Horseshoe Lattice Property-Structure Inverse Design Based on Deep Learning

Abstract

Lattice structures, characterized by their exceptional strength-to-weight ratios and energy absorption capabilities, have paved the way for pioneering designs in additive manufacturing (AM). To fully harness the potential of AM, robust inverse design methodologies are essential. In this study, a novel FEM-LSTM based lattice structure inverse design framework was proposed for horseshoe lattice structures characterized by Length (L), Radius (R), and Angle (A) to establish the structure-performance response. Using finite element analysis, a substantial dataset with distinct geometries and mechanical responses was meticulously furnished for training. Delving deeper into modeling, we developed an autoencoder framework anchored in long short-term memory (LSTM) networks, designed to adeptly decode the temporal intricacies of stress-strain attributes and seamlessly encode sequence characteristics. Compared to traditional GPR models and DNN models, the proposed model’s predictability increased by 9% and 7%, respectively, which is attributable to the exceptional capability of LSTM structure in handling time-series data. Our model, being versatile, can seamlessly integrate multiple stress-strain inputs, rendering precise geometric parameters that resonate with tailored design specifications. Such a streamlined approach effectively supplants the conventionally tedious iterative forward design and exhaustive simulation phases. In summation, the model emerges as a swift conduit for bespoke inverse design pertaining to lattice structures. And the paradigm of discerning time-series correlations through LSTM autoencoders holds vast potential across diverse time-dependent properties inherent to materials science.

1. Introduction

Lattice structures are defined in the literature as architecture formed by arrays of stochastic or non-stochastic structural units with edges and faces in space.¹⁾ With properties such as high strength-to-weight ratio, energy absorption capacity, and large porosity, lattice structures are gaining traction in various fields like aerospace and biomedical, serving as lightweight frameworks, shock absorbers, acoustic insulators, and suitable materials for medical implants.^2–4) They can be fabricated using traditional manufacturing techniques, including but not limited to investment casting,⁵⁾ water-jet cutting,⁶⁾ weaving,⁷⁾ and brazing.⁸⁾ However, in the conventional manufacturing process, the pores and wall thickness cannot be precisely controlled, which may lead to unstable properties of the lattice structure. In addition, the cells in the conventional process are produced individually and then further assembled or bonded, which can lead to inflexibility at the junction.^9,10) With the advent of additive manufacturing (AM) technology, it has become possible to manufacture components with complex geometries and diverse materials, including functionally graded materials and lattice structures.^9,11) Subsequently, this novel technology quickly attracted the attention of scholars worldwide, opening a new chapter for more elaborate and sophisticated lattice structure designs.¹²⁾

Generally, different lattice structures exhibit different performance advantages not only due to materials and size considerations. The horseshoe lattice structure, renowned for its exceptional stretchability and nonlinear stress-strain response (typically represented as J-shaped curves),¹³⁾ combines soft, pliable mechanics with high tensile strength and significant modulus enhancement at large strains. This amalgamation of properties makes it ideal for creating robust stretchable electronics.¹⁴⁾ Moreover, integrating lattice structures with bi- or tri-layer microstructures comprising different materials allows the design of microstructures for specific isotropic or anisotropic swelling¹⁵⁾ and thermal expansion patterns.¹⁶⁾ Owing to these properties, horseshoe lattice structures have found applications in stretchable electronic devices, including soft-shell packaging,¹⁷⁾ microstrip antennas,¹⁸⁾ and skin-integrated lactate/oxygen sensors.¹⁹⁾ It can be anticipated that horseshoe lattice structures will demonstrate the significant potential in wearable and implantable surface devices in the future.

