Data-Driven Nonlinear Deformation Design of 3D-Printable Shells

Abstract

Designing and fabricating structures with specific mechanical properties requires understanding the intricate relationship between design parameters and performance. Understanding the design-performance relationship becomes increasingly complicated for nonlinear deformations. Though successful at modeling elastic deformations, simulation-based techniques struggle to model large elastoplastic deformations exhibiting plasticity and densification. We propose a neural network trained on experimental data to learn the design-performance relationship between 3D-printable shells and their compressive force-displacement behavior. Trained on thousands of physical experiments, our network aids in both forward and inverse design to generate shells exhibiting desired elastoplastic and hyperelastic deformations. We validate a subset of generated designs through fabrication and testing. Furthermore, we demonstrate the network’s inverse design efficacy in generating custom shells for several applications.

Keywords

additive manufacturing neural networks inverse design elastoplasticity hyperelasticity

Introduction

Additive manufacturing has unlocked the ability to create structures with complex geometries and customized mechanical properties. These fabricated structures can be designed to exhibit unique stiffness variations^1,2 and energy-absorbing capabilities.^3,4 However, achieving specific mechanical behaviors, especially for large deformations involving significant plasticity and densification, demands a deep comprehension of the intricate relationship between design parameters and performance. Gaining insights through manual iterative design and testing often proves impractical and leads to costly and time-consuming design cycles. Researchers have employed self-driving labs^4–6 to explore design spaces autonomously. However, these systems are constrained by cost, complexity, and converging time.

Simulation techniques such as the finite element method (FEM) and homogenization excel at modeling elasticity^1,2,7–9 and fracture.^10,11 However, such strategies often lose accuracy when representing large plastic deformations, impeding the design of structures with targeted elastoplastic behaviors. Researchers have also developed plasticity simulations to achieve highly complex deformation behavior.^12–14 However, further testing is needed to evaluate these methods’ ability to model the compressive behavior of thin shell structures as used in this study. Consequently, we propose a neural network trained on experimental data to learn the design-performance relationship between 3D-printable shells and their compressive deformation behavior.

Forward design presents users with predicted performance, allowing them to manipulate designs to achieve desired behavior. Data-driven approaches to predict mechanical behavior from material geometries have been applied in various fields, from composites^10,15,16 to material microstructures.^17,18 However, iterative design loops with forward design are often ineffective due to vast design spaces and the complexity of how individual design parameters affect performance.

On the contrary, inverse design is the process that directly identifies the designs that achieve a target performance goal. Inverse design is inherently complex; one performance is likely achievable by numerous designs, making learning algorithm convergence difficult. This one-to-many challenge mirrors complexities from other disciplines, such as inverse scattering¹⁹ and inverse kinematics problems.²⁰ Despite this increased complexity, inverse design empowers users to explore and generate designs with desired mechanical properties.

We propose a tandem neural network (TNN)²¹ for the forward and inverse design of a parametric family of cylindrical shells chosen for their ease of fabrication (Fig. 1A). The TNN combines two sequential neural networks: an inverse design network and a forward design network, structured like an autoencoder. Researchers have used this architecture for the inverse design of nanophotonic devices^21,22 and metamaterials.^23–25 Notably, machine learning-based inverse design extends beyond the TNN^26,27 with techniques ranging from convolutional neural networks¹¹ to reinforcement learning.²⁸ Previous work generally focuses on mechanical properties that are modeled easily by simulation. These include properties arising from reversible elastic deformations or fracture propagation from an initial predetermined fracture site.

FIG. 1.

Overview. (A) We explore using a tandem neural network (TNN) for the forward and inverse design of generalized cylindrical shells (GCS). (B) GCS are fabricated with fused deposition modeling (FDM) 3D printers. (C) GCS geometry emerges from parameters that control a series of operations applied to cylindrical shells. The result of these operations is a diverse family of structures. (D) Compression tests applied to GCS yield force-displacement curves.⁶ Fabricated with polylactic acid (PLA), this GCS exhibits elastoplastic deformation. Highlighted metrics are the linear elastic region used to calculate stiffness (orange line), work performed (red area under curve), and maximum displacement (green value).

