Sketch-Guided Topology Optimization with Enhanced Diversity for Innovative Structural Design

Abstract

Topology optimization (TO) is a powerful generative design tool for innovative structural design, capable of optimizing material distribution to generate structures with superior performance. However, current topology optimization algorithms mostly target a single objective and are highly dependent on the problem definition parameters, causing two critical issues: limited human controllability and solution diversity. These issues often lead to burdensome design iterations and insufficient design exploration. This paper proposes a multi-solution TO framework to address them. Human designers express their stylistic preferences for structures through sketches which are decomposed into stroke and closed-shape elements to flexibly guide each TO process. Sketch-based constraints are integrated with Fourier mapping-based length-scale control to enhance human controllability. Solution diversity is achieved by perturbing Fourier mapping frequencies and load conditions in the neural implicit TO framework. Adaptive parallel scale adjustment is incorporated to reduce the computational cost for design exploration. Using the structural design of a wheel spoke as a case study, the mechanical performance and diversity of the generated TO solutions as well as the effectiveness of human control are analyzed both qualitatively and quantitatively. The results reveal that the sketch-based constraints and length-scale control have distinct control effects on structural features and have different impacts on the mechanical performance and diversity, thereby enabling fine-grained and flexible human controllability to better balance conflicting objectives.

1. Introduction

Topology optimization (TO) is a powerful tool for innovative structural design. By optimizing material distribution within a given design domain, it can generate conceptual designs that meet specified objectives and constraints, offering novel structures with superior engineering performance [1,2,3,4,5]. It has been widely applied in tasks like lightweight optimization and also shows great potential in other fields including heat transfer [6,7], fluid dynamics [8], photonics [9,10], etc.

An important feature of TO is that it eliminates the need for parameterization, avoiding the limitations of manually defined design parameters, and offering a broad, unconstrained design space. As such, it is an ideal generative design algorithm for the early conceptual design phase. Currently, TO algorithms for structural design can generate highly complex and optimized structural features, surpassing parametric design solutions in both mechanical strength and material efficiency [11]. However, they have two key limitations that hinder their broader application: a lack of efficient human control over the generated structural features and a limited diversity of solutions [12,13]. Since design exploration involves multiple complex and hard-to-quantify factors, a practical approach is to search for a diverse set of good solutions rather than a single optimized one, with cross-domain experts analyzing, filtering, and determining the final optimal design through iterations. In the collaboration between design experts and TO algorithms, enabling a human to directly and precisely control some structural features, such as making the minimal length scale of structures equal to a preset value, is crucial to reducing iterative processes. Meanwhile, solution diversity is key to exploring high-quality and novel designs.

Recently, studies focusing on enhancing human controllability and solution diversity for TO algorithms have attracted increasing attention. To address the first challenge, some studies [14,15,16,17,18] have focused on controlling the length scale of generated geometric features, as the length scale is critical for issues like local stress concentration, structural complexity, and manufacturability. Other approaches focus on guiding the TO process using specific geometric patterns, such as periodic cells [19,20,21,22], image content [23], or a reference design [24]. Among these, sketch-based TO guidance methods [12,25] demonstrate efficient controllability and are particularly intuitive for designers accustomed to expressing ideas through sketches. However, existing studies only provide a single way to achieve human control, limiting the flexibility in managing the fine-grained features and hindering the achievement of proper balance among conflicting design factors. Furthermore, though human control inevitably impacts design exploration freedom and reduces solution diversity, there are no existing studies that have tried to quantify this effect or explored methods to address it.

To address the second challenge, existing studies typically enhance solution diversity by perturbing the load condition parameters or adjusting the multi-objective weights [12,26]. Some studies [13,26,27,28] also propose data-driven approaches to enhance diversity by learning implicit representations from existing TO datasets, followed by random sampling in the learned latent feature space. However, studies on TO solution diversity remain lacking, and to our knowledge, no studies have quantitatively analyzed TO solution diversity or compared different diversifying approaches. Existing qualitative results indicate insufficient diversity, and show that diversifying solutions by merely perturbing hyper-parameters has a significant negative impact on the mechanical performances of structures.

To address the aforementioned challenges in enhancing collaboration between human designers and generative algorithms in innovative structural design, this paper proposes a novel multi-solution TO framework. This framework integrates two approaches to achieve human controllability: sketch-based constraints and length-scale control of geometric features. Free-hand sketches are decomposed into stroke elements and closed-shape elements, with two distinct constraints introduced for them accordingly: a stroke-based undirected structural boundary constraint and a closed-shape-based regional constraint. They are further integrated with length-scale control of geometric features to provide multifaceted control strategies. Each control strategy, with unique control characteristics for structural features, has a different impact on design quality and diversity, thereby enabling fine-grained and flexible human controllability. Additionally, the framework enhances solution diversity by introducing randomness through the perturbation of Fourier mapping frequencies and load condition parameters in the density-based neural implicit TO process. To reduce the computing cost caused by having multiple TO processes, this framework implements an adaptive parallel computing scale adjustment method. A case study of an innovative design of automobile wheel spokes is conducted, and comprehensive studies on human controllability and its impact on both design quality and diversity are carried out qualitatively and quantitatively.

The main contributions of this paper are summarized as follows:

• Novel sketch-based constraints, including a stroke-based undirected structural boundary constraint and a closed-shape-based regional constraint, are proposed to enable sketch-based TO control.
• Sketch-based constraints and length-scale control of geometric features are integrated, providing multifaceted control strategies to enable fine-grained and flexible control.
• A diversifying method based on random sampling of Fourier mapping frequencies in a neural implicit TO process is proposed, and quantitative studies on different diversifying methods as well as the effects of human control strategies on design quality and diversity are conducted.
• An adaptive parallel computing scale adjustment method is incorporated to improve the efficiency of multiple TO processes.

2. Related Works

2.1. Human Control in Topology Optimization

The main advantage of TO is that designers only need to specify the design objectives, constraints, and load conditions within the design domain, without the need to provide prior knowledge for structural design parameterization. Most existing TO algorithms, particularly the earlier developed ones [29,30,31], provide an optimal solution of structural topology for a TO problem according to the design factor of mechanical performance. However, as numerous factors need to be considered in engineering practice, the TO algorithms fail to produce satisfactory solutions directly, leading to laborious iterations between human designers and the algorithms [32,33]. This problem became more pronounced due to the immature post-processing techniques of TO solutions. Effective human control over TO solutions is the key to this problem, which is becoming a highlight topic in this research field.

Recent studies on this problem have focused on controlling the length scale of geometric features [14,15,16,17,18], as the length scale is very important not only for mechanical performance but also other factors like manufacturability. Traditional TO algorithms tend to generate excessively fine-grained structures, which can incur problems in addictive manufacturing process such as local collapse or difficulty in support material removal. Ha et al. [15] and Schiffer et al. [16] achieved interactive control of the feature sizes within a region of interest through a proximity-based weighting function and three-phase projection scheme, respectively. The latter further enabled control of multi-phase maximum and minimum solid feature sizes and/or void feature sizes. Li et al. [34] explicitly derived the numerical formulation of minimum structural feature size constraints for density-based TO algorithms, offering broad applicability and low sensitivity to algorithm parameters. He et al. [35] introduced a hole-filling strategy in the bidirectional evolutionary structural optimization (BESO) framework to control structural complexity by filling internal cavities and tunnels, respectively, with preset structures whose size parameters were adjustable. Chandrasekhar et al. [14] introduced a Fourier mapping layer in a density-based neural implicit TO framework to control the maximal and minimal geometric feature size. Lazarov et al. [36] provided a detailed review and comparative analysis of studies on length-scale control for density-based TO algorithms.

