Evolutionary design of nonuniform cellular structures

Materials and Design 141 (2018) 384–394

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Evolutionary design of nonuniform cellular structures with optimized Poisson's ratio distribution Yafeng Han, Wenfeng Lu ⁎ Department of Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1, 117975, Singapore

H I G H L I G H T S

G R A P H I C

A B S T R A C T

• Variable Poisson’s ratio was easily achieved by modifying interior angel of the traditional reentrant structure. • Cellular structures with various deformation properties was built by connecting unit cells with different Poisson’s ratio. • Evolutionary algorithm guaranteed the Poisson’s ratio distribution was optimum for the given objective deformation. • Application of “discrete cosine transform encoding” can significantly reduce the computational cost.

a r t i c l e

i n f o

Article history: Received 21 September 2017 Received in revised form 21 December 2017 Accepted 25 December 2017 Available online 27 December 2017 Keywords: Nonuniform cellular structure Poisson's ratio Evolutionary algorithm Re-entrant structure Additive manufacturing

a b s t r a c t For negative Poisson's ratio (NPR) cellular structures, most previous research focus on the design of unit cells, and then repeat the unit cell to construct uniform cellular structures. However, there is a disadvantage that these structures do not have much design freedom to achieve high-level functions, such as performing a desired deformation. As a solution, an evolutionary design method is proposed to develop nonuniform cellular structures. To conduct this method, the design domain is divided into finite unit cells with tunable Poisson's ratio (PR). With a given objective deformation, the value of each unit cell's PR is optimized using evolutionary algorithm (EA). In order to reduce the computational cost of the algorithm, discrete cosine transform (DCT) is applied to encode the structure for evolving. Considering the geometrical complexity of the optimized nonuniform cellular structures, additive manufacturing (AM) is chosen to build them physically. Both two-dimensional (2D) and threedimensional (3D) design cases were developed and analyzed to validate the proposed method. The computational and experimental results showed good conformation with each other. Most importantly, this novel design method brings huge potential to NPR cellular structures with high-level functions and much wider applications. © 2017 Published by Elsevier Ltd.

1. Introduction The Poisson's ratio (PR) of materials describes the negative ratio of transverse strain to axial strain under applied loading. In general, most materials possess positive PR, which means they contract when ⁎ Corresponding author. E-mail address: [email protected] (W. Lu).

https://doi.org/10.1016/j.matdes.2017.12.047 0264-1275/© 2017 Published by Elsevier Ltd.

stretched and expand when compressed. However, a bunch of materials have been proved to have negative Poisson's ratio (NPR) [1–3]. This unusual auxetic behavior can be understood through geometry of their microstructures and the geometrical deformations resulting from applied loads [4,5]. With enhanced mechanical properties such as shear modulus, plane strain fracture toughness and indentation resistance [6], NPR structures have great potential in variable applications, such as soft actuators [7], novel fasteners [8], shock absorbing materials [9], prostheses

