mathrm{e}}$ with density less than $\overline{\mathsf{\rho }}$ are defined as 0, while the design variables ${\mathsf{\rho }}_{\mathrm{e}}$ with density greater than $\overline{\mathsf{\rho }}$ are defined as 1. The volume-preserving projection method applies a specific threshold to convert the gray image obtained at each iteration as shown in Figure 2a into a binary image as shown in Figure 2b. The variable can be defined by the Equation (2).

$$\begin{array}{l}\widetilde{{\rho }_i }=\left\{\begin{array}{c}1,{\rho }_i>\overline{\rho }\\ 0,{\rho }_i\le \overline{\rho }\end{array}\right.\\ {\displaystyle \sum \widetilde{{\rho }_i}{v}_i}={\displaystyle \sum {\rho }_i{v}_i}\end{array}$$

(2)

This paper employs the fire-burning method (FBM) [20,36] to extract information about the holes from the binary image, including the hole number and the element set of each hole. The FBM can obtain multiple subdomains from a discrete design domain based on element adjacency information and properties (the density of elements), with each subdomain containing elements of the same density and each subdomain singly connected. The specific steps are as follows: Identify all elements with a density of 0 within the design domain Ω, and then divide all void elements into multiple void subdomains based on adjacency relationships (four adjacency). Subsequently, all hole subdomain information is obtained by eliminating the void subdomains adjacent to the design domain boundary. We use $H_j$, $j=\mathrm{1,2},\cdots m$ to represent the element set of all holes, where m represents the hole number. The Figure 2c shows the information of the number of holes and all elements contained in each hole in the structure extracted by the FBM, including three holes represented by $H_j$, where $j=\mathrm{1,2},3$.

To extract the set $B$ of boundary elements for an enclosed solid region as shown in Figure 2d, the Laplacian operator is employed to extract the boundary. The enclosed solid regions obtained after hole filling are subjected to convolution with the Laplacian operator. The four neighbor template H of the Laplacian operator is shown below:

$$H=\left[\begin{array}{ccc}0& -1& 0\\ -1& 4& -1\\ 0& -1& 0\end{array}\right]$$

(3)

The element ${\rho }_e$ does not belong to the boundary set B if the result of the convolution operation is ${\rho }_e^{\text{'}}=0$; otherwise, it belongs to the boundary set $B$. In order to judge the elements on the boundary of the design domain, this paper uses a 0 element to expand the design domain, and finally obtains the boundary of the closed solid area, which is represented by the red element in Figure 2d.

Each pixel in the image corresponds to each element in the finite element grid. In this article, the terms unit and pixel refer to the same square design unit, as the design domain is discretized by the units of the square. To obtain positional information for each element within the holes, in the following context, we assume a finite element grid with $n$ finite elements, and the corresponding centroid coordinates $C\in R^{n\times 2}$ are defined as:

$$C=\left[\begin{array}{cc}{x}_1& y_1\\ ⋮& ⋮\\ x_n& y_n\end{array}\right]$$

(4)

then the coordinates ${HC}_j$ corresponding to the elements contained in each hole $H_j$ are defined as:

$$H{C}_j=\left[\begin{array}{cc}{x}_1& y_1\\ ⋮& ⋮\\ x_{m_j}& y_{m_j}\end{array}\right]$$

(5)

where $m_j$ represents the total number of elements contained in the hole $H_j$.

3.2. Calculating the Area and Centroid of Holes

Holes in binary images usually have complex geometries. In order to regularize the shape of holes, we first obtain the basic geometric information of the hole shape, which includes the area, centroid, and elongation of each hole. We define the area of each pixel as 1, then the area $S{H}_j$ of the $j$-th hole $H_j$ is the total number of elements contained in $H_j$, and the centroid coordinates ($x_c$,$y_c$) of the hole $H_j$ are determined by the following equation:

$$\begin{array}{l}S{H}_j=m_j\\ x_c={\displaystyle \frac{1}{m}}{\displaystyle \sum _{j=1}^m{x}_j}\\ y_c={\displaystyle \frac{1}{m}}{\displaystyle \sum _{j=1}^m{y}_j}\end{array}$$