To achieve feasible lattice structure design, traditional forward design methods combine existing structural characteristic databases with experimental evaluation to verify the reliability of structures. Xiong et al.²⁰⁾ used finite element analysis (FEA) simulation to obtain different stress-strain data for different horseshoe-shaped lattice structures, and then constructed a Gaussian process regression (GPR) model to study the relationship between horseshoe geometric parameters and stress-strain, so that the structure can be inversely designed according to properties. Considering that traditional design methods significantly rely on intuition and experience, inverse design is an innovative research methodology that centers on reversing the design of a new structure with a desired performance from a predetermined performance or functional requirement, which eliminates inefficient trial and error. Notably, the rapid advancement of Machine Learning (ML) and Deep Learning (DL) techniques, known for their robust data generalization, has brought revolutionary changes in material design, with inverse design of lattice materials becoming more prevalent. Deng et al. showcased an energy absorption system for 2D mechanical metamaterials with specific nonlinear mechanical responses, achieved through the inverse design and optimization using combined MLP models and evolutionary strategies. Jiang et al.²¹⁾ introduced a deep learning-enhanced additive manufacturing framework for inverse design, leading to the customized design of an ankle brace with tailored mechanical properties. Kollmann et al. employed a variational autoencoders (VAE) model to generate 2D optimal metamaterial designs aimed at maximizing bulk modulus, maximizing shear modulus, or minimizing Poisson’s ratio.²²⁾ Mao et al. employed a Generative Adversarial Networks (GAN) to create 2D architected materials that achieve Hashin–Shtrikman upper bounds on isotropic elasticity.²³⁾

The current mainstream inverse-design process can be categorized into indirect, semi-direct, and direct methods. However, in practical applications, establishing effective and efficient structure-performance mapping for complex topological structure parameters often proves challenging.^24,25) Semi-direct Inverse Design employs modeling parameters to reduce the dimensionality of optimization problems, making them computationally more efficient and demonstrating higher potential for industrial applications.²⁶⁾ Bastek et al. proposed a DL framework based on fully connected structures for the inverse design of truss structures with completely customized anisotropic stiffness.²⁷⁾ The main design parameters of this model include topological design parameters, tensile tensors, rigid body rotation, the number of stretch cycles, and relative density. The research to some degree proves that DL models are capable of mapping desired mechanical characteristics to the material’s modeling parameters. For the exploration and optimization of lightweight metamaterials, Challapalli et al. introduced a reverse design framework utilizing GANs, with 27-vertex RVEs demonstrating different connectivities to depict the geometries of lightweight metamaterials.²⁸⁾ This dataset comprises 1500 different lattice unit cells, with their mass and compressive stress obtained through finite element simulations. However, the Semi-direct Inverse Design method applied in the field of lattice materials typically focuses on point-to-point relationships. The fundamental principle of this method is to map certain attributes or functionalities to design parameters, thereby realizing specific target geometries or structures. Deep learning models like Multilayer Perceptrons (MLP), Convolutional Neural Networks (CNN), Generative Adversarial Networks (GAN), and Variational Autoencoders (VAE) are employed in the process to comprehend and forecast these intricate mapping relations. The point-to-point approach might not entirely grasp the behavior of materials in more complicated, dynamic, or non-linear conditions, especially in contexts involving time-series data or necessitating consideration of long-term behavior.

Furthermore, considering the continuity and time relevance of mechanical response data, Recurrent Neural Networks (RNN), whose chain-like characteristics are pivotal in deep learning for processing time-based input features, have been introduced. Gorji applied Recurrent Neural Networks to the multiaxial constitutive model modeling of two-dimensional foam materials, demonstrating that RNNs can effectively predict the stress-strain response of complex anisotropic plasticity models (Yld2000-2d and HAH hardening) under any loading path.²⁹⁾ Nevertheless, in the context of long time series problems, the vanishing or exploding gradient problem greatly complicates the training of RNNs. Consequently, conventional recurrent networks usually incorporate gated transition functions to alleviate these issues, which is the essence of the Long Short Term Memory (LSTM) architecture. LSTM-based deep learning paradigms have been widely used in a variety of time-series related studies, such as molecular dynamics,³⁰⁾ constitutive relationship construction³¹⁾ and so on.

In this study, we present a new FEM-LSTM based lattice structure inverse design framework, designed to adeptly decode the temporal complexities of stress-strain attributes. By employing a topological design methodology for horseshoe lattice structures characterized by Length (L), Radius (R), and Angle (A), a database mapping structure to performance response was established. Moreover, comparisons with conventional deep neural networks and Gaussian process regression models were made to ascertain the FEM-LSTM model’s accuracy and generalization capability in predicting geometric parameters from stress-strain attributes. The impact of various stress-strain input conditions on the predictability of the model was discussed.

