Pixel design method for deformable structures based on gyroid and topology optimization

Triply periodic minimal surface (TPMS) is a class of periodic surface structure in three principal directions, which is infinite and non-self-intersecting. In the mathematical description, the mean curvature of TPMS is always zero, which means that the sum of the principal curvatures of each point on TPMS is zero. In addition, any sufficiently small patch taken from the TPMS has the smallest area among all patches manufactured within the same boundaries.^1–3 The first examples of TPMSs were presented by Schwarz in 1865,⁴ followed by his student Neovius in 1883.⁵ They described five types of TPMS. These structures were named as Schwarz primitive (P), Schwarz diamond (D), Schwarz hexagonal (H), Schwarz crossed layers of parallels, and Neovius (N). Then, gyroid (G) surface and more kinds of TPMSs were presented by Schoen in 1970.⁶ A series of research focused on the special performances and properties of some disciplines, such as mechanics,^7–9 optics,^10,11 thermology,^12,13 and chemistry.^14,15 Compared to other structures, the surfaces of TPMS are very smooth, which has no sharp edges or junctions as the conventional lattice structures.^16,17 Among TPMSs, gyroid is the most famous one, and several experimental and numerical studies investigated the mechanical properties of gyroid structures.^18–23 A better performance on energy absorption and compressive strength is demonstrated, indicating that gyroid structures hold promise as a multi-functional structure for a wide range of applications.^24–27 In addition, the development of additive manufacturing (AM) technology improves the application value and reduces the difficulties of fabricating TPMSs. Most of the current TPMS structures are fabricated by AM methods.^28–31 

These studies mainly focus on the impact of periodic arrangement. The method for structure optimization is rarely mentioned in them. Other researchers focused on the optimizing methods based on the cellular structures of TPMSs. Topology optimization is one of the hotspots in current research works.^32–44 These proposed methods focus on the stiffness adjustment and optimization of structures composed of TPMSs, which enhance the stability of structures under specific loadings. The deformation of the structure in these cases is not the emphasis of this research. However, studying the relationship between stiffness distribution and deformation, and further, using special stiffness distribution to guide deformation and achieve expected deformation effect has significant value in research and application, such as functionally graded material⁴¹ and programmable soft robotics.⁴² In general, large deformations may lead to structural failure of plastic materials. It is necessary to use hyperelastic materials^43,44 to prevent structural damage during loading. These aspects of research are rarely mentioned in studies about TPMSs.

In this work, we focus on the relationship between stiffness distribution and deformation in structures. Design method on controllable deformation of deformable structures is also discussed. The gyroid lattices with different relative densities are carried out, and a database containing lattices with variable stiffness is established. The hyperelastic UV-cured material is used to manufacture the soft gyroid lattices. Variable permutations of the gyroid lattices are generated, and their deformation characteristics under compression are analyzed to prove that different lattice combinations can lead to different deformation modes of the structures. According to the previous discussions, a pixel design method for deformable structures based on soft gyroid lattices and topology optimization is proposed. This method can be used to design functionally graded or programmable deformable structures.

In this equation, X = 2απx, Y = 2βπy, Z = 2πγz, α, β, and γ are constants related to the unit cell size in the x, y, and z directions, respectively. When this surface is infinite and periodic in 3D, it is referred to as a TPMS. As shown in Fig. 1(a), examples of TPMS lattices built-in solid-networks are provided. These lattices, which we used in this article, are all generated in MSLattice.⁴⁵ The solid-network TPMS lattices can be created based on the zero-thickness minimal surfaces. The volumes are divided by the minimal surface as the solid domain and the other as the void domain. This is done by considering the volume bounded by the minimal surface such that φ(x, y, z) > c or φ (x, y, z) < c to create a solid-network lattice.

Then, the lattices with the relative density φ of 20%, 40%, 60%, and 80% can be obtained as shown in Fig. 1(b). To analyze the mechanical properties of these gyroid lattices, the homogenization method is used through experimental compression test.

According to the stiffness gradient controlled by the relative density of the gyroid lattice, the specific stiffness distribution of a structure can be obtained by adjusting the composition of the gyroid lattices, which are filled in the structure. Then, the stiffness distribution is able to affect the conduction of stress in the structure during compression process. Macroscopically, a specific deformation mode is generated on the whole structure. To validate the predictions, we use four gyroid lattices with a size of 10 × 10 × 10 mm³ to construct a square structure measuring 20 × 20 × 10 mm³. The compositions of the lattices are shown in Fig. 2(a). The different composition modes, uniform composition (UC) and random composition (RC), are manufactured in STL. file format. In order to predict the deformation behavior, the finite element analysis (FEA) was carried out using Abaqus dynamic explicit code (Dassault Systemes, France), as shown in Fig. 2(b). The models are meshed using improved ten-node tetrahedron elements (C3D10M), and two rigid faces are set above and below the model, which is simulated to be compressed. The two faces are meshed with rigid elements (R3D4). The mesh details of the composition models are shown in Table I. One of the faces is set to fixed constraints, and the other face is set to displace in the y direction. The material is set using the five-constant Mooney–Rivlin model for hyperelasticity. The values of the constants for the model are shown in Table II. In addition, the standard quasi-static uniaxial compression test is carried out to verify the accuracy of simulation. The setup of the test is shown in Fig. 2(c).