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[10], air filters [11], biomaterials [12], piezocomposites [13], furniture [14,15], and even buildings [15]. Among different types of auxetic structures, cellular structures attract the most significant interest. Other than their better mechanical performance, cellular structures can also provide design freedom to obtain optimized functions for specific applications. One of the earliest articles in this area which established one kind of manufacturable reentrant unit cell based auxetic foams, was published in 1987 [1]. Later, several different auxetic cellular structures were developed, such as chiral honeycombs [16,17], origami based structures [18,19], etc. Especially, with the development of structural optimization technologies [20], numerous computational generated unit cells with prescribed PR have been introduced [20–23]. Besides, the elastic instability caused auxetic behavior of several soft porous structures were analyzed recently [24–27]. However, all these studies only focused on unit cell design and aimed at developing uniform cellular structures with NPR. One great disadvantage is that these kinds of uniform NPR structures are difficult to be optimized for high-level functions [28,29], such as satisfying a specific deformed shape or locomotion. To meet such high-level functional requirements, nonuniform cellular structures [30], which means the geometry and property of each unit cell in the structure is nonuniform and disordered, are needed. To build such structures, there are mainly two challenges need to be overcome: finding suitable unit cells with tunable PR, and optimizing PR distribution in the structure to achieve the objective deformation. Several studies have been conducted to develop unit cells with tunable PR. Li et al. [31] developed a bi-material re-entrant structure that has tunable PR from positive to negative by changing the temperature. Grima et al. [32] investigated the tunable auxetic property of structures made of magnetic components by applying external magnetic field. The most common method to achieve tunable PR is to change the geometry of unit cell [24,33–35], and the relationship between structural geometry and PR can be easily obtained through finite element analysis (FEA) [36]. The advantage of this method is obvious as PR of the unit cell can be simply manipulated without applying any extra field variables, such as temperature or magnetic field. As for optimization techniques, homogenization based topology optimization [37] methods have been applied to the design of nonuniform cellular structures [38–41]. With optimized material distribution, these optimized cellular structures possess higher stiffness or less material consumption compared with uniform structures. However, developing structures with optimized PR distribution is beyond the capability of topology optimization. One reason is that traditional topology optimization is based on the homogenization of unit cell Young's modulus and it is very difficult to manipulate the distribution of PR in the structure. Detailedly, the correlation between Poisson's ratio and global stiffness matrix is extremely complex and far from linear, different form Young's modulus. As a result, gradient of the sensitivity for PR optimization is not continuous, and the optimum is difficult to achieve through gradient-based approaches, such as Optimality Criteria (OC) methods or the Method of Moving Asymptotes (MMA) [37]. Another reason is that homogenization methods are generally limited to optimizing simple and overall parameters, such as deflection, stiffness or single-point displacement [28,42]. When facing high-level design requirements, such as performing an exact deformation, these approaches will become impractical [28,29]. As a solution, global optimization approaches are needed to optimize the structural PR distribution for a given deformation. With both high accuracy and convergence rate, EA [43] is one of the effective approach to design such nonuniform cellular structures. Evolutionary algorithm, also called genetic algorithm (GA) [44], has been widely applied in the structural design area. To introduce EA to structural optimization problem, encoding a physical structure to computer accessible data is the main procedure [45]. Direct encoding, in which each individual element (pixel or voxel) in the structure was represented by a gene in the genotype, was widely applied in early attempts [46,47]. However, direct encoding will lead to high cost of

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computation, especially when facing design problems with large number of variables. Thus, finding an efficient encoding method that can encode physical structure with minimum number of parameters becomes the main challenge for EA based structural design [28,29]. As summarized in [45], several encoding methods have been proposed, for example, graph structures, generative encodings, and constrained bit-wise encodings. These efficient representation methods make EA more practical for large-scale structural design. Another challenge to develop nonuniform cellular structure is at the manufacturing stage. With such a high geometrical complexity, traditional manufacturing methods are obviously inapplicable. The unique capabilities of additive manufacturing (AM) techniques [48] bring unprecedented design and manufacture freedom. Several nonuniform cellular structures [15,39,40] have already been manufactured with different AM processes. In this paper, an EA based optimization method is proposed to design nonuniform cellular structures with the ability of obtaining desired deformation. Reentrant honeycomb structure was modified and improved to achieve tunable PR, by changing its interior angle. In the range of achievable PR, EA was conducted to optimize the PR of each unit cell in the design domain. To reduce the computational cost of the algorithm, DCT method was chosen to encode the PR of each unit cell in the structure. Based on the optimization result, unit cells with corresponding interior angles were fitted into the design domain to build the final nonuniform cellular structure. High-level functional nonuniform cellular structures, which can obtain the desired deformation, can be generated using this method. This novel design method can hopefully expend the application of NPR structures with unprecedented deformation property. More significantly, the developed nonuniform cellular structures can perform desired deformation with only one material by manipulating structural PR distribution through geometrical change of unit cells, which makes it manufacturable for most AM processes. 2. The proposed method To achieve the objective of designing nonuniform cellular structures with optimized PR distribution, the available range of unit cell's PR should be analyzed firstly. In this available range, PR distribution can be optimized by EA to conform with the objective deformation. Finally, the nonuniform cellular structure can be constructed by mapping unit cells with different PR to the optimized PR distribution. As illustrated in Fig. 1, this method is mainly composed of three steps, which are elaborated below. (i) Unit cell analysis: Tunable PR of both 2D and 3D reentrant unit cells were achieved by changing their interior angles. NinjaFlex, which is an additive manufacturable rubber-like material, was chosen as the base material for better deformation property. As most rubber-like materials, the chosen material can be assumed as linear elastic when the strain is low (b 0.5) [9]. Considering the deformation is relatively large, geometrically nonlinear FEA was applied to analysis the PR of unit cells with different interior angles. The relationship between unit cell interior angle and structural PR can be calculated, and the range of available PR can be obtained. (ii) Evolutionary algorithm (EA): Within the range of PR, values of all unit cell's PR were evolved to guarantee the optimized structure could perform the objective deformation. Mainly, the evolutionary procedure contains two steps: encoding and evolving. The main purpose of encoding is to represent all the values of unit cell PR mathematically, for the sake of computational processes. To accelerate the computing of the algorithm, DCT encoding is applied in this study. The reason is that DCT has the ability of compressing the number of variables when representing a structure, without losing much accuracy. After encoding, evolving process was conducted based on the most common GA

