Issue 
Int. J. Simul. Multidisci. Des. Optim.
Volume 15, 2024



Article Number  12  
Number of page(s)  14  
DOI  https://doi.org/10.1051/smdo/2024011  
Published online  14 August 2024 
Research article
Topology optimization design of microstructures with zero Poisson's ratio
State Key Laboratory of Subtropical Building and Urban Science, School of Civil Engineering and Transportation, South China University of Technology, Guangzhou 510640, People Republic of China
^{*} email: ctpwei@scut.edu.cn
Received:
1
March
2024
Accepted:
14
May
2024
Mechanical metamaterials are materials that possess unconventional mechanical properties that are not found in homogeneous materials, achieved through specific artificial microstructures. Topology optimization is an effective design method for such materials. In this paper, a topology optimizationbased method for designing zero Poisson's ratio mechanical metamaterials is proposed. Firstly, a zero Poisson's ratio topology optimization objective function is constructed based on the energy homogenization method, and the optimal microstructure topology configuration and elastic coefficient matrix under different initial topologies and volume fractions are obtained through the boundary density evolution topology optimization method. To improve the saw tooth effect of the boundaries, the node density level set method is used to smooth the boundaries of the microstructure. Then, in finite element simulation analysis, it is demonstrated that the proposed method can effectively design microstructures with zero Poisson's ratio properties. It is also shown that microstructures with expected stiffness and zero Poisson's ratio properties can be obtained by changing the volume fraction and selecting materials with different stiffnesses. Finally, the unit cell is periodically arranged to form a multilattice metamaterial, and its zero Poisson's ratio mechanical performance in both X and Y directions is verified.
Key words: Boundary density evolution / topology optimization / zero Poisson's ratio / microstructure / metamaterial
© X. Li et al., Published by EDP Sciences, 2024
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
As industrial products become more diverse, complex, and highperformance, traditional structural design can no longer meet the various needs of industrial products. Topology optimization, a novel design method, has become a common approach in the field. The theory of topology optimization design of microstructure unit cells in composite materials was first proposed by Sigmund in the mid1990s [1]. It is an effective tool for seeking the optimal material distribution in the given design space. With breakthroughs in computer technology and the continuous improvement of mechanics theory, various topology optimization methods have gradually developed and become one of the leading design methods for supermaterial structures. Based on the idea that “structure determines performance” in materials science, supermaterials, a particular type of material, can obtain properties that surpass those already existing in nature through macro or microscale structural design [2,3]. As a new branch of research in the field of supermaterials, mechanical supermaterials try to obtain unconventional mechanical properties by employing specific artificial microstructures, such as negative Poisson's ratio [4,5] and zero Poisson's ratio materials [6]. Traditional materials usually exhibit positive Poisson's ratio effects, as shown in Figure 1a, where the material contracts transversely by a certain proportion under tension. However, for zero Poisson's ratio materials, as illustrated in Figure 1b, do not shrink or expand in another direction when stretched, making them widely researched due to their unique properties. Aerospace and defense industries, for instance, require the development of new materials with specific properties.
In Figure 1, L and Lʹ denote the length of the material before and after deformation, B and Bʹ denote the width of the material before and after deformation, and N denotes the axial tensile force on the material.
