Active composites (ACs), materials whose shape changes in response to external stimuli (heat, light, water, magnetic fields), offer significant potential for creating shape-morphing structures. The shape transformation is determined by the spatial distribution of active and passive constituent materials. 4D printing, allowing for complex voxel-level material encoding, provides unparalleled freedom in designing these shape-changing ACs. However, efficiently determining the optimal material distribution to achieve a desired 3D shape change presents a significant challenge. Current computational design strategies, including gradient-based topology optimization (TO) and gradient-free methods like finite element (FE)-based evolutionary algorithms (EA), face limitations. TO requires complex gradient derivations and struggles with material nonlinearities, while FE-EA suffers from high computational cost due to numerous FE calculations. This research aims to overcome these limitations by leveraging the power of machine learning (ML) to create a fast, accurate, and computationally efficient inverse design approach for 4D-printed AC plates. The ability to rapidly and accurately design these structures has broad implications for various applications, ranging from soft robotics to adaptive structures and biomedical devices. This is particularly relevant as the design space for 2D active structures with 3D deformation is significantly larger than that for simpler 1D structures. Existing ML-based materials design mostly focuses on optimizing mechanical properties rather than shape changes, leaving a substantial gap in methodologies for the design of 4D-printed ACs.

Previous research has explored computational design strategies for 4D printing, using both gradient-based and gradient-free methods. Gradient-based topology optimization (TO) has been employed to design shape-changing behaviors of ACs and optimize the compliance of soft actuators, but it often requires complicated gradient derivations and may struggle with nonlinearities. Gradient-free methods, such as FE-based evolutionary algorithms (EA), have successfully designed shape-change responses but suffer from high computational cost due to the numerous FE calculations needed to explore the large design space. Reduced-order forward models have been developed to address this computational burden, but accurately and efficiently exploring the design space and solving inverse problems remain challenging. Recent advancements in machine learning (ML) offer a promising avenue for developing fast, accurate, and computationally affordable predictive models that can be integrated with optimization algorithms. While existing works have used ML for materials design, primarily focusing on optimizing mechanical properties like strength and toughness, limited research exists on ML-based design of shape changes in ACs. The design for shape changes is particularly challenging due to the complex mapping from material distributions to shapes, high dimensionality of shape data, and variability of target shapes, demanding highly accurate ML models. Existing methodologies fall short in addressing the complex shape-change design of 4D-printed ACs, particularly for 2D active structures with 3D deformation which poses a significantly higher dimensional and intricate challenge than the previously studied 1D active beams with 2D shape change. Geometric mapping strategies, such as exploiting conformal mapping to obtain a spatially varying expansion field, have shown success in designing target surfaces but have limitations in accurately considering mechanics and handling non-developable surfaces. This paper aims to address the limitations of existing methods by combining ML with both GD and EA to create a highly efficient tool for designing 4D-printed active composite plates.

This work presents a machine learning-based approach for the efficient forward shape prediction and inverse design of voxelized active composite plates. Finite element (FE) simulations using Abaqus were conducted to generate a dataset of material distributions and corresponding actuated shapes. The dataset included both fully random and island designs. The simulations used two different boundary conditions (BCs): original and converted BCs. The original BCs simplified calculations and facilitated data augmentation, while the converted BCs proved beneficial for machine learning (ML) prediction by introducing spatially sequential dependency in the shape data. Data augmentation, leveraging symmetries in the geometry, increased the dataset size to 900,000 samples. A deep residual network (ResNet)-based ML model was trained using this dataset to predict the actuated shape (x, y, z coordinates) from the material distribution (represented as a 3D binary array). Different network architectures and training combinations of coordinates were explored to optimize the model's performance. The trained ML model demonstrated high accuracy (R² > 0.999 for x, y and R² = 0.995 for z, further improved to R² = 0.999 for z using symmetry averaging). The ML model significantly outperformed FE simulations in prediction speed. For inverse design, a global-subdomain design strategy was employed, combining ML with both gradient descent (GD) and evolutionary algorithm (EA). The ML model accelerated GD by allowing for efficient gradient computation via automatic differentiation (AD). For the global design step, both ML-GD and ML-EA were used. The subdomain design step, focusing on regions with larger errors, utilized ML-EA due to its effectiveness in exploring the reduced design space. A distance-weighted loss function was employed for the global step to guide the optimization from fixed to free boundaries. The optimization algorithms were used to minimize the error between the ML-predicted shape and the target shape. The accuracy of designs was quantified using the absolute distance between sampling points in the predicted and target shapes. A normal distance-based loss function was introduced for irregular target shapes. In experiments, grayscale digital light processing (g-DLP) printing was used to fabricate the designed AC plates. A photocurable resin, formulated with a volatile and a nonvolatile component, enabled shape morphing via differential shrinkage upon heating. Different material systems were also tested to showcase the general applicability of the design approach.