(6)

3.3. Calculating the Elongation of Holes

We utilize the minimum bounding rectangle (MBR) algorithm [37] to obtain the MBR of the hole shape. This method relies on the following principle: if a rectangle is the MBR of a convex polygon, then this rectangle is collinear with one edge of the convex polygon. By employing this algorithm, we can obtain the MBR for each hole, as indicated by the red line in Figure 3. By inputting the coordinate information of the hole element ${HC}_j$ into the algorithm, it will provide us with the coordinates of the four vertices of the MBR $\left[X_r,Y_r\right]$. The length $l$ and the width $h$ of the rectangle can be calculated through the coordinates of the four vertices of the minimum bounding rectangle.

$$\begin{array}{l}\left[X_r,Y_r\right]={\left[\begin{array}{cccc}{x}_1& x_2& x_3& x_4\\ y_1& y_2& y_3& y_4\end{array}\right]}^T\\ l=\max \left\{\sqrt{{\left(x_1-x_2\right)}^2+{\left(y_1-y_2\right)}^2},\sqrt{{\left(x_2-x_3\right)}^2+{\left(y_2-y_3\right)}^2}\right\}\\ h=\min \left\{\sqrt{{\left(x_1-x_2\right)}^2+{\left(y_1-y_2\right)}^2},\sqrt{{\left(x_2-x_3\right)}^2+{\left(y_2-y_3\right)}^2}\right\}\end{array}$$

(7)

The elongation rate $\epsilon =l/h$ of the hole shape is the ratio of the long axis $l$ to the short axis $h$.

3.4. Best Equivalent Shape Matching

We discovered that although the shapes of the internal holes in the resulting structure are irregular, the majority of shapes are close to triangular or quadrilateral. Considering these findings, we only chose deformable shape feature templates, namely ellipses, rectangles, and isosceles triangles in this paper. These templates possess the ability to adapt to more intricate hole shapes by adjusting their elongation ratios. Due to the discrete nature of digital images, all digital images are sensitive to image resolution, making it more challenging to describe the shapes of objects with low resolutions. In our process of measuring hole shapes, as the area of the hole decreases, the resolution of the hole’s shape reduces. Consequently, the error in measuring the hole’s shape increases. Therefore, in practical operations, the number of grids in the design domain should not be too small. This article suggests that the area of holes should not be less than 20 units.

3.4.1. Measurement Method for Shape Similarity of Equivalent Shapes

Methods for measuring the similarity of simple regular shapes have been proposed [33,38], which involve assessing the ratio between an object’s area and the area of a regular shape. For instance, rectangularity is a value reflecting how closely a shape resembles a rectangle. A traditional approach to measuring rectangularity involves comparing the shape’s area with its MBR. To avoid the influence of prominent boundaries on the MBR, we combine the concept of an equivalent rectangle, replacing the MBR with the equivalent rectangle of the object. The ratio of the overlapped area to the original shape area is termed as rectangularity.

• Rectangularity Measurement

Rectangularity $Re$ is defined as the ratio of the overlap area between the original hole shape and its equivalent rectangle to the area of the original hole shape as:

$$Re={\displaystyle \frac{A_R\cap A_O}{A_O}}$$

(8)

where the area of the hole shape is represented by $A_O$, as depicted in Figure 4a. $A_R$ refers to the area of the equivalent rectangle, shown as the red region in Figure 4b. The overlapping area between the hole shape and its equivalent rectangle is illustrated by the black portion in Figure 4b.

• Isosceles Triangle Measurement

The isosceles triangle measurement $Tr$ reflects the degree to which the shape approximates an isosceles triangle. The definition of $Tr$ is similar to that of rectangularity, as follows:

$$Tr={\displaystyle \frac{A_T\cap A_O}{A_O}}$$

(9)

where $A_T$ represents the area of the equivalent isosceles triangle as depicted in Figure 4 d in the red region; the overlapping area between the original shape and its equivalent isosceles triangle is illustrated in Figure 4d by the black section.

