# Reinforcement learning-enabled design of topological interlocking materials for sustainable multi-material additive manufacturing

Hichem Seriket, Oualid Bougzime, Yuyang Song, H. Jerry Qi, Frédéric Demoly

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Hichem Seriket, Oualid Bougzime, Yuyang Song, H. Jerry Qi, Frédéric Demoly. Reinforcement learning-enabled design of topological interlocking materials for sustainable multi-material additive manufacturing. Additive Manufacturing, 2025, 111, pp.104992. 10.1016/j.addma.2025.104992. hal-05455120

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Submitted on 12 Jan 2026

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Research paper

# Reinforcement learning-enabled design of topological interlocking materials for sustainable multi-material additive manufacturing

Hichem Seriket <sup>a,b ID</sup>, Oualid Bougzime <sup>a,b ID</sup>, Yuyang Song <sup>c</sup>, H. Jerry Qi <sup>d</sup>, Frédéric Demoly <sup>a,b,e ID</sup>,*

<figure><img src="image_3.png" /></figure>

## ARTICLE INFO

Keywords:

Computational design

Topological interlocking materials

Multi-material additive manufacturing

Reinforcement learning

Q-learning

Voxel-based modeling

## ABSTRACT

Additive manufacturing (AM) has significantly expanded the possibilities to design sophisticated shapes and structures with unique properties and materials to achieve unprecedented functionalities. A notable trend in AM is the integration of multiple materials within a single structure to achieve multifunctionality while minimizing part count. However, multi-material AM presents inherent challenges, particularly in terms of printability constraints and environmental considerations, such as the recyclability of composite structures. Although the current effort in hybrid AM offers a partial solution to address some of these challenges, material versatility and sustainable disassembly remain major barriers. This research aims to introduce a computational interlocking design strategy for multi-material AM on a voxel basis, thus enabling controlled material disassembly and reuse. Reinforcement learning, especially Q-learning, is employed to optimize and explore the spatial arrangement of topological interlocking materials in the three-dimensional design space, which facilitates modularity while maintaining structural stability. Implemented via a Python-based computational framework interfaced with a computer-aided design environment, this approach is validated across various structural configurations, including cubic, beam, and irregular shapes. Our findings demonstrate a path towards sustainable, reusable, and recyclable multi-material AM, offering new possibilities for circular manufacturing and resource-efficient design.

## 1. Introduction

For several years, the assembly of interlocking shapes has attracted significant interest in diverse fields such as toys and education [1,2], engineering and design [3,4], woodworking [5-7], architecture [8-10], and manufacturing [11,12], due to its versatility and ease of implementation. By enabling the creation of complex and robust structures from modular components, interlocking assembly strategies can eliminate the need for adhesives or mechanical fasteners [4]. This technique offers several advantages, such as the ability to easily assemble and disassemble structures, which ensures the recyclability and reusability of parts. Typically, these parts – termed as puzzles, bricks, or blocks in the literature [2,13-17] – are designed to fit together seamlessly, therefore forming a cohesive and stable structure that can withstand various loads and stresses [18,19]. This construction strategy not only

simplifies the assembly process but also reduces the need for time-consuming joining methods, thus making it an attractive option for sustainable construction practices.

This fabrication strategy is particularly aligned with current industrial stakes, which increasingly focus on rethinking products and the way we use and consume them in order to reduce environmental impact. The circular economy action plan, adopted by the European Commission in 2020, outlines a new agenda for sustainable growth, which introduces key regulatory requirements for industry [20]. Similarly in the US, the transition from a linear to a circular consumption model includes several action plans. Notably, the US Environmental Protection Agency (EPA) has initiated action plans centered on sustainable materials management and waste reduction [21]. In industries such as automotive manufacturing, these requirements demand adopting sustainable product design or ecodesign methodologies for reducing

<footnote><sup>a</sup> Université Marie et Louis Pasteur, UTBM, CNRS, Laboratoire Interdisciplinaire Carnot de Bourgogne ICB UMR 6303, 90010 Belfort, France</footnote>

<footnote><sup>b</sup> Université Bourgogne Europe, CNRS, Laboratoire Interdisciplinaire Carnot de Bourgogne ICB UMR 6303, 21000 Dijon, France</footnote>

<footnote><sup>c</sup> Toyota Research Institute of North America, Toyota Motor North America, Ann Arbor, MI 48105, USA</footnote>

<footnote><sup>d</sup> The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA</footnote>

<footnote><sup>e</sup> Institut universitaire de France (IUF), Paris, France</footnote>

<footnote>* Corresponding author at: Université Marie et Louis Pasteur, UTBM, CNRS, Laboratoire Interdisciplinaire Carnot de Bourgogne ICB UMR 6303, 90010 Belfort, France.</footnote>

<footnote>E-mail address: frederic.demoly@utbm.fr (F. Demoly).</footnote>
weight, enhancing durability and safety, simplifying repair through disassembly and reuse, and more [22–25].

