## 3 The Sciences of \"Admissible\" Things
### Modeling and Design
#### 3.4.1 Isolated Voxels  
The fundamental mechanical considerations of 4D shape-changing structures produced through AM require modeling that incorporate measurements, the initial and final stress analyses, and complex modeling [94]. However, as shown in Figure 8, the characteristics of an isolated object are determined not only by its shape but also by other parameters, such as those associated with energy stimulation, which influence its steady or time-resolved deformation(s) and sometimes material transport inside a voxel. [PERSON] et al. [95] developed a model for designing and synthesizing morphing mechanisms using materials capable of exhibiting preprogrammed complex changes. By integrating active and passive materials, their algorithm can encode the desired change into the material distribution of the mechanism. This approach paves the way for a new generation of material-driven machines that are lightweight, adaptive, damage-resistant, and easy to produce using AM. Currently, the modeling part of their work serves as a preliminary step towards advancement in this direction. Similarly, [PERSON] et al. [96] introduced a case study guiding the development of machine learning models to predict the behavior of nonlinear 4D printing scenarios. Developing a FE machine learning model to predict the bending angle of a flexible pneumatic actuator allows studying the influence of input variables on its bending. This initiative aims to run FE simulations to train data for machine learning. In addition, [PERSON] et al. [97] utilized multi-material printing capabilities to fabricate heterogeneous hinge modules. Different configurations can be encoded during AM due to the variable distribution and direction of modularly designed hinges. Following this  
Figure 8: Shape changes of a voxel when exposed to stimuli. experimental aspect, [PERSON] et al. [98] developed a design strategy to use the desirable 4D morphing of multi-material composites. Composites with bilayer laminates composed of shape memory polymer and elastomer were made with variable thickness ratios to control the self-bending of the composite material. FE simulations were used to understand the underlying processes of the composite materials and to generate accurate predictions for the experimental results. [PERSON] et al. [99] used similar foundations with simple thermoplastic structures. However, leveraging the foundational principles of simplified systems, [PERSON] et al. [100] capitalized on the understanding that the changes of active materials, when exposed to stimuli, exhibit nonlinear behaviors, which are challenging to characterize qualitatively and quantitatively, even with advanced numerical solvers. They propose:  
1. Developing standardized kinetic components in smart materials that exhibit transformation primitives, such as bending and twisting, to be used as active components for mechanical assemblies with rigid parts;
2. Introducing an open kinetic library, accessible for downloading data on kinetic components to incorporate into designs, as well as enabling users to upload and share their own data;
3. Running simulations based on empirical methods using kinetic components within assemblies. The authors have put forward two design proposals utilizing standardized kinetic components: an icosahedron and a mounting platform. Despite ongoing scientific exploration in this area, these elements delineate tangible constraints on the advancement of 4D printing applications. The abundance of possibilities poses a challenge in establishing a robust pathway forward. Until these barriers are adequately addressed, only a few specific and straightforward niches will be able to sustain activity. For instance, achieving the desired low response times by designers may necessitate producing very small objects or employing optical or electromagnetic stimulations with well-defined temporal bases and simplified modeling.