Tensile Properties of 3D‐Projected 4‐Polytopes: A New Class of Mechanical Metamaterial

In this article, we research the tensile behavior mechanical metamaterial based on the 3D projections of 4D geometries (4‐polytopes). The specific properties of these mechanical metamaterials can be enhanced by more than fourfold when optimized within a framework powered by an evolutionary algorithm. We show that the best‐performing metamaterial structure, the 8‐cell (tesseract), has specific yield strength and specific stiffness values in a similar range to those of hexagonal honeycombs tested out‐of‐plane. The 8‐cell structures are also cubically symmetrical and have the same mechanical properties in three orthogonal axes. The effect of structure is quantified by comparing metamaterial tensile strength against the Young's modulus of constituent solid material. We find that the strength‐to‐modulus value of the 8‐cell structures exceeds that of the hexagonal honeycomb by 76%. The 5‐cell (pentatope) and 16‐cell (orthoplex) metamaterials are shown to be more effective under tensile loading than gyroid structures, while 24‐cell (octaplex) structures display the least optimal structure‐properties relationships. The findings presented in this paper showcase the importance of macro‐scale architecture and highlight the potential of 3D projections of 4‐polytopes as the basis for a new class of mechanical metamaterial.

DOI: 10.1002/adem.202300251 In this article, we research the tensile behavior mechanical metamaterial based on the 3D projections of 4D geometries (4-polytopes). The specific properties of these mechanical metamaterials can be enhanced by more than fourfold when optimized within a framework powered by an evolutionary algorithm. We show that the best-performing metamaterial structure, the 8-cell (tesseract), has specific yield strength and specific stiffness values in a similar range to those of hexagonal honeycombs tested out-of-plane. The 8-cell structures are also cubically symmetrical and have the same mechanical properties in three orthogonal axes. The effect of structure is quantified by comparing metamaterial tensile strength against the Young's modulus of constituent solid material. We find that the strength-to-modulus value of the 8-cell structures exceeds that of the hexagonal honeycomb by 76%. The 5-cell (pentatope) and 16-cell (orthoplex) metamaterials are shown to be more effective under tensile loading than gyroid structures, while 24-cell (octaplex) structures display the least optimal structure-properties relationships. The findings presented in this paper showcase the importance of macro-scale architecture and highlight the potential of 3D projections of 4-polytopes as the basis for a new class of mechanical metamaterial.

Simulation Results
The 3D-projected 4-polytope geometries were developed using the simulation-based approach described in detail in the Experimental Section. The 3D projections of 5-, 8-, 16-, and 24-cell 4-polytopes were designed with the aim of maximizing specific stiffness. An optimization framework was used to automate exploration of the design space, which monitored incremental improvements in specific stiffness with each new structure generated. As the mass and hence the relative density of each iteratively generated design could not be the same due to the high number of parametric design variables, a specific property was chosen as the performance metric in this optimization study. Figure 1 illustrates the improvements in specific stiffness at 0% (no optimization), 25% of optimal, 50% of optimal, and 100% (full optimization) of each 3D-projected 4-polytope in tension. The maximum improvement in specific stiffness through our optimization framework is 121.79%, 72.45%, 163.44%, and 468.71% for 5-, 8-, 16-, and 24-cell metamaterials, respectively. The 8-cell metamaterial structure has a specific stiffness that at 0% optimization, is greater than the fully optimized 5-, 16-, and 24-cell structures. It has therefore a base cellular architecture that is already notably higher-performance when compared against the other 3D-projected 4-polytopes researched here. It also exhibits the lowest percentage increase from its unoptimized to its fully optimized states, while the 5-, 16-, and 24-cell, as a percentage improvement from a base structure, can be seen to benefit significantly more through the optimization framework, clarifying that adjustments of parametric design variables can yield a wide range of metamaterial architectures with improved properties of specific stiffness. Figure 2 shows the elastic strain energy density for each of the 3D-projected 4-polytope metamaterials at different levels of optimization starting at unoptimized (0%) to be fully optimized (100%). These plots show that strain energy density increases within each metamaterial structure as a function of increased levels of structural optimization. As such, each unit cell type (5-, 8-, 16-, and 24-cell) can be seen to develop a higher overall capacity to store elastic strain energy when loaded in tension, as the structure is progressively optimized. Since the capacity to store strain energy is also related to the overall tensile modulus of each structure, the unit cells with the most evenly distributed strain energy densities gain the highest stiffness. The plots furthermore enable the identification of regions within each metamaterial unit cell that contribute the most toward strain energy absorption and thus the overall stiffness. Nevertheless, our broader objectives were not only to optimize the structure for stiffness, but also for lightweightness. This was achieved using a coupled objective for mass minimization (cf. Experimental Section). When the apparent density is thence taken into account, we note that the fully optimized structures for each of the 3D-projected 4-polytope types have the highest specific stiffness values as previously shown in Figure 1. Visually, the most effective strain energy density distribution is in the fully optimized 8-cell structure, followed by the 5-and 16-cell structures. The structure with the most localized strain energy density, namely 24-cell, has the lowest capacity for storing strain energy and