Recent advances in additive manufacturing (AM) have demonstrated the ability to fabricate sophisticated shapes and structures while integrating multiple functionalities. Through the available AM techniques and printable materials, it is now possible to create property-graded and multi-material structures tailored for specific mechanical and functional needs. In this context, the use of interlocking materials has emerged as a promising solution, especially in addressing sustainability challenges with the fabrication of large or highly functional assemblies [12]. This strategy is also beneficial when printing dissimilar materials [26,27]. As an example, prior work in the related field of 4D printing – which combines AM with stimuli-responsive materials – has proposed converting voxel-based material distributions (comprising both active and passive materials) into interlocking blocks [4,26]. This approach not only facilitates the printing process but also enhances the reconfigurability of shape-changing structures. In such implementations, topological interlocking materials are printed separately using different AM techniques, then assembled post-fabrication. Other studies have further explored interlocking by printing two materials using a single technique [28,29], combining AM technique with formative methods [27], or using different AM techniques [26].

However, fully leveraging interlocking strategies in AM also requires advanced computational design methodologies. Various computational frameworks have been developed to generate interlocking blocks or topological interlocking materials to create 2D and 3D designs [6,11,17,30–33]. Unlike earlier approaches relying on Platonic solids-based interlocking [34,35], recent work has investigated pixel and voxel-based representations to design 2D structures and 3D interlocking geometries. Deep learning-based generative models have been applied to automate the creation of complex interlocking puzzles [17,36], while polyomino puzzles have introduced efficient methods for achieving assembly and disassembly in interlocking systems [32]. Moreover, voxel-based assembly algorithms have enabled the automatic layout of interlocking structures [37,38], therefore opening possibilities to control material distribution and topological configurations. A brute-force method has been introduced to test all possible configurations and orientations of interlocking shapes within a predefined volume, although it is computationally expensive [37]. A grammar-based approach has been developed to precompute all possible arrangements of two predefined blocks in contact, which reduces computation time at the cost of high storage requirements [39]. Another method involves reducing degrees of freedom (DOF) in multi-material assemblies by iteratively carving smaller blocks and exchanging voxels between neighboring blocks to progressively lock the structure [31]. The DOF reduction approach has proven particularly effective in rigid interlocking assemblies [40], as shown in works on topology optimization [41], damage detection [42], and dynamic analysis [43].

Despite these advancements – particularly those focused on interlocking 3D structures – significant challenges remain in effectively exploring the design space for fully interlocking parts in all three spatial directions while simultaneously ensuring feasible assembly and disassembly. This limitation highlights the need for more agile and parametric computational design approaches capable of identifying interlocking solutions that not only meet mechanical stability requirements but also support ease of fabrication, collision-free assembly, disassembly and reuse.

Therefore, this paper aims to develop a new computational framework that combines reinforcement learning (RL) with digital design to create topologically interlocking materials for multi-material AM. RL, a branch of machine learning, is particularly well-suited for complex, high-dimensional, and combinatorial design problems [44–48]. In our approach, an RL agent explores the evolving design space to learn effective interlocking strategies, thus making by interacting within an evolving design space, sequential decisions that carry long-term

consequences. This learning-based approach enables the discovery of counterintuitive solutions that can subsequently be evaluated from both assembly and disassembly perspectives. Ultimately, this work lays the foundation for advancing sustainable multi-material AM and 4D printing.

## 2. Computational interlocking design

### 2.1. Overview

The computation and exploration of topological interlocking materials within 3D design spaces present significant combinatorial challenges, particularly in addressing the limitations of multi-material AM, 4D printing, and sustainability requirements. In this work, we propose a computational design method based on RL, which operates at the voxel level to determine physically feasible interlocking blocks. Specifically, we implement a model-free Q-learning algorithm that learns voxel consolidation strategies through interaction with the design environment. Fig. 1 provides an overview of the proposed framework, which combines voxel-based modeling, interlocking computation and exploration, and assembly sequence planning. For the sake of clarity throughout the paper, the term “topological interlocking materials” will be referred to as interlocking blocks. These blocks are defined as spatial arrangements of voxels, generated through successive consolidation actions. As for voxels, they represent discrete units of volume in a 3D grid to hold material properties. The following sections describe each part of the framework in detail.

### 2.2. Design space voxelization

By considering an initial design space, the first step consists in generating a point cloud through 3D decomposition. This decomposition can be controlled by a resolution parameter in order to fine-tune the final shape of interlocking blocks. This point cloud represents the set of voxel centers with spatial coordinates where voxel sizes are equivalent to the defined resolution. Here, the voxelization provides voxel material and adjacency relations through which interlocking blocks are constructed via voxel consolidation actions. For cubic or parallelepipedal shapes, voxelization is straightforward and does not pose significant challenges. However, when working with complex geometries or irregular shapes, a more adapted voxelization strategy is required. Two primary approaches may be considered: (i) including voxels that lie entirely within the target geometry, or (ii) incorporating voxels that intersect with the boundary of the design domain. Fig. 2 illustrates the adopted voxelization strategy for two types of geometric configuration. Starting from an initial continuous design domain, a point cloud is generated by applying a spatial decomposition with a predefined resolution (Fig. 2a). For irregular geometries, selected voxels may extend beyond the strict interior of the object to cover the boundary region. This inclusion is essential to ensure an adequate representation of external functional surfaces that may be constructed or refined in subsequent stages of the design process (Fig. 2b).

