## 1 Introduction

For a given load, the addition of ribs or stiffeners to a plate or shell structure can significantly enhance its stiffness and vibrational characteristics. As a result, stiffened plates or shells find extensive application as primary or secondary load-bearing components in various fields including automotive, aerospace, and civil engineering structures. They offer notable advantages in terms of ease of manufacturing and a high rigidity-to-weight ratio. However, it should be noted that the layout, size, and shape of the stiffeners have a profound impact on the mechanical properties of such structures. Consequently, the establishment of a systematic design method holds great importance for engineers and researchers seeking to optimize the performance of stiffened plates or shells.

Topology optimization is a widely used method for optimizing initial designs by adjusting their geometric and material properties, considering a set of specified objectives and constraints. Over the past few decades, several topology optimization methods have been developed to determine the optimal layout of the structures. These methods include homogenization-based methods (Bendsøe and Kikuchi, 1988), density-based methods (Bendsøe and Bendsoe, 1989; Zhou and Rozvany, 1991), level set methods (Wang et al., 2003; Allaire et al., 2004), evolution methods (Xie and Steven, 1993), and feature-mapping methods (Guo et al., 2014; Zhang et al., 2017). Also, they have been successfully applied to numerous engineering structures (Liu et al., 2015a; Li et al., 2021a; Li et al., 2022).

For the design of stiffened plates or shell structures, one of the most commonly employed approaches in designing stiffened plates is the ground structure method (Locatelli et al., 2011; Duan et al., 2018). This method involves the initial establishment of a large number of stiffeners, followed by their layout determination using topology optimization techniques. Dugre et al. (Dugré et al., 2016) utilized this method to design pressurized stiffened panels. Ding and Yamazaki (Ding and Yamazaki, 2004; Dong et al., 2020) developed a stiffener design method inspired by the growth and branching patterns observed in natural trees. Bojczuk and Szteleblak (Bojczuk and Szteleblak, 2008) proposed a heuristic design strategy based on topology derivatives that consider the impact of stiffener shape and position on structural performance. To reduce the number of required beams, some researchers introduced nodal coordinates of the beams as design variables and optimized the position of beam nodes to obtain an optimal layout (Descamps and Coelho, 2014). Furthermore, apart from stiffener layout design, optimizing the size and shape of stiffeners can also enhance structural performance. Li et al. (Li et al., 2021b) proposed a topology optimization method that simultaneously optimizes the layout and cross-section of stiffeners based on the Giavotto beam theory. It should be noted that, these aforementioned methods primarily focus on the design of stiffener layout and do not address the design of stiffener size and height.

Another method for optimizing stiffened plates is through the use of topography optimization (Cheng and Olhoff, 1981) to determine the optimal layout. This approach focuses on obtaining an optimized thickness distribution rather than a clear stiffener layout. Consequently, several researchers have proposed different strategies to identify optimal stiffener layouts based on the thickness distribution (Gea and Luo, 1999; Rasmussen et al., 2004). To expand the design space, simultaneous optimization of the stiffener layout and heights is performed. In a notable contribution, Gersborg et al. (Gersborg and Andreasen, 2011) presented a parameterized interpolation formulation that utilized a three-dimensional solid model. Building upon this work, Liu et al. (Liu et al., 2015b) further enhanced the method by introducing a novel design variable type to circumvent the presence of grey elements, which is named as H-DGTP. The effectiveness of the approach was demonstrated through successful applications in designing large-aperture space telescopes (Liu et al., 2014) as well as in aerospace contexts (Hou et al., 2017; Zhou et al., 2020). However, in certain design problems, the optimized results frequently exhibit stiffeners that are excessively thick or thin, posing significant challenges for the manufacturing process. Additionally, the optimized outcomes may feature sharp chamfers and sudden changes in thickness, further complicating the practical implementation.