Research on human controllability also includes using specific geometric patterns [19,20,21,22]. Zhang et al. [19] used predefined texture patterns to guide the optimization, dividing the design domain into multiple groups, where the repetition of the control pattern within each group was managed independently. Meng et al. [20] achieved similar controllability by arranging predefined patterns to determine the design domain. Li et al. [21] derived specific geometric constraints for the internal regions of shell structures to make the generated structures match the predefined patterns. Zhang et al. [25] proposed using free-hand sketches for TO guidance by extracting multi-level morphological features from design images using convolutional neural networks and introducing a similarity constraint between the extracted morphological features of TO solutions and the sketch. Their proposed method generates structural designs that embody the high-level morphological semantics of the sketch. Zhang et al. [23] used aesthetic images as guidance, employing independent feature extraction neural modules for the aesthetic style and form content of the images, respectively, which were then used to apply separate constraints. Zhong et al. [37] proposed exploiting the text–image priors embedded in the CLIP model [38] to enable text-based control over the aesthetic styling of the generated structures. Li et al. [12] proposed combining drawing guidance with manual scoring to modify TO solutions interactively. The sensitivity numbers of BESO within the affected regions of drawn strokes were adjusted, while the sensitivity numbers could also be globally adjusted according to manual scoring feedback.

Overall, length-scale control of geometric features has been enabled by several existing methods with its effectiveness and reliability validated, while guidance with specific geometric patterns is relatively immature. A proper length-scale control method can be selected according to customized design requirements for a specific TO framework, such as a density-based solid isotropic material, with penalization (SIMP) or BESO algorithms. The key challenge here is to properly determine the length-scale parameters [14,16]. For guidance with specific geometric patterns, existing studies can be divided into two categories: leveraging domain knowledge, or reference designs based on similarity constraints. Methods of the first category are sensitive to the application scenario and highly dependent on expertise, which limits their usability. Methods of the second category encounter inevitable conflicts between the similarity constraint and other objectives like mechanical performance optimization. Current methods tackle this only by a single weight coefficient of the similarity constraint, which is insufficient to reach a proper balance.

To enhance the flexibility of human control over TO solutions and better balance the various design factors, this paper combines sketch-based constraints with length-scale control. The sketches are decomposed into stroke elements and closed-shape elements, with different constraint terms and penalty weights assigned, respectively, to enable fine-grained control of different structural features. As studies on TO solution diversity and the impact of human controllability on diversity are lacking, this paper fills this gap as well.

2.2. Deep Learning-Enhanced Topology Optimization

Deep learning (DL) [39] has shown great potential in learning latent features from large datasets, facilitating tasks like new content exploration in latent spaces and prediction in decision-making tasks. In the field of TO, integrating DL methods has emerged as a promising solution to some of the field’s tough challenges. Such studies can be broadly categorized into three types: (1) developing black-box TO solver models that replace traditional iterative TO algorithms [27,40,41,42,43]; (2) fusing DL with traditional algorithms [13,44,45,46]; and (3) using DL models to replace the finite element analysis (FEA) process, thus accelerating the iterations [47,48,49].

Among the first type of study, Yu et al. [42] used an encoder–decoder framework based on convolutional neural networks (CNNs) to establish a mapping from boundary conditions and optimization settings to the solutions of traditional TO algorithms. They further integrated conditional generative adversarial networks (cGANs) to improve image resolution of the outputs. Nie et al. [40] proposed a cGAN-based TO solver model, and incorporated multi-physical field simulation results with the load and boundary conditions to improve the quality of the generated designs. Maze et al. [41] replaced cGANs with a diffusion model [50,51] as the TO solver, and introduced a surrogate model to ensure structural performance and manufacturability. Zheng et al. [43] employed a U-Net model [52] as a 3D TO solver. Yoo et al. [27] proposed learning low-dimensional latent space from existing TO designs with a convolutional auto-encoder, then generated diverse designs via a sampling strategy based on design of experiment (DOE) in latent space.

Another branch of studies focused on fusing DL methods with iterative TO algorithms. Chandrasekhar et al. [44] encoded the density function of the Solid Isotropic Material with Penalization (SIMP) method using neural networks (NNs), combining the sensitivity analysis through automatic back-propagation of the NNs and independent FEA module. Chen et al. [45] introduced the FEA results of the initial material distributions as conditional input to the neural-encoded density function. Their results showed improved structural performance and higher computing efficiency. Wang et al. [46] embedded TO within a cGAN framework with loss terms of structural compliance and volume constraints to guide the optimization. Jang et al. [13] used reinforcement learning in an iterative fusion pipeline to improve design diversity.

A significant challenge in TO is the computational cost associated with the FEA. To address this, Qian and Ye [48] proposed a dual-model artificial neural network for forward computation and sensitivity analysis of the FEA, with discrepancy loss between the predicted and true FEA results. Yin et al. [49] introduced physics-informed neural networks (PINNs) to simulate the forward and sensitivity analysis of the FEA instead. Li et al. [47] proposed using a structure-mapping neural network to refine coarse-grained TO solutions into fine-grained ones at each iteration, thereby enabling an FEA to be conducted on low-resolution grids.

Although DL-based TO solvers beat traditional iterative algorithms in inference efficiency and design diversity, the generated designs mostly fail in satisfying mechanical constraints and manufacturability, making the DL-based solvers invalid as direct solutions to design problems. Using DL-based methods to replace FEA is effective in accelerating iterations while having a small influence on final outputs. One of the obstacles to using data-driven methods is the difficulty in establishing large datasets. Despite current advancements in incorporating DL, traditional iterative TO algorithms continue to play a critical role in ensuring the reliability and quality of the final designs. Fusion of them to fully leverage their respective strengths has great potential for achieving more efficient, high-quality, and diverse topology optimization.

3. Method

Figure 1 shows the proposed multi-solution TO framework for solving the typical structural design problem of minimizing compliance under volume constraint. Multiple parallel TO processes for solving the same problem are employed with consistent algorithm hyper-parameters. In each process, a SIMP-based method with a neural network that implicitly encodes the density function over the continuous domain is implemented. The volume constraint and sketch-based constraints are transformed into penalty loss terms and combined with the compliance loss term through a weighted sum, serving as the total training loss for the neural-encoded density function. The neural input signals are Fourier frequency components of sampled grid points. The high- and low-frequency thresholds of the input signals determine the length scale of the spatial geometric features. The Fourier sampling frequencies and load force magnitudes are randomly perturbed across TO processes to produce diverse design solutions. The TO processes are synchronized, and have their solution set filtered by removing redundant solutions at regular optimization iteration intervals after the first several iterations, which adaptively adjusts the parallel scale. Filtering is according to the similarity metrics in terms of geometry and design objective values.

The remainder of this section gives details of the proposed framework and the application of it in an innovative structural design scenario.

3.1. TO Process with Neural-Encoded Density Function

The mathematical formulation of solving the aforementioned design problem with the SIMP method is given in Equation (1). The optimizable variables of the design problem are the material density values of the discrete elements, $\mathit{\rho }=\left\{{\rho }_e\right\}$, where ${\rho }_e$ is the material density of the e-th element within the value range [0, 1]. ${\rho }_e=1$ represents that this element has solid material and ${\rho }_e=0$ represents that this element is void. The design objective is to minimize the total structural compliance C.