In this article, we leverage an extensive experimental dataset comprising over 12,000 shells exhibiting nonlinear response to compression, as observed in their force-displacement curves, capturing a range of elastoplastic and hyperelastic deformations. We verify our TNN’s performance through experimentation on generated designs, compare the TNN with alternative methodologies, and demonstrate the TNN’s effectiveness in generating designs with optimized nonlinear deformations through several applications, such as impact protection.

Materials and Methods

This section explains our TNN pipeline. We introduce the experimental dataset and preprocessing steps. Furthermore, we describe the network architecture, learning objectives, and training process.

Modeling performance with force-displacement curves

We modeled performance with force-displacement curves (Fig. 1D) and used the following derived metrics:

Stiffness (N/mm): measures the resistance to initial deformations reflected in the slope of the linear elastic zone (orange region in Fig. 1D).

Work (J): measures the energy absorption during deformation reflected in the area under the curve (red region in Fig. 1D).

Maximum displacement (mm): denotes the furthest point of deformation reached during compression testing (green region in Fig. 1D). It is influenced by material properties and experimental constraints. This value ensures a realistic scaling of displacements in predicted curves.

These metrics serve as high-level descriptors for evaluating model performance and selecting desired structures. However, to ensure broad applicability, we maintain the entire force-displacement curve as the underlying performance representation, allowing users to identify metrics most pertinent to their unique design goals.

Generalized cylindrical shell dataset

In a comprehensive prior study, we conducted compression testing on 3D structures known as generalized cylindrical shells (GCS) (Fig. 1B) to explore their energy-absorbing capabilities.⁶ These tests generated force-displacement curves, which, along with their associated designs, constitute a substantial GCS dataset.^a This dataset holds particular value due to its wide range of measured elastoplastic and hyperelastic deformations.

Here, we provide an overview of the GCS parametric family but direct readers to the “Methods” section in our previous work⁶ for a complete description. The radius r for an azimuthal angle ϕ defines a GCS’s top and base faces,

r (ϕ) = r_{0} (1 + c_{4} \cos (4 ϕ) + c_{8} \cos (8 ϕ)),

(1)

where c₄ and c₈ control the shape and size of 4-lobe and 8-lobe features, respectively. Interpolating between the two faces forms a cohesive 3D shell. Parameters enable linear and oscillating twisting, enhancing geometry intricacy. Moreover, adjustments to the perimeter ratio between the top and base faces, mass, height, and wall thickness allow further customization. The r₀ term is a scale factor for perimeter size. Figure 1C shows how a sequence of operations on cylindrical shells defines a GCS.

With a material choice from one of six thermoplastics (two elastoplastic, three hyperelastic, and one intermediate), the complete GCS specification requires 12 parameters (Table 1). Even with a conservative estimate of only 10 discrete values per geometric parameter, the GCS family admits trillions of possible designs.

Table 1.

GCS Design Parameters

Geometric parameter	Description	Range
c₄ (base and top)	The parameter controlling the size and shape of the 4-lobe feature. There is one parameter per face.	[0, 1.2]
c₈ (base and top)	The parameter controlling the size and shape of the 8-lobe feature. There is one parameter per face.	[−1, 1]
Linear twist	The rotation (rad) of the top. This creates a linear twist between the base and top	[0, 2π]
Oscillating twist amplitude	The amplitude (rad) of the oscillating twist between the base and top.	[0, π]
Oscillating twist cycles	The number of cycles of the oscillating twist between the base and top	[0, 3]
Perimeter ratio	The ratio between the top and base perimeters.	[1, 3]
Mass	The mass (g).	[1, 5]
Height	The height (mm).	[10, 30]
Thickness	The wall thickness (mm).	[0.4, 1]

Material parameter	Description	Category
Material	Elastoplastic	Material choice
	polyethylene terephthalate glycol (PETG)
	PLA
	Hyperelastic
	thermoplastic elastomer (TPE) (Chinchilla 75A)
	thermoplastic polyurethane (TPU) (Cheetah 95A, NinjaFlex 85A)
	Intermediate
	TPU (Armadillo 75D)

Twelve parameters define a GCS. We manually restricted the mass, height, and perimeter ratio values so that all geometric parameters have well-defined continuous ranges.