2. Methods

2.1 Horseshoe lattice design

The topological structure exerts a significant influence on its mechanical performance. However, conventional hyperelastic models prove inadequate in effectively capturing the inherent correlation between these two aspects.³²⁾ To address this issue, we have endeavored to resolve it through the utilization of quantifiable parameters for the rational design of the topological structure, coupled with the formidable data analysis capabilities of deep learning, thereby enabling inverse design of the structure. Through this methodology, we are able to leverage deep learning techniques for the precise design and optimization of structures. As depicted in Fig. 1(a), the horseshoe lattice unit is defined by four design variables: length L, radius R, width W, and angle A. For modeling convenience, we uniformly set the width to 1 mm. After arraying the units according to specific rules, a unique lattice structure with high stretchability is formed, as illustrated in Fig. 1(b). The ranges of L, R, and A are tailored to fit the intended application, and a 100 mm × 100 mm space is designated for generating horseshoe lattice structures. Once the design space is established, as presented in Table 1, the geometric parameter ranges are determined. We then adopt the Latin hypercube sampling method to generate each variable randomly and randomly combine different variable values to obtain different variable groups.³³⁾ Each group signifies a distinct unit cell structure. Subsequently, we apply CAD software to design the unit cell structure according to the variables of each group. After arranging these in a specific array, a horseshoe lattice structure is designed.

Fig. 1

The geometrical unit. (a) Parameters that make up the geometric unit. (b) Different cells can form different lattice structures according to certain regular arrays.

Table 1 Ranges for design variables of a horseshoe unit cell.

As shown in Fig. 2, different L, R, and A variables can model different horseshoe lattice structures. As can be seen in Fig. 2(a) and Fig. 2(c), the variable that has the most significant impact on the overall structure is A. It affects the degree of curvature of the overall structure and as A increases the curvature of the overall structure increases. R is the radius of the cell rounded, and the larger it is the more round the cell is. As shown in Fig. 2(b), when R is small, the arcs are shorter compared to the other three structures. L is the linear transition part connecting the two cell arcs, and Fig. 2(d) shows that the larger L is the more apparent the overall structure appears.

Fig. 2

(a)–(d) show that unit cells with different geometrical parameters on the design space will form different lattice structures.

2.2 Inverse design approach

As illustrated in Fig. 3, the data-driven integrated inverse design approach proposed in this study consist of three key steps:

Fig. 3

Schematic diagram of the data-driven integrated design method.

Step 1: Data acquisition—In this step, based on specific scenario requirements, we define an approximate range for geometric parameters. We then establish a sampling strategy in which each data point is subject to finite element simulation. This process generates a comprehensive dataset, enabling the machine learning model to effectively learn.

Step 2: Construct an appropriate ML model and adjust model parameters—Depending on the type of dataset, select a suitable model from the multitude of available machine learning algorithms. For instance, if the dataset mainly consists of images and the target task is image recognition, a Convolutional Neural Network (CNN) model is commonly employed. On the other hand, if your dataset displays robust sequential logic features, Long Short-Term Memory (LSTM) and Recurrent Neural Networks (RNN) are particularly effective in processing sequential sequences.

Step 3: Perform inverse design according to requirements—In this final step, the model takes the required mechanical properties as input and rapidly generates the corresponding geometric parameters.

With the model in place, it can conduct inverse design of structures in milliseconds, bypassing the time-consuming process of initial forward design, preliminary simulations for parameter estimation, and further meticulous adjustments.

