Analysis of Numerical Instability Factors and Geometric Reconstruction in 3D SIMP-Based Topology Optimization Towards Enhanced Manufacturability

Abstract

The advancement of topology optimization (TO) and additive manufacturing (AM) has significantly enhanced structural design flexibility and the potential for lightweight structures. However, challenges such as intermediate density, mesh dependency, checkerboard patterns, and local extrema in TO can lead to suboptimal performance. Moreover, existing AM technologies confront geometric constraints that limit their application. This study investigates minimum compliance as the objective function and volume as the constraint, employing the solid isotropic material with penalization method, density filtering, and the method of moving asymptotes. It examines how factors like mesh type, mesh size, volume fraction, material properties, initial density, filter radius, and penalty factor influence the TO results for a metallic gooseneck chain. The findings suggest that material properties primarily affect numerical variations along the TO path, with minimal impact on structural configuration. For both hexahedral and tetrahedral mesh types, a recommended mesh size is identified where the results show less than a 1% difference across varying mesh sizes. An initial density of 0.5 is advised, with a filter radius of approximately 2.3 to 2.5 times the average unit edge length for hexahedral meshes and 1.3 to 1.5 times for tetrahedral meshes. The suggested penalty factor ranges of 3–4 for hexahedral meshes and 2.5–3.5 for tetrahedral meshes. The optimal geometric reconstruction model achieves weight reductions of 23.46% and 22.22% compared to the original model while satisfying static loading requirements. This work contributes significantly to the integration of TO and AM in engineering, laying a robust foundation for future design endeavors.

1. Introduction

The combination of topology optimization (TO) and additive manufacturing (AM) is one of the most effective tools for structural weight reduction [1,2]. In the design process, the TO is at the initial conceptual design stage, and the quality of its structural configuration directly influences the performance of the final structure. At present, TO-based methodologies include the ground structure approach [3], the homogenization method [4], the variable thickness method [5], the variable density method [6], the evolutionary structural optimization [7], bi-directional evolutionary structural optimization [8], the moving morphable component [9], the level-set method [10], floating projection topology optimization (FPTO) [11,12,13], smooth-edged material distribution optimization for topology (SEMDOT) [14], and the independent continuous mapping method [15]. These methods exhibit distinct advantages in mitigating numerical instability issues (e.g., the variable thickness method demonstrates suitability for shell structure optimization with limited design freedom; the BESO enables bidirectional material addition/removal optimization and adapts well to elastoplastic material design). However, their integration within commercial finite element analysis (FEA) software remains limited, and they have not been extensively implemented in mainstream engineering platforms. Among them, the variable density method, due to its well-established theoretical framework, has been extensively utilized in major FEA software such as HyperWorks, Ansys, Abaqus, SolidThinking Inspire, and Nastran [16]. However, engineers inevitably encounter numerical instability issues (such as intermediate density, mesh dependency, checkerboard patterns, and local extrema) when implementing TO via existing commercial software. These issues are primarily caused by numerical instability or singular solutions in the field functions of the FEA software, and the numerical instability of TO directly affects the accuracy, convergence, and manufacturability of the numerical results [17] and thereby leads to features such as isolated bodies, thin bars, and disconnected branches that negatively affect the performance of the structure. Specifically, isolated bodies refer to features that are not connected to any other parts of the structure; thin bars refer to features that are close to the minimum focal diameter of the laser spot; and disconnected branches refer to features that are connected to each other only at one end of the structure.

Precise control of the structural configuration could help designers generate topological features with clear definitions and excellent performance. To achieve this goal, researchers have proposed a diverse range of methodologies such as the higher-order element method [17], the perimeter constraint method [18], the mesh filtering method [19], the optimality criteria method [20], and the method of moving asymptotes (MMA) [21] to control the structural configuration. Additionally, subsequent investigations have further analyzed the influence of various parameters on the structural configuration. Wang et al. [22], based on the solid isotropic material with penalization (SIMP) method, employed the volume preservation projection method to calculate the genus or number of holes, combined with the filtered function calculated using persistence homology and the design space progressive restriction method, to analyze the minimum compliance optimization problem of a two-dimensional static cantilever structure. The obtained results revealed that with the growth of the filter radius, the number of holes gradually decreased, with a mesh size of 100 × 160, a penalty factor of three, and a maximum number of iterations of 100. Since the filter radius was set as 1.1, many thin bars appeared in the structure. In addition, the influences of various mesh sizes and volume fractions on the structural configuration subjected to a five-hole constraint were also analyzed. Da Silveira and Palma [23] proposed a modified ordered SIMP method and applied a filter radius of 8.4 and an initial design variable of 0.2 to the minimum compliance optimization problem of a two-dimensional four-phase bridge; simultaneously, in the design of steel and aluminum cantilever beams, the filter radius was set equal to 1/100 of the structure length. A too-small filter radius leads to thin bars in the structural configuration. Chen [24], using HyperWorks and a three-dimensional node model, analyzed the effects of the volume fraction, penalty factor, dimensional constraints, symmetry constraints, and checkerboard pattern control on the TO configurations. The performed study revealed that the reasonable range for the volume fraction is 30% to 40%, and when the penalty factor was set as 6, the TO model could meet the structural performance, manufacturability, and ease of the processing requirements. The reasonable range for the minimum member size constraint is 3 to 4 times the average unit size, whereas the maximum member size constraint is 2 times the minimum member size constraint. Moreover, symmetry constraints could make the distribution of structural units more reasonable, and checkerboard pattern control could make the surface of the structural configuration smoother. However, existing investigations on the factors affecting TO essentially focus on two-dimensional and three-dimensional geometric models and analyze only a subset of the influencing factors, while the study of complex three-dimensional models is insufficient in practical engineering applications. In addition, the current proposed parameter standards are mostly absolute values, lacking generality and applicability.

Furthermore, due to the typically complex geometric configurations generated by TO, traditional manufacturing processes often find them difficult or impossible to fabricate. The AM enables the fabrication of complex geometric structures via layer-by-layer stacking of materials to form solid parts [25,26]. However, current AM technologies are still constrained by geometric limitations, and fully unrestricted fabrication is not yet achievable [27,28]. Therefore, it is necessary to perform geometric reconstruction on the structural configurations generated by TO [29,30] to meet the limitations of AM fabrication (e.g., sharp corners, closed internal holes, and overhang angles) [31,32,33,34]. Xiao et al. [35] conducted TO on antenna brackets based on the forming constraints and design rules of selective laser melting (SLM) and optimized the redesigned structure through shape optimization, ultimately achieving a weight reduction of 30.43% and a 50.18% increase in fundamental frequency. Seabra et al. [36] proposed a part design process that combines SLM and TO. In their investigation, with minimum compliance as the objective function and volume as the constraint, they utilized HyperWorks to design and verify an aircraft bracket, which ultimately achieved a 28% reduction in weight and a twofold increase in the safety factor. Shi et al. [37] performed geometric reconstruction on a heavy-load aerospace bracket after TO and optimized the reconstructed model through size optimization. They also performed loading tests on an aerospace bracket manufactured using the SLM. The obtained results revealed that the bracket achieved an 18% reduction in weight while meeting the mechanical performance requirements. Such investigations effectively combine the advantages of TO and AM, especially by fully considering the AM constraints during geometric reconstruction. However, these studies have not fully addressed the feature handling issues in TO structural configurations that may affect the performance of the structure, nor have they deeply explored the effects of TO numerical instability on the optimization results.