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Fig. 1. General processes of the proposed method.

programming. During evolving, fitness evaluation was done by comparing the difference between deformation of each individual structure and the pre-defined objective. Linear static FEA was conducted to analyze the deformation of each individual, during the process of EA. Mathematically, the optimization objective is to minimize the fitness value, and the algorithm will complete when fitness value satisfies the termination criteria. (iii) Results generation and tests: After encoding and evolving, the optimized PR distribution could be obtained. By mapping unit cells with different PR to the corresponding location in this distribution, the final nonuniform cellular structure can be constructed.

In the end, 2D and 3D beam design were demonstrated to validate this method, and several arbitrary shapes were defined as objective deformation. For 2D problem, two different transition methods (binary and continuous) were utilized to analyze the relationship between cellular continuity and structural deformation. Besides, efficiency and effectiveness of two different encoding methods (direct and DCT) were also compared. This novel method introduced EA based structural optimization to design nonuniform cellular structures. Compared with conventional topology optimization, this method may have higher computational cost. However, the ability of handling high-level functions makes the proposed method practical for the design of nonuniform cellular structures with optimized PR distribution. Unit cell analysis and evolutionary design procedures are elaborated in Sections 3 & 4. In Section 5, results of FEA and tensile tests of both 2D and 3D beams are demonstrated. Geometrical nonlinearity was considered during FEA tests, in order to get better accuracy for large deformation analysis.

will be an output-displacement (Dis_out) in the upward direction. Particularly, the X direction displacement of the right edge and the Y direction displacement of the upper edge were constrained to be uniform, aimed at eliminating the unit cell boundary effect and maintaining the geometric symmetry during deformation. With these two constraints, the PR of the unit cell ν can be directly calculated by ν ¼ −Dis out=Dis in

To achieve a broader range of ν and make sure there is no intersection between struts, the angle θ is varied from 120° to 240°. Assumed as linear elastic, 3D printable rubber-like material (Ninjaflex) with Young's modulus of 12Mpa and Poisson's ratio of 0.34 was chosen as the base material. Geometrically nonlinear FEA of 13 (θ with increment of 10° from 120° to 240°) unit cells was done in ABAQUS to find the relationship between PR ν, angle θ and nominal strain ε. Analysis results are illustrated in Fig. 3. From the results, it can be observed that unit

3. Unit cell analysis 3.1. 2D unit cell The purpose of unit cell analysis is to get the range of available PR, and find the relationship between unit cell geometry and its PR. For 2D problem, re-entrant honeycomb unit cell [1,50–52] is applied to construct the optimized structure. The specific geometries of the unit cell are shown in Fig. 2. Apart from good mechanical properties, re-entrant unit cell has the ability of achieving tunable PR ν by only changing interior angle θ. With the boundary conditions set as shown in Fig. 2, a uniform input-displacement (Dis_in) is applied on the right edge, and there

ð1Þ

Fig. 2. Geometries and boundary conditions of the selected unit cell.

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Fig. 3. Poisson's ratio of the 2D unit cells with different interior angle.

cell's PR ν is not constant with the change of nominal strain ε. Besides, there is no linear correlation between PR ν and angle θ when the strain is low (ε b 0.15). Only when the strain becomes larger (ε N 0.15), the relationship starts to be linear. While the strain is too large (ε N 0.3), available PR range will be very narrow. In order to get a linear relationship and wider range of available PR ν, the applied strain better lies in the range of 0.15 b ε b 0.3. With any applied strain in this range, there will be a one-to-one corresponding relationship between ν and θ. The optimized PR distribution can be mapped into a nonuniform cellular structure, based on this relationship. As for the Young's modulus of unit cell with different θ, there is no much difference. Specially, it should be emphasized that the θ = 180° unit cell has a positive PR. Commonly, this unit cell will have zero PR when analyzed without considering wall thickness. In this study, however, the thickness is considered, and the horizontal struts will become thinner when stretched. Therefore, θ = 180° unit cell has a positive PR. Besides, this is also why absolute values of all θ b 180° unit cells' PR are larger than their opponents. 3.2. 3D unit cell By modifying the 2D re-entrant structure, simple 3D unit cell can be modeled. The geometry of the 3D unit cell is shown in Fig. 4a. Geometries and boundary conditions of the unit cell at the X-Z cutting plane is presented in Fig. 4b. With an axisymmetric geometry, Y-Z plane