• Ellipticity Measurement

The ellipticity of the shape, $El$, reflects the degree to which the shape approximates an ellipse. The definition of ellipticity can be defined as:

$$El={\displaystyle \frac{A_E\cap A_O}{A_O}}$$

(10)

where $A_E$ denotes the area of the equivalent ellipse as shown in Figure 4c in the red region; the overlapping area between the original shape and its equivalent ellipse is represented in Figure 4c by the black section.

3.4.2. Method for Calculating Equivalent Shapes

A method for determining an equivalent rectangle has recently been proposed [34,38], which derives the equivalent rectangle of a target shape by obtaining its geometric parameters. We have expanded upon this concept by introducing the concepts of equivalent isosceles triangles and equivalent ellipses. The derived equivalent shapes of the target shape are expected to adhere to the following assumptions:

• The centroids of the two shapes should coincide;
• The areas of the two shapes should be identical;
• The two shapes should possess equivalent aspect ratios;
• The two shapes should have the maximum possible overlapping area.

To obtain the equivalent characteristic shapes of the target, such as rectangles, isosceles triangles, and ellipses, it is necessary to extract the geometric parameters of the target’s hole shape. This requires an initial computation of the hole shape’s centroid, area, and elongation ratio. To ensure that the elongation ratio of the shape of the hole computed using the MBR matches that of the equivalent regular shapes, it becomes essential to establish the relationship between the elongation ratio of the shape and the shape parameters of the equivalent forms (such as the length and width of a rectangle, the long and short axes of an ellipse, and the base and height of an isosceles triangle).

As shown in Figure 5a, the length of the equivalent rectangle is equal to the long axis $l$ of the minimum bounding rectangle of the hole, and the width of the equivalent rectangle is equal to the short axis $h$ of the MBR. As shown in Figure 5b, the long axis of the equivalent ellipse is $l/2$, and the short axis is $h/2$.

Note that when using the minimum bounding rectangle method to calculate the elongation ratio of a shape, the shape parameters of both the equivalent rectangle and ellipse can be uniquely determined, while there are two cases for the isosceles triangle. The first case is that the base of the isosceles triangle coincides with one side of the MBR as shown in Figure 5c; the relationship between them can be expressed as follows:

$$\begin{array}{c}a=h\\ b=l\end{array}$$

(11)

The second case is that the waist of the isosceles triangle coincides with one side of the MBR as shown in Figure 5d; their relationship can be mathematically expressed as follows:

$$\begin{array}{l}a=\sqrt{l^2-{\left({\displaystyle \frac{b}{2}}\right)}^2}\\ b=\sqrt{{\left(l-\sqrt{l^2-h^2}\right)}^2+h^2}\end{array}$$

(12)

where $b$ is the length of the base of the isosceles triangle, and $a$ is the height of the isosceles triangle.

After obtaining the ratio of the geometric parameters of the equivalent shape, we can determine the size of the shape parameters according to the principle that the area of the hole is the same as the area of the equivalent shape (Equation (13)). The method to determine the position of the equivalent shape involves aligning the centroid position of the equivalent shape with the computed hole position.

$$\begin{array}{l}{A}_R=A_T=A_E=A_0\\ A_R=a_R{b}_R\\ A_T={\displaystyle \frac{1}{2}}{a}_T{b}_T\\ A_E=\pi a_E{b}_E\end{array}$$

(13)

By calculating the overlap area between the equivalent shape $E{S}_{\theta }$ and the original shape $S_0$ at various directions $\theta$, the orientation $\widehat{\theta }$ of the equivalent shape is determined as the direction that maximizes the overlapping area, as illustrated in Figure 6. Therefore, the direction of the equivalent shape $\widehat{\theta }$ can be obtained using the following formula:

$$\widehat{\theta }=\max \left\{S_0\cap E{S}_{\theta },\theta \in \left[0,2\pi \right)\right\}$$

(14)

For the two equivalent isosceles triangles obtained, the one with a larger overlap area with the original shape is the final equivalent isosceles triangle. The matching equivalent shape (MES) for each hole is the one corresponding to the maximum value among the rectangle degree, isosceles triangle degree, and ellipse degree of the hole. Therefore, the MES can be obtained using the following formula:

$$MES=\max (Re,Tr,El)$$

(15)

The MES is shown in Figure 7, which calculates the values of $Tr$, $Re$, and $El$ for several examples of hole shapes. The specific algorithm for obtaining the MES is detailed in Algorithm 1.