2.2.1 Data acquisition

Import the horseshoe lattice into the finite element simulation (FEA) ABAQUS/STANDARD (Dassault Systèmes^®, 2020) software for tensile testing. In ABAQUS, two-dimensional beam structure modeling is utilized. The material used for simulation is Vero Cyan (Vero Cyan is a commercially available material, part of a series known as PolyJet Vero, produced by the 3D printing manufacturer Stratasys). It is a rigid, opaque, photosensitive resin with a Young’s modulus of approximately 1.7 GPa and a Poisson’s ratio of roughly 0.5. The corresponding mechanical response data can be found in previous studies.^34,35) In order to mitigate edge effects, the simulations utilized a square matrix comprising a multitude of horseshoe-shaped unit cells, each with dimensions of approximately 20 mm * 20 mm. Nodes on the horseshoe matrix’s bottom edge were fixed with an encastre boundary condition, while a tensile load was applied via displacement control of the top edge nodes. Boundary conditions are set on one side of the lattice structure to limit its degrees of freedom, while on the other side, the load is applied by point displacement. The simulation results, shown in Fig. 4, compare the stress-strain responses of two samples before and after stretching. It is observed that the horseshoe lattice structure with larger curvature maintains a lower stress level even after 50% deformation, exhibiting distinct J-shaped stress-strain curves. Upon completing the simulation, we acquired the stress-strain response of each unique structure. Post-processing was then conducted using Python to batch-process the stress-strain data from ABAQUS. Subsequently, the data was consolidated into a file using the Pandas library, thus completing the construction of our datasets.

Fig. 4

(a) and (d) show the finite element simulation results of two different structures. Specifically, A1 and B1 represent the simulation results of the structures under 25% strain, whereas A2 and B2 represent the results under 50% strain.

2.2.2 Model training

During the data collection phase, we created 300 unique samples, each representing a distinct structural design. These structures were simulated in ABAQUS/STANDARD (Dassault Systèmes^®, 2020), and a text file was compiled for each sample. Each file comprised 41 sets of stress-strain data, covering strain values from 0 to 50% and their corresponding stress values.

As depicted in Fig. 5, an auto-encoder model based on Long Short-Term Memory (LSTM) was developed. This model consists of an LSTM network for the encoding layer and a dense linear layer for decoding. The LSTM layer encodes features from the stress-strain curve, while the linear layer decodes these features to output geometric parameters: Length (L), Radius (R), and Angle (A). The model inputs these 41 sets of stress and strain, producing the corresponding geometric parameters as output. In this work, MSE (mean-squared error) is used as the loss function, which can be described as:

  

\begin{equation} \mathit{MSE} = \frac{1}{n} \sum_{i = 1}^{n}(Y_{i} - \hat{Y}_{i})^{2} \end{equation} (1)

where $\hat{Y}_{i}$ represents the model predicted value, and Y_i represents the true label value. Out of our 300 sets of data, 270 sets were used as training sets and 30 were used as test sets. Several manual attempts were made on the hyperparameters of the model and the following optimal hyperparameters were determined: epochs 10000, optimizer Adam, batch 30, number of LSTM hidden layers 1, number of LSTM hidden nodes 82, number of decoder linear layers 3, and the number of hidden nodes were 120, 30, and 3, respectively. As shown in Fig. 6, the model converges after 5000 epochs and the loss value reaches a satisfactory accuracy.

Fig. 5

The composition of the LSTM-Autoencoder model.

Fig. 6

Learning rate curve during training.

In addition, we used Deep Neural Networks (DNN) and Gaussian Process Regression (GPR) as comparison models. The hyperparameters for the DNN in this study are set as follows: learning rate: 0.001; number of layers: 4; number of neurons per layer: 100; activation functions: ReLU; batch size: 30; regularization techniques: L1/L2 regularization; regularization strength: 0.001; optimization algorithm: Adam. We employed a Gaussian Process Regressor (GPR) model with specific hyperparameters. The kernel function was set as the Radial Basis Function (RBF) with a length scale of 3 and optimization bounds between 0.01 and 1000, governing the smoothness of the function. We set the regularization parameter (alpha) to 0.1 to add a regularization term in the objective function, mitigating the risk of overfitting by penalizing larger parameter values. The number of times for the optimizer to run from different starting points (n_restarts_optimizer) was set to 20, implying that the optimization process was initiated from multiple points, with the best output among these runs taken as the final result.

3. Results

The distribution of the data set over the space of the three design parameters (L, R, and A) is shown in Fig. 7. The stress value corresponding to the maximum deformation is used as the index P for evaluating the tensile performance of the structure. The larger P is, the worse the tensile performance is, and conversely, the smaller P is, the better the tensile performance is. It can be seen from the figure that the dark blue points with small P-values are concentrated in the upper part of the three-dimensional space. The larger A region indicates that the curvature of the structure has a critical effect on its tensile properties.

Fig. 7

Dataset in the design space.