Olympic et al. [7] first designed a zero Poisson's ratio hexagonal honeycomb structure in 2010, derived the expressions of Young's modulus and shear modulus equivalent to the structure through theoretical analysis, and analyzed the effect of honeycomb parameters on its onedimensional mechanical properties. Grima et al. [8] improved the existing hexagonal honeycomb structure in 2010 by proposing a semi concave honeycomb structure, and verified its zero Poisson's ratio characteristics through finite element analysis. Lu [9] and Li et al. [10] analyzed the equivalent elastic modulus and shear modulus of the honeycomb material proposed by Olympic et al. [7], derived the theoretical calculation formula for the equivalent elastic modulus in the plane of the zero Poisson's ratio honeycomb core, and compared the calculation results with the finite element experimental results to verify the correctness of the formula. Simone et al. [11] designed a hybrid zero Poisson's ratio honeycomb structure in 2017 by alternating orthogonal hexagonal and concave hexagonal honeycomb structures. Through compression tests, it was found that the compressive strength of the structure is twice that of the original structure, under the condition of constant inplane stiffness. Gong et al. [12] proposed a quadrangular starshaped zero Poisson's ratio honeycomb structure that can realize deformation in two orthogonal directions. Cheng et al. [13] classified cells based on the type of straight arms and proposed a cross shaped mixed zero Poisson's ratio honeycomb cell, and analyzed the mechanical properties of the structure. Chang et al. [14] designed inverse from the nature of the material, first selected the value of Poisson's ratio of the microstructure, and then through the continuous optimization of the microstructure finally obtained the microstructure with the target Poisson's ratio. Hamzehei et al. [15] introduced a new 4D zero Poisson's ratio metamaterials fabricated through the 3D printing technology, and through the experiments of the new material and the traditional 3D folded metamaterials were compared concerning the energy absorption ability, which proved the mechanical properties of the new material. Chen et al. [16] built inplane negative Poisson's ratio and outofplane zero Poisson's ratio 3D structures based on 2D parallelogram honeycomb by orthogonal splicing and verified them by using finite element simulations and experiments. Gaal et al. [17] constructed a new zero Poisson's ratio structure by using two parallel rigid rods and flexible springs connecting the rigid rods. They discussed the possibility of extending the model to threedimensional structures with compression in any direction, and also discussed the advantages and disadvantages of these models.
From the analysis of the existing research results, it can be seen that there are many mature research works on zero Poisson's ratio structural design, but most of them are based on honeycomb structure, and there are fewer studies combining both topology optimization and zero Poisson's ratio microstructural design.
Clausen et al. [18] proposed a simplified design consisting of a set of parameter hyperellipsoids, where the length of each hyperellipsoid is controlled by the design point. Then, shape optimization was used to fine tune the hyperellipsoid design. A series of honeycomb structures with Poisson's ratios ranging from −0.8 to 0.8 were optimized, and experimental verification was conducted to verify that the structure maintains a constant Poisson's ratio even under large deformations; However, the drawback of this method is that it requires optimization of specific initial shapes, which has significant limitations. And topology optimization does not require any restrictions on the initial shape of the material, so it has a wider range of applications and is more in line with practical engineering needs. For example, Yang et al. [19] used a multi evaluation point functional primitive topology optimization method to design a zero Poisson's ratio structure. They defined two sub structures, positive and negative Poisson's ratios, in the same topology structure to achieve the design of a zero Poisson's ratio structure. Then, they verified the zero Poisson's ratio effect of the structure through simulation. The results showed that the Poisson's ratio of the function primitive configuration designed with maximum flexibility was closer to zero, but the zero Poisson's ratio effect of a combination structure with positive and negative Poisson's ratios was unidirectional.