shares the same parameters. Different from the 2D unit cell analysis, the input displacement (Dis_in) is applied along Z-axis at the top of the unit cell. Considering the symmetry of both geometries and boundary conditions, output displacements in X and Y direction will be the same (Dis_out). The PR of this 3D unit cell ν can still be calculated by Eq. (1). This 3D unit cell shares the same base material (with different color) as the one of 2D cell. By changing the interior angle θ form 120° to 240°, PR of each 3D unit cell was also analyzed in ABAQUS. Fig. 5 demonstrates the nonlinear analysis results. We can see that, for different nominal strain (from 0 to 0.5), unit cell PR ν is always linearly related with θ, and the only change is the range of available PR. With this relationship, design of 3D nonuniform cellular structure can be processed. 4. Evolutionary algorithm In the proposed method, EA is the tool of finding the optimum PR distribution that guarantees the cellular structure obtaining the desired deformation. The formulation for the proposed nonuniform cellular structure optimization problem is given in Eq. (2). Different form the FEA method used for unit cell analysis in Section 3, linear static FEA was applied to analyze each individual structure during EA. As shown in Eq. (2), the structure is divided into n unit cells, and each unit cell is assumed as isotropic and linear elastic. As a result, the global stiffness matrix K is only related to unit cell's Young's modulus E and Poisson's

Fig. 4. Details of 3D unit cell: (a) the general geometry; (b) specific geometries and boundary conditions on X-Z cutting plane.

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Fig. 5. Poisson's ratio of the 3D unit cells with different interior angle.

ratio ν. Considering E of unit cells do not have much difference with the change of interior angle θ, only unit cells' PR matrix ν(θ)n is necessary for solving the individual FEA. Based on the correlation between ν and θ form Fig. 3 or Fig. 5, the optimization objective becomes finding the optimum θ distribution in the structure, with minimum fitness value (FitV). For the implementation of FEA, the continuous objective deformation is discretized into j unit cells' displacements at the corresponding edge. Consequently, FitV of any individual can be calculated by summing up the absolute differences between objective displacement (ObjDi) and individual displacement (IndDi) of the ith unit cell on the edge, form i = 1 to i = j.

j Minimize : FitV ¼ ∑i¼1 ObjDi −IndDi K E; νðθÞn ; F Subjected to : ð j ≤nÞ

ð2Þ

The main reason why linear static FEA was applied is that only the final deformed boundary shape of each individual is needed to conduct fitness evaluation. Even though the unit cell's PR keeps changing during the loading process, the final PR value is fixed and only related with interior angle θ and the applied strain ε, as demonstrated in Fig. 3 and Fig. 5. With the assumption of uniform Young's modulus, the structure's final deformation shape can be obtained through simple linear static FEA, without considering material or geometrical nonlinearity. This is also why uniform displacement was applied in both 2D and 3D demonstration examples. Besides, linear static FEA can also be easily programed and integrated with EA in Matlab. Without the necessity of any low efficiency data exchanging between different software, the computational efficiency, which is the main concern of EA based structural optimization [45], can be improved dramatically. Encoding and evolving are the two general steps to apply EA to structural design [45]. To solve the nonuniform cellular structure design problem, encoding, which translates physical phenotype (all elements' PR) to mathematical genotype (design variables), is of extreme

Fig. 6. The evolving process of structural PR distribution.

importance. Considering EA is a searching-based optimization method, the computational cost will be extremely high if the number of variables is large. Therefore, finding a highly efficient encoding method is extraordinarily significant when facing complex design problems. As for evolving process, the existing GA programming can be easily modified to solve the problem. 4.1. Encoding For the proposed nonuniform cellular structure design problem, PR of each unit cell in the structure need to be encoded for the processing of computer. The simplest way is “direct encoding”, which means that each unit cell's PR is set as one variable. However, the computational cost could be extremely high when facing large structural design problems. How to represent all the unit cells' PR with less variables becomes the main challenge of encoding. Here we draw a lesson form image compression technique. A generative method, discrete cosine transform (DCT) is applied to do the encoding [28]. DCT is one type of discrete Fourier transform, and it has the ability of translating any arbitrary matrix to a same dimensional matrix of frequency components. The other way around, inverse DCT can translate the frequency matrix to the original matrix. Most significantly, DCT can compress the most meaningful information of the original matrix into its lowest order frequency coefficients. This ability of using small number of frequency coefficients to represent large number of variables gives DCT widely application in jpeg image compression. In this paper, this method is applied to compress the numerical representation of both 2D and 3D structures' PR distribution. Because the higher order frequency components are eliminated, sharply changed deformation will be hard to represent. Fortunately, the elastic deformation of a macro-scale structure is always continuous and at low order, which