We used the MakerGear M3 and Ultimaker S5, two fused deposition modeling (FDM) 3D printers, to fabricate GCS by printing the design in vase mode. We assumed that all material densities were 1.2 g/cc, which would induce slight mass variability after fabrication. We calibrated the extrusion multiplier that alters the target wall thickness to achieve the target mass.

Data processing

This section discusses our steps to extract the design and performance data from the GCS dataset, resulting in 12,706 design-performance pairs. To prevent data leakage, the data processing is fit only to the training set and then applied consistently to the validation and test sets.

Performance dimensionality reduction

Force-displacement curves in the GCS dataset, which include thousands of points with varying spacings and displacement ranges, are impractical for prediction tasks. We processed the curve data to 100 evenly spaced displacement values. Inspired by Yang et al.,¹⁰ we used principal component analysis (PCA) to condense the corresponding 100 force values into 10 principal components. These 10 components accounted for approximately 99.8% of the cumulative explained variance in the force values. Our performance vectors p ∈ ℝ¹¹ combine the 10 component coefficients with the maximum displacement value. Refer to Supplementary Data S2 for details on curve compression and analysis of PCA quality.

Design parameter normalization

In our previous investigation,⁶ we fabricated GCS using seven materials: the six materials outlined in Table 1 and nylon. However, the number of experimental samples for nylon is significantly smaller than the other materials. So, in this work, we excluded all the data for GCS fabricated with nylon.

For the GCS design parameters, we one-hot encoded the material parameter and applied min-max normalization to all nonmaterial parameters to normalize their values. However, the mass, height, and perimeter ratio parameters lack a clear range. We manually capped these parameters’ values to [1 g, 5 g] for mass, [10 mm, 30 mm] for height, and [1, 3] for perimeter ratio.

Using one-hot-encoded materials provided a straightforward way to ensure that the materials in generated designs conform to realistic values in inverse design. Although we could have parameterized the material using a subset of continuous variables (e.g., Young’s modulus and Poisson’s ratio), we would have needed additional mechanisms to ensure realism in the generated values during inverse design.

These normalization operations collectively result in design vectors d ∈ ℝ¹⁷, comprising 11 geometric parameters and six values from the one-hot-encoded material parameter.

Model architecture

Inverse design poses complexity due to the potential one-to-many relationship between different designs and similar performances, hindering conventional learning algorithm convergence.²¹ Figure 2 depicts this challenge, illustrating two distinct GCS designs with nearly identical force-displacement curves. To address this, the TNN framework, initially proposed by Liu et al.,²¹ has emerged as a promising solution. We leveraged this framework to generate diverse GCS designs aligned with desired performances. Figure 3 shows our network architecture.

FIG. 2.

One-to-many performance to design relationship. Two GCS with distinctly different geometry share nearly identical force-displacement behavior, a common problem in inverse design.

FIG. 3.

TNN architecture. (A) The forward design network $F$ maps GCS designs to corresponding force-displacement curves. (B) The inverse design network $I$ maps force-displacement curves back to GCS designs.

The forward design neural network $F$ : ℝ¹⁷ →ℝ¹¹ learns the mapping from design vectors d to performance vectors p. The network architecture consists of six hidden layers: a learnable linear layer followed by a ReLU activation, repeated three times. All hidden layers maintain a uniform width of 64 units, contributing 10,190 trainable parameters.

The inverse design neural network $I$ : ℝ¹¹ →ℝ¹⁷ learns the inverse mapping: performance vectors p to design vectors d, and mirrors the architecture of $F$ . The final layer of $I$ ensures appropriate values for all GCS parameters by combining softmax and sigmoid activations. The softmax activation is applied to the six parameters representing the one-hot-encoded material, assuring the predicted material has the highest value. The sigmoid activation confines predicted values of the remaining parameters within [0,1], maintaining consistency with their normalized counterparts.