### 2.3. Model-free Q-learning

The voxelized design space provides a discrete and structured environment that is well-suited for RL, especially model-free Q-learning. One of the key advantages of Q-learning lies in its ability to operate without the need for extensive pre-labeled data. As such, it can be effective in sparse-reward and discrete-state environments, as demonstrated in prior studies [49,50]. In the proposed framework, each design state is represented through a sequence of voxel consolidation actions. These actions are evaluated using a reward function that encapsulates multiple criteria, including changes in structural DOFs, compactness, and assemblability. Through iterative interactions with the design environment – following the classical observation-action-reward cycle –
<figure><img src="image_4.png" /><figcaption>**Fig. 1.** Proposed computational design approach integrating design space voxelization, of interlocking block determination and exploration using Q-learning, and assembly sequence planning for multi-material AM.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Fig. 2.** Representation strategies for computation (point cloud) and visualization (voxels) of (a) regular shapes and (b) complex shapes.</figcaption></figure>

the Q-table is continuously updated. This learning process enables the agent to refine its decision-making policy over time, which progressively optimizes the structural integrity and interlocking performance of the generated blocks (Fig. 3).

Here, the efficiency and convergence of the learning process is driven by critical hyperparameters such as the learning rate, discount

factor, and exploration rate. Initially, a high exploration rate is employed to ensure a broad sampling of the solution space. This rate is then gradually reduced to favor exploitation of the most promising voxel consolidation strategies discovered. This adaptive balance between exploration and exploitation enhances the ability to generate stable interlocked structures.
<figure><img src="image_6.png" /><figcaption>Fig. 3. Model-free Q-learning applied to the computation of interlocking blocks in a voxel-based 3D design space.</figcaption></figure>

To evaluate the impact of consolidation actions on structural stability, we consider the DOF of each element as an indicator of potential movement. In the initial stage, the design environment is composed solely of individual voxels. Each voxel is assumed to possess 6 independent DOFs, corresponding to possible translational motions along the ±x, ±y, and ±z axes (here voxel rotations are excluded). As voxels are aggregated through consolidation actions, they form interlocking blocks, and the DOFs of these blocks are no longer considered individually but collectively. A block's DOF is reduced when its constituent voxels are geometrically constrained by adjacent voxels or blocks. Specifically, if any voxel within a block is obstructed in a given direction due to physical contact with neighboring elements, that entire block is considered restricted along that axis, which results in a loss of one DOF. This aggregation mechanism ensures that the structural configuration reflects increasing mechanical interdependence and constraint as consolidation progresses. Consequently, the total DOF of the system decreases, indicating a transition from a loosely connected set of elements to a more integrated and mechanically stable interlocking structure.

In the proposed Q-learning framework, the state s encodes the current configuration of all voxels and blocks within the three-dimensional design space. This configuration is formally represented as a sorted list of spatial coordinates of each active entity in the grid. Initially, the environment comprises solely individual voxels; as the learning progresses, consolidation actions lead to the construction of interlocking blocks. The state representation dynamically evolves to reflect changes in spatial arrangement and connectivity, which are critical for assessing mechanical stability. An action a corresponds to a consolidation operation between two adjacent entities (i.e., voxels or blocks), either extending an existing block or initiating a new one. From the second step onward, the agent selects actions from an expanding set of candidate consolidations involving both voxels and blocks. These actions reduce the total DOF of the system. The reward r associated with each action is defined as

$$r = c_1 \Delta \text{DOF} + c_2 f_{\text{dis}} + c_3 f_{\text{compact}} + c_4 f_{\text{size}} - p, \quad (1)$$

where ΔDOF represents the reduction in the system's total DOF resulting from the consolidation action. The binary term $f_{\text{dis}}$ indicates whether the resulting block can be disassembled via a single translational motion (1 if true, 0 otherwise). Directional checks are performed on the boundary voxels of the block, and physical collisions are detected by grid-occupancy tests on ±x, ±y, and ±z axes. The compactness measure $f_{\text{compact}}$ is defined as the ratio of the number of voxels in a block to the volume of its bounding box, providing a normalized

metric for spatial efficiency. The term $f_{\text{size}}$ evaluates how close the current block size (i.e., in terms of current voxel count, $V_{\text{current}}$) is to a predefined target (i.e., maximum voxel number, $V_{\text{max}}$), typically given by

$$f_{\text{size}} = 1 - \frac{|V_{\text{max}} - V_{\text{current}}|}{V_{\text{max}}}. \quad (2)$$

A penalty $p$ is imposed if the resulting block fails to meet critical criteria such as being mechanically secured or non-removable. The coefficients $c_1, c_2, c_3$, and $c_4$ are empirically tuned to prioritize the influence of DOF minimization while integrating additional geometric and functional criteria. A sensitivity analysis of these coefficients is provided in Supplementary Materials. This formulation enables the agent to discover stable, disassemblable, and printable interlocking solutions in a highly combinatorial design space.