Fig. 7. Structural geometries and boundary conditions.

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4.2. Evolving

Table 1 Parameters of three different methods for the 2D structural design. Transition strategy

Binary state

Encoding method

Binary, direct

Real-value, direct

Continuous state Real-value, DCT

Variable number Variable values Population size Mutation rate

12 × 6 −1.40 or 1.82 100 0.2

12 × 6 −1.40 to 1.82 100 0.2

6×3 −1 to 1 50 0.2

Fig. 8. Evolving processes of binary state structures.

The evolving process in the proposed design method can be implemented with a common GA programming, as shown in Fig. 6. Before the evolving process begins, an initial population of genotypes should be created randomly. These individuals are the first generation of design solutions. As illustrated in Eq. (2), fitness evaluation can be easily achieved by comparing the deformation of each individual with the objective deformation. Individuals in the first generation always have high FitV, and then selection, crossover and mutation become necessary. Parents selection is carried out using liner-ranking method [53], which ranks all the individuals by their FitVs in an increasing order. Then, individuals will be selected as parents with a probability based on the same linear ranking function. Each parent is paired with a random mate to reproduce the next generation. When two parents are selected for crossover, each gene of the offspring is randomly chosen from either parent. The new generation will continue the iteration until the individual that satisfies the termination criteria (FitV do not changes in 50 generations) is generated. To cover as large design space and avoid local minimum, mutation is one of the most useful procedures. Mutation means that each genotype of an individual has a user-definable mutation probability (mutation rate) to be modified. In this research, the mutation rate is set relatively high as 0.2, to eliminate possible local minimum.

makes this encoding method extremely applicable for structural design [28]. Both encoding and decoding of the proposed EA optimization can be performed with this translation, and the processes are as:

5. Demonstration examples


Encoding : dct νðθÞn ; m ¼ f ðaÞm
0 Decoding : νðθÞn ≈ νðθÞn ¼ idct f ðaÞm ; n ðm≤nÞ Subject to : ðm≤nÞ

2D beam with size of 120 × 60 mm2 is chosen as the instance sample to illustrate this nonuniform cellular structure design method. Geometries and boundary conditions of the design can be simplified as demonstrated in Fig. 7. By using the 2D unit cell in Fig. 2 to map the design domain, this beam has a resolution of 12 × 6 pixels. A uniform displacement of 20mm was applied horizontally at the right edge as shown in Fig. 7, and the nominal strain ε of unit cell is (20/120 = 0.167). By drawing a vertical line at ε = 0.167 in Fig. 3, the generated thirteen cross points with curves are the corresponding PR ν for these thirteen unit cells with different interior angles. The unit cell with θ = 120° has the minimum PR ν = − 1.40, and the unit cell with θ = 240° has the maximum PR ν = 1.82. Through curve fitting of these thirteen points, linear dependency had the best fitness and the relation is established in Eq. (4).

ð3Þ

where, ν(θ)n is the phenotype matrix of n unit cells' PR. f(a)m is the encoded genotype matrix with m frequency components, and a is the amplitude of each component. In this study, −1 to 1 in dimensionless units is selected as the value range of a. The range is arbitrary, and can be linearly scaled without effecting the result. By zero-padding (n − m) higher order frequency components to f(a)m, an approximate phenotype matrix ν(θ)n′ can be obtained by applying inverse DCT. The deviation between ν(θ)n′ and ν(θ)n′ will be small enough, as long as m is not too small and the variables in phenotype matrix do not change extremely. As a result, a structure with n unit cells can be represented by only m parameters without losing much accuracy. After DCT encoding, evolving process can be conducted more efficiently with less genotype components.

5.1. 2D beam

ν ¼ −0:0269  θ þ 5:0459

ð4Þ

With the known relationship between interior angle and Poisson's ratio, genetic algorithm can be processed. Three different construction

Fig. 9. Evolving process of the binary state, 2nd order deformation design: form random generated PR distribution matrix (a) to optimized PR distribution (b).