hence the lowest specific stiffness. This is due to the fact that the high strain energy density levels concentrated in a single location tend to cause early local failures within the structure under tensile loading, lowering therefore, the limit of elastic proportionality in these structures. As such, we note that the optimal structures are the ones that are able to share strain energy most effectively throughout the larger volume of a unit cell. This can therefore be seen as a fundamental design consideration in high stiffness mechanical metamaterials. In the case of our 3D-projected 4-polytopes, this can in turn be directly correlated to the geometrical complexity of each of the structures. The best-performing 8-cell metamaterial has the lowest levels of geometrical complexity, followed by the 5-and 16-cell, whereas the 24-cell, which is the least optimal in terms of performance within the range of linear elasticity, is geometrically most complex. Structures with a higher level of geometrical complexity tend to have higher numbers of sharp features, such as corners and slender edges, and this results more points within the structure where with high localized strain energies. Figure 3 provides experimental values of specific stiffness plotted against the specific yield strength for each of the fully optimized 3D-projected 4-polytope metamaterials. Also included in the figure are experimental values for gyroids and for hexagonal honeycomb structures tested out-of-plane. The data points represent the arithmetic mean value of five experimental samples tested in tension with vertical and horizontal error bars showing the full ranges of experimental values for specific stiffness and specific yield strength, respectively. All of the samples were manufactured using the same 3D prototyping technique as discussed in the Experimental Section. The experimental results follow a similar trend to the predicted simulation results, with the 8-cell metamaterial exhibiting the highest specific stiffness (0.89 MNm kg À1 ) out of the 3D-projected 4-polytope metamaterials, followed by the 5-, 16-, and 24-cell structures, the arithmetic means of which were 0.46, 0.40, and 0.39 MNm kg À1 ,  respectively. This is in line with the predictions made from the simulation results where the cells with highly distributed strain energy density were found to have a higher capacity for storing strain energy and hence an overall higher stiffness. Specific yield strength values for the 3D-projected 4-polytope metamaterials follow a slightly different trend, with the 8-cell structure still exhibiting the highest value of specific strength (6.71 kNm kg À1 ), but this is then followed by the 5-, 24-, and then 16-cell structures, the arithmetic means of which were 4.60, 4.48, and 3.13 kNm kg À1 , respectively. The higher specific strength of the 24-cell metamaterial as compared to the 16-cell structure is presumably a result of the central alignment of the 24-cell metamaterial unit cells enabling a higher level of elastic energy absorption prior to failure and suggesting that the unique geometrical features in individual unit cells of these metamaterials play an important role in determining the load bearing to the point of yield. Moreover, as shown by the simulation results summarized in Figure 2, the elastic strain energy density is more localized in the 16-cell structure than it is in the 24-cell structure.

Experimental Results
In addition, it can be observed that all 3D-projected 4-polytope metamaterials have higher specific stiffness and strength values than the gyroid structure. When compared to the hexagonal honeycomb loaded in the out-of-plane direction, we note that the arithmetic means for each 3D-projected 4-polytope metamaterial type is lower in terms of specific stiffness, and that only the upper experimental range of the 8-cell structure overlaps with the arithmetic mean value of the honeycomb. Nevertheless, the 8-cell structure has the highest specific yield strength when compared to any of the six structures presented here, including that of the hexagonal honeycomb. Here, the mean value is 1.05% higher than that of the honeycomb. Similarly to the hexagonal Color maps for each of the 3D-projected 4-polytope metamaterials. Each column illustrates the progress in optimization at the following different stages: i) 0% (unoptimized), ii) 25%, iii) 50%, iv) 75%, and v) 100% (fully optimized). The color legend represents the elastic strain energy density values in J cm À3 and the cells are loaded in tension to 2% strain. www.advancedsciencenews.com www.aem-journal.com honeycomb, the 8-cell has thin-walled features aligned along the direction of loading and hence such an arrangement contributes toward the high specific stiffness and high specific yield strength values. Moreover, the 8-cell structure was developed using an optimization approach which implicitly reduces stress concentration points by adjusting geometrical features within the unit cell. The 8-cell structure unlike the honeycomb has cubic symmetry (i.e., identical mechanical properties in three orthogonal axes). The honeycomb structure in comparison is essentially a 2D structure that is extruded in the third dimension, and which has high stiffness and strength in only the out-of-plane direction. The experimental specific stiffness and specific yield strength results discussed in the preceding paragraph are also summarized in Table 1 and 2, respectively. The tables provide the arithmetic means, experimental ranges, medians, standard deviations (SDs), and coefficients of variance. Additionally, the experimental results are compared against the simulation outputs using percentage differences and Z-score (Z ) values. As shown in Table 1, the simulated specific stiffness results are 0.87%, 13.24%, 13.77%, and 14.07% higher than the results obtained experimentally for the 8-, 16-, 5-, and 24-cell structures, respectively. The Z-score values follow a similar trend and are between 0.20 and 9.12, with the 8-cell having the lowest Z-score, while the 16-cell has the highest. The results summarized here indicate that the simulation predicts the specific stiffness of the 8-cell structure with a good level of accuracy.