At the core of the algorithm lies the Q-value update mechanism, which employs the temporal difference learning framework. The Q-learning update rule is defined as

$$Q(s, a) \leftarrow Q(s, a) + \alpha \left( r + \gamma \max_{a'} Q(s', a') - Q(s, a) \right), \quad (3)$$

where $s'$ is the next state resulting from applying action $a$, $\alpha$ is the learning rate ($0 < \alpha \le 1$), and $\gamma$ is the discount factor ($0 \le \gamma \le 1$). The term $\max_{a'} Q(s', a')$ represents the highest estimated Q-value over all possible actions $a'$ from the subsequent state $s'$, effectively capturing the best future reward achievable if the optimal policy is followed. Action selection is guided by an epsilon-greedy (ε-greedy) policy. With probability $\epsilon$, the agent selects a random action to encourage exploration of new structural possibilities in the design space; otherwise, it exploits the highest estimated Q-value. The value of $\epsilon$ decays over time (e.g., $\epsilon \leftarrow 0.995\epsilon$), ensuring that the learning strategy shifts from exploration to exploitation as training progresses. To prevent endless training loops and stabilize convergence, we introduce a “patience” criterion: if no significant improvement in cumulative reward is observed over a predefined number of consecutive episodes, the learning process is halted automatically. This safeguard – although disabled for the results shown to allow full convergence – remains available to reinforce robustness when dealing with highly noisy or very large design spaces. Algorithm 1 presents the pseudo-code of the developed model-free Q-learning for the generation of interlocking blocks.

## 2.4. Assembly sequence planning

Following the Q-learning process, once a valid interlocking configuration has been established over one episode, each interlocking
## Algorithm 1 Generation of Topological Interlocking Materials

1. **Initialize** Q-table as empty
2. **Set** learning rate $\alpha$, discount rate $\gamma$, exploration rate $\epsilon$, and decay factors
3. **for** each episode **do**
4. Reset environment and set $S \leftarrow$ initial state
5. $total\_reward \leftarrow 0$
6. **for** each step until maximum steps reached **do**
7. Identify all adjacent pairs in state $s$
8. **if** random probability $\le \epsilon$ **then**
9. Select a random action $a$
10. **else**
11. Select action $a$ that maximizes $Q(s, a)$
12. **end if**
13. Execute $a$ and observe new state $s'$
14. Compute reward $r$ based on:
$$r = c_1 \Delta DOF + c_2 f_{dis} + c_3 f_{compact} + c_4 f_{size} - p$$
15. **Update** Q-table:
$$Q(s, a) \leftarrow Q(s, a) + \alpha \left( r + \gamma \max_{a'} Q(s', a') - Q(s, a) \right)$$
16. $total\_reward \leftarrow total\_reward + r$
17. $s \leftarrow s'$
18. **if** DOF $\le$ target or max iterations reached **then**
19. **Break**
20. **end if**
21. **end for**
22. Decay exploration rate: $\epsilon \leftarrow \epsilon \times exploration\_decay$
23. Record episode statistics
24. **Save model:** Save Q-table
25. **end for**

block, when generated, is constrained to possess exactly one allowable translational motion along a principal axis direction ($\pm x, \pm y, \pm z$). This criterion ensures the feasibility of physically separating blocks from the assembly without violating the mechanical interlock condition. Therefore, at the end of each episode, the algorithm concatenates a step-by-step guide to ensure disassembly of the interlocking blocks. The assembly sequence is then derived from the inverse of the recorded disassembly operations associated with the consolidation actions of the block. This step can also be addressed in a different way. If we consider the final interlocking assembly, it is possible to generate the disassembly sequence iteratively by evaluating the local DOF of each interlocking block in potential translations along the $\pm x, \pm y$ or $\pm z$ directions, therefore updating the entire configuration after each removal. In this context, the assembly sequence can also be considered as the inverse of the disassembly sequence.

## 3. Results and discussion

In this section, we present and analyze the results of the proposed RL-based computational design framework. Our evaluation considers multiple performance aspects, including design generation, voxel consolidation, and computational efficiency throughout the Q-learning training process.

### 3.1. Geometric case studies

To demonstrate the ability of the proposed framework in computing and exploring interlocking structures that adhere to designer-specified geometric constraints, four case studies – each representing varying levels of shape complexity – are presented. These comprise: (1) a cube with a side length of 30 mm, (2) a structural beam with dimensions

<table><caption>**Table 1**

Voxel counts and voxel sizes for the studied geometric shapes.</caption><thead><tr><td>Shape</td><td>Voxel count</td><td>Voxel size (mm)</td></tr></thead><tbody><tr><td>Cube</td><td>216</td><td>5</td></tr><tr><td>Structural beam</td><td>1,280</td><td>2.5</td></tr><tr><td>Arch</td><td>615</td><td>2</td></tr><tr><td>Stanford bunny</td><td>1,037</td><td>2</td></tr></tbody></table>

of 100 mm × 20 mm × 10 mm, (3) a semicircular arch defined by an outer radius of 20 mm, an inner radius of 10 mm, and a depth of 10 mm, and (4) a Stanford bunny enclosed within a bounding box of 26 mm × 42 mm × 36 mm [51,52] (see Fig. 4a).

To initiate the computation and exploration of interlocking blocks, each design space is converted into both point cloud and voxel-based representations. The point cloud is used for Q-learning-based reasoning, while the voxel grid supports CAD-based visualization and editing. A custom component – called Voxelizer – has been developed using C# language in Grasshopper®, a plugin for the commercial CAD software Rhinoceros® (Version 7) to support this critical step. As shown in Fig. 4b, this component is compatible with any input mesh or B-Rep geometry and allows designers to specify the desired grid resolution. The output can be generated either as a voxel list or as point list (voxel centers) to generate the related representations, as shown in Fig. 4c and Fig. 4d respectively. Table 1 summarizes the characteristics of the generated design spaces as a function of the specified voxel grid resolution.