For a specific design problem to be solved, the user-defined hyper-parameters include the applied force $\mathit{F}$, the prescribed volume $V^{*}$, the element stiffness matrix for solid material element ${\mathit{K}}_e^0$, the material’s Young’s modulus $E_0$, the element volume $v_e$, and the penalization exponent p. In Equation (1), the first constraint, $\mathit{K}\left(\mathit{\rho }\right)\mathit{u}=\mathit{F}$, is the static equilibrium equation, where $\mathit{K}(·)$ is the finite element stiffness matrix as a function of the element densities $\rho$ assembled from the element stiffness matrix ${\mathit{K}}_e\left(\mathit{\rho }\right)$ computed by Equation (2). $\mathit{u}$ is the displacement field computed from the static equilibrium equation. The second constraint represents the volume constraint.

$$\begin{array}{cc}\hfill \underset{\rho }{min}& C={\mathit{F}}^{\top }\mathit{u}={\mathit{u}}^{\top }\mathit{K}\left(\mathit{\rho }\right)\mathit{u}\hfill \\ \hfill \mathrm{subject}\mathrm{to}& \mathit{K}\left(\mathit{\rho }\right)\mathit{u}=\mathit{F}\hfill \\ \hfill & \sum _e{\rho }_e{\nu }_e\le V^{*}\hfill \end{array}$$

(1)

$$\begin{array}{c}\hfill \mathit{K}\left(\mathit{\rho }\right)=\left[{\mathit{K}}_e\left(\mathit{\rho }\right)\right]=\left[{\mathit{K}}_e^0{E}_0{\rho }_e^p\right]\end{array}$$

(2)

In the SIMP method, the element densities are continuous within the range [0, 1], and are pushed to 0 or 1 with increasing penalty value (typically initialized as $p=1$ and gradually increased), thereby generating an approximately binary topology design solution. The element densities are optimized on the basis of an FEA of structural compliance conducted on fixed sample grids using domain-specific methods such as Optimality Criteria (OC) or Method of Moving Asymptotes (MMAs). This approach limits the resolution of the solution and requires an analytical sensitivity analysis. This study adopts the neural-encoded density function proposed by Chandrasekhar and Suresh [44], which leverages the high non-linearity of neural networks to represent a continuous density function over the design field in the SIMP method. This transforms the discrete element density optimization into continuous density function learning, alleviating the limitation of the solution resolution. This also allows different sample grids to be applied for various FEA objectives, for better balance between FEA accuracy and computation cost. Additionally, a sensitivity analysis can be automatically performed using the back-propagation of the neural network. The neural-encoded density function is denoted as ${\rho }_w(·)$, where w is the neural weights.

3.2. Length-Scale Control

This study adopts the Fourier mapping method [14] to achieve structural feature length-scale control. The input coordinates of sample points are transformed into Fourier domain frequency signals, which are filtered using high- and low-frequency thresholds corresponding to the maximum and minimum allowable feature sizes, $l_{max}$ and $l_{min}$, respectively. For each sample point $p=(x,y)$, the frequency signals are computed as shown in Equation (3), and are then passed through the leaky ReLU activation function and the sigmoid normalization function before being fed into the neural network. The relationships between the frequency thresholds and the allowable feature sizes are shown in Equation (4).

$$\begin{array}{cc}\hfill {\mathit{FT}}_{\mathit{f}}\left(p\right)=& \left[cos\left({\displaystyle \frac{\pi }{h}}\left(f_1^{x}x+f_1^{y}v\right)\right),sin\left({\displaystyle \frac{\pi }{h}}\left(f_1^{x}x+f_1^{y}v\right)\right)\right.\hfill \\ \hfill & cos\left({\displaystyle \frac{\pi }{h}}\left(f_2^{x}x+f_2^{y}v\right)\right),sin\left({\displaystyle \frac{\pi }{h}}\left(f_2^{x}x+f_2^{y}v\right)\right)\hfill \\ \hfill & ⋮\hfill \\ \hfill & \left.cos\left({\displaystyle \frac{\pi }{h}}\left(f_{n_f}^{x}x+f_{n_f}^{y}v\right)\right),sin\left({\displaystyle \frac{\pi }{h}}\left(f_{n_f}^{x}x+f_{n_f}^{y}v\right)\right)\right]\hfill \end{array}$$

(3)

$$\begin{array}{c}\hfill f\in \left[{\displaystyle \frac{h}{l_{max}}},{\displaystyle \frac{h}{l_{min}}}\right]\end{array}$$

(4)

where $\mathit{f}=\{f_1^x,f_1^y,f_2^x,f_2^y,f_{n_f}^x,f_{n_f}^y\}$ is the set of Fourier frequencies, $n_f$ is the number of frequencies, ${\mathit{FT}}_{\mathit{f}}\left(p\right)$ is a $2n_f$-length vector after Fourier mapping, and h is the mesh edge length.

Structural compliance is evaluated by finite element analysis using a uniformly sampled grid. The forward computation of sample densities, static equilibrium equation, and computation of final compliance are given in Equations (5)–(7), respectively.

$$\begin{array}{c}\hfill {\rho }_e{\left(\mathbf{w},\mathit{f},p\right)}_{p\in e}={\rho }_{\mathbf{w}}\left({\mathit{FT}}_{\mathit{F}}\left(p\right)\right)\end{array}$$

(5)

$$\begin{array}{c}\hfill J_e={\mathit{u}}_e^{\top }{\mathit{K}}_e{\mathit{u}}_e\end{array}$$

(6)

$$\begin{array}{c}\hfill J=\sum _e{\rho }_e^p{J}_e\end{array}$$

(7)

where e is the analysis element, $J_e$ is the compliance of element e, and J is the total compliance.

In the experiment of this study, an FEA is conducted with one sample point located at the center of element. When higher FEA accuracy is required, it is straightforward to use multiple sample points within an element, taking advantage of a continuous density function. The number and distribution of Fourier frequencies have a significant influence on design solutions; this will be analyzed and used to enhance solution diversity in Section 4.

3.3. Sketch-Based Constraints

Free-hand sketches are intuitive for designers to express ideas. Some aspects of sketch images such as line thickness or shading variations are irrelevant to the desired structural topology, and the sketch can be simply decomposed into open-stroke elements and closed-shape elements. Corresponding constraint terms tailored to the unique characteristics of strokes and closed shapes are introduced. Strokes reflect the designer’s intended structural boundaries without imposing constraints on material distribution in adjacent regions. During the early design ideation, designers typically conceptualize characteristic lines that capture key stylistic or visual features rather than defining precise shapes. To accommodate this, a stroke-based constraint is introduced, ensuring that the structural boundaries of the generated design align with the sketched strokes while maintaining a degree of freedom in neighboring regions to facilitate further exploration of the design space. The proposed constraint is an undirected structural boundary constraint, where the loss is computed based on the similarity between current structural boundaries and the sketched strokes, as described in Equation (8).

$$\begin{array}{c}\hfill L_{edge}=\sum _{i,j}{\left({\mathit{RI}}_{i,j}-{\overline{\mathit{E}}}_{i,j}\right)}^2·{\mathbb{I}}_{[{\mathit{RI}}_{i.j}>0.5]}\end{array}$$

(8)

where $\mathit{RI}$ represents the sketch image normalized to [0,1], $\overline{\mathit{E}}$ represents the current structural boundaries within the design domain, and ${\mathbb{I}}_{[·]}$ is the indicator function.