Therefore, this study takes the complex structure of a metallic secondary load-bearing gooseneck chain in civil aviation as the research target, investigates minimum compliance as the objective function and volume as the constraint, employs the solid isotropic material with penalization method, density filtering, and the method of moving asymptotes. It systematically and comprehensively examines how factors like mesh type, mesh size, volume fraction, material properties, initial density, filter radius, and penalty factor influence the TO results and proposes relevant recommended standards. In addition, the present investigation addresses the issues of feature management that may affect the structural performance in TO-generated configurations and the AM constraints, proposing geometric reconstruction guidelines and optimization paths in the TO post-processing stage. These measures aim to assist and maximize the advantages of combining TO and AM, improving the practicality and efficiency of the design results.

The main contributions of the authors in this paper can be itemized as follows: (i) systematically and comprehensively examine how factors like mesh type, mesh size, volume fraction, material properties, initial density, filter radius, and penalty factor influence the 3D TO results and propose relevant recommended standards for a metallic secondary load-bearing gooseneck chain in civil aviation; (ii) innovatively introduce a geometric reconstruction approach based on the marching cubes algorithm, PolyNURBS modeling, and SolidWorks modeling (MPS); and (iii) provide theoretical guidance for TO-based designs oriented toward engineering applications, with substantial practical implications.

The remainder of this paper is organized as follows: Section 2 introduces the TO mathematical model based on the SIMP. Section 3 analyzes the key factors affecting numerical stability, including volume fraction, mesh type, mesh size, material properties, initial density, filter radius, and penalty factor. Section 4 proposes specific guidelines and routes for geometric reconstruction to improve the manufacturability of TO. Section 5 presents and discusses the simulation results. Finally, Section 6 provides the conclusions of the study.

2. Topology Optimization of the Mathematical Model Based on SIMP

The variable density method expresses the relative density of a unit and its material elasticity modulus through a continuous variable density function, transforming the discrete optimization problem into a continuous one. Common interpolation models for the variable density method include SIMP [38,39] and the rational approximation of material properties (RAMP) model [40]. This paper adopts the extensively utilized SIMP interpolation model, which drives intermediate-density units toward 0 or 1 through a penalty factor. Mathematically, it is stated by

$$E_i=E_i(x_i)=x_i^{p}E, 0\le x_i\le 1$$

(1)

where E represents the elasticity modulus of the solid material; p denotes the penalty factor; i is the unit number; and $x_i$ signifies the density of the unit, with a value of 0 indicating no material and 1 indicating full material.

2.1. Density Filter Method

This paper utilizes the widely validated density filter method [41] to alleviate checkerboard patterns and mesh dependence in TO. This approach can be mainly formulated as follows:

$${\tilde{x}}_i={\displaystyle \frac{{\displaystyle \sum _{j\in N_i}{H}_{ij}{x}_j}}{{\displaystyle \sum _{j\in N_i}{H}_{ij}}}}$$

(2)

where ${\tilde{x}}_i$ represents the filtered unit density vector, $N_i$ is the set of elements j for which the center-to-center distance $\Delta (i,j)$ to element i is smaller than the filter radius r, and $H_{ij}$ denotes the weight coefficient, a function of the distance between adjacent units, as presented below:

$$H_{ij}=\max (0, r-\Delta (i,j))$$

(3)

where $\Delta (i,j)$ denotes the distance between the centers of units $i$ and $j$, and $r$ signifies the filter radius. The filtered density could define a modified density field by updating Equation (1) as

$$E_i({\tilde{x}}_i)={\tilde{x}}_i^{p}E$$

(4)

2.2. Objective Function and Constraints

This paper aims to minimize compliance with the volume as a constraint and density as a design variable. The main goal is to find the material density distribution that minimizes the structural deformation under predefined boundary conditions and to provide a general deformation measure called compliance, as presented in the following formula:

$$c(\tilde{x})=F^{T}U(\tilde{x})$$

(5)

where $c(\tilde{x})$ denotes the compliance vector, $F$ represents the nodal force vector, and $U(\tilde{x})$ signifies the nodal displacement vector. Combined with the volume constraint, the minimum compliance problem can be mathematically defined as

$$\begin{array}{l} \mathrm{find} x={[x_1,x_2,\dots ,x_e,\dots ,x_n]}^{\mathrm{T}}\\ \mathrm{minimize} c(\tilde{x})=F^{T}U(\tilde{x})\\ \mathrm{subject} \mathrm{to} v(\tilde{x})={\tilde{x}}^{T}v-\overline{v}\le 0\\ x\in \chi , \chi =\{x\in {\Re }^n:0<x_{\min }\le x\le 1\}\end{array}$$

(6)

where density $\tilde{x}$ is given in Equation (2), $n$ is the number of units after discretization, $v={[v_1,\dots ,v_n]}^{\mathrm{T}}$ represents the unit volume vector, and $\overline{v}$ denotes the volume constraint for the predefined design region (i.e., volume fraction). The vector $F$ depends on the design variables and the nodal displacement vector, where $F=K(\tilde{x})U(\tilde{x})$. To avoid singularity in the global stiffness matrix $K(\tilde{x})$, x_min is set equal to 0.001. The computational formula is given in the following form:

$$K(\tilde{x})={\displaystyle \frac{E}{1-{\mu }^2}}k(\tilde{x})$$

(7)

where $k(\tilde{x})$ represents the unit stiffness matrix.

2.3. Method of Moving Asymptotes (MMA)

The MMA algorithm, proposed by Svanberg [42], adjusts the curvature of the convex linearization method to alleviate the issue of local extrema and obtain the global optimal solution. In the current design $x^{(k)}$, the MMA algorithm approximates the minimum compliance problem in Equation (6) as a linear programming problem:

$$\begin{array}{l} \mathrm{find} x={[x_1,x_2,\dots ,x_e,\dots ,x_n]}^{\mathrm{T}}\\ \mathrm{minimize} -{\displaystyle \sum _{i=1}^n[\frac{{(x_i^{(k)}-L_i^{(k)})}^2}{x_i-L_i^{(k)}}\frac{\partial c}{\partial x_i}({\tilde{x}}^{(k)})]}\\ \mathrm{subject} \mathrm{to} v(\tilde{x})={\tilde{x}}^{T}v-\overline{v}\le 0\\ x\in {\chi }^{(k)}\end{array}$$

(8)

where ${\chi }^{(k)}=\left\{x\in \chi \left|0.9L_i^{(k)}+0.1x_i^{(k)}\le x_i\le 0.9U_i^{(k)}+0.1x_i^{(k)},i=1,\dots ,n\right.\right\}$. The lower and upper asymptotes, denoted by $L_i^{(k)}$ and $U_i^{(k)}$, are iteratively updated to reduce oscillations or improve convergence rates.

3. Analysis of the Influencing Factors in Topology Optimization

After geometric simplification, this study utilizes the general module of ABAQUS to complete key steps such as unit type selection, mesh generation, boundary condition setting, and load application. Based on this and in combination with the TO mathematical model outlined in Section 2, a secondary development based on TOSCA is carried out to analyze the key factors affecting numerical instability in TO. Specifically, this study examines the effects of various factors, including volume fraction, mesh size, material properties, initial density, filter radius, and penalty factor, on numerical stability.

3.1. Geometric Model Simplification

In theoretical TO research, hexahedral meshes have been extensively employed due to their high computational accuracy and numerical stability. However, in practical engineering applications, to reduce the complexity of mesh generation for complex geometries, certain geometric features that affect mesh generation often need to be removed or simplified. For instance, small chamfers can be removed, or concave regions can be filled before meshing with hexahedral units, or more flexible tetrahedral meshes can be used directly. To investigate the specific influence of different factors on the TO results under hexahedral mesh conditions, this study utilizes the original model of the gooseneck chain by removing and filling chamfered and concave regions, as illustrated in Figure 1.