### 3.2. Interlocking block generation and exploration

To achieve interlocking block computation and exploration, the proposed Q-learning-based framework has been implemented within a custom Python code – that is related to the aforementioned CAD system – to rigorously manage geometric data and interlocking constraints. The implementation leverages the generated point/voxel grid and logs every consolidation action in an assembly log. The point cloud generated using the Voxelizer Grasshopper® component is recorded through a CSV file that is parsed to generate a point cloud, and the grid is instantiated exactly according to the provided coordinates without added margins. This precise grid construction preserves the fidelity of the geometric constraints during both training and inference. Moreover, the proposed Q-learning process is tightly coupled with the voxel consolidation logic. Here, the Q-value update is executed directly during the construction of interlocking blocks. Each consolidation operation removes aggregated voxels from the grid, updates the environment state, and records the action in an internal log for subsequent assembly sequence reconstruction, thus supporting downstream applications such as fabrication path planning or robotic assembly. As the agent explores different action sequences, it can learn to favor configurations that minimize global DOF while maximizing compactness, within the limits of a designer-defined block size. This drives the system towards mechanically robust assemblies.

The training proceeds episode by episode until either a user-specified maximum number of steps is reached or the global DOF of the assembled structure falls within a target range between 1 and 3. This target range was carefully selected as follows: a DOF = 0 would correspond to a fully constrained configuration, whereas higher DOF values indicate excessive movement potential, resulting in inadequate assembly stability. Therefore, targeting a residual DOF, such as $1 \le$ DOF $\le$ 3, reflects our design strategy by achieving near-complete assembly securing while preserving minimal global mobility. Table 2 summarizes the set of Q-learning hyperparameters and parameters adopted through experience for each of the four geometric case studies presented in this work. The values are adapted to reflect the complexity of each design space, ranging from regular shapes (i.e., the cube and structural beam) to more intricate geometries (i.e., the arch and Stanford bunny).
<figure><img src="image_7.png" /><figcaption>**Fig. 4.** Geometric case studies and their conversion to suitable representations: (a) Rough geometric description of a cube, a structural beam, an arch, and a Stanford bunny; (b) partial design flow using a custom Grasshopper® component to generate specific representations; (c) point cloud presentation and (d) voxel-based representation of the case studies. All scale bars: 1 cm.</figcaption></figure>

<table><caption>**Table 2**
Q-learning hyperparameters and parameters used for each geometric shape.</caption><thead><tr><th>Type</th><th>Name</th><th>Cube</th><th>Structural beam</th><th>Arch</th><th>Stanford bunny</th></tr></thead><tbody><tr><td rowspan="5">Hyperparameter</td><td>Learning rate α</td><td>0.1</td><td>0.1</td><td>0.1</td><td>0.1</td></tr><tr><td>Discount factor γ</td><td>0.9</td><td>0.9</td><td>0.9</td><td>0.9</td></tr><tr><td>Exploration rate ε</td><td>0.9</td><td>0.9</td><td>0.9</td><td>0.9</td></tr><tr><td>Exploration decay</td><td>0.99</td><td>0.99</td><td>0.995</td><td>0.995</td></tr><tr><td>Episodes</td><td>500</td><td>300</td><td>100</td><td>10</td></tr><tr><td rowspan="4">Parameter</td><td>Maximum steps</td><td>5,000</td><td>20,000</td><td>4,000</td><td>30,000</td></tr><tr><td>Maximum block size</td><td>90</td><td>110</td><td>100</td><td>200</td></tr><tr><td>Target DOF</td><td>3</td><td>3</td><td>3</td><td>3</td></tr></tbody></table>

The hyperparameter values (α, γ, ε) were determined through a heuristic trial-and-error process supported by systematic grid searches. Specifically, we developed a Python script that automatically explored a predefined range of candidate values for α and ε (keeping γ fixed), evaluating each configuration against key performance metrics: cumulative reward, convergence speed, and resulting block count. This automated tuning enabled precise identification of the optimal settings (α = 0.1, ε = 0.9 with decay 0.99), thus striking an effective balance between early exploration and later exploitation.

The application of the proposed Q-learning-based framework to the selected case studies demonstrates its ability to consolidate complex design spaces from a voxel basis into mechanically balanced structures. The resulting structures typically consist of reasonable number of interlocking blocks while exhibiting low total DOFs, which reflects robust mechanical interlocking. As for the cube geometry, Fig. 5a illustrates the distribution of the number of computed interlocking blocks across training episodes, all while satisfying the target final DOF constraint. This performance is further corroborated by the observed convergence in DOF reduction of representative training episodes over successive steps (Fig. 5b).