To address these concerns, this study proposes a novel topology optimization design method for stiffened plates that incorporates min-max length scale control. In this study, we adopt the robust filtering techniques introduced by Sigmund and Wang (Sigmund, 2007; Wang et al., 2010) in the H-DGTP (Liu et al., 2015b) framework to effectively regulate both the length scales of the stiffeners and the reinforcement gaps. The utilization of this method is motivated by several key advantages. Firstly, it circumvents the issues associated with gray scale solutions, yielding clear-cut black and white outcomes. Secondly, the robust filtering approaches provide a means to account for spatial manufacturing tolerances, which are commonly encountered in machine manufacturing processes. Thirdly, it allows for the inclusion of chamfers to mitigate material concentration effects and obtain well-defined layouts of the stiffeners. Furthermore, to ensure a balanced distribution of materials and obtain a clear stiffener layout, we incorporate the maximum length scale constraint proposed by Guest (Guest, 2009a).

The structure of this paper is organized as follows: Section 2 provides a comprehensive review of the formulation of the H-DGTP method, and then list incorporation of the robust formulation and the maximum length scale constraint for stiffeners. In Section 3, we present the numerical implementation of the proposed method, which includes sensitivity analysis and flow charts to outline the optimization process. Section 4 presents a detailed analysis of the numerical results obtained from the optimization process. Finally, in Section 5, we offer concluding remarks and provide a concise summary of the key findings presented in this paper.

## 2 The proposed method

In this section, we present the key details of the proposed method. Firstly, we provide a concise description of the explicit parameterization (H-DGTP) approach for designing the layout and heights of stiffeners. Subsequently, we outline the application of the robust formulation and the maximum length scale constraint within the H-DGTP framework. These enhancements aim to achieve a more precise and refined stiffener design.

### 2.1 Formulation of the H-DGTP method

The H-DGTP method, initially introduced by Liu et al. (Liu et al., 2015b), presents a novel formulation for describing the topology of stiffened plates using a 3D model. An illustrative representation of the method is depicted in Figure 1. This method utilizes two types of design variables. The first type, denoted as *L*_{j}, is defined on the base surface and represents the density of a stiffener. A value of *L*_{j} = 1 indicates the presence of a stiffener, while *L*_{j} = 0 indicates the absence of a stiffener. The second type of design variable, denoted as *h*_{j} = [0–1], corresponds to the height of a stiffener. To ensure applicability to non-uniform meshes, a base surface is introduced, allowing for control of the minimum length of the stiffeners by adjusting the mesh size within the base surface.

According to the definitions provided earlier, the density of arbitrary element in the *j*th row of elements can be expressed as:

where *h*_{j} defines the height of the *j*th stiffener and serves as a design variable, *H* represents the Heaviside function, and

where

### 2.2 Minimum length scale control of the stiffener

Numerous length scale control methods have been proposed by researchers (Guest et al., 2004; Guest, 2008; Guest, 2009b; Wang et al., 2011; Zhang et al., 2014). Among them, an implicit length scale control approach commonly employed is density filtering with a projection, which can be traced back to the pioneering work of Guest (Guest et al., 2004). In this method, the projection threshold value is set to zero, and the minimum length is twice the filtering radius. Subsequently, Wang et al. (Sigmund, 2007; Wang et al., 2010) introduced a robust formulation to ensure stable optimization convergence, allowing for implicit length scale control when the eroded, intermediate, and dilated designs share the same topology. Numerical examples have demonstrated the effectiveness of this approach in generating visually pleasing and unambiguous 0–1 results, with precise control over the minimum length in both solid and void phases. As the H-DGTP method employs an individual design variable *L*_{j}, the robust formulation can be seamlessly applied in this study to achieve minimum length scale control for the stiffeners.

Following the idea in robust filtering approaches, design variable *L*_{i} can be projected towards designs:

where the subscript * denotes *d*, *i*, and *e*, which means

where *i*,

with *i* and *j* respectively. For another variable

The density of any one element in the *j*th row of elements is then modified as:

The design problem is formulated as a min/max problem:

where **K** is the global stiffness matrix and is assembled by SIMP interpolate:

where **K**_{e} is the (global level) element stiffness matrix of the solid element. *N* is the number of elements and the penalization power *p* = 3 is introduced to yield distinctive “0–1″ designs.

Qian et al. (Qian and Sigmund, 2013) derived the analytical formulas for predicting the minimal length scale *b*_{min} as a function of the projection threshold

It should be noted that in order to avoid confusion and facilitate reading, the parameters related to the minimum and maximum sizes are explained below:

It is worth noting here, for

### 2.3 Maximum length scale control of the stiffener

In actual topology optimization processes, when the load conditions are too simplistic or when excessive material usage is employed, it often leads to the accumulation of a significant amount of material in local regions. As a result, clear reinforcement structures cannot be obtained. To achieve a clear reinforcement structure, we introduce the topology structure maximum size constraint formulation proposed by Guest (Guest, 2009a).