The choice of the mesh size substantially affects the accuracy and efficiency of the FEA. When the mesh size is too large, the accuracy of the computational results may be compromised. Conversely, a mesh size that is too small could significantly increase the computational time and lead to excessively high computational costs. Therefore, choosing an appropriate mesh size is crucial to ensure computational accuracy while controlling costs. For this purpose, a mesh sensitivity analysis is performed on the simplified model (model after feature removal and filling), the results of which are demonstrated in Figure 2. From Figure 2a, it can be seen that when the hexahedral mesh size approaches 1.75 mm, the maximum stress value gradually converges to roughly 800 MPa. Similarly, Figure 2b demonstrates that when the tetrahedral mesh size approaches 4 mm, the maximum stress value converges to approximately 851 MPa. The stress discrepancies between hexahedral and tetrahedral meshes arise from the combined effects of shape function accuracy, material anisotropy, singularity handling mechanisms, and convergence efficiency.

3.2. Default Settings

The boundary conditions and load settings in the ABAQUS general module are illustrated in Figure 3. The operational requirements were appropriately determined based on the results of an overall FEA of a specific civil aircraft structural component. In establishing the finite element model for the secondary metal gooseneck chain, the actual connection method was simplified to the bolt-hole fixed constraints, based on its loading and connection conditions. The load application points were set at the center of the gooseneck chain’s ear hole ring, and the load magnitude and direction were then determined based on the components along each coordinate axis. To ensure the safety of the design results, a 1.5× scaling factor for F was utilized, with the following load values: F_x = 331.50 N, F_y = 35,989.50 N, and F_z = −3846 N. The default size of 14 mm is taken for the mesh utilized in the finite element software, and the material properties are set as E = 110.3 GPa and $\mu$ = 0.33.

In the TO module, the non-design regions include the ear flanges and the eight bolt connection holes at the bottom, whereas the remaining areas are set as design regions. The symmetry constraints were imposed on the X-axis at the bottom. The SIMP interpolation model was utilized in the optimization process, with the initial density set to the default value of 0.5. The filter radius was taken as 1.3 times the average unit edge length, and the penalty factor was set to the default value of 3. All subsequent analyses in this paper are based on these default parameters unless other values are explicitly specified for them. The computer used was a Lenovo Legion laptop with the following configurations: Intel^® Core^TM I7-10750H, CPU@2.60GHz, and RAM16.0GB (Intel, Santa Clara, CA, USA).

3.3. Volume Fraction

Before performing TO, it is necessary to define the structure’s volume fraction ($\overline{v}$), which is the ratio of the volume after TO to the original volume, as illustrated in the following formula:

$$\overline{v}={\displaystyle \frac{v_{to}}{v_0}}\times 100\%$$

(9)

where $v_{to}$ denotes the volume after TO and $v_0$ signifies the original volume. The smaller the $\overline{v}$ level, the greater the weight reduction.

With other parameters held at default values, this study analyzes the impact of varying volume fractions ($\overline{v}$) on the TO structure configuration. The results are presented in

Table 1. TO structure configurations obtained for the hexahedral and tetrahedral meshes at various volume fractions ($\overline{v}$).

$\overline{\mathit{v}}$ (%)	Hexahedral Mesh	Tetrahedral Mesh	$\overline{\mathit{v}}$ (%)	Hexahedral Mesh	Tetrahedral Mesh
10			60
20			70
30			80
40			90
50			100

. According to Table 1, it can be observed that for hexahedral meshes, the TO results at various volume fraction values are constrained by the structural geometry. As the volume fraction lessens, more material is removed; in the case of a volume fraction equal to 10%, only the non-design regions remain. In contrast, when the volume fraction is set as 60%, 70%, or 80%, the optimization results demonstrate relatively optimal structural configurations, but undesirable features, such as isolated bodies and disconnected branches, still affect structural performance. Furthermore, for a volume fraction of 60%, the minimum optimized volume is still 15.85% larger than the original unoptimized volume and cannot achieve the weight reduction goal. Compared to hexahedral meshes, the tetrahedral meshes exhibit a substantial reduction in isolated bodies and disconnected branches in the TO results. Meanwhile, the optimal structural configuration is obtained in the case of $\overline{v}$ = 60%, but it still fails to achieve the weight reduction objective.

In summary, the volume fraction substantially affects the TO structure configuration; however, regardless of whether hexahedral or tetrahedral meshes are utilized, the current settings for $\overline{v}$ have not resulted in an ideal structure that meets the weight reduction goal. This is primarily due to the involvement of the mesh size and its related parameters in the TO procedure, and the default mesh size does not usually satisfy the optimization requirements. Therefore, further analysis of the influence of the mesh size on the structure configuration is required. In addition, it is necessary to check whether the mesh sizes determined by the static mesh sensitivity analysis are suitable for the TO. For the $\overline{v}$ setting, this study refers to the experimental validation results from Ref. [43] and utilizes the ratio of the finite element model volume to the original model volume as the default input value of $\overline{v}$, i.e., 41.75%.

3.4. Mesh Size

Table 2. TO structure configurations obtained for the hexahedral and tetrahedral meshes in the presence of various mesh sizes ($\overline{v}$ = 41.7%).

Mesh Size (mm)	Hexahedral Mesh	Mesh Size (mm)	Hexahedral Mesh	Mesh Size (mm)	Tetrahedral Mesh	Mesh Size (mm)	Tetrahedral Mesh
14		3.5		14		4.5
10		3		10		4
6		2.5		8		3.5
5.5		2		6		3
5		1.75		5.5		2.5
4.5		1.5		5		2.25
4		1.25

presents the TO structure configurations obtained under various mesh sizes for both hexahedral and tetrahedral meshes. Figure 4 illustrates the effects of the mesh size on the compliance and computation time. According to Table 2 and Figure 4a, it is clear that as the hexahedral mesh size reduces, the number of branching features in the TO results substantially rises, and the overall structure configuration alters significantly. As the mesh size is taken as 10 mm, the structure configuration connects the first and second columns of the bolt connection holes. As the mesh size places in the range of 3–6 mm, the structure configuration gradually alters to connect the first column to the third and fourth columns of the bolt connection holes. At a mesh size of 5 mm, although the structure configuration has fewer features that affect the structure performance and seems to be relatively ideal, the computational accuracy has not converged. As the mesh size continues to reduce, the structure configuration undergoes more remarkable changes. For the mesh size in the interval of 1.25–2.5 mm, the structure configuration is stabilized and connected to the first and second columns of the bolt connection holes. According to Table 2 and Figure 4b, it can be seen that compared with the hexahedral meshes, the tetrahedral meshes do not exhibit a substantial increase in branching features as the mesh size decreases. The structural configuration regarding the connections at the bolt connection holes is almost consistent with the result of the hexagonal mesh and maintains a stable connection to both the first and second columns throughout the bolt connection holes. For the case of a mesh size of 4 mm, the structure configuration does not exhibit any features that affect the structural performance of the civil aircraft and presents an ideal optimization result.

In summary, the structural configurations obtained under the default mesh size conditions do not meet the requirements of the force transmission path. Unlike hexahedral meshes, tetrahedral meshes offer more stable structural configurations across various size conditions, making them preferable for TO. In addition, based on the results of the static mesh sensitivity analysis in which the computational accuracy converges to 1%, the determined mesh size is also applicable to the TO mesh size requirements. Therefore, in the subsequent analyses, the hexahedral mesh size of 2 mm and the tetrahedral mesh size of 4 mm are utilized as default values to further examine the influence of other factors on the TO results.

3.5. Material Properties

It is well known that changes in material properties remarkably affect the results of static analysis, but further investigation is still required regarding the impact of material properties on the structure configuration in the TO.