In the field of topology optimization, one of the commonly used microstructure design methods is the SIMP method [20], based on which previous researchers have investigated microstructures with limiting elastic properties [21], negative Poisson's ratio microstructures [22], microstructures with maximized bulk modulus and maximized shear modulus [23], and so on. While using the SIMP method to directly design zero Poisson's ratio structures, the whole optimization process will be unstable, and additional constraints must be added or the algorithm must be improved. Huang et al. [24] used the SIMP method to take the original zero Poisson's ratio configuration as a prototype, divided the whole structure into two parts, the design domain and the nondesign domain, and designed a new zero Poisson's ratio configuration with the goal of volume minimization and the stiffness constraints as the condition, which improved the outofplane bending flexibility of the overall structure without reducing the transverse shear rigidity and bending stiffness, and the shortcoming of this method is that it is necessary to specify the initial configuration of the structure. Wang et al. [25] used the SIMP method and added weight factors to establish a multi stiffness topology model, and obtained a new zero Poisson's ratio honeycomb structure. It not only meets the inplane stiffness requirements for deformation, but also gives the structure a specific loadbearing capacity, that is, high out of plane bending stiffness. Due to the need to consider the sensitivity of horizontal changes and vertical loadbearing capacity, a unidirectional zero Poisson's ratio microstructure was designed. Based on the homogenization method and SIMP method, Chen et al. [26] designed topological box structures with positive and negative Poisson's ratio effects, and then combined the positive and negative Poisson's ratio box structures to design a zero Poisson's ratio microstructure. The advantage of this is that it avoids the oscillation or convergence problems caused by directly optimizing the Poisson's ratio to zero, and can better perform comparative research and mechanism analysis. The disadvantage is that the zero Poisson's ratio structure cannot be designed directly, and due to this special design method, the zero Poisson's ratio effect of the designed zero Poisson's ratio structure is also unidirectional.
In summary, research combining topology optimization with zero Poisson's ratio microstructure design is limited. Direct design of zero Poisson's ratio structures requires further constraints or improved algorithms, including previous studies using multivalued point function primitive topology optimization methods to design zero Poisson's ratio structures composed of positive and negative Poisson's ratio cell walls; Alternatively, the SIMP method can be used to design zero Poisson's ratio structures with new mechanical properties on the original zero Poisson's ratio configuration; And using weight factors to establish a multistiffness topology model for designing zero Poisson's ratio honeycomb structures; Alternatively, it is possible to design positive and negative Poisson's ratio box structures, and then design a zero Poisson's ratio microstructure by combining the positive and negative Poisson's ratio box structures. However, there are still some limitations, such as high dependence on the initial design and unidirectional deformation of the structure.
In this paper, a zero Poisson's ratio topology optimization objective function was constructed based on the energy homogenization method. By using the boundary density evolution topology optimization method, microstructures with zero Poisson's ratio properties under different initial topologies and different volume fraction conditions are obtained. Finite element analysis was used to verify that the method proposed in this paper can effectively design microstructures with bidirectional zero Poisson's ratio properties.
The section arrangement of this article is as follows: In Section 2, a theoretical model for topology optimization of zero Poisson's ratio microstructures is proposed based on the energy homogenization method and boundary density evolution method. Section 3 presents numerical examples of topology optimization of zero Poisson's ratio microstructures under different initial design domains and different volume fractions. In Section 4, finite element simulation analysis is conducted on the examples, indicating that the method proposed in this paper can effectively design microstructures with expected stiffness and zero Poisson's ratio properties.
Fig. 1 The tensile deformation state of materials with positive and zero Poisson's ratio. 
2 Topology optimization model for zero Poisson's ratio microstructures
2.1 Material interpolation model
In the topology optimization method based on element density, the element density is generally used as the design variable. At this time, the density corresponding to the material element is x_{e}, and Young's modulus of the element is usually defined by interpolation. In this paper, the same material interpolation model as the SIMP method [20] is used, and the elastic modulus of the element is expressed as:
where E_{0} is Young's modulus of the material, E_{min} is a very small Young's modulus assigned to the void region to prevent singularity of the element stiffness matrix, and p is the penalty factor used to improve the stability of the algorithm.
2.2 Periodic boundary conditions
Due to the periodic arrangement of microstructures, it is necessary to ensure the stress and displacement continuity of the microstructure unit cell boundary. Under the assumption of periodicity and given strain , the displacement field of the basic unit can be written as the sum of the macroscopic displacement field and the periodic fluctuation field :
However, in practical situations, since is unknown, the equation cannot be directly imposed on the boundary. This general expression needs to be transformed into explicit boundary conditions. Xia et al. [23] proposed a unified periodic boundary condition for microstructure unit cells with paired and parallel boundaries based on the small deformation assumption as follows:
in the above equation, k+ and k− represent the positive and negative directions along the y_{j} direction, respectively. Since the boundaries of the periodic microstructure unit cell are parallel, the of the two can be eliminated by subtracting them as follows:
for a periodic microstructure cell, is a constant in equation (4). Given , the value on the righthand side of the equation is a constant.