Algorithm 1: Algorithm to obtain the MES of a hole.

Input the coordinates of hole elements ${HC}_j$. Calculate the hole centroid coordinates ($x_c$,$y_c$), area $S{H}_j$, and elongation $\epsilon .$ Determine the proportion of shape parameters for isosceles triangles, rectangles, and ellipses based on the relationship shown in Figure 5. Then, determine the shape parameters according to Equation (13). Align the centroid of the equivalent shape with that of the hole. Subsequently, compute the coordinates of the shape (such as vertices for isosceles triangles, vertices for rectangles, boundary coordinates for ellipses) based on the parameters of the equivalent shape. Employ a binary mapping method to identify elements encompassed by the equivalent shape (as discussed in Section 5.1). Determine the angle $\widehat{\theta }$ of the equivalent shape according to Equation (14). Calculate the $Re,TrandEl$ according to Equations (8)–(10). Obtain MES according to Equation (15). End

4. Hole Number Control

Changing the number of holes within the appearance target image can control the quantity of holes while regulating their shapes. When determining the appearance target image, we can employ a method that involves filling redundant holes to guide the desired number of holes in the optimized structure. With $m$ as the current number of holes within the structure, we set a limit on the maximum hole count as $\overline{m}$ which is defined as $m$ ≤ $\overline{m}$. After binarizing the gray image of the structure, we judge the relationship between the number of holes $m$ and our desired hole number $\overline{m}$. If $m>$ $\overline{m}$, we fill one of the smallest area holes in the binary image with a solid object. It should be emphasized that no matter how large the gap between $m$ and $\overline{m}$ is, we only fill one hole at a time. For example, we define $\overline{m}=1$. By filling the smallest hole in Figure 8a, we obtain the appearance target image as shown in Figure 8b. If the number of holes meets the specified criteria, the hole filling operation is not required. By gradually filling holes, we can ultimately achieve control over the number of holes in the structure.

5. Appearance Target Image and Minimum Scale between Holes Control

5.1. Generate Appearance Target Image

The method involves mapping geometric characteristic shapes onto a fixed grid using a binary density-based mapping approach. This technique uses a fixed grid for analysis, thus avoiding the need for re-meshing. The mapping of geometric characteristic shapes onto the fixed grid involves determining if an element lies within the interior of the characteristic shape. If the centroid of an element is within the characteristic shape’s interior, it is considered part of the characteristic shape; otherwise, it is not. As illustrated in Figure 9a, the MES for each hole is determined. The application of the above method involves mapping the matched equivalent shapes onto the binary image after hole filling, shown in Figure 9b, to obtain the appearance target image as depicted in Figure 9c.

However, it is crucial to consider interference between holes in the appearance target image, as demonstrated in Figure 10d. We address this issue by employing a hole shrinking strategy. By reducing the hole’s area and consequently diminishing the equivalent shape’s area, we successfully avoid interference among equivalent shapes in the appearance target image. Similarly, we also utilize a hole shrinking strategy to solve interference between holes and the boundaries of solid elements in the appearance target image. Additionally, this strategy enables us to control the minimum scale between holes.