Table 3. TO structure configurations obtained in the presence of various material properties ($\overline{v}$ = 41.75%, a hexahedral mesh size of 2 mm, and a tetrahedral mesh size of 4 mm).

No.	Material Properties	Hexahedral Mesh	Tetrahedral Mesh
1	E = 110.3 GPa; $\mu$ = 0.31
2	E = 110.3 GPa; $\mu$ = 0.33
3	E = 118 GPa; $\mu$ = 0.33
4	E = 120 GPa; $\mu$ = 0.342
5	E = 209 GPa; $\mu$ = 0.269

presents the TO structure configuration corresponding to various material properties [43] under the conditions of a hexahedral mesh size of 2 mm and a tetrahedral mesh size of 4 mm. Figure 5 demonstrates the influence of various material properties on the compliance and computation time for both hexahedral and tetrahedral meshes. The results provided in Table 3 and Figure 5 reveal that regardless of whether a hexahedral or tetrahedral mesh is used, when the material properties undergo significant changes, the TO structure configuration remains almost unchanged and has only a slight impact on the compliance value. This phenomenon can also be explained by Equation (7). In TO calculations, the global stiffness matrix is closely related to the elastic modulus (E) and Poisson’s ratio ($\mu$). The changes in these two factors lead to changes in the values of $K(\tilde{x})$, resulting in fluctuations in the compliance value. Furthermore, as demonstrated in Equation (1), the TO utilizes unit density as a design variable in the range of [0.001, 1]. The given elastic modulus (E) exhibits a transcending or descending trend according to a proportional relationship. This indicates that, in TO, changes in the material properties of the same material type do not substantially influence the results of structural configuration optimization. Therefore, the typical material properties of the same material type can be employed as input values in the TO procedure. In the following analyses, the material parameters E = 110.3 GPa and $\mu$ = 0.33 will be utilized as default material properties. These parameters were validated through finite element simulations and experimental comparisons, providing reliable and conservative results for the analysis.

3.6. Initial Density

The initial density, x_initial, and material properties are crucial factors assigned to each unit during the initial calculation of TO. These factors include consistent values for E, $\mu$, and density. In TO, x_initial is typically defaulted as $\overline{v}$. For instance, in the case of $\overline{v}$ = 40%, each unit is assigned a value of x_initial = 0.4. Although the mature MMA is employed in this study to reduce the impact of the initial parameters on the TO results and ensure a global optimum solution rather than a local optimum, further research is still required to examine whether various values of x_initial affect the global optimal solution.

Table 4. TO structural configurations obtained with various initial densities x_initial ($\overline{v}$ = 41.75%, hexahedral mesh size = 2 mm, tetrahedral mesh size = 4 mm, E = 110. 3 GPa, and $\mu$ = 0.33).

xinitial	Hexahedral	Tetrahedral	xinitial	Hexahedral	Tetrahedral
0.1			0.6
0.2			0.7
0.3			0.8
0.4			0.9
0.5			1.0

presents the TO structural configurations obtained with various x_initial values under the conditions of a hexahedral mesh size of 2 mm and a tetrahedral mesh size of 4 mm. Figure 6 analyzes the influence of various x_initial values on the compliance and computation time. For the hexahedral mesh, the TO structural configurations for different values of x_initial remain consistent along the primary load path, with only slight variations in the secondary load path. Using the compliance value of x_initial = 0.1 as a baseline, the maximum compliance variation rate is 1.42%, whereas the maximum computation time variation rate is 22.44%. In the case of 0.5 ≤ x_initial ≤ 1, the compliance variation rate and the computation time variation indicate two opposite trend behaviors. The structural configuration, compliance, and computation time all demonstrate superior performance at x_initial = 0.5. For the tetrahedral mesh, the TO structural configurations remain essentially unchanged for different x_initial values. Using the compliance value of x_initial = 0.1 as a baseline, the maximum compliance variation rate is obtained as 0.72%, and the maximum computation time variation rate is attained as 23.66%. In the case of x_initial > 0.5, compliance reaches near convergence, with the optimal compliance observed at x_initial = 0.5. The structural configuration exhibits no substantial features affecting performance, and the computation time is at the optimal level.

In summary, the impact of x_initial on the structural configuration and compliance is minimal. Compared to the hexahedral mesh, the influence of x_initial on the tetrahedral mesh is even weaker. This indirectly validates the effectiveness of the MMA algorithm in mitigating local extrema issues in the TO procedure. Furthermore, for both hexahedral and tetrahedral meshes, a value of x_initial = 0.5 yields the ideal structural configuration. As a result, it is recommended to set x_initial = 0.5 as the default input value for the TO.

3.7. Filter Radius

As illustrated in Equations (3) and (4), the main principle of the density filtering method is to modify the density of the central unit using the density information of all units within the filter radius (r). Specifically, the weighted average density of units within the filter radius replaces the density of the central unit. The size of r determines the number of units involved in density filtering. When r is smaller than the average edge length of the unit, only the central unit falls within the filter range, and the density filter method fails to function effectively. This can lead to issues such as checkerboard patterns and mesh dependency, which in turn negatively affect the TO structural configuration.

Table 5. TO structural configurations obtained for various values of the filter radius (r) ($\overline{v}$ = 41.75%, hexahedral mesh size = 2 mm, tetrahedral mesh size = 4 mm, E = 110.3 GPa, $\mu$ = 0.33, and x_initial = 0.5).

r (mm)	Hexahedral Mesh	r (mm)	Hexahedral Mesh	r (mm)	Tetrahedral Mesh	r (mm)	Tetrahedral Mesh
0.5		3.4		0.25		6.4
1.0		3.6		0.5		6.8
2.0		3.8		1.0		7.2
2.2		4.0		2.0		7.6
2.4		4.2		4.0		8.0
2.6		4.4		4.4		8.4
2.8		4.6		4.8		8.8
3.0		4.8		5.2		9.2
3.2		5.0		5.6		9.6
				6.0		10

presents the TO structural configurations obtained with various filter radii (r) for hexahedral mesh sizes of 2 mm and 4 mm. Figure 7 presents the analysis results of the effect of different r values on compliance and computation time. The predicted results provided in Table 5 and Figure 7 reveal that for the hexahedral mesh, in the case of r = 0.5, the number of units in the filter range is small, and the efficiency of the density filtering method is limited. This leads to a large density discrepancy between adjacent units, which yields excessive branching in the structural configuration. With the increase in r, the number of neighboring units used to adjust the central unit’s density grows, and the density discrepancies among units reduce. The penalty factor applied to the unit densities further reduces the branching, resulting in a clearer and more stable structure. In the case of 4.6 ≤ r ≤ 5, the compliance value converges, and the structural configuration remains consistent. For the case of 0.25 ≤ r ≤ 2, several performance-affecting features appear in the tetrahedral mesh. With the growth of r, the structural configuration becomes clearer, and the number of performance-affecting features lessens. In the case of 4 ≤ r ≤ 6.8, the compliance value converges, and the structural configuration stabilizes, with no performance-affecting features remaining. However, in the case of r > 7.2, the number of units becomes too large, resulting in overly small density discrepancies among units after filtering, which makes the overall density distribution too uniform and introduces features that affect structural performance.

In conclusion, r has a significant effect on the structural configuration. For hexahedral meshes, the recommended filter radius is between 4.6 and 5, i.e., 2.3 to 2.5 times the average unit edge length. For tetrahedral meshes, the recommended filter radius is between 5.2 and 6, i.e., 1.3 to 1.5 times the average unit edge length. The suggested coefficient for tetrahedral meshes is smaller than that for hexahedral meshes because, for the same r, a tetrahedral mesh has more units than a hexahedral mesh, and therefore, the density filter method has a greater effect. In the subsequent analysis of other factors, the default value of r is set to 2.4 times the average unit edge length for hexahedral meshes and 1.4 times for tetrahedral meshes.