2.3 Material equivalent elasticity matrix
The homogenization method can establish the connection between the macro and micro scales and is the basis for the optimal design of microstructures. The basic idea of the homogenization method is given in Figure 2. It is assumed that the macrostructure is composed of composite materials whose equivalent properties can be calculated by the homogenization method [23].
In Figure 2, Γ represents the composite elastomer, f represents the body force, Γ_{t} represents the surface force applied, Γ_{u} denotes a given displacement boundary condition, x is a macroscopic variable, y is a local microscopic variable, and the local structure at a macroscopic point x can be viewed as consisting of a periodic arrangement of single cells.
According to the theory of asymptotic homogenization, the effective elastic tensor of periodic materials can be obtained as follows:
where Y represents the area of the design domain of the periodic microstructure cell, is the element test strain field under periodic boundary conditions, with two independent vectors. is the unknown periodic fluctuation strain field generated by applying the element test strain, which can be obtained by solving the following homogenization equation:
The energy homogenization method [24] uses the principle of average stress and strain. By directly applying the unit test strain to the boundary of the basic unit, corresponding to the superimposed strain field in equation (5) can be derived, and the equivalent elastic tensor of the material microstructure can be rewritten as:
In finite element analysis, the basic unit is discretized into N elements, and equation (7) can be approximately represented as:
where is the unit displacement solution for the corresponding unit test strain ϵ^{0 (kl)}, k_{e} is the element stiffness matrix. In the case of twodimensional structures, it can be noted that the subscript correspondence relationship is 11→1, 22→2, 12→3, and equation (8) can be expanded as:
where Q_{ij} is the strain energy of a single cell and can be expressed as:
are the sums of element mutual energies :
Fig. 2 Diagram of the idea of the homogenization method. 
2.4 Zero Poisson's ratio microstructure topology optimization model
This article presents a topology optimization design for materials with zero Poisson's ratio. According to the definition of Poisson's ratio, it can be calculated as follows:
If the above formula is directly used as the objective function, the entire optimization process may not be stable. Therefore, this paper proposes an equivalent expression for zero Poisson's ratio as follows:
where c is the objective function for optimization, and G_{f} is the iteration step.
During the entire optimization process, as the number of iterations increases, the values of and become smaller and smaller, while the values of and become larger and larger, which makes the Poisson's ratio of the microstructure closer to zero. Therefore, the mathematical model for topology optimization of zero Poisson's ratio microstructures can be expressed as:
where c is the optimization objective function, is the microstructure's equivalent elastic tensor, V(x_{e}) is the volume of the microstructure, V_{0} is the volume constraint, and x_{e} is the density of the element.
2.5 Sensitivity filtering and density filtering for boundary density evolution optimization method
To ensure the existence of a reasonable solution and avoid checkerboard patterns in topology optimization problems, some constraints must be applied to the optimization process. A common method is sensitivity filtering [27], which can be expressed as follows:
where N_{e} denotes the set of all elements within a straight line distance from the center of elements e within the filtering radius; γ is a lower bound given to avoid 0 in the denominator; and H_{ei} is a weight factor denoting the weight that each element i occupies within the filtering radius, defined as:
where Δ(e,i) represents the distance between the center points of element e and element i.
To improve the instability of the SIMP method in optimizing zero Poisson's ratio structures and to enhance computational efficiency, this paper adopts the boundary density evolution optimization method with boundary density hierarchical filtering [28], which can perfectly solve the problem of gray elements and converge quickly. The specific algorithm is as follows:
where a_{0} is an intermediate parameter, G_{f} is the density filter grading factor, is the updated design variable, loop is the number of loops, and a_{1} is the element number corresponding to the design variable less than a_{0} (loop) in the current loop step.