The proposed maximum length scale constraint requires that the length scale all structural members be less than *b*_{max}. The scheme proposed here enforces this constraint by passing a circular test region

where *x* and *y* follows as:

**FIGURE 2**. The test region of radius *e*. Elements with centroid located within this region belong to set

Constraint (12) is reformulated in discretized form as:

where

To reduce the number of constraints, the maximum length control of the above equation can be condensed by the p-norm aggregation function:

where *p* is the index factor whose value is chosen as 100 in this paper.

The volume of voids measurement in the test region

where, the exponent *q* dictates the degree to which elements with intermediate volume fractions may count towards the volume of voids requirement, where *i*-field) is typically selected to impose constraints. However, numerical examples demonstrate that when the maximum size constraint is applied to the *i*-field, the resulting structure fails to control the dimensions of the stiffener spacing. Therefore, it is recommended to apply the maximum size constraint to the dilated field (*d*-field). The magnitude of

where *ψ* = 5%.

## 3 Numerical implementations

### 3.1 Sensitivity analysis

In order to apply the gradient-based solver to handle the topology optimization problem, the sensitivity of the compliance objective function with respect to *L*_{j} and *h*_{j} can be calculated. Based on the chain rule, the sensitivity can be formulated as follows:

It is worth noting that

where

According to Eq. 2,

Finally, the sensitivity of the maximum size constraint function to the design variable is presented:

where

### 3.2 Flow chart

In order to explain the proposed algorithm process more clearly, we express it in the form of pseudocode as shown in Table 1.

It is worth noting that the counters *i*, *k* and *j* refer to iteration number, continuation step and iterations since last continuation steps, respectively. The number of different *β* value is given by *k*_{max}. In order to make the minimum length scale constraint easier to be implemented, a maximum size constraint is applied to the dilated field. In the intermediate design, the outer diameter *b*_{max,i} = *b*_{max}, inner diameter *b*_{min,i} = *b*_{min} of the annular test domain. For the dilated design, the outer diameter of the annular test area is *b*_{max,d} = *b*_{max}+0.6*b*_{min}, inner diameter is *b*_{min,d} = *b*_{min}+0.3*b*_{min}.

## 4 Numerical examples

Based on the above presented topology optimization method, numerical examples in three dimensions are presented. For all examples, the material of the structure is isotropic with Young’s modulus *c*). Design variables are updated using the Method of Moving Asymptotes.

### 4.1 Example 1: design of a simply supported plate

The first example is a simply-supported plate design problem. The dimensions and load/boundary conditions for the design domain are shown in Figure 3. The design domain is discretized by

First, the design problem is optimized with different filter radius (*r*_{min} = 2,3,4,5,6,7), as shown in Figure 4. As can be seen, the proposed method successfully generates stiffener plate for all these examples, and besides, with the increase of filter radius, the minimum length scale gets bigger, demonstrating that the proposed method is capable of controlling the length scale of the optimized stiffener plate. Here it should be noted that the compliance gets larger, i.e., the structural performance becomes worse, when we use bigger filter radius. This is because the length scale constraint becomes stronger, and also the manufacturability gets better. Thus we should select a proper filter radius to balance the manufacturability and structural performance well when we use the proposed method in practice.

As discussed above, different filtering radii result in different minimum size of the structure, and finally generate different topology optimization results. Due to the fact that the height field of the stiffener is also filtered with fully same filtering radius of the density field in the above results, the height of the optimized stiffener is gradient with a certain slope. Now, let us keep the density filtering radius fixed, i.e.,

### 4.2 Example 2: design of a cantilever beam

In this section, we consider a design problem in 3D. The design domain shown in Figure 6 is a cuboid of size

First, the robust formulation considering minimum length scale constraint is applied to solve this design problem with *r*_{min} = 2 and *η* = 0.25. In this example, the stiffener plate structures with uniform and gradient stiffener height are both considered. The optimized structures are shown in Figure 7 (a) and (b), respectively, and also, the design fields for case (b) are given in Figure 8. As can be seen from the results, the height for the optimized structures (a) and (b) are indeed uniform and gradient, which shows that the proposed method can generate structure with uniform or gradient stiffener.