The results of the FEA (finite element analysis) simulation and the compression tests are shown in Figs. 2(d) and 2(e). It can be observed that similar deformations of the models occurred in simulation and experiments. Due to the uniform distribution of the material in the UC model, the deformation of the model is symmetrical during compression. While the stiffness distribution is inhomogeneous like the RC model, the deformation is asymmetric. As V₀ increases from 2 to 6 mm, the model is compressed in the vertical direction. Meanwhile, a tilt to the right has occurred on the model. According to these results, the impact on the stiffness distribution of the whole structure is relatively obvious by adjusting the composition of the gyroid lattices, which can macroscopically alter the deformation of the structure.

According to the results of our previous study,⁴⁶ the average normalized stiffness coefficient (ANSC) distribution in the design domain Ω can be obtained by pixelating the result of the material volume factor θ distribution obtained by topology optimization. This process is achieved by calculating the mean value of θ within each pixel block. The variable-stiffness gyroid lattices with a hyperelastic material are used to fill the structure based on the ANSC distribution.

The flow chart of the proposed method is shown in Fig. 3. The gyroid lattice and its mechanical properties have been discussed. The lattice database is established based on the gyroid lattice with varying relative density. In addition, the experimental results prove that using lattice arrangement to guide the deformation of the deformable structure manufactured by a soft material is feasible. Then, we focus on the topology optimization related to deformable structures and their performance on local and global deformations. According to the ANSC distribution, the gyroid lattices can be filled into the design domain and the remapped deformable structure can be obtained. To validate this prediction, topology optimization is performed on various continuous structures for specific objectives and applications.

The uniaxial compression tests are performed on the specimens of the cloaking structure filled with the gyroid lattices base on topology optimization (with cloaking) and the structure filled with the same relative density (φ = 60%) lattices (no cloaking), respectively. The setup of the tests is designed as shown in Fig. 4(b). The loading force is applied on the specimens gradually. A home-made deformation detection sensor is used to measure the deformation degree of the specimens. The sensor is composed of a bending beam, a strain gauge, and a signal convertor, as shown in Fig. 4(c). The strain gauge is pasted on the lower surface of the bending beam. Under compression, the lower surface of the bending beam and the strain gauge will bend downward, which makes the resistance of strain gauge change. Furthermore, the signal convertor transforms the value of the resistor to voltage signal directly. The voltmeter is used to detect the variation of the voltage signal. The voltage measured increases with deformation of the specimens. The experimental results are shown in Figs. 4(d)–4(f). It can be seen that while the loading force reaches 62.8 N, the deformation of the structure without cloaking design measured by the sensor begins to increase rapidly. While the cloaking structure is under the same loading, the strain is almost unchanged, and after the loading force reaches 93.7 N, the measured deformation begins to increase. Compared with the no cloaking structure, the deformation of the objective domain in the structure with cloaking design increases more slowly when the loading force becomes larger. This phenomenon proves the performance of the cloaking design with the proposed method.

The complete compliant plier is manufactured by 3D printing as shown in Fig. 6(a). A compression test is conducted on the obtained plier to assess its deformation performance. The setup of the test is shown in Fig. 6(b). The force F is gradually applied to the handle of the plier, and the distance of jaw X is measured by the calipers. The test results are shown in Fig. 6(c). The value of X decreases by 20% under a loading of 34.7 N. This plier can be used to grip objects with diameters smaller than the jaw. The pen, glue bottle, and paint tube are gripped up as shown in Fig. 6(d).

In conclusion, a design method for controllable deformation and functional grading of deformable structures is focused on. The homogenization method is used by a compressing test, and an obvious gradient of the stiffness is formed between two lattices generated with different relative densities. A database that contains soft gyroid lattices with variable stiffness is established. Variable permutations of the gyroid lattices are fabricated with the hyperelastic material. Their elastic characteristics under compression are investigated and validated through FEA simulation and experiments. The experimental and simulation results are anastomotic and prove that different lattice combinations can lead to different deformation modes of the structures. Based on the discussions herein, a pixel design method for deformable structures based on soft gyroid lattices and topology optimization related to deformation is proposed. Two applications, a cloaking metamaterial, and a compliant plier are designed by the proposed method and tested. According to the results, deformation of the measured point on the cloaking structure decrease by 93.1%, 84.6%, and 81.2% under loading forces of 100, 120 and 140 N compared to the structure without a cloaking design. In addition, the weight of the structure is reduced by 13.2% due to the design. For the designed compliant plier, the distance of the jaw decreases by 20% under a loading of 34.7 N, which is useful to grip objects smaller than the jaw. These results demonstrate the practicality and feasibility of the proposed method.

This work was supported in part by the National Natural Science Foundation of China under Grant Nos. 52275009 and 52207038, in part by Postgraduate Research & Practice Innovation Program of Jiangsu Province under Grant No. SJCX23_0062.