The impact of varying parameters L, R, and A on the P-value is visually represented in Fig. 8. In the graph, a color closer to red indicates a higher P-value, while darker colors represent lower P-values. A comparative analysis of the graphs clearly shows that the angle A exerts the most significant impact on the structure’s tensile properties. Larger A angles correspond to lower P-values, denoting enhanced tensile properties. Conversely, smaller L and R values reduce the structure’s curvature, leading to a sharper appearance and subsequently poorer tensile performance.

Fig. 8

The effect of different L, R, and A on P-values.

In order to verify whether the LSTM-Autoencoder based model can perform better in sequential feature extraction. It was compared with a deep neural network (DNN) consisting of all linear layers and a Gaussian process regression model (GPR), a traditional machine learning regression algorithm.^14,25) DNN is a deep neural network composed of linear layers, while GPR is a machine learning model with a strong ability to fit continuous data. After a certain number of iterative trainings, the models have reached convergence. The training results are shown in Fig. 9, where is the square of the Pearson correlation coefficient, which is a quantity that indicates the strength of the linear correlation between two values. The value range is: [0,1]. The larger the value, the stronger the correlation. Its mathematical expression is as follows,

  

\begin{equation} R^{2} = 1 - \frac{\displaystyle\sum (\hat{y}_{i} - \bar{y})^{2}}{\displaystyle\sum (y_{i} - \bar{y})^{2}} \end{equation} (2)

$\hat{y}_{i}$ represents the model predicted value, y_i represents the true value, and $\bar{y}$ represents the mean of the true value. The closer R² is to 1, the better the model and its prediction. As can be seen in Fig. 9, the MSE of the three models are all at a low error level, and the R² is above 0.9, indicating that the three models have an excellent fitting performance on the training set. However, on the test set, it can be seen that the MSE of the GPR model has been higher than 1, and the R² has also dropped to 0.824, indicating that the GPR model cannot predict well when unfamiliar data is used as input.

Fig. 9

MSE and R² of each model on the dataset.

Thirty new sets of test data were set up to validate the model’s ability to predict the three decision parameters L, R, and A. The sampling methodology used to generate the 30 sets of data and the dataset was using the same Latin hypercube sampling. The correlation scatter plots of the three models are shown in Fig. 10, and the R² in Fig. 10 indicates that our model has a significant advantage over the other two control models. The model shows a significant improvement in the prediction accuracy of all three parameters, L, R, and A compared to the other two models. In addition, all three models predicted parameter A very accurately, demonstrating that the machine model also takes into account the greater influence of parameter A on the stress-strain performance of the structure.

Fig. 10

(a)–(c), (d)–(f), and (g)–(i) represent correlation scatter plots of the true values against the outputs of our model, the DNN model, and the GPR model, respectively.

In order to further verify the fit of the whole stress-strain curve the model was utilized to take the stress-strain data of the four samples as inputs. Subsequently, three predicted parameters given by the model were obtained and the new structure was reconstructed by these three decision parameters. Finally, the predicted stress-strain curves were obtained after further simulation. The stress-strain curves predicted by the model and the real stress-strain curves of the four samples are shown in Fig. 11. It can be seen that the re-simulated stress-strain curve for a given structure is very close to the real value when the model receives new data.

Fig. 11

(a)–(d) each display the test results of four different structures. The red curve represents the actual stress-strain values of the structures, while the black curve represents the stress-strain values of the structures as predicted by the model.

Considering that a range of stress-strain requirements are usually required in practical application scenarios, it means that our input may be one or more sets of stress-strain. Taking the stress and strain requirements of this area as input, the model will automatically output the corresponding geometric parameters. For example, in a certain application scenario, the stress in the region must not exceed 0.01 MPa when the structural strain in the region is required to be 0.2. Therefore, taking this group of stress and strain as input, the model outputs the corresponding geometric parameters in milliseconds. Our model can take multiple sets of stress and strain requirements as input and output as many structural parameters that meet the mechanical performance requirements. In our work, one, two and three sets of groups of requirements are taken as inputs and then three sets of corresponding structural parameters are obtained. These geometric parameters were modelled and the finite element (FEA) simulation was re-run to obtain their stress-strain curves. The results are shown in Fig. 12, where the red points are the required values we have set and the green curves are the stress-strain curves generated from the finite element analysis (FEA) simulation of the corresponding geometrical parameters given by the model. As can be seen that the stress values and the corresponding strains are all close to the required values. In Fig. 12(d), four sets of requirement variables are set. As the number of sets of requested values increases, the accuracy slightly decreases. However, if a confidence interval is set to allow for a 15% error, it is found that most of the required values fall within the confidence interval.