The ResNet-based ML model achieved excellent accuracy in predicting the actuated shape from the material distribution, significantly outperforming FE simulations in terms of speed. The global-subdomain design strategy effectively addressed the challenges of the large design space, producing optimal designs for a wide range of target shapes. Both ML-GD and ML-EA yielded accurate designs, with ML-EA demonstrating particular effectiveness for irregular targets. The distance-weighted loss function in the global step further improved design accuracy. Specifically, for FE-derived target shapes, excellent agreement was observed between the target shapes, ML-predicted optimal shapes, FE-simulated shapes, and experimentally obtained shapes. For algorithmically generated target shapes, including developable and non-developable surfaces, the approach successfully achieved optimal designs validated through experiments, demonstrating its applicability across various material systems, actuation mechanisms, and length scales. Furthermore, the use of a patch representation and normal distance-based loss function allowed for the accurate design of irregular target shapes, such as a crumpled piece of paper and a surgical mask. The experimental results strongly validated the designed shapes for both regular and irregular targets, including results from multiple material systems and length scales. The time cost for inverse design using ML-GD and ML-EA was drastically reduced compared to conventional FE-based methods. ML-GD took approximately 3 minutes, while ML-EA took approximately 12 minutes for the global step and 0.6 minutes for the subdomain step. In contrast, the estimated time cost for FE-EA was 2200-5600 hours, and for FE-GD, 11-28 hours. The ML model, while requiring initial FE simulations for training, significantly reduced the overall design time for subsequent designs.

This study successfully demonstrates a highly efficient and accurate ML-based approach for the inverse design of 4D-printed active composite plates. The integration of ML with GD and EA addresses the limitations of traditional computational design methods, significantly reducing computational cost and improving design accuracy. The ability to handle a vast design space and achieve high-fidelity predictions allows for the creation of complex, shape-changing structures that would be impractical to design using conventional methods. The results of this study have significant implications for the design and fabrication of advanced 4D-printed structures, particularly for applications requiring precise shape control and complex geometries. The findings directly address the challenges of efficiently designing for complex 3D shape changes in active composite plates, significantly advancing the field of 4D printing and its applications. The presented methodology opens the door for exploring even more complex designs with a larger number of voxels and multiple materials, further broadening the applications of 4D-printed shape-morphing structures.

This research presents a novel ML-enabled approach for efficient forward prediction and inverse design of 4D-printed active composite plates. The method successfully achieves highly accurate designs for both regular and irregular target shapes across various material systems and length scales. The significant reduction in computational time compared to conventional methods positions this approach as a highly effective tool for the intelligent design and fabrication of advanced 4D-printed shape-morphing structures. Future work may focus on expanding the design space, improving printing accuracy, and developing physics-informed ML models to further enhance efficiency and accuracy.

While the ML model exhibits high accuracy, its performance is dependent on the quality and quantity of the training data. The accuracy of the FE simulations used to generate the training data can also impact the overall accuracy. The current design space (15x15x2 voxels) limits the complexity of the designs that can be achieved. Some unsmooth features were observed in the printed AC sheets due to variations in the curing distance during DLP printing. The method's applicability to non-square initial shapes or those with finer features requires further investigation.