A different trend is observed when comparing specific yield strength results. The simulation output suggests higher values are possible, with the percentage difference ranging between 32.26% and 35.33%, while the Z-score values are between 3.63 and 11.42. This difference in simulation-predicted and experimentally tested behaviors can also be observed in Figure 4. Here, axial stress-strain curves obtained from simulations are plotted in dashed lines while the upper and lower experimental testing bounds are shown in solid lines. The simulations predict the sample stiffness with high accuracy, especially at low strain values and in each of the 3D-projected 4-polytope metamaterials, the simulation results lie closer to the upper bound of the experimental results. As the tensile strain values increase, the gradients of the simulation and experimental curves start diverging indicating that the unit-cell structures undergo plastic deformation at lower strain values than predicted by the computational models. Consequently, the yielding strength values of the metamaterials presented here are also higher than those of the simulation results.
The main reasons for such discrepancies are due to 1) manufacturing-related limitations and 2) a low number of neighboring unit cells in the tensile samples. As discussed in the Experimental Section, the experimentally measured mass was found to be higher than that calculated from the CAD models used in the simulations by 11.62%, 0.95%, 12.67%, and 13.80% for the 5-, 8-, 16-, and 24-cell structures, respectively. Higher mass directly affects the apparent density values leading to lower overall predictions of the specific properties when compared with simulation results. The 8-cell experimental samples were only 0.95% higher in mass than the equivalent CAD models while also having the most accurate simulated specific stiffness predictions with a difference of 0.87% between simulations and experiments. The structure with the highest noticeable difference between simulated and experimentally obtained specific stiffness results, the 24-cell (14.07%), also has the highest sample mass variation (13.80%). Higher sample mass can result from additional resin deposition within a unit cell during the manufacturing process. Any unwashed resin within the unit cell may accumulate at specific locations, such as corners and  pockets. The additional resin partially cures during the postcuring process. This unwanted material may cause asymmetrical deformation of a structure under loading, giving rise to stress concentrations and leading to premature yielding. This issue is further pronounced by the variations in material properties due to uneven polymer cross-linking during post-curing. As the samples are exposed to UV light, the level of polymerization is affected by the geometry of the metamaterial structure as well as its surface-to-volume ratio. As such, polymer cross-linking levels vary between individual samples as well as between different metamaterial sample sets. This variability can be observed in Figure 4 when comparing the upper and lower bounds of the experimental results. In addition to these manufacturing-related limitations, the experimental results are affected by the low number of neighboring unit cells in the tensile samples. The 2 Â 2 unit-cell arrangement in the cross section of the tensile sample was chosen to ensure that the samples could be tested using the 32 mm sized tensile testing grips; however, such an arrangement is far from ideal for a unit-cell-based metamaterial.
In fact, simulation results by ref. [46] suggest that the ratio of sample cross section to unit-cell size should be higher than 10 to adequately homogenize a mechanical metamaterial. As the simulations do not account for manufacturing imperfections and have unit-cell boundary conditions that are representative of an infinite-size sample, the simulation outputs compute an idealized metamaterial response. Thus, high prediction accuracy is observed for stiffness values at low strains while the stress levels within the structure are low, and geometrical as well as material-property-related imperfections do not play a key role in the deformation of the metamaterial structure. As strain increases, the significance of the mentioned imperfections exacerbates causing the unit-cell geometries to deform in an asymmetrical manner, therefore causing premature local point yielding of the structure. Consequently, this affects the observed experimental stress-strain behavior where the experimental samples tend to have a semi-linear part of the curve after the yield point which is different to the strain-softening type of behavior predicted by the simulations. Such behavior is believed to be a superposition of asymmetrical elastic deformation of the whole unit cell as well as the local point yielding within the structure due to sample imperfections.