2.6 Optimal criterion algorithms
For optimization problems, such as the one presented in equation (17), the optimal criterion algorithm is typically chosen due to its stability and efficiency. The design variables are updated using the strategy outlined in reference [27]:
where m is the movement limit value, η is the numerical damping factor and B_{e} is determined by the optimal conditions as follows:
where the Lagrange multiplier λ must satisfy the volume constraint condition, and its specific value can be determined using the dichotomy algorithm.
The sensitivity of the objective function ∂c/∂ x_{e} can be calculated using the adjoint method [27]:
where k_{0} is the element stiffness matrix of elements with unit Young's modulus. The structure is uniformly meshed into finite elements and the volume v_{e} of each element is set to 1, therefore the sensitivity of the volume constraint to the design variable is:
2.7 Boundary smoothing
If finite element results are output directly, the boundary of the microstructure will appear jagged, which is not convenient for the subsequent 3D printing of metamaterials. It is necessary to smooth the boundary of the microstructure. This article mainly uses element density to convert to node density and then uses the node density level set to display the boundary of the microstructure. Node density can be obtained by interpolation from element density. Taking the structure of the four elements shown in Figure 3 as an example of a twodimensional plane problem, the formula for converting element density to node element density is as follows:
where x_{i}(i =1,2,3,4) represents the density of each element, and D_{i}(1,2,3,...,9) represents the density of each node.
The boundary of the microstructure can be defined by utilizing a node density level set as follows:
in which D_{i} represents node density, Ω represents the solid design domain, Γ represents the boundary design domain, D\Ω represents the hollow design domain, and S represents the level set value.
After using the node density level set to perform smoothing on the microstructure, there may be some errors between Poisson's ratio of the microstructure models and the original results. Therefore, it is necessary to finetune the level set values of the node density to obtain a microstructure model with a Poisson's ratio closer to zero. The details are shown in the example later.
Fig. 3 Example diagram of a fournode element in twodimensional plane. 
3 Numerical example of zero Poisson's ratio microstructure design
This section focuses on the topology optimization design of microstructures with zero Poisson's ratio. Specific examples are given for different initial topologies and volume fractions, and the influence of different initial topologies and volume fractions on the designed microstructures with zero Poisson's ratio is discussed.
3.1 Zero Poisson's ratio microstructure designs in different initial design domains
Assuming a material volume fraction of 0.5 and Young's modulus of 1 and Poisson's ratio of 0.3 for the solid material, and using a finite element mesh of 100 × 100, the microstructure is topologically optimized. Table 1 shows the optimization results for three different initial design domains.
It is easy to see from Table 1 that the initial design domain of the material has a certain influence on the topological structure of the final optimized structure. In the actual optimization process, there are many different local optimal solutions due to topological optimization, and there are also many possible results for the topological optimization of the microstructure. However, the resulting Poisson's ratio is close to zero in all cases.
Figure 4 shows the iteration optimization curves of Poisson's ratio for microstructures with different initial design domains. It can be seen from the figure that Poisson's ratio iteration curves of the microstructures corresponding to different initial design domains are different during the optimization process. However, as the number of iterations increases, the Poisson's ratio finally converges to a value close to zero, and a microstructure with zero Poisson's ratio characteristics is generated. It can be seen that when the iteration step reaches about 60 steps, the Poisson's ratio begins to remain around 0, reflecting the stability of the algorithm.
The optimization results for three different initial design domains.
Fig. 4 Iteration optimization curves of Poisson's ratio for microstructures with different initial design domains. 
3.2 Zero Poisson's ratio microstructure designs with different volume fractions
To further discuss the influence of volume fraction on the topological optimization results, the volume fractions are set to 0.4, 0.5, and 0.6, respectively. The optimization results are shown in Table 2.