The structural boundaries are extracted by a differentiable convolution operator on the density distribution map, where the convolution kernel is the edge-detecting Sobel operator, as shown in Equations (9)–(11). The density distribution map is obtained by forward computation of the density function at uniformly sampled grid points. In experiments, the sample grid resolution is taken as the image resolution of the sketch for convenience.

$$\begin{array}{cc}\hfill \mathit{E}& =\sqrt{{\left(\mathit{I}\ast {\mathit{K}}_x\right)}^2+{\left(\mathit{I}\ast {\mathit{K}}_y\right)}^2+ϵ}\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill {\mathit{E}}^{*}& ={\displaystyle \frac{\mathit{E}-min\left(\mathit{E}\right)}{max\left(\mathit{E}\right)-min\left(\mathit{E}\right)}}\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill \overline{\mathit{E}}& ={\mathit{E}}^{*}·{\mathbb{I}}_{\left[\right(i,j)\notin \mathbf{P}]}\hfill \end{array}$$

(11)

where $\mathit{I}$ is the density distribution map, ∗ is the convolution operator, ${\mathit{K}}_x$ and ${\mathit{K}}_y$ are the Sobel kernels along the x- and y-directions, respectively, $ϵ$ is a small constant to avoid division by zero (in experiments, $ϵ$ = 1 $\times {10}^{-6}$), and $\mathit{P}$ is the non-design field within the density distribution map.

The sketch-based constraint is transformed into a penalty loss term for the neural density function. Since the steps of structural boundary extraction and constraint loss computation are differentiable, sensitivity analysis for this constraint term can be automated through the back-propagation of the neural network. In experiments, morphological dilation is applied to preprocess the sketch image to improve the saliency of strokes, and the weight coefficient of this constraint is gradually increased as similar to the volume constraint.

The closed-shape elements not only contain contour lines, which reflect the intended structural boundaries, but also signify the visual integrity of the drawn shape by the closure of the contours. Based on this, a direct way to apply a constraint is to make the closed shape directly represent a cavity or a solid. This study presents a more flexible constraint that drives the material distribution within the closed-shape region to be nearly uniform while allowing some variation. It is transformed to a penalty loss term, thereby enforcing a soft constraint to achieve visual integrity. The calculation of this loss term is shown in Equation (12).

$$\begin{array}{c}\hfill L_{area}=\sum _{i,j}{\left({\mathit{RIF}}_{i,j}-{\mathit{I}}_{i,j}\right)}^2·{\mathbb{I}}_{[{\mathit{RIF}}_{i.j}>0.5]}\end{array}$$

(12)

where $\mathit{RIF}$ is the preprocessed sketch image with the closed-shape region flood-filled and normalized to [0, 1].

The weight coefficient for this loss term is a useful control parameter to adjust the saliency of the drawn closed shapes in the generated structure. When it is set as small, the generated structure may have small branches within the closed shape which are not visually obvious but significantly reduce the total compliance. Also, the preserved freedom of exploration within the closed shapes promotes solution diversity. When it is set as large, the closed-shape-based constraint approximates the hard boundaries of the design field, which is suitable when local structural details have been determined.

In application, designers can first conceptualize key stylistic structural features, sketch them, and make the algorithm explore diverse designs by setting a small weight coefficient for the closed-shape-based constraint. In subsequent interactive iterations between designers and the TO algorithm, structural parts are gradually designed and the constraint weight coefficient can be set to be larger for structural shapes in those parts to preserve them, until the structural design is complete.

The total loss function for updating the neural-encoded density function is given in Equation (13).

$$\begin{array}{c}\hfill L={\displaystyle \frac{J}{J_0}}+\alpha {\left({\displaystyle \frac{{\sum }_e{\rho }_e{v}_e}{V^{*}}}-1\right)}^2+\beta ·L_{edge}+\gamma ·L_{area}\end{array}$$

(13)

where $J_0$ is the initial total compliance value for normalization, $\alpha$ is the weight coefficient for the volume constraint, and $\beta$ and $\gamma$ are weight coefficients for the stroke-based constraint and closed-shape-based constraint, respectively.

3.4. Parallel Computing for Multi-Solution TO Framework

In the proposed TO process, perturbing the load force magnitudes, the initial weights of the neural network, as well as the Fourier sampling frequencies are all useful ways to generate multiple diverse designs. To account for the low interpretability of the neural network, this study perturbs the Fourier sampling frequencies $\mathit{f}$ and the load force magnitudes $\mathit{F}$ across multiple TO processes. As will be proved by the experiments in Section 4, this can provide sufficient design diversity.

A larger parallel scale increases the likelihood of discovering more innovative and high-performance designs, but it also increases the computing cost and produces more redundant solutions that are identical or highly similar to each other. To balance this conflict and remove redundant solutions, an adaptive parallel scale is employed. Multiple synchronized parallel TO processes are run, and a similarity measure for each pair of solutions within the current solution is set after a fixed number of iterations are conducted. A randomly selected solution is removed from pairs with high similarity to reduce the parallel scale before continuing the optimization iterations.

Similarity indexes include differences in compliance values and the image-based Structural Similarity Index Measure (SSIM) and pixel-wise mean absolute error (MAE), as computed by Equations (14) and (15). The SSIM index is better at capturing the structural characteristics that align with human visual perception, while the MAE index better reflects the data-level difference.

$$\begin{array}{cc}\hfill SSIM\left(x,y\right)& ={\displaystyle \frac{\left(2{\mu }_x{\mu }_y+C_1\right)·\left(2{\sigma }_{xy}+C_2\right)}{\left({\mu }_x^2+{\mu }_y^2+C_1\right)·\left({\sigma }_x^2+{\sigma }_y^2+C_2\right)}}.\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill MAE\left(x,y\right)& ={\displaystyle \frac{1}{N}}\sum _{i=1}^N\left|x_i-y_i\right|\hfill \end{array}$$

(15)

where x and y represent the density distribution maps of two solutions, ${\mu }_x$ and ${\mu }_y$ are the average density values, ${\sigma }_x^2$ and ${\sigma }_y^2$ are density variances, ${\sigma }_{xy}$ is the covariance of density distribution maps, $C_1$ and $C_2$ are small constants to avoid division by zero, and N is the number of pixels.

In early iterations of the TO process, the number of gray cells with intermediate density values is large, which makes the structural boundaries blurry. As the iterations progress and the SIMP penalty factor increases, the densities tend to binarize and structural boundaries become more clear and stable. As design diversity is mostly cultivated in the early iterations, a parallel scale is adapted only after the early iterations.

A conflict between human preferences expressed through sketches and the compliance minimization objective is very likely to occur. Guided by sketch-based constraints, some generated structural features may be detrimental to compliance minimization, thus are unreasonable for multi-objective optimization. Recognizing such structural features, i.e., the region of conflict, is important to ensure the quality of solutions.