The specific stiffness and strength results presented so far were engineered using a computational approach which aimed to maximize the total stored elastic strain energy in each of the 3D-projected 4-polytope metamaterials. However, recognizing how such an approach affects other mechanical properties allows us to compare 3D-projected 4-polytope metamaterials in more detail, and to better understand the mechanical performance  www.advancedsciencenews.com www.aem-journal.com of these structures. Table 3 summarizes the Young's modulus, yield and tensile strengths, and modulus of resilience and toughness values for each of the 3D-projected 4-polytope metamaterials, as well as for the gyroid and hexagonal honeycomb structures. The standard deviation (SD) and coefficient of variance (CoV) values are also included in Table 3 for each sample. Stress-strain plots representative of the experimental results are summarized in Table 3 to provide a better insight into the experimental results. The 8-cell structure has the highest Young's modulus of 213.93 MPa, which is 17.62% higher than that of the hexagonal honeycomb structure tested in the out-ofplane direction with the value of 181.88 MPa. The 5-and 16-cell structures have values of 86.99 and 60.41 MPa, respectively, surpassing the gyroid which has Young's modulus of 27.12 MPa. The structure with the lowest value of 25.13 MPa is the 24-cell; however, the 24-cell also has the lowest apparent density of 63.48 kg m À3 out of the six experimentally tested samples as presented in Table 4. The CoV values for all of the samples are between 1.67% and 13.39% showing a high level of consistency between the experimental sample results. A similar trend is observed when analyzing yield and tensile strength results. The 8-cell structure has the highest values of 1.62 and 3.67 MPa for yield and tensile strength, respectively, which are 26.56 % and 76.44% higher than those of the hexagonal honeycomb with 1.28 and 2.08 MPa for the yield and tensile strength results, respectively. Following the 8-cell and the hexagonal honeycomb, the highest yield strength values in descending order were obtained for 5-cell, 16-cell, 24-cell, and gyroid structures, respectively. The tensile strength results follow a similar trend to those for yield. Following the 8-cell and hexagonal honeycomb structures, the highest tensile strength values, in descending order, were obtained for the 5-cell, 16-cell, gyroid, and 24-cell structures, respectively. Although the trend is similar, it should be noted that the 24-cell structure yields at 16.67% higher stress in comparison to the gyroid structure; however, it has a 32.99% lower tensile strength value. These results are expected since the 3D-projected 4-polytope metamaterials were designed using computational methods to maximize the elastic strain energy-storage capacity, which directly correlates to the overall stiffness of the structure. The ability to absorb elastic strain energy can also be evaluated by comparing the modulus of resilience values for the experimental samples. The experimental results suggest that the 8-cell structure has the highest modulus followed by the hexagonal honeycomb and then the 5-cell, with values of 5.30, 4.67, and 4.29 kJ m À3 , respectively. When expressed as a percentage difference with respect to the honeycomb value, the 8-cell has a 13.49% higher, while the 5-cell has an 8.14% lower modulus of resilience, suggesting that both the 8-and 5-cell metamaterials are highly suited for applications where elastic energy absorption is desirable. Both the 16-and 24-cell structures outperform the gyroid by 88.54% and 91.67%, respectively, in terms of the modulus of resilience. Lastly, as the tensile strain increases and the plastic deformation range is reached, all of the 4-polytope metamaterials fail at the tensile strain range between 0.018 and 0.034, which is higher than that of the honeycomb, which has a strain to failure value of 0.014, and lower than the gyroid, which has a strain to failure value of 0.042, as shown in Figure 5. The modulus of toughness is highest for the 8-cell with a value of 44.12 kJ m À3 , which is 70.48% greater than that of the gyroid. The relatively high toughness of the 8-cell is a consequence of the high Young's and plasticity moduli, while the gyroid has the second highest toughness due to the sample failing at high strains, rather than that due to it having a high modulus. The rest of the experimental samples have toughness values that are within the range of 11.56 and 17.45 kJ m À3 .
To compare the performance of the 4-polytope-based metamaterials against other metamaterial structures, performance indices, namely normalized Young's modulus (E=E 0 ), normalized strength (σ=σ y ), and tensile strength/Young's modulus (σ t =E 0 ) were plotted against the relative density of the structures in Figure 5a-c, respectively. Here, E 0 and σ y are Young's modulus and yield strength of the constituent material, respectively. The figure also includes the data for G shellular and P shellular structures tested axially in compression rather than tension presented in the research publications by Akbari et al. [15] and Nguyen et al. [17] . Additionally, the plots are overlaid with the generic performance ranges of honeycombs in compression (grey), foams (light blue), and natural materials (light red) using the data made available in ref. [17]. As shown in Figure 5a, 3D-projected 4-polytopes tend to have higher normalized Young's modulus values when compared against the P shellular structures with the datapoints for 5-, 8-, 16-, and 24-cell structures all situated above the predicted P shellular performance, as illustrated by the red dashed line spanning across the full range of relative densities. When compared against the G shellular structures, also referred to as gyroids and indicated in this figure by the yellow dashed line, all 3D-projected 4-polytopes (with the exception of the 24-cell) exhibit higher normalized Young's modulus values than that are observed for G shellular structures. It can be noted that in general, the 4-polytope-based structures tend to lie closer to the generic honeycomb range with 5-, 8-, and 16-cell having relative densities in the range of 12.85% to 20.69% while the relative density of the 24-cell is 5.45%. The overall best-performing structure is the 8-cell, showing the highest values of normalized Young's modulus. Figure 5b summarizes the performance in terms of normalized strength for the 3D-projected 4-polytope metamaterials, gyroid structures, and honeycomb structures tested in tension, and for P shellular [17] structures tested in compression. Similarly to the plot shown in (a), Figure 5b also includes the generic performance ranges for honeycombs in compression, foams and natural materials. Since there can be significant observable differences between the strengths of mechanical metamaterials in compression and tension, we do not include P shellular or G shellular in compression in this plot. This is evidenced in Figure 5b by the noticeable differences between honeycombs in compression and those tested in tension, a scaling law is shown by a dotted dark