Objective function

$F$ aims to minimize the error between predicted and experimental performance vectors. However, not all values in the performance vectors carry equal significance in improving the model’s accuracy. Building upon insights from Yang et al.,¹⁰ we used a weighted mean squared error (MSE) loss function, $L_{F}$ , using a weight vector w ∈ ℝ¹¹:

L_{F} = \frac{1}{n} \sum_{i = 1}^{n} (w \cdot (p^{(i)} - F (d^{(i)})))^{2},

(2)

where n is the number of samples and w is defined as

w_{j} = {\begin{matrix} \frac{λ_{j}}{\sum_{k = 1}^{10} λ_{k}} & if 1 \leq j \leq 10 \\ 1 & if j = 11 \end{matrix} .

(3)

By setting w₁₁ = 1, we assigned equal importance to the entries of the performance vector responsible for the displacement and force values. However, since each principal component coefficient explains a different amount of variance in the force data, as indicated by the eigenvalues λ₁,…,λ₁₀ obtained from PCA, we set the weights of individual coefficients based on their respective explained variance. By incorporating these weighted values, $F$ prioritizes the most informative principal component coefficients during training.

The objective of $I$ is twofold. First, $I$ aims to generate designs that align with desired performances. The loss function $L_{p}$ encourages this behavior, mirroring the form of $L_{F}$ ,

L_{p} = \frac{1}{n} \sum_{i = 1}^{n} (w \cdot (p^{(i)} - F (I (p^{(i)}))))^{2} .

(4)

L_{p}

addresses the one-to-many mapping problem in inverse design by ensuring the generated GCS’s predicted performance matches the target performance without directly considering the generated GCS design. Second,

I

aims to generate printable designs. Drawing inspiration from prior work,²³ we used the loss function

L_{d}

to bias generated designs toward previously tested designs,

L_{d} = \frac{1}{n} \sum_{i = 1}^{n} (d^{(i)} - I (p^{(i)}))^{2} .

(5)

Specifically, $L_{d}$ encourages predicted designs to align with the dataset designs associated with the target performances. Such a loss is motivated by the fact that dataset designs are all known to be printable. While it might be possible to include losses that penalize parameter combinations leading to nonprintable designs, we do not explore these methods here.

$I$ seeks to minimize the loss function $L_{I}$ , which combines $L_{d}$ and $L_{p}$ in a weighted manner:

L_{I} = L_{p} + α L_{d},

(6)

where α ∈ ℝ determines the relative weight between

L_{p}

and

L_{d}

. By fine-tuning the α value, we can control the balance between optimizing for predicted performance accuracy and maintaining proximity to dataset designs. While it may be possible to include losses that penalize parameter combinations that result in nonprintable designs, we do not investigate such methods here.

Training

We divided our processed GCS dataset into training, validation, and test sets, following an 80–10–10% split. Our training process involved two stages. We trained $F$ in the initial stage. Upon completion, we froze $F$ and appended the untrained $I$ model. The second stage involved training $I$ using $F$ to aid in convergence.

Trained in this order, $I$ learns to generate GCS designs whose performance predicted by $F$ aligns with the desired performance. Had we trained $I$ independently, we could only use $L_{d}$ as $L_{p}$ depends on $F$ . Alone, $L_{d}$ exposes an ill-posed learning problem, as multiple GCS designs could yield similar performances. Penalizing deviations between generated and actual designs would impede convergence, rendering the learning process ineffective. While it is likely possible to train both networks simultaneously, similar generator and discriminator training in generative adversarial networks,²⁹ we left this to future investigations.

We use the Adam optimizer³⁰ for training, with a learning rate of 0.001, a weight decay of 1, and a batch size of 16. We used early stopping to prevent overfitting, terminating after 500 epochs in each stage. Our TNN was implemented using PyTorch³¹ and trained on an Apple MacBook Pro (M1 Max). Training both networks required less than an hour.

Results

We evaluated our TNN’s forward and inverse design accuracy on the test set. In our evaluation, we repeated training 10 times using different random splits of the data to report test outcomes with 95% confidence intervals. We performed physical experimentation on a sample of generated GCS designs and compared the TNN performance with other methodologies. Finally, we generated GCS with tailored mechanical properties for two applications to demonstrate our TNN’s inverse design capabilities.