The specific operational steps for the hole shrinking strategy are outlined as follows:

• Obtain the grayscale image, as depicted in Figure 10a, using classical topology optimization methods.
• Process the obtained image through binarization to achieve Figure 10b.
• Fill the redundant holes. At this point, we defined $\overline{m}=6$.
• After determining the MES for all holes, obtain the results depicted in Figure 10c.
• Map the MES onto the binary image to obtain the appearance target image, as shown in Figure 10d, where interference between holes becomes apparent.
• Before performing hole shrinking on interfering holes, expand the holes in the appearance target image by k pixels using morphological dilation operations to generate an interference assessment map, as illustrated in Figure 10e. This step aims to prevent the holes being too close to or touching each other.
• Identify all interfering holes based on the interference assessment map and replace the equivalent shapes of interfering holes in the appearance target image with their original hole shapes. Use morphological erosion operations to reduce the size of the holes by one pixel. This process results in an updated appearance target image shown in Figure 10f.

5.2. Appearance Constraints

After obtaining the final target appearance map, we define the distance between the grayscale image of the current design domain and the appearance target image as the appearance constraint $A\left(\rho \right)$. Specifically, the target for each element within the design domain is the corresponding element density on the appearance target image. The distance between each element and its corresponding target is defined as $d\left({\rho }_i,{\rho }_{mi}\right)$. To improve convergence, the constraints on the boundary elements are relaxed. A simple sum aggregation function is used to ensure a well-defined first-order derivative, resulting in the appearance function. This appearance function measures the distance between all elements in the design domain and the appearance target. Guided by the appearance target image, the number, shape, and spacing of the holes in the structure can be controlled. In each iteration, the appearance target image is updated based on the grayscale image, ultimately achieving a balance between structural compliance and the appearance target. $A\left(\rho \right)$ is defined as follows:

$$\begin{array}{l}A(\rho )={\displaystyle \frac{1}{n}}{\displaystyle \sum \left\{\begin{array}{c}\mathrm{d}\left({\rho }_i,{\rho }_{mi}\right),i\notin B\\ 0,i\in B\end{array}\right.}\\ \mathrm{d}\left({\rho }_i,{\rho }_{mi}\right)={\left({\rho }_i-{\rho }_{mi}\right)}^2\end{array}$$

(16)

where ${\rho }_i$ represents the gray density of the$i$-th element in the gray image, and ${\rho }_{mi}$ represents the corresponding gray density of the $i$-th element in the appearance target image.

The derivative of the appearance constraints concerning gray density is defined as follows:

$${\displaystyle \frac{\partial A\left(\rho \right)}{\partial {\rho }_i}}=\left\{\begin{array}{c}{\displaystyle \frac{2\left({\rho }_i-{\rho }_{mi}\right)}{n}},i\notin B\\ 0,i\in B\end{array}\right.$$

(17)

6. Results and Discussion

In this section, the optimization model minimizes the structural compliance under volume constraints and hole appearance constraints. The templates used to control the regularity of holes include rectangles, isosceles triangles, and ellipses. The convergence criterion adopted in this paper is consistent with the convergence criterion of the classical topology optimization algorithm [39], and the maximum number of iterations is set to 300. The elastic material is set to a Young’s modulus and a Poisson’s ratio of 1 and 0.3, respectively.

6.1. Example 1

The cantilever beam problem illustrated in Figure 11a uses a design domain divided into 160 × 100 elements; the final design volume fraction is constrained to 0.5, with a filter radius of 3. The left edge of the design domain is completely fixed, and an external force is applied at the midpoint of the right edge. The control parameter k for the minimum distance between holes is set to 1, and there is no constraint on the maximum number of holes, with $\overline{m}$ set to a large value (for example, $\overline{m}=100$). The appearance constraint $A^{*}$ is set to $10\% A_0$, where $A_0$ corresponds to the appearance function associated with the structure in Figure 11b.