blue line to represent the honeycomb in tension. A point of note here is that 8-and 16cell structures in tension lie on the scaling law line for honeycomb in tension, in terms of both strength and relative density. However, the 24-cell and gyroid in tension lie above the line, indicating that when compared against their relative densities, these structures are stronger in tension than honeycombs, and 8-, 16-, and 5-cell 3D-projected 4-polytopes. Figure 5c shows the tensile strength normalized by Young's modulus of fully cured 3D printing resin, and is plotted against the relative density of the experimental samples summarized in Table 4. This normalization approach allows for the visualization of the strength gain due to the metamaterial structure, rather than its constituent material properties. [47] The structure with the highest tensile strength to modulus ratio is the 8-cell indicating that the tesseract arrangement is highly effective for applications requiring high tensile stiffness and strength, as previously shown using stiffness and tensile strength values (cf. Table 3). In a descending order of performance, the 8-cell is followed by the hexagonal honeycomb, 5-cell, 16-cell, and gyroid structures with the 24-cell having the lowest strength gain as a result of its architecture. The figure also includes trendlines showing scaling laws calculated using the methodology presented in ref. [48], σ t E 0 ¼ ð ρ ρ 0 Þ n , where σ t is the tensile strength; E 0 and ρ 0 are Young's modulus and density of the constituent material, respectively; ρ is the density of the sample; and n is a positive integer empirically found for each structure. The values for n are 3.92, 4.01, 3.61, 2.76, 3.22, and 3.83 for 5-cell, 8-cell, 16-cell, 24-cell, gyroid, and honeycomb, respectively. Figure 6 shows the representative failure modes observed in each tensile specimen including the 3D-projected 4-polytopes as well as gyroid and hexagonal honeycomb structures. The most common failure mode for the 5-cell structure is within the middle section of a single unit cell (marked in red dashed line in Figure 6) with the crack forming in a direction perpendicular to the loading axis and extending through the inner as well as outer thin-walled structure of the unit cell. The failure path occurs at the location of the unit cell that was predicted to have the highest localized elastic strain energy density, which is the middle section of the unit cell, as shown in Figure 2. Similar behavior is observed for the 8-cell structure with the failure occurring in the center of the unit cell and the crack extending perpendicularly to the loading axis. The 16-cell failure appears at the interface between the neighboring unit cells with the weakest line following the valleys along the unit-cell boundary, in line with the simulation predictions shown in Figure 2. The 16-cell has the two regions with the highest concentration of strain energy, which is at the boundary of the unit cell where the experimental tensile samples tend to fail, and on the outer walls in the middle of the unit cell. The experimental results show that the weakest part of the 24-cell is within the slender tapers conjoining unit cells. The failure occurs within the unit cell rather than at the boundary between the neighboring cells with the failure path forming around the drain hole located at the tapered conjoins. This location was also predicted to have the highest localized strain energy density within the 24-cell as shown in Figure 2. Failure within the gyroid structure appears along the 45°shear plane with the crack forming a long failure path by spiralling . Experimental results for 4-polytope metamaterials, gyroid, and honeycomb samples compared against P and G shellular structures. [15,17] The figure represents a) normalized Young's modulus (E=E 0 ), b) normalized strength (σ=σ y ), and c) tensile strength/Young's modulus (σ t =E 0 ) values plotted against relative sample density. G shellular results are only shown in (a); generalized performance indices for honeycombs, foams, and natural materials are shown in (a) and (b) while scaling law trendlines for 4-polytope metamaterials, gyroid, and honeycomb samples are included in (c). All the data for P and G shellular structures and generic performance indices for foams, honeycombs, and natural materials were obtained using axial compressive rather than tensile testing.
www.advancedsciencenews.com www.aem-journal.com within the internal gyroid geometry. Lastly, the hexagonal honeycomb structure fails along the perpendicular plane at a random location within the tensile sample. The representative failure modes indicate that the 3D-projected 4-polytopes tend to fail along the plane perpendicular to the axis of loading, which is expected considering the cubically symmetrical nature of these unit cells. It is also worth noting that apart from the 16-cell, all other 4-polytope projections failed within the unit cell rather than at the boundary where the neighboring unit cells are conjoined. The experimental results are in excellent agreement with the simulation outputs as the failures occur within the areas of each unit cell where the elastic strain energy concentration tends to be the highest. Moreover, this represents the overall suitability of the manufacturing approach presented in this work regardless of the manufacturing limitations previously discussed in this section.

Conclusions
In this article, we designed and optimized novel metamaterial structures under tension using 3D projections of 4D geometries (4-polytopes) as a basis structure. This is a new class of parametrically optimized cubically symmetrical mechanical metamaterials and we showed that they exhibit superior properties of specific stiffness and strength when compared to more conventional structures such as gyroids and honeycombs. While gyroids are also cubically symmetric, each of the four 3D-projected 4-polytope metamaterials outperformed the gyroid in terms of both specific stiffness and specific yield strength. Under the optimization framework, the specific stiffness properties of the initial 4-polytope projections were improved by 122%, 72%, 163%, and 469% for 5-cell (pentatope), 8-cell (tesseract), 16-cell (orthoplex), and 24-cell (octaplex) metamaterials. However, not all 4-polytope projections yielded promising final properties. In addition to the gyroid, the experimental results were also benchmarked against the well-known and commonly used hexagonal honeycomb structure tested in the out-of-plane direction. The optimized 8-cell (tesseract) structure exhibited a higher specific yield strength than the honeycomb and a marginally lower average specific stiffness. Our results demonstrate that by coupling evolutionary-algorithm-based optimization methods with parametric design, we can enhance the mechanical performance of mechanical metamaterials without compromising mass. While the focus of this paper is specific stiffness and strength under tensile loading conditions, the design and optimization framework presented here can also be used to optimize for a range of other mechanical properties under different loading conditions.