Fig. 12

(a)–(d) display the stress-strain values of structures as predicted by the model under both single and multiple stress-strain requirements.

4. Discussion

The results indicate that all three models can inversely output geometric parameters based on given stress and strain values. The Gaussian Process Regression (GPR) model, however, exhibits a relatively lower accuracy, as evidenced by discrepancies observed between the predicted values and the actual data. This might be attributed to GPR’s preference for processing continuous data, whereas our regression task involves data sampled at discrete intervals. In contrast, it can be observed that both Deep Neural Networks (DNN) and the LSTM-Autoencoder model exhibit higher accuracy in processing complex data. Furthermore, the LSTM-Autoencoder model achieves a ∼7% higher prediction accuracy than the DNN model, which is attributable to the exceptional capability of LSTM structure in handling time-series data. In practical applications, a significant challenge arises when input data significantly diverges from our dataset’s range, leading to reduced predictive accuracy, which is primarily due to the limited external generalization capability of the neural network structure. To address this issue, creating a more extensive training dataset based on the FEM-LSTM framework proposed in this paper may be a viable solution.

The trained model has proven effective for inverse design problems, enabling quick and accurate design of structures to meet specific stress-strain requirements in various applications. The LSTM-Autoencoder approach is not only applicable to stress-strain-related models in materials science, but it also extends to the study of physical properties exhibiting significant time-series characteristics. With continuous improvements in the inverse design concept, dataset expansion, and advancements in horseshoe lattice structures for wearable electronics, our inverse design methodology promises wide applicability for a range of customized scenarios.

With improvements in data capture capabilities of Additive Manufacturing (AM) processes, machine learning (ML)-based surrogate models are poised to supersede simulation-based or certain traditional surrogate models, thereby enhancing computational efficiency. Additionally, the benefits of machine learning in composite design can extend to other material properties, such as strength and the Poisson ratio. Establishing a more extensive process-structure-property relationship model can support and address a broader spectrum of design problems. The novel model achieved higher accuracy compared to traditional models, and subsequent experimental validation might be a potential path for future research. Further, we will also explore the practical implementation and empirical assessment of our proposed method to verify its performance in the real world.

5. Conclusions

With the advancement of Additive Manufacturing (AM), there is a growing need for composite design involving gradient functional materials and metamaterials. This trend has spurred the development of numerous process-structure-property (PSP) surrogate models, which require high accuracy and computational efficiency for effective application. In this paper, we introduced an integrated deep learning method to address the inverse design challenge in lattice structure design, boasting superior computational speed and accuracy compared to conventional surrogate models. The FEM-LSTM based lattice structure inverse design framework uses a topological design methodology for horseshoe lattice structures characterized by Length (L), Radius (R), and Angle (A) to establish the structure-performance response. By employing continuous stress-strain data with sequential characteristics as training data, the LSTM-Autoencoder model’s predictive accuracy was validated. Compared to traditional GPR models and DNN models, the proposed model’s predictability increased by 9% and 7%, respectively, which is attributable to the exceptional capability of LSTM structure in handling time-series data. Variations in L and R primarily affect the curvature and roundness of the cell within the lattice structure. However, their influence on the tensile properties (P-value) is less significant compared to the Angle (A). The Angle (A) is found to be the most critical factor affecting the structure’s tensile properties. Larger angles lead to greater curvature, which enhances the tensile properties indicated by lower P-values.

The methodologies presented here can be applied more broadly to inverse design in lattice structures and to explore complex interactions among processes, structures, and properties. Future research should aim to broaden the PSP framework to tackle more intricate design problems. Simultaneously, emerging advancements in Machine Learning (ML) and Deep Learning (DL) present new opportunities for lattice design. Applying these cutting-edge technologies will enable the creation of novel structures more aptly suited to diverse application scenarios.

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