Figure 11b illustrates the structure obtained in the first phase of our optimization algorithm without appearance function constraints. In the absence of constraints on the hole number, the structure from the first phase aligns with that of classical topological optimization. In the second phase, under the influence of the appearance constraints, the final optimized result is generated, as shown in Figure 12c. The variation of the objective function, the compliance, during the optimization process is displayed in Figure 13a. Classical topological optimization converges by the 81st iteration, following which the appearance constraint is introduced. During subsequent iterations, it can be observed that the objective function c rapidly increases initially, then gradually decreases until convergence. Simultaneously, the variation of the appearance constraint A throughout the iterations is represented by the blue line in Figure 13b, while the change in volume fraction is depicted by the orange line in Figure 13b. Overall, after the regularization of the shapes of the three holes in the structure, the structural flexibility increases from 39.24 to 39.71. The optimization results for when the appearance constraints $A*$ are set at $50\%A_0$ and $25\%A_0$ are illustrated in Figure 12a,b, respectively.

For different structural features of hole shapes, we can choose various equivalent shape templates to better fit the hole shapes. These templates can be simple regular shapes or complex shapes. We can also increase or decrease the number of templates. By selecting different combinations of templates, we can flexibly handle the hole shape optimization problem in topological optimization. For example, under the initialization conditions of example 1, we separately used equilateral triangles, rectangles, and ellipses as the shape templates to constrain the regularization of the holes, resulting in the optimization results shown in Figure 14a–c, respectively.

6.2. Example 2

As shown in Figure 15a, the classic short cantilever beam problem involves a design domain discretized into 160 × 100 elements, with a final design volume fraction constrained to 0.5 and a filtering radius of 3. The left edge of the design domain is completely fixed, and an external force is applied to the bottom right corner of the cantilever beam. The minimum distance control parameter between holes k is set to 1, and there is no control over the maximum number of holes. The appearance constraint $A^{\mathrm{*}}$ is set to $10\mathrm{\%}{A}_0$. Figure 15b shows the structure obtained in the first stage of our optimization algorithm under the aforementioned conditions. Figure 16b displays the final optimized result.

Next, we take $k=0,1,2,3$ to demonstrate the control effect of the minimum distance between holes. From Figure 16, it can be observed that using the hole shrinking strategy proposed in this paper can effectively resolve the interference between holes and control the minimum distance between them. The changes in compliance, volume fraction, and appearance constraint during the iteration process under different $k$ values are shown in Figure 17. In the early stages, the appearance constraints A rapidly oscillate with iterations, reflecting the gradual resolution of interference between holes in the appearance target image through the hole shrinking strategy.

To verify the effect of hole number constraints, we added a hole number constraint and took $k=1$ and $\overline{m}=\mathrm{0,1},\mathrm{2,3},\mathrm{4,5},6.$ The optimized results under different hole quantity constraints are shown in Figure 18. Figure 19 show the changes in volume fraction and appearance constraint during the optimization process for $\overline{m}=2$.

6.3. Example 3

This example concerns the MBB beam, a widely used benchmark example in topology optimization. Due to the symmetry of the MBB beam, research typically analyzes only half of the beam. However, in order to regularize the shape of the holes, the entirety of the beam was chosen as the design domain in this study. The design domain was divided into 240 × 40 elements, with a filter radius of 3. The volume fraction was limited to 0.5, and the boundary conditions and loads are shown in Figure 20a.

The classical topology optimization results are displayed in Figure 20b. After introducing an appearance constraint represented by$A^{*}$ = $10\%A_0$, the optimized results are depicted in Figure 20c. In this example concerning the MBB beam, when calculating the shape appearance target image, we applied a hole interference control strategy too. We set the control parameter for the minimum distance between holes as k = 1. The hole interference control strategy to address the interference between holes led to gradual filling of the holes at the bottom-left and bottom-right corners during the optimization process.

7. Conclusions

In this work, three adaptable equivalent geometric templates are constructed to measure and match hole shapes. Additionally, the control of hole shape, hole number, and the minimum scale between holes has been integrated through the utilization of an appearance target image. Numerical examples demonstrate the effectiveness of the proposed method. This method can obtain high-performance structures with specified constraints on hole shape, number, and spacing. Nevertheless, it is concluded that controlling holes often results in compromised structural performance. The appearance constraints can be extended to three-dimensional cases where the user-desired appearance target model needs to be researched further.