Experimental Section
Design of 3D-Projected 4-Polytope Unit-Cell Structures: The four metamaterial architectures presented in this paper were developed using regular convex 4-polytopes as baseline geometries. The specific 4-polytopes used were 5-cell (pentatope), 8-cell (tesseract), 16-cell (orthoplex), and 24-cell (octaplex), and each is shown as a Schlegel diagrams in Figure 7a. The Schlegel perspective reduced these 4-polytopes from 4D to 3D and thus enabled the visualization of these geometries as projections in 3D space. Each of the wireframe structures in Figure 7a were then used to create a single metamaterial unit cell by using the edges and vertices of the wireframe as the contours for the thin-walled structures shown in (b) and (c), as semitransparent and solid unit-cell models, respectively. As the wireframes only presented points in 3D space, rather than a solid-body geometry, the thin-walled features were created to closely follow the boundaries of the projected 4-polytopes. In a similar manner to the wireframe representations, the thin-walled unit cells maintained geometrical cubic symmetry and all associated symmetry planes. This approach to metamaterial design provided flexibility such that solid features within the unit cell could be easily adjusted while following the silhouette of the projected 4-polytope structures. To exploit this flexibility, the solid features were parametrized as shown in Figure 8 and annotated in Table 5 to adjust the overall geometry of a unit cell and allow the generation of different thin-walled structures that are developed based on the same 4-polytope projections. This parameter adjustment also enabled changing of the perspective depth while still abiding by the geometrical definition of 4-polytopes and maintaining the same symmetry planes within the unit cell. For example, the change in the perspective depth for an 8-cell unit cell enlarged or shrank the geometrical primitive, the inner cube, within the structure. As the result, this affected the mechanical properties of the structure while also allowing the generation of an infinite number of different unit cells stemming from the same 4-polytope. Additionally, to ensure the manufacturability of the designs using a low-forcestereolithography (LFS) method, drain holes were introduced to allow the resin to circulate within the structure as shown in Figure 8. The external www.advancedsciencenews.com www.aem-journal.com dimensions of the unit cells were constrained by a bounding box of a cubical shape with an edge length of 15 mm and these were kept the same for each of the generated designs. This ensured that only the internal geometry of a 3D-projected 4-polytope structure could be altered, while the boundary conditions, shown in Figure 9, were kept constant. As such, regardless of the geometrical variations through optimization, unit cells could still be stacked together with the bounding boxes of each unit cell forming a regular square tessellation. An assembly of the unit cells stacked in the aforesaid linear manner ensured that each metamaterial block had the same properties along the three principal axes of a metamaterial array. Simulation Setup: Computational-unit-cell models were developed for each of the 3D-projected 4-polytope structures using a commercially available FE analysis package (Abaqus by Dassault Systémes) to determine the elastoplastic mechanical behavior and properties of the structures under tension. Standard (implicit) solver was employed for this problem. The material properties used in the models are representative of the coupon samples manufactured using Formlabs Clear V4 fully cured resin and tested in tension with Young's modulus value of 1.93 GPa, Poisson's ratio of 0.38, and density of 1.164 g cm À3 . A yield strength, σ y , of 59.7 MPa is reached under loading and the material experiences a strain-softening phase, which is defined as σ y ¼ 34.83 Â ε À0.142 y , where ε y is yield strain. As all of 3D-projected 4-polytopes analyzed in this paper have cubic symmetry, only one quarter of each unit cell was modeled to make the simulations more efficient and to reduce the computational time. To achieve that, symmetry boundary conditions were prescribed to the inner X (orange dashed line) and inner Z (red dashed line) planes as shown in Figure 9. To simulate tensile loading, the bottom of each unit cell was set as an encastre boundary condition and the top was prescribed a tensile displacement at an equivalent tensile strain of 0.04. To represent the effect of neighboring unit cells deforming in the same manner, another set of symmetry boundary conditions was used at the outer surfaces  www.advancedsciencenews.com www.aem-journal.com (parallel to the X and Z planes) of the quarter unit cells. The structures were discretized using a free (unstructured) meshing technique as it was found to be more reliable for auto-generated structures rather than swept and structured meshes. Tetrahedral meshing elements (C3D10) were used and the mesh density was chosen following a mesh convergence (performed for each of the four 3D-projected 4-polytope structures). A mesh growth rate of 1.05 was selected to discretize the features using evenly sized elements. Additionally, a remeshing condition following the parameters described here was introduced to address any meshing difficulties arising due to the automated generation of the 3D-projected 4-polytope structures during the optimization stage. The edges of each structure were seeded to maintain a minimum constraint of three elements across the thinnest feature while the element number was set to increase with increasing feature size. Such an approach was found to be effective in evaluating the through-thickness response of the thin-walled structures while maintaining an optimal mesh size that would not compromise computational results. The implicit solver utilized the Newton-Raphson method and was chosen as it enabled the use of   Figure 4, adapted from ref. [40]. 5 Figure 9. Summary of the boundary conditions applied to 3D-projected 4-polytope metamaterial quarter unit-cell models (left side) and the full unit cell illustrations (right side). The outer surfaces of each model were prescribed symmetry boundary conditions about the X and Z planes located at the interface between the neighboring cells. Orange and red dashed lines on the quarter models (left side) mark the inner X and Z symmetry planes within the unit cells, respectively, which were prescribed symmetry boundary conditions. An encastre BC was used at the bottom and tensile displacement BC resulting in 0.04 strain was applied to the top of each quarter model.