It can be seen from Table 2 that the optimization results for different volume fractions are similar to those for different initial design domains, that is, the topological structures of the microstructures are different for different volume fractions, but their Poisson's ratio values are all close to 0.
Figure 5 presents the iteration optimization curves of Poisson's ratio for microstructures with different volume fractions. As can be observed from Figure 5, similar to Figure 4, the Poisson's ratio of the microstructure also stabilizes around 0 when the iteration steps reach around 60, further indicating the stability of the algorithm.
Optimization results of zero Poisson's ratio microstructures with three different volume fractions.
Fig. 5 Iteration optimization curves of Poisson's ratio for microstructures with different volume fractions. 
4 Experiment simulation and analysis
To validate the zero Poisson's ratio characteristic of the microstructure, the designed microstructure model is imported into ABAQUS for analysis. The element chosen is quadrilateral, and the elastic modulus and initial Poisson's ratio are consistent with the conditions during optimization. The boundary conditions are set by fixing the horizontal displacement of the middle vertical nodes, as well as the vertical displacement of the middle horizontal node. Then, the top and bottom boundaries are loaded, and the equivalent Poisson's ratio is calculated as the ratio of the average displacement of the left and upper adjacent edges.
Figure 6 shows the microstructure simulation model.
Fig. 6 The microstructure simulation model. 
4.1 Analysis of mechanical properties of zero Poisson's ratio microstructures with different initial design domains
After using node density level sets to smooth the boundaries of microstructures, there will be some errors between the Poisson's ratio of the microstructure model and the original results. Therefore, this error can be reduced by adjusting the level set values.
Table 3 shows the deformation cloud maps and corresponding equivalent Poisson's ratios (EPR) of microstructure simulation models with different initial design domains before and after modification of the level set values. As shown in Table 3, the number outside the parentheses below each deformation cloud map represents its level set value, and the number inside the parentheses represents the volume fraction of the microstructure.
The conclusions that can be drawn from Table 3 are as follows:
If the level set values are not finetuned, the microstructure's equivalent Poisson's ratio is also close to 0, but with a slightly larger error.
The equivalent Poisson's ratio of the model and the optimized finite element results have a certain error due to the smooth treatment of the model boundary. Finetuning the level set values can reduce this error. Table 3 shows that, upon adjusting the level set value, the volume fraction of the optimized structure with initial topology of four, five, and nine holes accounts for 94.2%, 94.3%, and 92.9% of the original volume, respectively. However, the Poisson's ratio of the structure is only 25%, 6.3%, and 9.5% of that of the original structure, respectively. By adjusting the level set value of the original structure, the error due to the boundary smoothing treatment can be reduced, resulting in a Poisson's ratio that is closer to the optimization result.
The deformation patterns of the microstructures with three different initial topologies, as shown in the deformation cloud maps, all meet the deformation mode of zero Poisson's ratio materials. This means that when compressed or stretched in one direction, there is neither stretching nor contraction in the other direction. Different initial topologies have almost no effect on the equivalent Poisson's ratio of the material, as seen from their equivalent Poisson's ratios. This indicates that the method proposed in this paper is effective in designing microstructures with zero Poisson's ratio characteristics.
Deformation cloud maps of microstructures with different initial design domains.
4.2 Analysis of mechanical properties of zero Poisson's ratio microstructures with different volume fractions
The preceding section discussed how different initial design domains affect the microstructural characteristics of zero Poisson's ratio in a design. This section focuses on the effects of varying volume fractions on the mechanical properties of microstructures with zero Poisson's ratio. Table 4 displays the deformation cloud maps and equivalent Poisson's ratios of microstructures with different volume fractions, before and after modifying the level set values. The number outside the parentheses below each deformation cloud map represents its level set value, while the number inside the parentheses represents the volume fraction of the microstructure. From Table 4, it is evident that the mechanical properties of microstructures under different volume fractions still conform to the zero Poisson's ratio requirement. Similar to the case of different initial design domains, varying volume fractions have little impact on the zero Poisson's ratio of the microstructure, which remains close to zero. Moreover, finetuning the level set values can reduce errors caused by boundary smoothing processing.
