# Sketch-Guided Topology Optimization with Enhanced Diversity for Innovative Structural Design
## 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 conceptualdesign 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-quantity 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.