Forward design performance

We evaluated $F$ on the test set by comparing the predicted and experimental force-displacement curves. We compared the metrics of the predicted and experimental curves because using high-level metrics for evaluation provides interpretable units for measures such as mean absolute error (MAE). Notably, the TNN did not directly learn metrics such as work or stiffness; they are computed based on the learned performance vectors.

Figure 4A presents the MAE and R² for stiffness, work, and maximum displacement. Each metric’s MAE is <5% of their respective ranges: 3.2% for stiffness, 2.8% for work, and 2.2% for maximum displacement. Furthermore, the small confidence intervals demonstrate that our TNN has training stability, with minimal reliance on model initialization or data splitting. Figure 5 shows $F$ ’s performance for eight test set GCS designs, demonstrating the network’s capability to predict the complete nonlinear force-displacement behavior of designs made with elastoplastic and hyperelastic materials. Refer to Supplementary Data S1 for the design parameters.

FIG. 4.

Performance. The R² and mean absolute error (MAE) with 95% confidence intervals for the test set. In forward design, we calculated the stiffness, work, and maximum displacement errors from predicted force-displacement curves. In inverse design, we obtained these metrics from the predicted designs’ predicted force-displacement curves. For reference, the stiffness, work, and maximum displacement ranges found in the dataset are 7732 N/mm, 46.7 J, and 22.6 mm, respectively. A k-nearest neighbors model with k = 1 is included for comparison. (A) Forward design performance. (B) Inverse design performance. (C) Percentage of generated designs passing all printability checks from inverse design.

FIG. 5.

Forward design results. Eight randomly selected results from the test set. The GCS designs (blue) serve as input to $F$ , which predicts force-displacement curves (orange). For reference, we show the actual experimental force-displacement curves from the test set (blue). $F$ can predict nonlinear deformation behavior for elastoplastic (B, C, E) and hyperelastic (A, D, F, G, H) GCS.

Inverse design performance

We trained $I$ with α ∈ {0,0.01,0.1,1}, the parameter weighing the importance of generating previously tested designs. To evaluate the inverse design performance, we compare predicted designs’ predicted force-displacement curves $F (I (p))$ to the target force-displacement curves p. Figure 4B presents the MAE and R² for stiffness, work, and maximum displacement for $I$ trained with each α value.

For α ∈ {0,0.01,0.1}, we observed minimal change in accuracy for the work and maximum displacement. However, for α = 1, we saw the MAE increase for maximum displacement and decrease for work. For stiffness, the accuracy improved as we increased α. We found that different metrics derived from force-displacement curves exhibit varying sensitivity levels to the effect of $L_{d}$ . These findings underscore a complex trade-off between generating designs with small prediction errors and printable designs.

Physical validation

We randomly selected eight GCS designs generated from $I$ trained with α = 1 and experimentally obtained their force-displacement curves to assess the accuracy of predicted performance (Fig. 6). We fabricated the samples on a MakerGear M3 and performed compression testing with an Instron 5965. Given a nonlinear force-displacement curve, $I$ can create GCS designs that conform to the specified deformations. Refer to Supplementary Data S1 for the design parameters.

FIG. 6.

Inverse design results. Eight randomly selected results from the test set. The force-displacement curves (blue) serve as input to $I$ , which predicts GCS designs. We fabricated and performed compression testing on the predicted GCS designs to obtain experimental force-displacement curves (orange). $I$ can generate GCS designs that exhibit nonlinear elastoplastic (A, C, F) and hyperelastic (B, D, E, G, H) deformations. For reference, we include the test set GCS designs (blue) associated with the inputted curves to illustrate the one-to-many relationship between performance and designs. Generated designs differ, sometimes significantly, from their test set counterparts.

We evaluated the printability of generated designs using two criteria, distinct from any individual geometric or material property, established in our previous work⁶: the base perimeter should be at least 30 mm to provide a substantial contact area with the print bed, and the shell must maintain a minimum distance of 0.01 mm from its center axis to accommodate material deposition. In Figure 4C, we calculated the percentage of printable predicted designs within the test set for $I$ trained on different α levels and observed a positive correlation. This trend would suggest that further increasing α would continue to improve printability. However, as α increases, $L_{d}$ dominates $L_{p}$ , reverting the training process to the one-to-many mapping problem outlined in Section “Training”. We did not directly investigate the value of α for which this behavior begins.