To achieve this, we propose a sensitivity-based method to automatically recognize the region of conflict in structures. Given the TO framework’s total loss minimization, the elements with positive sensitivity values for the sum of compliance loss and sketch-based penalty loss, denoted as $L_t$, should have their density values decreased under the virtual condition where sketch-based constraints are not applied. Therefore, among the solid elements in the generated structures, the ones with positive sensitivity values can be recognized as the region of conflict. The computation is expressed in Equation (16). The sensitivity can be conveniently computed via gradient back-propagation of neural networks, and the autograd function of PyTorch 1.5 is implemented in the experiment. During iterations of adaptive-scale parallel solving, solutions with large conflict regions are also removed.

$$\begin{array}{c}\hfill F=\{{\rho }_e|{\displaystyle \frac{\partial L_t}{\partial {\rho }_e}},{\rho }_e>0.8\}\end{array}$$

(16)

Algorithm 1 is the pseudo-code of the multi-solution TO framework with adaptive parallel scale.

Algorithm 1: Pseudo-code of multi-solution TO framework with human controllability and adaptive parallel scale.

Input: $V\ast$, RI, $l_{min}$, M Parameter: ${\rho }_{\mathbf{w}}$ Output: $\left\{T{O}_i\right\}$ Initialize $E_0,K_e^0,t_{SSIM},t_{MAE},t_{comp},t_{vol},N,N_0,n,\alpha ,\beta ,\gamma ,p=1$ Compute RIF from RI For each of M individual TO process do:    Randomly initialize neural weights $\mathit{w}$    Sample $\mathit{f}$ according to Equation (1); ${\mathit{F}}_b$, grid points ${\mathit{p}}_{FEA}$ for FEA, ${\mathit{p}}_{OP}$ for output    Compute $J_0$ For each iter of N iteration do:    For each of n parallel steps do:       For each i of M individual TO process do:          NN forward for $p_{\mathbf{w}}\left({\mathit{FT}}_{\mathit{f}}\left({\mathit{p}}_{OP}\right)\right)$, generate current design $T{O}_i$          NN forward for $p_{\mathbf{w}}\left({\mathit{FT}}_{\mathit{f}}\left({\mathit{p}}_{FEA}\right)\right)$, compute $J_i,L_i,V_i=\sum p_e{v}_e$          NN backward, update $\mathit{w}$       Update $p=min(p_{max},p+p_{\mathrm{Inc}})$; $\alpha =min({\alpha }_{max},\alpha +{\alpha }_{\mathrm{Inc}})$          $\beta =min({\beta }_{max},\beta +{\beta }_{\mathrm{Inc}})$; $\gamma =min({\gamma }_{max},\gamma +{\gamma }_{\mathrm{Inc}})$    If iter $>N_0$ then:       For each pair ${(T{O}_i,T{O}_j)}_{i\ne j}$ do:          Compute $SSIM(i,j)$, $MAE(i,j)$          If $SSIM(i,j)>t_{SSIM}$ and $MAE(i,j)<t_{MAE}$ and $|J_i-J_j|<t_{\mathrm{comp}}$ and $|V_i-V_j|<t_{\mathrm{vol}}$ then:             delete $T{O}_j$ For each $T{O}_i$ do:    If $T{O}_i$ has large region of conflict then:       delete $T{O}_i$ Return TO solutions $\left\{T{O}_i\right\}$

4. Experiments

4.1. Experiment Settings

This paper uses the innovative design of automotive wheels as a case study to validate the proposed multi-solution TO framework. The focus of the design is the disk-view 2D structural pattern of the spokes, which can be solved as a 2D structural TO problem. Typical design requirements for wheel hubs include being lightweight and having a high stiffness, using the structural yield (compliance) under specific static loading conditions as the stiffness metric. Accordingly, the TO problem is to minimize compliance under the total material volume constraint. Since the design of automotive wheels also requires aesthetic considerations, a balance between structural performance and aesthetics is desired, making this case highly suitable for validating the TO framework, aiming at enhancing solution diversity and human controllability.

Specific definitions of the design field and boundary conditions are shown in Figure 2. Structural compliance under both normal force and shear force is used as the stiffness metric. The normal force simulates the weight of the vehicle and wheel hub itself. It distributes on the outer rim within a 30° central angle range. The shear force simulates the ground friction force on the wheel, acting at the midpoint of the normal force distribution. Within a square domain, density values are set as 1 (solid) for the wheel rim and axle regions, variable (actual design field) for the spoke region, and 0 (void) for the other region. A fixation constraint is applied at the center of the axle.

The rotational symmetry of the wheel spoke is handled explicitly: the annulus design field is evenly split into several fan units, with one (e.g., the topmost) designated as the reference unit. After sampling grid points for density computation, the points outside the reference unit are mapped to their central symmetric counterparts inside the reference unit, based on angular intervals defined by the fan angle. The counterparts are then taken as the actual input to the density function.

The content of this section illustrates the following: (1) an experiment for generating innovative structural designs by the TO framework; (2) an experiment for human controllability; (3) an analysis of the impacts of human control on both the design quality and diversity; (4) a comparative analysis with existing methods; and (5) an analysis of adaptive-scale parallel computing. The first experiment is used to verify the ability of the proposed method to enhance structural design exploration, and a qualitative evaluation of the generated results is conducted. The second experiment is used to verify that our method enables designers to control structural features precisely and flexibly. The third experiment is conducted to quantitatively analyze the impacts of human control parameters on design quality and diversity, in order to provide insights for designers to better fine-tune them. The fourth experiment is to verify the superiority of the proposed method over existing ones. The fifth experiment is to verify the usefulness of an adaptive scale in improving computing efficiency. Figure 3 gives a summary of the experiment design.

4.2. Experiment for Generating Innovative Structural Designs

The TO framework is tested using the hyper-parameters summarized in

Table 1. Hyper-parameters of multi-solution TO framework.

Parameter	Value	Parameter	Value	Parameter	Value
Desired volume	0.35	$N_0$	30	${\alpha }_{max}$	100
NN layers	1 × (100,)	n	5	${\beta }_{max}$	1
$n_{EEA}$	180	M	20	${\gamma }_{max}$	0.1
$l_{min}$	9	${\alpha }_{Inc}$	0.3	$p_{max}$	6
$l_{max}$	80	${\beta }_{Inc}$	5 × ${10}^{-4}$	E	1
$n_f$	1000	${\gamma }_{Inc}$	5 × ${10}^{-5}$	Nu	0.3
$n_f$	30	$p_{Inc}$	0.04

to generate innovative structural designs of automobile wheels. $\alpha$, $\beta$, and $\gamma$ are initialized as 0. ${\alpha }_{Inc}$, ${\beta }_{Inc}$, and ${\gamma }_{Inc}$ are the increment values in each TO iteration. $n_{FEA}$ is the grid sampling resolution for FEA computation. An adaptive parallel scale is not implemented by setting $N_0$ equal to N. Sketch images are preprocessed by applying a morphological dilation operator of kernel size (2,2). Figure 4 illustrates the loss curves and intermediate solutions during the iterative TO process. The two sketch-based constraint penalty losses take effect on the density distributions from the beginning of the iterations. As it iterates, the volume constraint has a stronger effect, with a decreased penalty loss value, the compliance gradually converges, while the sketch-based constraint penalty losses exhibit some degree of fluctuation. To ensure the comparability of results across multiple TO processes, a fixed number of iterations is set in the experiments. The convergence of each solving process is verified according to the relative changes in the loss terms for compliance and volume constraint. Over the last 30 iterations, if the relative changes in both losses remain below a threshold, the conversion of the process is validated.

Figure 5 and Figure 6 show the generated innovative structural designs constrained by two examples of free-hand sketches. Using a parallel scale of $M=20$ without an adaptive scale, a total of 20 designs are generated for each sketch image; from them, 8 randomly selected designs are displayed.