**FIGURE 7**. The stiffener plate design considering minimum length scale constraint with **(A)** uniform stiffener height (*c* = 2,826.5512), and **(B)** gradient stiffener height (*c* = 2,672.4581).

Now, let us consider the cases where the maximum and minimum length scale constraints are simultaneously applied for the design problem with equal height of stiffener. Figure 9 shows the results with maximum length scale constraint parameter *r*_{max} = 3.5, and with the minimum length scale constraint parameter *r*_{min} = 2. Here, (a) and (b) are the results when the maximum length scale constraint is applied on the dilated and intermediate fields, respectively. As can be seen, if the constraint is added to the dilated structure, the length scale of the void parts can also be controlled. Therefore, it is recommended to add the maximum length scale constraint to the dilated structure for the proposed method.

**FIGURE 9**. The designs with the maximum and minimum length scale constraints considered simultaneously, **(A)** the maximum length scale constraint is added to the dilated structure, *c* = 3642.6962; **(B)** the maximum length scale constraint is added to the intermediate structure, *c* = 3300.3941.

The influence of the lower and upper limits of length scale is studied for this example. First, the cases with *r*_{min} = 4 and *r*_{max} = 6.5 is provided for the equal-height stiffener, as shown in Figure 10 (b). Here for comparision, Figure 10 (a), i.e., the design without maximum length scale constraint, is also given. As can be seen, the minimum and maximum length scales become bigger compared to the result with *r*_{min} = 2 and *r*_{max} = 3.5, demonstrating the proposed method can control the length scales accurately. In addition, the cases with gradient-height stiffener when *r*_{min} = 4 and *r*_{max} = 6.5 are also given, as shown in Figure 11.

**FIGURE 10**. The design problem with *r*_{min} = 4 and *r*_{max} = 6.5 for the case with equal-height stiffener, **(A)** without maximum length scale constraint, *c* = 2,894.3987; **(B)** with maximum length scale constraint on dilated design, *c* = 3421.4256.

**FIGURE 11**. The design problem with *r*_{min} = 4 and *r*_{max} = 6.5 for the case with gradient-height stiffener, **(A)** without maximum length scale constraint, *c* = 2,719.2780; **(B)** with maximum length scale constraint on dilated design, *c* = 3402.4494.

## 5 Conclusion

In order to achieve an optimal layout for a clear reinforcement structure, this paper applies size control algorithms from topology optimization to the design of stiffeners. This allows for control over both the maximum and minimum sizes of the stiffeners, as well as the spacing between them. In the proposed method, the robust topology optimization formulation and a maximum length-scale constraint are introduced into the H-DGTP method, generating a new topology optimization method for the design of stiffener plate considering min-max length-scale constraint. Compared to existing methods, the proposed approach not only provides a clear layout of stiffeners, but also is capable of optimizing the stiffener’s height, and besides is capable of controlling the maximum and minimum length scales of the optimized structures. Specially, when the upper and lower length scales are set to be close, the thickness of the stiffeners can be optimized to be uniform, which better meets engineering requirements. Numerical examples show that the combination of the robust filter approach and maximum length scale constraint enables precise control of both structural features and gap widths, while effectively avoiding acute angles, demonstrating the effectiveness of the proposed method.

## Data availability statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

## Author contributions

SW: Writing–original draft, Writing–review and editing.

## Funding

The author(s) declare that no financial support was received for the research, authorship, and/or publication of this article.

## Conflict of interest

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

## Footnotes

^{}1国家重点基础研究 (973) 计划项目 (2011CB610304), 国家科技支撑计划 (2011BAF11B03) 以及高档数控机床与基础制造装备科技重大专项 (2012ZX04010-011) 资助。

^{2}stliu@dlut.edu.cn

## References

Allaire, G., Jouve, F., and Toader, A.-M. (2004). Structural optimization using sensitivity analysis and a level-set method. *J. Comput. Phys.* 194, 363–393. doi:10.1016/j.jcp.2003.09.032

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