www.advancedsciencenews.com www.aem-journal.com bottom of the unit cell (location of the encastre boundary condition (BC) in Figure 9). Parametric Optimization: The 3D-projected 4-polytope geometries could be modified in an automated manner by parametrically manipulating the variables mentioned in Table 5. This was achieved by defining the geometrical features of the CAD models in a Python script and having parametric design variables as inputs to the code that can be linked to the optimization algorithm and automatically iterated. As a result, this enabled further exploration of the design space by means of an optimization algorithm. To explore this space in an efficient manner, manufacturing constraints were incorporated which were namely 1) minimum wall thickness and 2) minimum drain hole size. The former was limited by the peel-off force required to remove the sample from the resin tank after a new layer is 3D printed, while the latter was constrained by the viscosity of the resin and was required to ensure acceptable drainage of the hollow chambers within the 3D-printed structure. Additionally, the design variable arrangements resulting in suction cups, and concave features increasing the peel-off force and causing the prints to fail, were also omitted from the computational analysis. The parametric design variables listed in Table 5 were given upper and lower variable boundaries by specifying the range in which each parameter could be adjusted by the optimization algorithm. The starting parameter values for the initial unoptimized design were set to the midrange value of the specified variable adjustment range. Individual steps involved in producing tensile 4-polytope samples and the overall manufacturing approach are further discussed in Experimental Section.
The 4-polytope-projections, namely the 5-, 8-, 16-, and 24-cell, were optimized to achieve the highest specific stiffness value E ρ , where E is Young's modulus of the unit cell in tension and ρ is the apparent density of the unit cell. This was possible through the combination of a FE model of a unit cell, previously described in this section, with a parametric design optimization framework powered by a genetic algorithm (GA). Within this framework, parametric design variables served as the inputs to the unit-cell simulations while the outputs obtained from the FE analyses were fed to the optimization algorithm to enable exploration of the design space. When launched, the GA generated a set of inputs that were predicted to favor the objective function and used simulation outputs such as the strain energy (U ) and the total mass (m) of the generated unit cell as the indicators of success. This was repeated multiple times with different inputs until the structure with the highest specific stiffness was generated. The function describing this objective in the simplest form is shown as Equation (1), where σ is the first Piola-Kirchhoff stress, V is the total volume of the unit cell, and U e is the total elastic strain energy. A single objective GA was used instead of brute force methods to perform informed predictions of favorable input parameters based on stored outputs from preceding simulations. The objective function defining the specific stiffness was calculated using the elastic strain energy output (U e ) obtained from a single-unit-cell simulation. This is shown in full in Equation (2), where V e is the volume of a mesh element, and n is the total number of mesh elements while σ ij and ε ij are the stress tensor and the elastic strain tensor of a mesh element, respectively. This procedure was run iteratively to evaluate a wide range of metamaterials and correlate individual design parameters to their effect on the final specific stiffness À E ρ Á of the structure. The algorithm then used this information to select the best performing structures in tension and used crossover and mutation operations to generate a new population of structures. This execution loop was run until one of the stopping conditions was met and a near-optimum design solution was generated with the highest specific stiffness À E ρ Á as compared with all structures evaluated in the simulation. A GA flow diagram is shown in Figure 10.
The parameters chosen for setting up the GA and used in the optimization framework are summarized in Table 6. The computational complexity of the problem, the availability of parallel computing power and the number of 4-polytope parametric design variables (genes in GA) were taken into account to choose a population size of 28. To cap the maximum optimization cycle running time, the absolute number of generations was chosen as 50. Additionally, if a better solution was not found in the last 20% of the total generations executed, the cycle was stopped to save computational time. The optimization cycle for each of the unit cells reached the stopping criteria under 532 design iterations (raw optimization data was made available through the Edinburgh Data Share website). Crossover and mutation probabilities were chosen as 0.9 and 1=q, respectively, where q is the number of parametric design variables in the metamaterial design. Crossover and mutation distribution indexes were set to www.advancedsciencenews.com www.aem-journal.com 10 and 20, respectively. The algorithm parameters were chosen to suit the generic optimization problem based on ref. [49] and therefore parameter tuning was carried out as it was considered beyond the scope of this publication.