The sketch in Figure 5 contains sharp, angular strokes, reflecting a bold and austere style, while the sketch in Figure 6 displays droplet-shaped forms, reflecting a soft and graceful style. Qualitatively, it is evident that the generated solutions embody the visual styling features of those sketches. Also, they exhibit large differences from each other in terms of structural topology and geometric features, verifying the effectiveness of the proposed TO framework in generating innovative structural designs that conform to free-hand sketches and are diverse.

In most of the generated structures, a cavity that closely matches the closed-shape contour is generated, though subtle differences between them appear in some solutions. This is due to the closed-shape-based constraint being assigned with a large weight coefficient (${\gamma }_{Inc}=0.1{\beta }_{Inc}$). In a special case of solution 4 in Figure 6, the contour is matched by a solid structure rather than a cavity. This is caused by the undirected boundary constraint suppressing the closed-shape-based constraint. These experimental results show that the former constraint is beneficial for promoting solution diversity while achieving an effective sketch-based constraint. Also, the competition between two different constraint terms further facilitates the diverse design exploration.

Figure 7 presents more examples of generated solutions. Among them, two sketch images that only contain stroke elements also achieve effective human control. In solution 2 of Figure 7, the drawn strokes are implicitly reflected by several contour segments of different structural branches, demonstrating the flexibility of the undirected boundary constraints. Such flexibility benefits the generation of solutions with appealing visual effects. The compliance values vary by a large amount across the solution sets generated by different sketches, which indicates that the rationality of sketches has a decisive impact on the generated designs’ quality. The volume ratios have very small variance, which is due to the increment in the volume constraint weight being much greater than that of the sketch-based constraints.

When random perturbation is applied to more hyper-parameters of the TO algorithm, more diverse and innovative structural designs can be generated. Here, experiments generating more broadly diversified innovative structural designs of wheel hubs are conducted. The desired volume, load force magnitude, and Fourier sampling frequencies are all randomly perturbed. Meanwhile, the generated TO design images are further converted into 3D models to facilitate intuitive evaluation by designers and subsequent detailed simulation analysis. The 2D-to-3D conversion steps are illustrated in Figure 8.

The input image is first preprocessed by pixel inversion, hub region filling, denoising with Gaussian smoothing, and binarization based on Otsu’s method. Then, the contours within the design field are extracted and closed. After that, NURBS fitting is carried out on contour segments, which are split according to curvature. Finally, the NURBS curves are exported in the DXF format, and 3D models are obtained through extrusion and solid cutting operations.

Figure 9 showcases the innovative structural designs generated under the control of six different sketch examples, as well as the corresponding 3D design models. The obtained diverse innovative designs not only satisfy the sketch constraints but also have good performances in terms of the basic compliance metric. Their variance in shape is very useful to stimulate the creative thinking of designers. In practice, designers can first evaluate the generated solutions qualitatively in terms of aesthetic appeal, innovativeness, etc., then choose the ones with low compliance and human preference to further carry out a multi-criteria mechanical simulation analysis.

4.3. Experiment of Human Controllability

Figure 10 presents the experimental results of human controllability over the generated structural features by integrating the minimum allowed structural feature size $l_{min}$ and the relative weight coefficient of the closed-shape-based constraint over the stroke-based constraint $\lambda$ (${\gamma }_{Inc}=\gamma ·{\beta }_{Inc}$). It is intuitive for designers to take control by simply fine-tuning these two hyper-parameters, as their control effects are precisely targeting at specific structural features and are independent from each other.

When the control parameter $\lambda$ is set to be small, the drawn closed-shape region contains many branching structures. In this case, if the value of $l_{min}$ is also small, the generated structure is highly complex, with low manufacturability, the drawn shape is not visually prominent, though the compliance is small; if $l_{min}$ is larger, the structural complexity becomes lower, the hand-drawn shape is more prominent, but compliance increases significantly. When $\lambda$ is set to be larger, there are fewer branching structures within the drawn shape region, leading to a result similar to a hard constraint on the design field boundary. The shape becomes more prominent, especially when the shape has a large area, which significantly affects the aesthetic styling. $l_{min}$ has a bigger impact on compliance than $\lambda$. These experimental results verify that the two control parameters have different effects on the structural compliance, and mechanical and aesthetic performance, enabling human designers to achieve a better balance between multiple factors in a flexible control manner.

The flexibility of human controllability is valuable for practical application, as solving conflicts between different design factors is the main challenge in conceptual structural design. Mechanical and aesthetic performances are conflicting in most cases. Meanwhile, increased structural complexity lowers the manufacturability, as overly narrow structural branches may be unfeasible to produce. Structural complexity also impacts aesthetic appeal, with simpler and regular forms generally considered more aligned with modern aesthetic standards.

4.4. Impacts of Human Control on Design Quality and Diversity

The qualitative analysis of the experimental results as above has verified the effectiveness of the proposed TO framework to generate conceptual structural designs while providing flexible human controllability. The results also indicate that human control may conflict with the compliance minimization objective of the TO algorithm, leading to an increase in the compliance minimization. In the following, an ablation study is presented that quantitatively analyzes the impact of the three human control approaches on both the solution quality and diversity.

The design quality metrics include the total structural compliance and volume. Although multiple aspects in mechanics, fatigue life, heat transfer, manufacturability, etc., are necessary to verify the engineering applicability of design, it is too costly and time-consuming to carry out a detailed analysis for all design in the conceptual stage. Thus, using compliance and volume criteria can effectively find the initial designs that meet the basic requirements.

Diversity metrics include the image-based average SSIM index and MAE index between pairs of solutions, denoted as $\overline{SSIM}$ and $\overline{MAE}$, respectively, and the coefficients of variance for the compliance and volume ratio, denoted as $C{V}_{comp}$ and $C{V}_{vol}$. Their computation formulas are given in Equations (16)–(18).

$$\begin{array}{c}\hfill \overline{SSIM}={\displaystyle \frac{{\sum }_{i\ne j}SSIM(i,j)}{M}}\end{array}$$

(17)

$$\begin{array}{c}\hfill \overline{MAE}={\displaystyle \frac{{\sum }_{i\ne j}MAE(i,j)}{M}}\end{array}$$

(18)

$$\begin{array}{c}\hfill C{V}_{\ast }={\displaystyle \frac{\sigma (\ast )}{\mu (\ast )}}\times 100\%\end{array}$$

(19)

where $\sigma (\ast )$ is the standard deviation of the metrics’ values, and $\mu (\ast )$ is the mean of the metrics’ values.

In the ablation experiment of the stroke-based constraint, parameter ${\beta }_{Inc}$ is selected from [0, 5 $\times {10}^{-5}$, 2 $\times {10}^{-4}$, 5 $\times {10}^{-4}$, 8 $\times {10}^{-4}$]; a closed-shape-based constraint is not applied ($\lambda$ = 0). In the ablation experiment of the closed-shape-based constraint, as contour lines are essential to make the drawn shape visually prominent, ${\beta }_{Inc}$ is set to 5 $\times {10}^{-4}$, and $\lambda$ is selected from [8 $\times {10}^{-5}$, 4 $\times {10}^{-4}$, 2 $\times {10}^{-3}$, 1 $\times {10}^{-2}$, 5 $\times {10}^{-2}$]. In the ablation experiment of the minimal allowed feature size, $l_{min}$ is selected from [3, 6, 9, 12], ${\beta }_{Inc}$ is set to 5 $\times {10}^{-4}$, and $\lambda$ is set 0.05. The other hyper-parameters are the same as in Table 1. The sketch images of Figure 5 and Figure 6 are utilized as case 1 and case 2.