Fig. 6 presents representative interlocking solutions selected from successful converged training episodes for each geometry. From the

left to the right, each subfigure illustrates the computed interlocking model (as-computed view), the Q-learning-generated interlocking block (as-exploded view) decomposition suitable for AM purposes, and the corresponding 3D-printed prototype (as-built view). In the case of the cube geometry, Fig. 6a presents one of the best-performing episodes yielded an interlocking structure composed of 8 blocks with a total DOF of 3. For the more elongated structural beam, an increased number of training episodes enabled the generation of 8 blocks achieving a DOF of 3 (Fig. 6b). Similarly, the irregular geometries – i.e., the semicircular arch and the Stanford bunny – were successfully decomposed into 9 and 6 blocks, respectively, with corresponding DOF values of 3, demonstrating the framework’s ability to handle both regular and complex geometrical forms.

Furthermore, in the case of freeform structures, additional processing is required to preserve functional surfaces. The voxel-based representation, inherently constrained by the chosen grid resolution, may extend slightly beyond the original shape’s boundary. To reconcile this and ensure the fidelity of the final assembled geometry, a Boolean operation can be applied between the voxel-based interlocking solution and the original surface mesh. This step refines the shape appearance and ensures consistency with the initial design intent. Fig. 7 illustrates the external shape refinement of the Stanford bunny through the Boolean Intersect function.
<figure><img src="image_8.png" /><figcaption>Fig. 5. Results of the interlocking computation for the cube geometry. (a) Distribution of the consolidated blocks and final DOFs over training episodes, (b) DOF convergence of representative training episodes across steps.</figcaption></figure>

## 3.3. Material and fabrication method

All computed interlocking blocks were 3D-printed using the digital material FLX9895 from Stratasys – a rubber-like composite with Shore A95 hardness – via PolyJet™ (material jetting technique) on a Stratasys Objet260 Connex3 system. Once printed, they were assembled manually. Fig. 8 illustrates the resulting assembly sequence from the computational design framework to build the interlocking cube geometry. Although this demonstration focuses on a single-material configuration, preliminary experiments with various multi-material combinations (e.g., VeroWhite™, VeroClear™, and hybrid blends) confirmed basic printability and assembly feasibility, as shown through the multi-material beam in Fig. 6b.

## 3.4. Performance and computational metrics

To provide deeper insight into the learning process, it is essential to analyze the learning dynamics to investigate how the agent optimizes its merging decisions and achieves efficient convergence. To facilitate this analysis, various performance and computational metrics were monitored throughout the training phase. As a practical example, we trained our RL agent on a cube composed of 216 voxels over 500

episodes, each including 5,000 steps, chosen for its moderate complexity while remaining computationally feasible. Fig. 9 illustrates the evolution of the step number, time, and cumulative reward over training episodes for the cube geometry. The results of training performance related to other case studies are detailed in **Supplementary Material (Figure S1 and Figure S2)**. This training history is governed by the exploration-to-exploitation policy, therefore ensuring that the agent refines its policy while still adapting to new states. The continuous increase in cumulative reward suggests that the agent progressively learns to optimize its decision-making strategy, thus demonstrating strong generalization capabilities within the given task constraints. The presence of occasional fluctuations in reward underscores the dynamic nature of the environment but does not hinder overall learning progression. This reinforces the robustness of the training methodology and the effectiveness of the RL framework in optimizing sequential decision-making tasks.

The analysis of episode duration provides further insight into the efficiency of the learned policy (Fig. 9b). The learned policy demonstrates strong efficiency by maintaining episode durations within a well-controlled range. The majority of episodes are completed within 20 to 95 s, highlighting the agent's ability to execute voxel consolidation optimally and refine its decision-making strategy. This consistency suggests that the agent effectively minimizes unnecessary iterations,
<figure><img src="image_9.png" /><figcaption>**Fig. 6.** RL-based interlocking block generation and physical prototyping for the four case studies with increasing complexity. From the left to the right: computed interlocking assembly, Q-learning-generated interlocking blocks, and 3D-printed assembled structures. (a) Cube, (b) structural beam, (c) arch, and (d) Stanford rabbit. The middle column illustrates the consolidated blocks resulting from the learning process, while the right column confirms the feasibility of physical assembly through AM. All scale bars: 1 cm.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Fig. 7.** Functional surfaces refinement of voxel-based interlocking assembly and its application to the interlocking Stanford rabbit.</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>Fig. 8. Assembly planing of the interlocking cube geometry. (a) Exploded view highlighting the bill of materials and (b) assembly sequence.</figcaption></figure>

leading to a more structured and predictable learning process. Additionally, this bounded time frame ensures that computational overhead remains manageable, therefore reinforcing the feasibility of deploying the trained policy in real-world applications where time efficiency is a critical factor.

Moreover, we analyzed the cumulative reward alongside the number of step per episode, respectively shown in Fig. 9c and Fig. 9a. Episodes with high cumulative rewards and fewer steps indicate that the agent effectively identifies optimal consolidations. The strong correlation between increasing cumulative rewards and decreasing steps demonstrates the agent's ability to refine its decision-making over time. This improvement highlights the efficiency of the learned policy in reducing unnecessary actions while maximizing rewards, leading to a well-optimized strategy for sequential decision-making. We also observed that if more time is needed to reach a solution (i.e., if many steps are performed), the cumulative reward remains low. This reflects the inherent nature of Q-learning, which penalizes late-discovered solutions. However, an increased number of steps leads to longer episode durations and lower cumulative rewards, thereby teaching the agent to avoid action sequences that delay the achievement of the target DOF. Consequently, with continued training, the agent learns to choose actions that expedite the consolidation task to achieve the target structure.