The conclusions that can be drawn from Table 4 are as follows:
Table 4 shows that the mechanical properties of microstructures with different volume fractions still satisfy zero Poisson's ratio performance. Similar to the case of different initial design domains, the different volume fractions have almost no effect on the zero Poisson's ratio performance of the microstructures, which remains close to zero. By adjusting the level set values, the structures with volume fractions of 0.4, 0.5, and 0.6 were finetuned to 93.6%, 94.3%, and 91.3% of the original volume fractions, respectively. The Poisson's ratios were also adjusted to 4.2%, 6.3%, and 38.1% of the original values. Similar to the results in Table 3, demonstrating that adjusting the level set values effectively reduces the error caused by boundary smoothing treatment. This leads to a Poisson's ratio that is closer to the optimized value.
From the viewpoint of the deformation mechanism, most of the traditional zero Poisson ratio honeycomb structures are only unidirectional zero Poisson's ratio effect, and their deformation mechanism is to form the zero Poisson's ratio effect through the deformation rotation of the rods inside the structure. The microstructured metamaterials with bidirectional zero Poisson's ratio effect obtained by the optimization of the method in this paper are mainly formed by some horizontal and vertical rods through the nodes of specific shapes, and the characteristics of the structural composition are similar to the structure designed by Anders Clausen et al. [18]. All of them are formed by the crossing of horizontal and vertical rods after bending and deformation.
Table 5 presents a comparison chart of microstructure deformation before and after.
As can be seen from Table 5, when the pressure is applied to the vertical direction, some locally curved places of the original structural rods will be changed from the original curve to the shape close to a straight line because of the force, and some nodes will produce a certain degree of rotation, and the combined effect of both cancels out the deformation of the structure originally to be outwardly extended, so in the deformation of the force, it is mainly the rods and the nodes that bear most of the deformation, so that the structure has a zero Poisson's ratio effect and the color of the deformation cloud graph shows that the vertical rod bears most of the loads throughout the stressing process, so this paper's microstructures have a large stiffness at the same time as they have a zero Poisson's ratio effect.
To investigate the effect of volume fraction on the stiffness of the microstructure, further analysis was carried out on the stress and deformation of the microstructure. The change in stiffness was measured using equivalent stiffness, which is equal to the ratio of vertical load to vertical displacement. Figure 7 shows the equivalent stiffness of microstructures with different volume fractions before and after adjusting the level set values. The blue part represents the equivalent stiffness of the microstructure before the level set values are modified, and the yellow part represents the equivalent stiffness after modification.
As can be seen in Figure 7, the stiffness of the microstructure decreases with decreasing volume fraction. However, after the modification of the level set value, the stiffness of the structure changes but its zero Poisson's ratio property remains stable. Therefore, by choosing the appropriate volume fraction of the material and adjusting the level set value, zero Poisson's ratio metamaterials with desired stiffness can be designed.
Deformation cloud maps of microstructures with different volume fractions.
Comparison of microstructure before and after deformation.
Fig. 7 Equivalent stiffness maps of microstructures in Table 4 with different volume fractions before and after modification of the level set values. 
4.3 Mechanical performance analysis of zero Poisson's ratio metamaterials
To demonstrate that the multicellular structure obtained through the method proposed in this paper also has zero Poisson's ratio characteristics, single cells are arranged and combined to form multicellular metamaterials. Then, finite element analysis is used to analyze it in both the X and Y directions. Due to the symmetric constraints on the structure, the structural constraints remain unchanged, and the load changes from vertical to horizontal. Tables 6 and 7 show the microstructural deformation cloud maps and equivalent Poisson's ratios of singlecell and multicellular structures with volume fractions of 0.4 and 0.5 under compression in the Y and X directions, respectively.