3.8. Penalty Factor

According to Equation (1), the SIMP interpolation model of the variable density theory transforms the discrete problem into a continuous optimization problem by introducing intermediate-density units. Bendsøe and Sigmund [6] proved the physical existence of this model in 1999; however, in engineering applications, due to the difficulty of intermediate-density units, it is typically necessary to avoid their formation or at least minimize their quantity. For this purpose, a penalty should be applied to intermediate-density units to ensure that their density approaches 0 or 1. Bendsøe and Sigmund [44] proposed a formula to calculate the minimum value of the penalty factor as follows:

$$\begin{array}{l}p\ge \max \{{\displaystyle \frac{2}{1-\mu }},{\displaystyle \frac{4}{1+\mu }}\} (\mathrm{in} 2\text{-}\mathrm{D})\\ p\ge \max \{15{\displaystyle \frac{1-\mu }{7-5\mu }},{\displaystyle \frac{3}{2}}{\displaystyle \frac{1-\mu }{1-2\mu }}\} (\mathrm{in} 3\text{-}\mathrm{D})\end{array}$$

(10)

In the case of $\mu$ = 0.33, the minimum values of p in the second and third dimensions are obtained as 3 and 2, respectively. Bendsoe and Sigmund [44] recommended that the minimum value of p should be greater than 3 for both 2D and 3D cases, but the maximum value of p was not specified. Therefore, further analysis of p is required to verify the minimum value and also determine the maximum value.

Table 6. TO structural configurations obtained based on various penalty factors (p) ($\overline{v}$ = 41.75%, E = 110.3 GPa, $\mu$ = 0.33, x_initial = 0.5; hexahedral mesh size = 2 mm, r = 4.8, tetrahedral mesh size = 4 mm, and r = 5.6).

p	Hexahedral Mesh	Tetrahedral Mesh	p	Hexahedral Mesh	Tetrahedral Mesh
1.0			4.0
2.0			5.0
2.5			6.0
3.0			9.0		-
3.5			10		-

presents the TO structural configurations obtained with different values of p for hexahedral meshes with a size of 2 mm and tetrahedral meshes with a size of 4 mm. Figure 8 demonstrates the effect of the variable p on the compliance and computation time. According to Table 6 and Figure 8, it can be seen that for hexahedral meshes, in the case of p = 1, the penalty effect disappears, which leads to an unmanufacturable structure configuration. As p increases, the penalty of unit densities is strengthened, and the units with medium density gradually converge to densities of 0 or 1, and the number of branches in the structure configuration decreases. The branches connected to the third column of the bolt connection holes are gradually eliminated; the load transfer path becomes clearer, and the compliance converges. However, in the case of 6 < p < 9, the results do not converge, which leads to calculation failure. In the case of p ≥ 9, certain branches in the gooseneck of the chain are removed, forming large triangular voids, and new branches appear under the voids. For the tetrahedral mesh, in the case of p = 1, the penalty effect disappears, and the structure configuration cannot be manufactured. Compared with the hexahedral mesh, the tetrahedral mesh achieves a better penalty effect at smaller values of p. In the case of 2.5 ≤ p ≤ 3.5, the obtained structure configuration is nearly identical to that of the hexahedral mesh in the range of 3 ≤ p ≤ 4 with the compliance convergence. As p increases further, some branches in the structure configuration are eliminated. However, in the case of p > 6, the calculation results do not converge, which leads to computational failure.

In conclusion, in the case of p ≥ 6, the penalty factor becomes too large, causing calculation failure due to non-convergence. Therefore, the maximum value of p should be set below 6. For the hexahedral mesh, it is recommended to set p in the range of [3,4], and for the tetrahedral mesh, the recommended p range is [2.5, 3.5].

4. Geometric Reconstruction of Topology Optimization Results

Even with the recommended standards for various influencing factors proposed in Section 3, some features that affect the performance remain in the obtained TO structural configurations, and these should be further eliminated. In addition, even though SLM technology provides high degrees of freedom and surface quality, certain manufacturing limitations still exist, and the presence of these features may lead to manufacturing failures. Therefore, this chapter proposes geometric reconstruction criteria and a geometric reconstruction approach based on the manufacturing limitations of the SLM technology and the features that affect the structural performance observed during TO. In the next chapter, the FEA of the geometric reconstruction model is carried out through orthogonal experiments. The feasibility of the criteria and geometric reconstruction approach is analyzed to further promote the effective integration of the SLM and TO technologies.

4.1. Geometric Reconstruction Criteria

Based on the manufacturing constraints of SLM technology [27,28,45,46] and undesirable features such as isolated bodies and disconnected branches affecting the structural performance in the TO results from Section 3, the following geometric reconstruction criteria are established:

(1)

Removal of isolated body features: Isolated body features are primarily caused by intermediate-density units generated by the variable density method of TO. These features do not contribute to structural performance improvements, but they can lead to mesh failure in the FEA and thereby an increase in ineffective branches during AM; hence, they must be eliminated.

(2)

Material removal from thin branches: Thin branches can be considered thin-walled features. Unless the structure design specifically requires thin-walled features, such a feature generally increases the need for a support material and could damage the surface quality during support material removal; thus, such features should be eliminated.

(3)

Material removal and connection of disconnected branches: The presence of disconnected branches suggests that they may become part of the optimal load transfer path. These disconnected branches should be extended and connected to ensure that the structural performance meets the loading requirements. If additional weight reduction is feasible, the elimination of this feature should be validated to see whether the loading requirements can still be met, thereby maximizing the benefits of weight reduction.

(4)

Transition treatment of sharp corner features: Metal-based AM technologies typically use lasers as energy sources, and their laser spots are approximately circular with a minimum focal diameter. For instance, EOS devices have laser spot diameters ranging from 40 to 100 µm, and the SLM solution devices range from 70 to 115 µm. Therefore, unless fine sharp corner features must be retained, sharp corners should undergo transition treatment to accommodate the physical characteristics of the laser spot and improve manufacturing quality.

(5)

Removal of closed internal hole features: During TO, intermediate-density units may lead to the creation of closed internal holes. Metal AM technologies using powder spreading or powder delivery methodologies are ineffective at removing powder particles inside internal holes, thus requiring secondary processing to eliminate them. This post-processing reduces structural performance and damages the surface quality. Therefore, the generation of closed internal hole features should be avoided as much as possible.

(6)

Ensuring machining allowance: Although SLM technology offers higher manufacturing precision than other metal AM technologies, regions requiring higher precision than the SLM capability (e.g., mating surfaces or fixed connection areas) must be reserved with a machining allowance to allow for subsequent surface finishing to improve surface smoothness.

(7)

Shape optimization of stress concentration regions: TO is conducted based on the existing geometric model, and its optimization results are constrained by geometric boundary conditions. For stress concentration areas placed at geometric boundaries, shape optimization should be performed during post-processing to lessen stress concentrations and enhance mechanical performance.

4.2. Geometric Reconstruction Approach

Based on research on geometric reconstruction methods in current engineering applications and extensive validation with numerous examples, the geometric reconstruction approach based on the marching cubes (MC) algorithm [47,48]–PolyNURBS modeling–SolidWorks Modeling (MPS) was proven to be feasible and efficient. The specific steps of this approach can be itemized in the following form:

Step 1:

Adjust the MC algorithm’s isosurface threshold: By adjusting the isosurface threshold of the MC algorithm, the lower bound of the density value for intermediate-density units to be retained is controlled. The intermediate-density units above this lower bound are taken as the solid material, whereas those below it are considered the void material. This process lessens or eliminates the undesirable features resulting from TO under a given volume constraint that impacts structural performance.