The results of our RL framework demonstrate its effectiveness in optimizing sequential decision-making tasks. The agent exhibits a clear learning progression, as seen in the increasing cumulative reward, the controlled decay of the learning rate and exploration rate, and the systematic reduction in DOF (see Fig. 5b). The majority of episodes are completed within an efficient time frame. Furthermore, the correlation between high cumulative rewards and reduced step counts confirms that the agent successfully refines its strategy, minimizing unnecessary actions while maximizing efficiency.

### 3.5. Inference

Once training the model on the 6 × 6 × 6 cube (216 voxels) is completed (see Sections 3.2 and 3.4), the learned Q-table can be deployed to autonomously guide the voxel consolidation process in previously unseen three-dimensional design spaces. During inference, the agent leverages the stored Q-values to make consolidation decisions according to the same optimization criteria defined during training, i.e., structural stability, interlocking efficiency, and disassemblability. Because no further Q-table updates are required, inference is vastly

faster than training and can be repeated in a fully reproducible way by fixing the pseudo-random seed. To verify the consistency of the learned policy, we performed inference runs using five distinct seeds (0, 1, 7, 42, 123) with a low exploration rate ($\epsilon = 0.1$), ensuring exact replication of results.

We conducted inference on four distinct design spaces, including the trained cube (see Fig. 8) and three new geometries: a downsized cube (4 × 4 × 4 grid, 64 voxels), a scaled-up cube (8 × 8 × 8 grid, 512 voxels), and a solid sphere (radius 20 mm, 257 voxels). During training of the original cube geometry, a typical episode took on average ≈ 19.43 s (see Fig. 9b). In contrast, each inference run completed in only ≈ 6.25 s on average across the five selected seeds, while reliably achieving final DOF ≤ 3 and producing from 13 to 17 interlocking blocks (see Table 3). This corresponds to an ≈ 4× speed-up, highlighting the efficiency of policy execution relative to full Q-learning updates. Then, we evaluated inference on smaller and larger versions of the cube. For the 64-voxel cube, inference times ranged from 0.10 s to 0.57 s (mean ≈ 0.33 s), with final block counts between 10 and 24, and DOF ≤ 3 across seeds. For the 512-voxel cube, inference times increased to 447–555 s (see Table 3), confirming a roughly linear scaling with voxel count.

Finally, we applied the cube-trained model to a solid sphere (radius 20 mm, 257 voxels). Although this geometry was never seen during training, the model discovered valid interlocking assemblies in 45–56 s per seed (mean ≈ 48 s), whereas retraining a model on the solid sphere averaged 122.60 s per episode over 100 episodes. Final DOF values remained ≤ 3 and block counts between 7 and 9, thus demonstrating strong transferability to novel geometries. Additional solid spheres of 123 voxels (radius 15 mm) and 925 voxels (radius 30 mm) were also tested (see Supplementary Material, Table S1 and Table S2). Inference times rose approximately linearly with voxel count. While assembly quality (final DOF and block count) stayed acceptable until voxel counts exceeded ≈ 1000, where Q-table size began to affect lookup speed. It is worth noting that inference with fixed random seeds offers a fully reproducible, computationally efficient workflow for creating interlocking assemblies, which accommodates both familiar and entirely new geometries with minimal overhead.

### 3.6. Limitations and scalability

Although Q-learning enables efficient design exploration, the size of the Q-table grows nearly exponentially with the number of states (voxel grid size), which limits the maximum manageable resolution without resorting to compression or approximation techniques. In addition, the
<figure><img src="image_12.png" /><figcaption>Fig. 9. Training metrics evolution for the cube case. (a) Number of step per episode, (b) episode duration in seconds, and (c) evolution of the cumulative reward.</figcaption></figure>
<caption>**Table 3**</caption>

Inference results over five random seeds ($\epsilon$ = 0.1) using the model trained on the 216-voxel cube geometry. *Average training time per episode*: 19.43 s (216-voxel cube), 0.15 s (64-voxel cube), 504.80 s (512-voxel cube), 122.60 s (257-voxel solid sphere).