The experimental results of the ablation study are shown in Figure 11, Figure 12, Figure 13 and Figure 14. Quantitative results of several TO algorithm runs are given in

Table 2. Quantitative experimental results of ablation study.

Case 1
Setting	${\mathit{\mu }}_{\mathit{comp}}$	${\mathit{\mu }}_{\mathit{vol}}$	${\mathit{CV}}_{\mathit{comp}}$	${\mathit{CV}}_{\mathit{vol}}$	$\overline{\mathit{SSIM}}$	$\overline{\mathit{MAE}}$
${\beta }_{Inc}$ = 0	2.5831	0.3892	0.0204	0.0074	0.6461	0.1385
${\beta }_{Inc}=5\times {10}^{-4}$	3.1602	0.3970	0.0687	0.0111	0.6069	0.1634
${\beta }_{Inc}=5\times {10}^{-4}$, $\lambda$ = 0.05	3.1263	0.3948	0.0523	0.0073	0.7585	0.0841
${\beta }_{Inc}=5\times {10}^{-4}$, $\lambda$ = 0.05, $l_{min}$ = 12	3.4158	0.3952	0.0798	0.0068	0.7170	0.1058
Case 2
Setting	${\mathit{\mu }}_{\mathit{comp}}$	${\mathit{\mu }}_{\mathit{vol}}$	${\mathit{CV}}_{\mathit{comp}}$	${\mathit{CV}}_{\mathit{vol}}$	$\overline{\mathit{SSIM}}$	$\overline{\mathit{MAE}}$
${\beta }_{Inc}$ = 0	2.5979	0.3884	0.0230	0.0079	0.6457	0.1386
${\beta }_{Inc}=5\times {10}^{-4}$	2.9896	0.3919	0.0939	0.0094	0.6199	0.1579
${\beta }_{Inc}=5\times {10}^{-4}$, $\lambda$ = 0.1	2.8207	0.3894	0.0558	0.0070	0.7235	0.1111
${\beta }_{Inc}=5\times {10}^{-4}$, $\lambda$ = 0.1, $l_{min}$ = 12	2.9787	0.3928	0.0565	0.0138	0.8138	0.0659

. An increase in ${\beta }_{Inc}$ leads to obviously larger compliance. The average compliance value is 2.62 when this constraint is not applied, but increases to 3.15 and 2.87 for case 1 and case 2, respectively, given ${\beta }_{Inc}$ = 5 $\times {10}^{-4}$. Importantly, an increase in ${\beta }_{Inc}$ does not reduce the solution diversity but slightly improves it. A possible reason is that this constraint makes it easier to escape local optima through interfering with the compliance minimization objective while avoiding limiting the material distribution in the regions surrounding the strokes, thereby exploring the design space more divergently. Therefore, the undirected structural boundary constraint is very beneficial to preserve freedom of design exploration under sketch-based human control.

In contrast, the experimental results reveal that the closed-shape-based constraint reduces the solution diversity but has no obvious impact on the compliance or volume ratio. From an analysis of the results in Figure 10, the main reason for the reduction in the solution diversity is that a large $\lambda$ makes the drawn shape region approximate a non-design field. Stable compliance and volume ratio values indicate that the “remaining” design field is still sufficient for exploration. It is verified that by decomposing sketches into strokes and closed-shape-based elements with distinct control features, the solution quality and diversity can also be better balanced.

The experimental results also reveal that an increase in the minimal allowed feature size leads to larger compliance and reduced diversity, which aligns with the conclusions drawn from the qualitative analysis. Based on the control features of the three control parameters, a proper approach is to first focus on drawing strokes and simply rely on a stroke-based constraint to promote diverse exploration; subsequently, add closed shapes and increase this constraint weight incrementally to optimize compliance while restricting the material volume. The minimal allowed feature size should be preferably set to a large value according to multifactor requirements in practice.

4.5. Comparative Analysis with Existing Methods

The proposed TO framework is compared with the sketch-guided TO algorithm (baseline 1) by Oh et al. [26] and the interactive TO algorithm (baseline 2) by Li et al. [12]. Baseline 1 directly utilizes the entire reference sketch image and a similarity constraint between the TO solution and the sketch. In order to avoid using existing design datasets, to ensure comparability between methods, load condition parameter perturbation is adopted for baseline 1, replacing its data-driven approach of enhancing diversity. Baseline 2 combines the free-hand sketch and human scoring to achieve controllability, and achieves solution diversity by perturbing the sensitivity numbers of the soft-skilled BESO algorithm according to human input. To avoid the uncertainty of human input, the random perturbation of density values is employed as an approximate strategy, with three different perturbation magnitudes for analysis. The proposed TO framework without load condition parameter perturbation is also compared.

While traditional TO algorithms, such as SIMP or BESO, are unable to offer design diversity, existing multi-solution TO algorithms achieve this with three typical methods: simply perturbing the design parameters like load force magnitudes, data-driven feature learning, and adjusting the optimizable variables based on interactive human input. Data-driven methods offer strong design diversity, but both the diversity and quality of the solutions depend on the size and quality of the training dataset. As such, quantitative comparison between these two types of methods is unfeasible. Due to the uncertain effect of the subjectivity of human input, quantitative comparison between these types of methods and parameter perturbation-based ones is also unfeasible. Therefore, this paper only conducts quantitative comparative experiments using the two latest parameter perturbation-based methods, with the same optimizable parameters.

For each compared method, sketch example case 1 is used as the human input, and the perturbation range of the load force parameter $F_x$ is set as [0.3, 1.5]. The evaluation criteria for the methods are the same as above, with the average of 20 repeated runs for each method being the evaluation metric.

Table 3. Comparative experiment results for proposed TO framework and alternatives.

Setting	${\mathit{\mu }}_{\mathit{comp}}$	${\mathit{\mu }}_{\mathit{vol}}$	${\mathit{CV}}_{\mathit{comp}}$	${\mathit{CV}}_{\mathit{vol}}$	$\overline{\mathit{SSIM}}$	$\overline{\mathit{MAE}}$
Density perturb (amp = 0)	3.1164	0.3960	0.0151	0.0033	0.9731	0.0097
Density perturb (amp = 0.1)	3.3117	0.4019	0.0204	0.0047	0.9084	0.0207
Density perturb (amp = 0.4)	3.8991	0.4382	0.0192	0.0049	0.7358	0.0663
Load cond perturb	3.0322	0.3955	0.0243	0.0067	0.8700	0.0399
Fourier freq perturb	3.0322	0.3955	0.0298	0.0067	0.8641	0.0411
Proposed	3.0322	0.3955	0.0317	0.0075	0.8339	0.0516

presents the results of the comparative experiment. In terms of solution quality, the proposed TO framework, along with baseline 1, outperforms baseline 2 by a large margin, verifying that perturbations in the load condition or Fourier sampling frequencies have a very similar impact on the solution quality, but perturbing the density values significantly reduces the quality. The proposed method achieves the best performances, as indicated by the metrics in bold. In terms of diversity, the proposed framework outperforms baseline 1, as well as outperforming baseline 2 when it has a small perturbation amplitude (<0.4). It only slightly lags behind baseline 2 when the perturbation amplitude is large (=0.4) in terms of morphological diversity. For baseline 2, as the amplitude increases, the compliance and volume increase significantly, even violating the volume constraint, as indicated by the ${\mathit{\mu }}_{\mathit{comp}}$ metric shown in bold. While baseline 2 can generate the most diverse solutions ($\overline{SSIM}=0.7358$, $\overline{MAE}=0.0663$), it excessively sacrifices solution quality and struggles to reasonably balance the conflict between solution quality and diversity by only adjusting the perturbation amplitude. The above comparative analysis verifies the contribution of the proposed framework in achieving both high design quality and diversity.