Step 2:

Use PolyNURBS for solid fitting: The PolyNURBS method is commonly utilized to perform solid fitting on the TO structural configuration. During such a process, the pre-established geometric reconstruction criteria should be fully taken into account to ensure that the fitted structure meets the design requirements and optimizes performance.

Step 3:

Use SolidWorks for parametric modeling: In SolidWorks, parametric modeling is performed for non-design regions (such as ear flanges and bolt connection holes) to complement the structure generated by TO and complete the engineering design.

5. Simulation Experimental Results and Discussions

5.1. Simulation Protocol Based on the Orthogonal Experiment

To further examine the weight of various factors on the compliance and computation time, the maximum stress after geometric reconstruction, and the weight reduction ratio, an appropriate experimental design is necessary. If each factor is set with three levels, a full factorial design requires 27 experiments, which is both time-consuming and labor-intensive. To optimize the experimental process and enhance efficiency, commonly used experimental design methodologies include the Plackett–Burman method [49], the response surface method [50], and an orthogonal experimental design [51]. This study adopts the orthogonal experimental design method, which is suitable for multi-factor multi-level analysis, to reduce the number of experiments, decrease the computational time for statistical analysis of the experimental data, and quickly identify the optimal combination and structure of the experiment. In an orthogonal experimental design, the notation L_t(l^u) is commonly utilized to represent the orthogonal table, where L represents the code for the orthogonal table, t denotes the total number of experiments, l is the number of factor levels, and u signifies the number of factors.

Based on the analysis in Section 3, the main factors influencing the TO structural configuration are mesh size, filter radius, and penalty factor, and the recommended standards for each factor have been defined. Therefore, three levels for each factor are set (as illustrated in

Table 7. Factors and levels utilized in the orthogonal experimental design.

Mesh Type	Level No.	Factor Level
		C	D	E
Hexahedral	1	1.75	4.6	3
	2	2	4.8	3.5
	3	2.5	5	4
Tetrahedral	1	3.5	5.2	2.5
	2	4	5.6	3
	3	4.5	6	3.5

), and a three-factor, three-level L₉(3³) orthogonal table is designed (see

Table 8. Orthogonal table accounting for both hexahedral and tetrahedral meshes.

No.	Hexahedral			Tetrahedral
	C	D	E	C	D	E
1	1.75	4.80	4.00	3.5	6.0	3.0
2	1.75	5.00	3.50	3.5	5.2	2.5
3	1.75	4.60	3.00	3.5	5.6	3.5
4	2.00	5.00	4.00	4.0	5.2	3.0
5	2.00	4.80	3.00	4.0	6.0	3.5
6	2.00	4.60	3.50	4.0	5.6	2.5
7	2.50	5.00	3.00	4.5	5.2	3.5
8	2.50	4.60	4.00	4.5	6.0	2.5
9	2.50	4.80	3.50	4.5	5.6	3.0

). Here, C represents the mesh size; D denotes the filter radius; E stands for the penalty factor, and C1 signifies the first level of the mesh size factor C. Other symbols can be identically defined.

5.2. Structural Configurations of Topology Optimization and Results of Geometric Reconstruction

Table 9. TO structural configurations and geometric reconstruction results from orthogonal experiments via hexahedral and tetrahedral meshes.

No.	Hexahedral Mesh		Tetrahedral Mesh
	Structural Configuration	Geometric Configuration	Structural Configuration	Geometric Configuration
1
2
3
4
5
6
7
8
9

presents the TO structural configurations and their geometric reconstruction results obtained from orthogonal experiments via hexahedral and tetrahedral meshes. According to Table 9, it can be seen that, based on the recommended standards for influencing factors proposed in Section 3, the orthogonal experiments resulted in high-performance TO structural configurations. Subsequently, the obtained TO structural configurations were reconstructed according to the geometric reconstruction criteria and the geometric reconstruction approach by the FEA on the reconstructed models using a tetrahedral mesh size of 2 mm. The obtained results are provided in

Table 10. FEA results of the geometric reconstruction obtained from orthogonal experiments.

Mesh Type	No.	Factor Level			Compliance (mJ × 104)	Maximum Stress (MPa)	Number of Iterations	Computation Time (s)	Weight Reduction Ratio (%)
		C	D	E
Hexahedral	1	1.75	4.80	4.00	6.94	896.2	36	1166	20.95
	2	1.75	5.00	3.50	6.95	1095.7	37	1195	22.38
	3	1.75	4.60	3.00	6.94	1055.3	37	1189	22.38
	4	2.00	5.00	4.00	7.02	975.2	35	931	22.35
	5	2.00	4.80	3.00	6.94	959.6	38	900	21.31
	6	2.00	4.60	3.50	7.00	915.7	36	862	20.94
	7	2.50	5.00	3.00	7.00	957.7	38	460	20.64
	8	2.50	4.60	4.00	7.03	986.7	35	447	21.77
	9	2.50	4.80	3.50	7.04	948.8	36	442	23.46
Tetrahedral	1	3.5	6.0	3.0	6.88	1108.4	40	1232	21.03
	2	3.5	5.2	2.5	6.87	1080.9	49	1490	21.18
	3	3.5	5.6	3.5	6.87	998.5	38	1164	22.22
	4	4.0	5.2	3.0	6.86	928.0	40	1070	20.29
	5	4.0	6.0	3.5	6.91	969.8	37	837	20.36
	6	4.0	5.6	2.5	6.86	975.1	46	1017	21.24
	7	4.5	5.2	3.5	6.90	963.1	38	580	19.98
	8	4.5	6.0	2.5	6.90	972.7	46	699	20.18
	9	4.5	5.6	3.0	6.89	924.5	39	595	20.79

. The analyses reveal that the maximum stress of the reconstructed gooseneck chain primarily occurs in the stress singularity and stress concentration areas around the ear flange and the bolt connection hole. Since the maximum stress values in these areas do not reflect the actual situation and are not the focus of the present study, after removing the stress singularity and stress concentration regions, the neck of the gooseneck chain became the main maximum stress region. Compared with the hexahedral mesh, the recommended standards for influencing factors based on the tetrahedral mesh lead to more stable structural configurations in orthogonal experiments that eventually converge to the global optimal solution, thus confirming the effectiveness of the recommended standards.

5.3. Range and Variance Analyses

The results of the FEA reflect the range of indicator test values when varying the factor levels, as indicated by the R range. This range represents the difference between the maximum and minimum values of the average indicator test values for each level, as presented in the following formula:

$$R=\max \left\{k_m\right\}-\min \left\{k_m\right\} m=1,2,\dots ,l$$

(11)

where m represents the factor level number and k_m denotes the average test value related to level m, which is evaluated by

$$k_m={\displaystyle \frac{K_m}{s}}$$

(12)

in which K_m is the sum of the test values corresponding to level m on any column and s denotes the number of times each level appears on any column.

The R-value can help determine the primary and secondary factors affecting the test indicators, where a larger R indicates a greater influence of that factor on the test indicators. A trend chart, in which the factor level is plotted on the horizontal axis and the k_m values on the vertical axis, can provide a more intuitive understanding of the primary and secondary factors.

Variance analysis is an effective statistical test for experimental data. It determines the extent to which each factor affects the test indicators by calculating the sum of the squared deviations (SS_T) and the F-test value.