<table><thead><tr><th>Geometry</th><th>Seed</th><th>Step</th><th>Block count</th><th>Time<sub>inference</sub> (s)</th></tr></thead><tbody><tr><td rowspan="5">216-voxel cube</td><td>0</td><td>607</td><td>17</td><td>5.53</td></tr><tr><td>1</td><td>967</td><td>17</td><td>7.94</td></tr><tr><td>7</td><td>692</td><td>9</td><td>6.26</td></tr><tr><td>42</td><td>493</td><td>19</td><td>5.07</td></tr><tr><td>123</td><td>670</td><td>13</td><td>6.43</td></tr><tr><td rowspan="5">64-voxel cube</td><td>0</td><td>32</td><td>24</td><td>0.10</td></tr><tr><td>1</td><td>87</td><td>20</td><td>0.18</td></tr><tr><td>7</td><td>339</td><td>16</td><td>0.47</td></tr><tr><td>42</td><td>504</td><td>10</td><td>0.57</td></tr><tr><td>123</td><td>271</td><td>10</td><td>0.31</td></tr><tr><td rowspan="5">512-voxel cube</td><td>0</td><td>8,400</td><td>15</td><td>447.69</td></tr><tr><td>1</td><td>7,847</td><td>22</td><td>529.34</td></tr><tr><td>7</td><td>7,321</td><td>34</td><td>539.65</td></tr><tr><td>42</td><td>9,157</td><td>14</td><td>555.15</td></tr><tr><td>123</td><td>8,191</td><td>12</td><td>500.73</td></tr><tr><td rowspan="5">257-voxel solid sphere</td><td>0</td><td>4,967</td><td>9</td><td>47.98</td></tr><tr><td>1</td><td>5,348</td><td>8</td><td>52.43</td></tr><tr><td>7</td><td>4,892</td><td>8</td><td>45.58</td></tr><tr><td>42</td><td>10,000</td><td>9</td><td>56.08</td></tr><tr><td>123</td><td>5,345</td><td>7</td><td>42.78</td></tr></tbody></table>

current implementation does not leverage any parallelism; each episode runs strictly sequentially, significantly increasing training times as the geometry becomes more complex. Finally, voxelization granularity is constrained by memory and computational cost (in practice, grids larger than 200 × 200 × 200 become difficult to handle), imposing a tradeoff between geometric fidelity and computational load.

## 4. Conclusion and future work

This research introduces a novel AI-driven computational framework that integrates RL with digital design to generate topological interlocking materials for multi-material AM. By leveraging model-free Q-learning, the proposed method autonomously explores complex design spaces to produce structurally stable, disassemblable, and reusable building blocks. Therefore, this framework addresses key challenges in multi-material AM by ensuring efficient space-filling, high structural stability, and scalability for digital fabrication workflows. The capability to autonomously generate geometrically intricate yet mechanically viable designs underscores the potential of RL-based design for a broad range of applications, including soft robotics, structural electronics, and 4D printing, among others. Further research can be done to strengthen the training process by evaluating interlocking structures through finite element (FE) simulations and mechanical testing. Preliminary numerical investigation, described in **Supplementary Materials**, have provided promising results in terms of mechanical cohesion, constraint distribution, stiffness, and energy storage. These analyses will provide valuable insights into interface behavior and mechanical performance, as initiated in previous research work [41,53–55]. Additional efforts will be made to implement a robotic platform capable of handling interlocking blocks and ensuring product assembly in a more automated manner. To achieve this, particular attention will be given to the handling features of the interlocking blocks, as well as potential geometric deviations arising either from the block printing process or from the generated assembly sequence.

To further advance the computational design pipeline, upcoming work will integrate graph neural networks (GNNs) and deep Q-learning to tackle inverse design challenges in multi-material AM and 4D printing. This includes the optimization of mechanical performance and material distribution while accounting for physical constraints. GNNs

will provide a powerful framework for capturing the spatial and mechanical interdependencies within topological interlocking materials by representing the structure as a graph — where nodes correspond to interlocking blocks or material regions, and edges encode their interactions. This representation enables precise prediction and optimization of stress distributions, deformation behaviors, and overall mechanical performance. Simultaneously, deep Q-Learning will be employed to optimize the material distribution and block configuration by formulating the design process as a sequential decision-making problem. By utilizing neural network function approximators that eliminate the need to store prohibitively large tabular Q-tables, the learning capability will be increased over state spaces with millions of voxel configurations. This approach directly addresses the exponential growth limitation of classic Q-tables, allowing us to scale the framework to significantly larger and more complex geometries. In this way, the learning algorithm will iteratively explore design configurations, using reward signals derived from FE simulations where critical performance metrics can be evaluated. Further investigation will involve the use of both active and passive materials, ranging from metals to polymers and smart materials, to achieve real-world demonstrators. These developments aim to support a wide array of practical applications, including modular construction, automotive and aerospace.

## Computational setup

All numerical computations were performed on a system equipped with a 12th Gen Intel Core i9-12950HX 2.30 GHz processor, 64 GB of RAM, and an NVIDIA RTX A5500 Laptop GPU with 16 GB of VRAM. The code was implemented in Python 3.9.18. The training of the Q-Learning agent, which is responsible for generating interlocking block structures, was executed iteratively due to the dependency between consecutive block formations. Parallelization was not applied in this context as the construction process requires sequential actions.

## CRediT authorship contribution statement

**Hichem Seriket:** Writing – original draft, Software, Methodology, Investigation, Conceptualization. **Oualid Bougzime:** Writing – original draft, Software, Methodology. **Yuyang Song:** Writing – review & editing, Validation, Supervision, Methodology. **H. Jerry Qi:** Writing – review & editing, Validation, Supervision, Methodology, Conceptualization. **Frédéric Demoly:** Writing – review & editing, Writing – original draft, Validation, Supervision, Methodology, Investigation, Funding acquisition, Conceptualization.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgments

F. Demoly would like to acknowledge the support from Toyota Motor North America. This work was also supported by the IUF, France.

## Appendix A. Supplementary data

Supplementary material related to this article can be found online at https://doi.org/10.1016/j.addma.2025.104992.

## Data availability

Data will be made available on request.
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