4.6. Analysis of Adaptive-Scale Parallel Computing

To test the effectiveness of the adaptive parallel scale strategy for the multi-solution TO framework, comparative experiments are conducted with and without this strategy. Hyper-parameters $N_0=10$ and $N=30$ when they are implemented, and $N_0=N=30$ when they are not implemented, with solution set filtering only after the iteration ends. The initial parallel scale M is set to 50 with the adaptive parallel scale, and selected from [50, 40, 30] without it. The sketch example of case 2 is utilized as the human input for all cases. The threshold values for the similarity metrics are fixed during the iterations, which ensures that solution pairs are retained as long as their difference is large according to the absolute criteria. Those values are first empirically set to ensure that about 50% of the solutions from the initial solution set are retained, then adjusted to make each similarity metric effective for detecting similar pairs. We set $t_{SSIM}$ = 0.9, $t_{MAE}$ = 0.02, $t_{comp}$ = 0.1, and $t_{vol}$ = 0.01.

A comparison of the experimental results is shown in

Table 4. Comparative experimental results for adaptive parallel scale method.

Setting	Time (s)	Num	${\mathit{\mu }}_{\mathit{comp}}$	${\mathit{\mu }}_{\mathit{vol}}$	${\mathit{CV}}_{\mathit{comp}}$	${\mathit{CV}}_{\mathit{vol}}$	$\overline{\mathit{SSIM}}$	$\overline{\mathit{MAE}}$
N/Ad M = 50	3060.7	50	2.7101	0.3887	0.0264	0.0058	0.8428	0.0488
filtered	/	23	2.7320	0.3895	0.0320	0.0063	0.8110	0.0601
N/A M = 40	2448.6	40	2.7164	0.3888	0.0253	0.0060	0.8453	0.0477
filtered	/	19	2.7391	0.3897	0.0313	0.0065	0.8170	0.0578
N/Ad M = 30	1836.4	30	2.7190	0.3888	0.0273	0.0058	0.8473	0.0472
filtered	/	16	2.7415	0.3894	0.0330	0.0065	0.8241	0.0555
Ad N = 50	1810.4	16	2.7267	0.3891	0.0278	0.0054	0.8041	0.0617
filtered	/	16	/	/	/	/	/	/

. The number of retained solutions after filtering with the adaptive parallel scale is 16, the same as the number of obtained solutions without this strategy and a smaller initial parallel scale ($M=30$). We highlight the results obtained using the two experimental settings in bold in Table 4, as their comparison validates the superiority of adaptive-scale parallel computing. However, this strategy leads to a shorter computing time and improved solution diversity, due to divergent exploration primarily occurring in the early optimization iterations, making a larger initial parallel scale more beneficial.

As described in Section 3, due to the conflict between sketch-based constraints and the compliance minimization objective, the generated structures may include some unreasonable features. Figure 15 showcases two examples that include unreasonable features generated under the guidance of the sixth sketch example. In Figure 15a, a very thin protruding branch is generated guided by the stroke-based constraint, but it is not connected to the other side of the structure. From the von Mises stress map, which is useful to visualize the distribution of equivalent stress within the structural field and is computed by the finite element analysis steps as in the TO algorithm, it is clear that this branch is not contributing as a load path, which suggests that it is an unreasonable design. In Figure 15b, the protrusion part on the support structure also does not contribute as load-bearing and exists solely to conform to the drawn shape of the sketch.

Figure 16 shows examples of recognized regions of conflict using the sensitivity-based method. The regions of conflict are marked in red. The thin protruding branch in the second solution is recognized as unreasonable. It is clear that this feature would be the first to be removed if only compliance minimization and the volume constraint were applied. This solution is removed from the solution set, while the first solution with a smaller region of conflict is retained.

5. Conclusions

This paper proposes a novel sketch-constrained multi-solution topology optimization algorithm, designed for human–machine collaborative structural conceptual design exploration scenarios. It allows designers to express preliminary structural design features through hand-drawn sketches, which form constraints for the topology optimization algorithm. The algorithm divides the sketch content into open-line elements and closed-shape elements, and introduces line-based undirected structural boundary constraints and closed-shape-based regional material distribution constraints. The former imposes no material distribution restrictions on the areas on either side of the lines, enabling the exploration of many novel forms that integrate the sketch’s line elements into the topology, creating unique visual effects. Additionally, compared to unconstrained topology optimization, it does not reduce the morphological diversity of the design solutions. The latter drives uniform material distribution within the shape regions, which, while somewhat reducing the design exploration space and diversity, enhances the visual prominence of the sketch’s shape elements. Importantly, when line constraints are already in place, this approach does not reduce structural performance, allowing designers to gradually refine local structural designs while exploring other parts of the structure in a high-quality, divergent manner. Furthermore, the mutual suppression between the two sketch constraints can further enhance design diversity.

In contrast, existing sketch constraint methods only directly use the overall sketch similarity constraint to control the topology optimization results. The experiments show that the proposed method provides designers with a finer and more flexible approach to controlling algorithm results through hand-drawn sketches, allowing for better balancing of structural performance, lightweight, visual effects, design exploration diversity, and novelty. Additionally, the design diversity offered by the proposed method is significantly stronger.

This algorithm also integrates Fourier mapping-based feature size control, allowing designers to combine it with sketch constraints for flexible control over both the local features and overall complexity of the generated structure. By introducing randomness through perturbations in the Fourier sampling frequency and adopting a parallel optimization strategy based on adaptive scale adjustment, the algorithm generates diversified and innovative design solutions. The experiments show that the proposed Fourier sampling frequency perturbation method outperforms existing boundary condition parameter perturbation and design domain density perturbation methods both in terms of design solution quality and diversity. Based on the proposed parallel optimization strategy, the multi-solution topology optimization algorithm in this paper offers comprehensive advantages in manual controllability, design solution quality, diversity, and computational efficiency, providing an effective solution for current topology optimization algorithms that are unsuitable for structural innovation design exploration.

The research and experimental process also identified several issues that need further resolution. First, the conflict between the sketch constraints reflecting human preferences and the algorithm’s structural performance optimization objective has not been fully mitigated. Although the proposed algorithm uses sensitivity-based method to detect and remove unreasonable designs from the solution set, it is better to automatically generate solutions in which the material in each part of the structure not only satisfies the sketch constraints but also contributes to mechanical performance. Also, finite element calculations are the most time-consuming step in topology optimization algorithms, and exploring ways to avoid or accelerate the finite element computation process is also a valuable area for future research. Further, how to help designers quickly learn the basics of the control characteristics of hyper-parameters should be studied when the design structure becomes more complex and more hyper-parameters are included according to constraints like manufacturability. To improve the applicability of the proposed method in engineering design practice, future study will focus on involving the manufacturing constraints and consider the complex effects of material performance variance in different working conditions.