Manufacturability-aware deep generative design of 3D metamaterial units for additive manufacturing

Abstract

Mechanical metamaterials are artificial structures that possess exceptional mechanical properties that are not naturally occurring. The complex geometrical and topological features of these metamaterials pose significant challenges to both structure design and manufacturing, despite the recent rapid development of additive manufacturing (AM) techniques. Thus, an effective framework for designing 3D metamaterials with desired mechanical properties, while also ensuring AM manufacturability, is urgently needed. In this paper, an AM manufacturability-aware deep generative model-based design framework is proposed for designing 3D metamaterial units for target properties. To accomplish this, we propose using Variational Autoencoder (VAE) as the feature extractor, which maps the 3D metamaterial geometries to a low-dimensional latent feature space. The latent feature space is concurrently linked to discriminators/regressors to predict manufacturability metrics and mechanical properties. We demonstrate that the proposed design framework is capable of designing high-performance metamaterial units with various user-defined manufacturability metrics. To showcase the effectiveness of the proposed design framework, three design cases with different objective functions are presented, and the final optimal designs are validated by comparing them to state-of-the-art designs or the optimal designs obtained by topology optimization methods.

Appendix

1.1 Details of the microstructure family template-based method

This method is used to generate the first group of microstructure samples in the database. This method is a four-step process:

Step 1: Define rectangular bars in the continuous cubic spatial domain $\left[ {0,l} \right]^{3}$, where $l$ represents the length of the spatial domain. The rectangular bars are defined by the coordinates of the pair of reference points $[loc_{i} , loc_{j} \left] = \right[\left( {locx_{i} , locy_{i} ,locz_{i} { }} \right), \left( {locx_{j} , locy_{j} ,locz_{j} } \right)]$ and the side length of the square cross-section of the bar ($h$). Therefore, each bar corresponds to 7 design variables. To avoid overly complicated structures, we only create 1 or 2 bars in this step.

Step 2: Voxelate the spatial domain $\left[ {0,l} \right]^{3}$. In this work, we use $l = 48$, resulting in a $48 \times 48 \times 48$ voxel domain. The bars created in Step 1 are also voxelated. The voxels of the rectangular bars are referred as “original bar voxels” in the following steps.

Step 3: Map the original bar voxels by the midpoint of the grid along each axis sequentially, following the relationship $loc^{\prime} = l - loc$. Through this mirroring process, the original bar voxels on one side of each axis’s midpoint are reflected to the opposite side, creating a symmetric arrangement in all three directions.

Step 4: Record the indices of each voxel to enforce a hierarchical relationship between $locx_{i} ,{ }locy_{i} ,and locz_{i}$ coordinates. This step map the bar voxels $\left( {locx_{i} ,{ }locy_{i} ,locz_{i} { }} \right)$ obtained by previous step to new locations $\left( {locx_{i} {^{\prime}},locy_{i} {^{\prime}},locz_{i} {^{\prime}}} \right)$ by permuting their coordinates while adhering to the relationship $locx_{i}{\prime} > locy_{i}{\prime} > locz_{i} {^{\prime}}$. This step enforces a cubic symmetry in the structure.

After completing the abovementioned steps, the rectangular bars defined within the spatial domain $\left[ {0,l} \right]^{3}$ are mirrored to form a voxelated, cubic, symmetric metamaterial unit. For the metamaterial units generated by 1 bar, there are 7 design variables; for the metamaterial units generated by 2 bars, there are in total 14 design variables. Latin hyper sampling (LHS) is used to assign values to the parameters of each bar. In our LHS table, the values regarding the coordinates $(locx_{i} , locy_{i} ,locz_{i} , locx_{j} , locy_{j} ,locz_{j}$) are generated in the range of $\left[ {0,l} \right]$ and the parameter of side length $h$ is generated in the range of $\left( {0,l} \right]$ (Fig. 12).

Fig. 12

Detailed generation process of the first metamaterial unit database

1.2 Deep generative model that trains VAE and supervised learning model separately

Here, we present an alternative modeling approach that trains the VAE and supervised learning models separately. The VAE is firstly trained to obtain a low-dimensional latent feature space that solely depends on the geometries of the 3D metamaterial units. Then, we use supervised learning models to establish the relationship between the latent feature variables and the mechanical properties and manufacturability metric values. The accuracy of this model is evaluated based on the reconstruction accuracy and the accuracies of the supervised learning models. The metrics are introduced in Eqs. 10–12. The results are presented in Table 4. Compared to the proposed integrated manufacturability-aware model (Table 1), the strategy of training the VAE and the supervised learning models separately show slightly higher accuracy in structural reconstruction but lower accuracies in predicting properties and manufacturability metric values. The lower prediction accuracies may lead to infeasible or low-performance designs found by optimization.

Table 4 Accuracies of the alternative modeling approach on reconstruction of 3D metamaterial units and prediction of properties and manufacturability metric values

1.3 Convergence test to determine the dimensions of latent feature space

See Table 5.

Table 5 A convergence test to show the loss values of the proposed manufacturability-aware deep generative model on the test dataset

1.4 Hyperparameters of the manufacturability-aware deep generative model

See Table 6

Table 6 The detailed structure of the proposed manufacturability-aware deep generative model

1.5 Multi-objective design optimization: manufacturability metric values and true manufacturability of the optimal design candidates

See Table 7 and Fig. 13

Table 7 Manufacturability constraints and true manufacturability metric values of the optimal design candidates

Fig. 13

True manufacturability metrics values and the predicted values by the supervised learning model