More specifically, SS_T represents the total variance of the indicator test values from the overall mean, reflecting the total discrepancy between the test results. Mathematically, it is formulated as

$$S{S}_T={\displaystyle \sum _{m=1}^l{\displaystyle \sum _{g=1}^{t_m}{(y_{mg}-\overline{y})}^2}}$$

(13)

where t_m denotes the number of experiments at factor level m, y_mg represents the test result for the g-th experiment at factor level m, $\overline{y}={\displaystyle \frac{1}{t}}{\displaystyle \sum _{m=1}^l{t}_m{\overline{y}}_m}$ signifies the arithmetic average of all test results, and ${\overline{y}}_m={\displaystyle \frac{1}{t_m}}{\displaystyle \sum _{g=1}^{t_m}{y}_{mg}}$ denotes the arithmetic average of all test results at factor level m.

In addition, we need to define a statistical quantity for factors, that is, the so-called F_fac. As illustrated in the following, it is calculated as the ratio of the mean square between levels to the mean square within levels:

$${\mathrm{F}}_{fac}={\displaystyle \frac{M{S}_{fac}}{M{S}_e}}={\displaystyle \frac{\frac{S{S}_{fac}}{d{f}_{fac}}}{\frac{S{S}_e}{d{f}_e}}}$$

(14)

where df_fac = l − 1 represents the degree of freedom between levels, df_e = t − l denotes the degree of freedom within levels, $S{S}_{fac}={\displaystyle \sum _{m=1}^l{t}_m}{({\overline{y}}_m-\overline{y})}^2$ denotes the sum of squares of deviations between levels, and $S{S}_e={\displaystyle \sum _{m=1}^l{\displaystyle \sum _{g=1}^{t_m}{(y_{mg}-{\overline{y}}_m)}^2}}$ signifies the sum of squares of deviations within levels.

Based on the obtained F_fac values, significance level analysis should be performed via the accompanying probability p-value from the F-distribution table. The case of p < 0.01 indicates that the factor possesses a very significant effect on the indicator, whereas the case of 0.01 < p < 0.05 reveals a substantial effect; and finally, the case of p > 0.05 indicates that the factor has no significant effect on the indicator [52].

Figure 9 demonstrates the trend graph of the orthogonal experiments, and

Table 11. Range and variance analysis results of the orthogonal experiments.

Indicator			Hexahedral Mesh			Tetrahedral Mesh
			C	D	E	C	D	E
Compliance (mJ × 104)	km	k1	6.943	6.990	6.960	6.873	6.877	6.877
		k2	6.987	6.973	6.997	6.877	6.873	6.877
		k3	7.023	6.990	6.997	6.897	6.897	6.893
	R		0.08	0.017	0.037	0.024	0.024	0.016
	SST		0.010	0.001	0.003	0.001	0.001	0.001
	p		0.107	0.675	0.301	0.232	0.232	0.342
	Optimal level		C1	D2	E1	C1	D2	E1(E2)
	Primary and secondary factors		C > E > D			C = D > E
	Optimal solutions		C1E1D2			C1D2E1(C1D2E2)
Maximum stress (MPa)	km	k1	1028.667	998.833	1003.800	1062.600	990.667	1009.567
		k2	950.167	934.867	986.733	957.633	966.033	986.967
		k3	964.400	1009.533	952.700	953.433	1016.967	977.133
	R		78.5	74.666	51.1	109.167	50.934	32.434
	SST		10,495.042	9781.336	4060.749	22,953.002	3892.696	1659.376
	p		0.591	0.608	0.789	0.136	0.482	0.686
	Optimal level		C2	D2	E3	C3	D2	E3
	Primary and secondary factors		C > D > E			C > D > E
	Optimal solutions		C2D2E3			C3D2E3
Weight reduction ratio (%)	km	k1	21.903	21.697	21.443	21.477	20.483	20.867
		k2	21.533	21.907	22.260	20.630	21.417	20.703
		k3	21.957	21.790	21.690	20.317	20.523	20.853
	R		0.424	0.21	0.817	1.16	0.934	0.164
	SST		0.319	0.066	1.053	2.161	1.671	0.049
	p		0.944	0.988	0.835	0.017	0.021	0.426
	Optimal level		C3	D2	E2	C1	D2	E1
	Primary and secondary factors		E > C > D			C > D > E
	Optimal solutions		E2C3D2			C1D2E1
Computation time (s)	km	k1	1183.333	849.667	832.667	1295.333	1046.667	1068.667
		k2	897.667	833.000	836.000	974.667	925.333	965.667
		k3	449.667	848.000	862.000	624.667	922.667	860.333
	R		733.666	16.667	29.333	670.666	124	208.334
	SST		820,576.222	505.556	1547.556	675,120.889	30,104.889	65,106.889
	p		0.001	0.387	0.659	0.003	0.067	0.032
	Optimal level		C3	D2	E1	C3	D3	E3
	Primary and secondary factors		C > E > D			C > E > D
	Optimal solutions		C3E1D2			C3E3D3

lists the range and variance analysis results of the orthogonal experiments. From the R and SS_T values in Figure 9 and Table 11, it can be seen that for the hexahedral mesh, the impact weights of the factors on the compliance, maximum stress, weight reduction ratio, and computation time follow the following order: C > E > D, C > D > E, E > C > D, and C > E > D, respectively. In addition, for the optimal solution for each indicator, lower values for compliance, maximum stress, and computation time indicate better optimization results. Therefore, the optimal factor level for each indicator is the one with the minimum k_m value. The optimal solutions for compliance, maximum stress, the weight reduction ratio, and computation time also follow the following order: C1E1D2, C2D2E3, E2C3D2, and C3E1D2, respectively. For the tetrahedral mesh, the impact weights of the factors on the compliance, maximum stress, weight reduction ratio, and computation time follow the following order:

C = D > E, C > D > E, C > D > E, and C > E > D, respectively. The corresponding optimal solutions are C1D2E1(C1D2E2), C3D2E3, C1D2E1, and C3E3D3.

5.4. Comprehensive Balance Method Analysis

Table 11 presents that for various performance indicators, the influence of different factors is significantly different, making it difficult to uniformly determine the priority order of three factors among the four indicators. In addition, the importance of various indicators is contradictory. Therefore, it is necessary to comprehensively investigate all factors and indicators to extract the optimal solution. For multi-indicator orthogonal experiment analysis methods, the comprehensive balance method and the comprehensive scoring method are commonly utilized. However, due to the difficulty in reasonably selecting scoring criteria and weights for each indicator, this study employs the comprehensive balance method to examine the experimental results [53]. The core of the comprehensive balance approach is to identify the primary and secondary influencing factors for each indicator and their optimal solutions through an intuitive analysis of individual indicators. Then, domain knowledge and practical experience are combined to perform a comprehensive analysis and comparison of the indicators to ultimately derive the best overall solution.

In the comprehensive balance method analysis, the following principles must be followed: (1) Priority principle: When the influence of a factor on different indicators is significantly different, this factor should be prioritized as the main influencing factor at its optimal level. (2) Majority principle: When the influence of a factor on different indicators is similar, the principle of “minority follows majority” should be followed, and the level that often appears as the optimal level should be selected. (3) Weight principle: When different indicators have different levels of importance, such as when comprehensively judging the optimal levels of factors, the more important indicators should be prioritized. (4) Efficiency principle: When the discrepancy in the levels of a factor is small, the principle of improving efficiency and reducing consumption should be followed to select the appropriate level.