According to Table 6, it is evident that the equivalent Poisson's ratio of the structure increases to some extent as the single cell is converted to a multicell structure. The increase in Poisson's ratio is due to the change in the structure size and the number of finite element cells when converting from a unicellular structure to a multicellular structure, which leads to a certain amount of error. Nevertheless, the overall Poisson's ratio of the structure remains close to zero.
Table 7 shows that periodic microstructures with zero Poisson's ratio can be combined to form metamaterials with zero Poisson's ratio properties in both directions. The slightly different results in both directions are because the whole model has a slight error in the optimization process and is not ideally perfectly symmetric.
Deformation maps and Poisson's ratios of singlecell and multilattice structures with zero Poisson's ratio at different volume fractions in the X direction.
Deformation maps and Poisson's ratios of single cell and multilattice structures with zero Poisson's ratio for different volume fractions in the Y direction.
5 Conclusion
This paper proposes a topology optimization objective function for zero Poisson's ratio, and obtains microstructures with zero Poisson's ratio performance under different initial topologies and volume fraction conditions through the boundary density evolution topology optimization method. Due to the existence of errors between the microstructure model after boundary smoothing and the initial optimization result, a more reasonable microstructure model can be obtained by finetuning the node density level set value to reduce this error. Mechanical performance analyses of microstructure models under different conditions show that the initial design domain and volume fraction have almost no effect on the zero Poisson's ratio performance of the microstructure. The proposed method can effectively design microstructures with stable zero Poisson's ratio performance. Meanwhile, the examples also show that by selecting appropriate volume fractions and materials, microstructures with expected stiffness and zero Poisson's ratio performance can be designed. When the microstructure unit cell is combined into a multicell metamaterial through periodic arrangement, the overall Poisson's ratio of the entire structure still tends to zero, indicating that the proposed method can effectively design 2D metamaterials with bidirectional zero Poisson's ratio performance. Subsequently, the topology optimization design of 3D zero Poisson's ratio metamaterials will be further studied.
Funding
This research was supported by the National Natural Science Foundation of China (12072114), Guangdong Basic and Applied Basic Research Foundation (2023A1515012830), and the Guangdong Provincial Key Laboratory of Modern Civil Engineering Technology (2021B1212040003).
Conflicts of interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Data availability statement
All data in this article can be obtained under reasonable request.
Author contribution statement
Xueping Li: Conceptualization, Validation, Methodology, Supervision, Writing – review & editing. Jiajun Zhu: Data curation, Investigation, Methodology, Writing – original draft. Peng Wei: Conceptualization, Methodology, Supervision, Writing – review & editing. Cheng Su: Conceptualization, Methodology, Supervision.
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Cite this article as: Xueping Li, Jiajun Zhu, Peng Wei, Cheng Su, Topology optimization design of microstructures with zero Poisson's ratio, Int. J. Simul. Multidisci. Des. Optim. 15, 12 (2024)
All Tables
Optimization results of zero Poisson's ratio microstructures with three different volume fractions.
Deformation maps and Poisson's ratios of singlecell and multilattice structures with zero Poisson's ratio at different volume fractions in the X direction.
Deformation maps and Poisson's ratios of single cell and multilattice structures with zero Poisson's ratio for different volume fractions in the Y direction.
All Figures
Fig. 1 The tensile deformation state of materials with positive and zero Poisson's ratio. 

In the text 
Fig. 2 Diagram of the idea of the homogenization method. 

In the text 
Fig. 3 Example diagram of a fournode element in twodimensional plane. 

In the text 
Fig. 4 Iteration optimization curves of Poisson's ratio for microstructures with different initial design domains. 

In the text 
Fig. 5 Iteration optimization curves of Poisson's ratio for microstructures with different volume fractions. 

In the text 
Fig. 6 The microstructure simulation model. 

In the text 
Fig. 7 Equivalent stiffness maps of microstructures in Table 4 with different volume fractions before and after modification of the level set values. 

In the text 
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