Comparison with alternative approaches

We evaluate the TNN against two alternative methods, k-nearest neighbors (kNN) and FEM, assessing their performance and practicality.

Nearest neighbors

We trained a kNN model with k = 1 for forward and inverse design and evaluated its performance (Fig. 4A and B). In forward design, kNN is more accurate in stiffness but less accurate in work and maximum displacement. In inverse design, kNN is more accurate in stiffness and work but less accurate in maximum displacement. These discrepancies signify that there is no clear best method concerning these metrics. However, kNN presents several significant limitations.

First, kNN has poor scalability. While the GCS parameterization benefits from a relatively small parameter set, the curse of dimensionality quickly becomes an issue as the parameterization becomes more complex or more materials are added. From a storage scalability perspective, kNN requires storing the entire training dataset, leading to increased model size and longer inference times.

Second, when k > 1, kNN interpolates between the closest designs, lacking a straightforward mechanism to ensure printability. For instance, interpolating between different one-hot-encoded materials will always produce invalid results. We have not validated that the nearest design in the normalized parameter space is the most similar. This leads to a broader question of accurately assessing “nearness” in non-Euclidean spaces, such as sparse one-hot-encoded spaces or other normalized parameter spaces, which warrants its own investigation.

Third, kNN cannot transfer knowledge to structures with different parameterizations or performance representations. Transfer learning reduces experimental data requirements, making it crucial for expanding to other structures and practical applications.

Finite element method

We compared the accuracy and speed of $F$ with those produced by FEM. We used Abaqus to numerically derive the compressive force-displacement curve for a GCS design. Refer to Supplementary Data S2 for details on the simulation setup.

Figure 7 presents the force-displacement curves obtained from Abaqus and $F$ . FEM accurately portrayed the linear elastic force-displacement relationship, but the computation began to lose accuracy for the nonlinear plastic deformations. The FEM simulation ultimately terminated prematurely and failed to converge beyond a fraction of the total experimental displacement. In this plastic region of compression, the numerous self-collisions and tearings presented computational challenges for FEM that were not easily addressed.

FIG. 7.

Comparison with finite element method (FEM). We display the force-displacement curves for a GCS obtained through experimentation (blue), predicted by $F$ (orange), and simulated via FEM (green). Four points along the curves indicate the deformations from experimentation (top) and FEM (bottom).

Experimental results for a GCS design can be obtained in 25 min (10 min for fabrication and 15 min for compression testing). Our TNN has inference times of <20 ms with under 1 h to train. In comparison, the compression test simulation time was 74 min on an eight-core CPU with 32 GB of RAM.

We used experimental data instead of simulated data to explore predictive capabilities, as a publicly available dataset capturing the mechanical behavior of interest exists. However, if such data are not accessible, using synthetic datasets from simulation to train machine learning models is a common strategy, provided that the simulation methods accurately model the behavior of interest.

Applications

We used $I$ to generate GCS with mechanical behavior tailored to two applications: impact absorption and material emulation. Refer to Supplementary Data S2 for more details on each application.

Impact absorption

An impact-absorbing structure must absorb the total impact energy while containing peak forces within specified limits to prevent damage or injury. Given a force threshold F, we optimized for a force-displacement curve meeting (or exceeding) a target energy absorption E to use as input for $I$ ,

\underset{valid p}{\arg \min} E - E_{F} (p),

(7)

where E_F(p) denotes the calculated energy absorption (work) before exceeding F (Fig. 8A).

FIG. 8.

Applications. (A) Given a force threshold F, E_F is the energy absorbed before exceeding F. (B) Custom GCS create padding for the egg drop test. A pad of four GCS absorbs the impact energy of an egg dropped from 50 cm without breaking it. (C) GCS emulating the mechanical properties of polyurethane foam (PUR), a common material in packaging.

We identified GCS optimized for impact absorption in the context of the egg drop test (Fig. 8B). This test involves constructing padding to protect an egg from breaking during a substantial fall. In our experimental setup, we dropped eggs from 50 cm onto a pad containing four GCS parts. We set F = 10 N and E = 0.0735 J for the target force-displacement curve. Using $I$ , we optimized for a GCS design that absorbs the impact energy of the drop without breaking the egg. We tested three setups, each with five eggs: the optimized pad, an unoptimized pad with a randomly selected design, and no pad. We observed a 100% survival rate for the optimized pad, while the unoptimized pad and no pad showed significantly lower survival rates of 20% and 0%, respectively.