Manufacturing: The experimental samples were manufactured using LFS 3D printers from Formlabs. A photoreactive thermosetting Clear V4 resin was used to manufacture the tensile samples using the arrays of the unit cells previously discussed in the Experimental Section. The cells were stacked to create a systematic array of 2 Â 2 Â 4 with 16 unit cells in total and external sample dimensions of 30 Â 30 Â 120 mm. As shown in Figure 11, the unit cells in the middle section of the tensile sample ("test unit cells") are representatives of the 3D-projected 4-polytope geometries designed using the optimization framework, while the cells adjacent to the middle section ("transition structure") have artificially higher wall thicknesses in comparison to the central unit cells. The ends of the sample were further reinforced by filling in the cavities with a two-part epoxy (RS Epoxy 406-9592) to ensure that the strength of the "gripping area" was sufficient to withstand the grip forces applied by the tensile testing machine. Such a design approach with three different areas within each tensile sample was chosen to mimic the "dog bone" geometry of tensile coupons and hence maximize the chances of failure occurring within the central part of the specimen. The 3D printer layer height was set to 25 μm. Following the manufacturing process, the samples were washed with isopropyl alcohol in a Form Wash, to ensure the removal of as much excess resin as possible prior to post-curing of the samples (30 min at 6°C in a Form Cure UV chamber). As suggested by the manufacturer, the samples were post-cured to increase cross-linking within the polymer and hence increase the material quality of the 3D-printed samples. All of the manufactured samples were found to have high dimensional accuracy with deviation between the CAD models and the experimental samples being recorded at less than 0.42%. The mass variation, Δw, was found to be 0.1162, 0.0095, 0.1267, and 0.1180 for 5-, 8-, 16-, and 24-cell, respectively, and was calculated as Δw ¼ w sample w CAD À 1 , where w sample is the mass of the 3D-printed sample and w CAD is the mass calculated using the volume of the CAD models and the density of the cured Clear V4 resin. The main reason for variations in the sample mass when compared to the CAD models was found to be due to leftover uncured or partially cured resin within the samples. As the internal geometry of the unit cells was highly complex and the majority of the features were relatively small, some of the resin was inaccessible and could not be removed during the washing stage. As such, this residual resin cures, fully or partially, during the post-curing stage.
To minimize the extent of residual resin without affecting the mechanical properties of the samples, the washing procedure was adjusted for each specimen type. In addition to the metamaterial samples, sets of gyroid and hexagonal honeycomb tensile samples were manufactured in an identical manner to carry out an experimental comparison of the mechanical properties of these more common structures against the 3D-projected 4-polytope structures. A sinusoidal wave with an amplitude of 3.75 mm and a period of 30 mm was used for the gyroid structure, while the wall thickness was chosen as 0.85 mm. The apparent density of the "test unit cells" within the gyroid tensile sample was 108.82 kg m À3 while the relative density was 9.34%. The honeycomb tensile samples were made using hexagonal unit cells with a diameter of 8.5 mm and a wall thickness of 0.55 mm which Table 6. Summary of the parameters used in genetic algorithm setup, adapted from ref. [40]. q is the number of parametric design variables (genes). Figure 11. Tensile specimen geometry designed to recreate the tensile behavior of a "dog bone"-shaped coupon (black solid line). Main specimen areas: gripping (in purple), transition structure (in orange), and dedicated test unit cells (red dashed line). www.advancedsciencenews.com www.aem-journal.com resulted in "test unit cells" with apparent and relative densities of 192.89 kg m À3 and 16.56%, respectively. The hexagonal geometry was aligned so that the out-of-plane direction of the cells was in line with the tensile loading direction, to obtain out-of-plane properties of the honeycomb sample. Both, the gyroid and the hexagonal honeycomb tensile samples, were designed to have dedicated testing, transition, and gripping areas as shown for the 4-polytope metamaterial samples in Figure 11. Representative tensile specimens of all of the 3D-printed metamaterial structures, gyroid structures, and hexagonal honeycomb structures are shown in Figure 12.
Mechanical Testing Procedure: An Instron 8802 servo-hydraulic test machine mounted with a 250 kN load cell and tensile grips for samples up to 32 mm in diameter was used for testing specimens in tension. An Imetrum 2D digital image correlation system (resolution of 1400 Â 1000 at 17.8 fps) was used for taking two axial strain measurements across the four outermost faces of the unit cells situated in the middle of 2 Â 4 metamaterial array within the tensile specimen. This 2D image correlation technique allowed for measurements to be taken in the axial direction, which were then averaged to obtain the final strain measurement values. Five specimens of each 3D-projected 4-polytope metamaterial, gyroid, and honeycomb structure, manufactured in the same manner, were tested in tension at a ramp rate of 10 mm min À1 to obtain experimental results. This ramp rate was chosen to ensure Hookean behavior of the cured resin under tensile deformation.

Supporting Information
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