Based on Figure 9, Table 11, and the principles of the comprehensive balance method, the experimental results have been comprehensively analyzed with the following specific analysis steps:

5.4.1. Hexahedral Mesh

Factor C: According to the analysis of R-values, factor C is the most important factor for the three indicators—compliance, maximum stress, and computation time—but it is a less significant factor for the weight reduction ratio indicator. Therefore, the main factors should be given priority when determining the optimal level. From the k-value analysis, the values of k₁, k₂, and k₃ of factor C for compliance and the weight loss ratio are not substantially different, whereas, for maximum stress, the values of k₂ and k₃ demonstrate a slight difference. From the analysis of p-values, factor C exhibits a very substantial impact on computation time. Considering the importance of factor C for different indices and the need to reduce computational costs, C3 is selected.

Factor D: According to the analysis of R-values, factor D represents a minor factor for compliance, the weight loss ratio, and computation time, but it is a secondary factor for maximum stress. In determining the optimal level, priority should be given to the maximum stress indicator. From the analysis of k-values, the k₂ value is selected for all indicators. From the analysis of p-values, factor D possesses no significant impact on any of the four indicators. Based on the majority principle and the importance of factor D for different indices, D2 is selected.

Factor E: According to the analysis of R-values, factor E represents the main factor for the weight reduction ratio, secondary for compliance and computation time, and a minor factor for maximum stress. From the analysis of k-values, the values of k₁, k₂, and k₃ for factor E are not substantially different for compliance. From the analysis of p-values, factor E has no significant effect on any of the four indicators. Based on the importance of factor E for different indicators, E2 is selected.

Based on the above analysis, the optimal solution for the hexahedral mesh is obtained as C3D2E2, which corresponds to a mesh size of 2.5 mm, a filter radius of 4.8 mm, and a penalty factor of 3.5. This optimal solution corresponds to test 9 in Table 10, and the finite element stress map is presented in Figure 10.

5.4.2. Tetrahedral Mesh

Factor C: Based on the analysis of R-values, factor C is the most important factor for all four indices. From the analysis of p-values, factor C exhibits a very important effect on the computation time and a significant impact on the weight reduction ratio. Based on the influence of factor C on various indices and the engineering objective of weight reduction, C1 is selected.

Factor D: According to the analysis of R-values, factor D represents the most crucial factor for compliance, maximum stress, and the weight reduction ratio, but it is a minor factor for computation time. From the analysis of k-values, k₂ is selected for compliance, maximum stress, and the weight reduction ratio. From the analysis of p-values, factor D has a significant effect on the weight reduction ratio and computation time. Based on the impact of factor D on different indicators and the engineering goal of weight reduction, D2 is selected.

Factor E: According to the analysis of R-values, factor E is a minor factor for compliance, maximum stress, and the weight reduction ratio, and it is a secondary factor for computation time. From the k-value analysis, the values of k₁, k_2, and k₃ for factor E for the compliance and weight reduction ratio are not significantly different. From the p-value analysis, factor E has a significant impact on the computation time. Based on the effect of factor E on various indicators and the need to reduce the computational cost, E3 is selected.

Based on the above analysis, the optimal solution for the tetrahedral mesh is C1D2E3, which corresponds to a mesh size of 3.5 mm, a filter radius of 5.6 mm, and a penalty factor of 3.5. This optimal solution corresponds to test 3 in Table 10, and the finite element stress map is illustrated in Figure 11.

To validate the reliability and feasibility of the optimal solution obtained using the comprehensive balance method, a comparative study was conducted with the gooseneck chain’s finite element model, which was verified experimentally. The specific results are provided in

Table 12. Feasibility verification of the best overall solution.

Parameter	Original Model	Model in Ref. [43]	The Best Overall Solution
			Hexahedral Test 9	Tetrahedral Test 3
Volume (mm3)	281,544	226,899	215,427	218,913
Weight reduction ratio (%)	0	19.38	23.46	22.22
Maximum stress (MPa)	1332.3	1094.9	948.8	998.5

, and the weight reduction ratio is calculated based on the volume of the original model.

According to Table 12, it can be seen that the optimal solutions for both hexahedral and tetrahedral meshes obtained using the comprehensive balance method satisfy the static loading requirements and outperform the experimentally verified finite element models in terms of the weight reduction ratio and maximum stress. In particular, the optimal solution for the hexahedral mesh has the highest weight reduction ratio and the lowest maximum stress, indicating the potential for further weight reduction and providing more space for subsequent iterative optimization. This result also confirms the feasibility and benefits of the recommended standards for TO factors, geometric reconstruction criteria, and the geometric reconstruction approach.

5.5. Validation of Recommended Criterias of TO: A Case Study of an Aircraft Bracket

To validate the effectiveness of the proposed recommendation criteria, this study employs an aircraft bracket as a verification case [54] and utilizes both hexahedral and tetrahedral meshes for discretization modeling. In the Abaqus general module, boundary conditions and loading configurations are established, as illustrated in Figure 12; loading points are positioned at the center of the aircraft bracket’s ear hole ring with force components of F_y = 25,980.76 N and F_z = 15,000 N, while fixed constraints are applied to four bolt connection holes. Material properties are defined as E = 110.3 GPa and $\mu$ = 0.33. Within the TO module, non-design regions encompass the ear rings and bolt holes, with the remaining areas designated as design domains and subjected to X-axis symmetry constraints at the base.

Figure 13 demonstrates the mesh generation and TO structural configurations of the aircraft bracket developed under the recommended criteria. Specifically, Figure 13b displays the optimization results employing 2 mm hexahedral meshes with the initial density set to 0.5, the filter radius equivalent to 2.4 times the average element edge length (4.8 mm), and the penalty factor configured to 3.5. Figure 13d presents the optimization results employing 3 mm tetrahedral meshes, with the initial density set to 0.5, the filter radius scaled to 1.4 times the average element edge length (4.2 mm), and the penalty factor configured to three. Experimental verification confirms the complete elimination of isolated bodies, disconnected branches, and thin bars in both optimized configurations, substantiating the proposed criteria’s effectiveness in enhancing topology optimization manufacturability.

6. Conclusions

This paper focuses on minimum compliance as the objective function while using volume as the constraint. It employs the SIMP interpolation model of the variable density method, along with density filtering and MMA, to analyze the factors influencing topology optimization (TO) for a metallic secondary load-bearing gooseneck chain utilized in civil aviation. Key factors examined include mesh type, mesh size, material properties, initial density, filter radius, and penalty factor, with recommended standards proposed for each. In addressing challenges associated with TO, such as isolated bodies and disconnected branches, which can impair structural performance as well as limit additive manufacturing (AM), this paper introduces geometric reconstruction criteria and a reconstruction approach based on the marching cubes algorithm, PolyNURBS modeling, and SolidWorks modeling. This research not only advances the integration of TO and AM technologies in practical engineering applications but also lays a substantial theoretical foundation and provides guidance for future engineering design.

Nevertheless, certain limitations are acknowledged in this study. First, some features such as isolated bodies, disconnected branches, and thin bars in TO results may compromise structural performance, and their elimination through optimization procedures (rather than post-processing geometric reconstruction) warrants further investigation. Second, while compliance was adopted as the objective function with volume constraints, practical engineering applications necessitate incorporating additional multi-physics constraints such as fatigue stress. Furthermore, although parametric analysis via single-factor methods identified dominant influences of the mesh size, filter radius, and penalty factor, their synergistic interaction mechanisms remain insufficiently resolved, requiring systematic exploration through sensitivity analysis to elucidate coupling relationships. Additionally, the proposed penalty factor recommendations derive from conventional SIMP frameworks prove inapplicable to non-penalization algorithms (e.g., FPTO and SEMDOT). Finally, while fixed high-resolution meshes ensured computational accuracy, they substantially increased computational costs; future studies may implement adaptive meshing or anisotropic mesh refinement strategies to enhance their efficiency (potentially reducing computational load by 30–50%).