Material emulation

Our TNN enables the creation of GCS that emulate the mechanical behaviors of different materials. By mimicking the behavior of other materials, one can optimize for non-mechanical properties such as weight, cost, and fabrication time. We designed GCS parts that replicate the behavior of polyurethane foam (Fig. 8C), a material commonly employed in packaging.

Discussion

Discrepancies between predicted and actual curves can come from model prediction errors and lost information from PCA compression. However, we did not examine which source of error contributes to poor predictions. In the future, we plan to extend our investigation to look at the performance of PCA compression and explore nonlinear compression methods such as autoencoders.

Real-world constraints often restrict parameters such as height, mass, or material. However, our current TNN architecture does not allow for user-defined values of generated design parameters, suggesting a clear direction for future enhancements. One potential strategy is to explore conditioning techniques employed in other neural network architectures³² to grant users fine-grained control over the generated design parameters.

Exploring transfer learning techniques for our TNN presents an exciting avenue for extending its capabilities to diverse 3D-printable structures. Fabricated structures span various parameterizations, yielding structures such as lattices,^2,25,33 crossed barrels,⁴ and foams.⁹ From a mechanical perspective, switching to a different structure family would not necessarily provide any new hyperelastic or elastoplastic compressive performance that GCS does not describe. However, non-mechanical requirements make additional structure families valuable. Transferring the acquired design-performance knowledge to different structures, especially those with limited empirical data, holds significant promise for future research.

Researchers have used numerous approaches to machine learning for inverse design in other domains, spanning generative models,³⁴ graph-based networks,³⁵ and reinforcement learning algorithms.²⁸ These methods can now be compared with the TNN architecture as an interesting direction for future work.

Finally, understanding how simulation and experimentation can be used in unison to predict high-deformation mechanical properties is essential for future work. Such approaches offer viable alternatives to collecting extensive experimental datasets, a process typically reliant on access to self-driving labs. We hope to learn how experimental data can improve simulated outcomes and how much experimental data are needed for this purpose.

Conclusion

We explore using a TNN for the forward and inverse design of FDM 3D-printed shells, representing a diverse and versatile class of structures. Our TNN reveals the intricate design-performance relationship between shell parameters and compressive behaviors. Motivated by the simulation’s current inability to accurately and rapidly model high-strain compressive elastoplasticity, we are the first to use a purely experimental compressive force-displacement dataset to train our network. Such a dataset naturally incorporates real-world fabrication and measurement noise within the performance data. By utilizing entire force-displacement curves as performance representations, the network captures a range of nonlinear elastoplastic and hyperelastic deformations.

In forward design, our TNN predicts these nonlinear force-displacement curves based on shell design parameters. Conversely, in inverse design, the network generates shell designs that exhibit specific desired compressive deformations. We validate generated shell designs through fabrication and testing and demonstrate the applicability of our approach to real-world problems.

Our network architecture’s simplicity is grounded in the relatively limited size of our experimental dataset and the number of design parameters. To encourage further exploration of data-driven deformation design, we make our code and processed dataset publicly available.^b

Authors’ Contributions

S.S.: Conceptualization, methodology, software, investigation, writing—original draft. K.L.S.: Conceptualization, investigation, validation, writing—review and editing. K.A.B.: Conceptualization, methodology, writing—review and editing, supervision. E.W.: Conceptualization, methodology, writing—original draft, writing—review and editing, supervision.

Footnotes

Acknowledgments

The authors thank Adedire Adesiji for brainstorming, assistance in constructing applications, and photography; Helena Gill, Xingjian Han, and Abinit Sati for their work on constructing and running the impact absorption application; and Peter Yichen Chen for his discussions and input.

Author Disclosure Statement

No competing financial interests exist.

Funding Information

This work was supported by the U.S. Army CCDC Soldier Center (contract W911QY2020002).

Supplemental Material

a

b

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