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Fig. 10. Test results of binary state structures with 1st order (first column), 2st order (second column), and “sine” wave deformation (third column): (a–c) are the FEA results of the binary state structures, (d–f) are the deformation results of the binary state structures under stretch.

Fig. 11. Comparison between test results and objective deformation: (a) 1st order deformation; (b) 2nd order deformation; (c) “sine” wave deformation.

Fig. 12. Evolving processes of both DCT and direct encoding algorithms: (a) 1st order deformation; (b) 2nd order deformation; (c) “sine” wave deformation.

methods were tested to find a better solution with higher efficiency and effectiveness, as listed in Table 1. To analysis the effect of structural continuity to the final deformation, two different transition strategies were applied to build these structures: binary state and continuous state. For binary state design, PR of each unit cell can only be set to minimum (− 1.40) or maximum (1.82). While, continuous state design can achieve gradually changed PR form minimum to maximum. Correspondingly, both binary and real-value encoding methods were applied in the EA. Considering the complexity of continuous state structures, two different encoding methods were conducted to test whether encoding

has influence on time consuming of the EA. First method is direct encoding, which sets all 12 × 6 unit cells' PR as design variables. Another method, DCT encoding used a frequency matrix with 6 × 3 elements to represent this 12 × 6 structure. Because the size ratio between the physical and frequency matrices is 2:1, features smaller than approximately 2 pixels will be difficult to represent, and thus less likely to be evolved. In addition, − 1 to 1 was set as the value range of frequency components. Table 1 summarizes the details of these three different algorithms. Three different deformations, “1st order”, “2nd order” and “sine” wave shapes, were selected as objectives to demonstrate the proposed

Fig. 13. Evolving process of the continuous state, 2nd order deformation design: from random generated frequency matrix (a) to optimized result (b). After inverse DCT, the optimized structural PR distribution is shown in (c).

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Fig. 14. Test results of continuous state structures with 1st order (first column), 2st order (second column), and “sine” wave deformation (third column): (a-c) are the FEA results, (d-f) are the deformation results under stretch, (g-i) demonstrate the comparison of objective and test results.

method. By evolving PR of each unit cell, the optimized cellular structures with nonuniform Poisson's ratio distribution were generated. Computational and experimental analysis were done to validate the result structures. As can be seen from Fig. 8, all three binary state structural optimization were accomplished in b80 generations. Average time cost of algorithms is 18.3 Seconds. From the random generated 1st generation individual to the final optimized PR distribution, the evolving process can find the optimum PR distribution generation by generation. Fig. 9 is a demonstration example that shows the change of structural PR distribution during the evolving process of the 2nd order deformation design. To reduce the influence of fixed ends and get better results, several straight struts were added to both end of these structures when carrying out FEA and tensile tests. Considering the geometry of the generated nonuniform cellular structures is complex, it will consume much time and cost to manufacture them by traditional processes. Soft material based fused deposition modeling (FDM) technique, which is one type of AM processes, was applied to manufacture these structures. The deformations of both results are shown in Fig. 10. Comparison between both test results and objective deformations are illustrated in Fig. 11. As can be seen, the 2nd order deformation tests have the best conformation with objective. While, deformation of 1st order and “sine” wave tests have some small errors. Particularly, the experimental result of “sine” wave deformation has a broken strut at the one of the high stress area. This caused a remarkable discontinuity in the deformed structure, and made the deformation largely different from FEA result (Fig. 11c). As for continuous state structures, both direct and DCT encoding methods were applied. Fig. 12 collects the evolving processes of the aforementioned design problems. For these three different objective deformations, we can see that fitness values of both direct and DCT encoding methods do not have much difference after 250 generations evolving. However, the population size of direct encoding method

(100) is twice that of DCT encoding (50), so the FEA iteration count will also be twice larger for direct encoding. Considering that the majority time of this evolutionary based structural optimization algorithm is consumed during FEA of each individual, the computational cost of direct encoding will be much higher. The tested average computing time of direct encoded ((Tdir = 208.5s)) and DCT encoded algorithms (TDCT = 124.2s) had proved this. The results output is also a timeconsuming process, and this maybe the reason why TDCT was slightly longer than half of Tdir. Most importantly, DCT encoding has much higher convergence rate at the early stage. This property makes DCT encoding extraordinarily suitable for large design problems. Fig. 13 is a demonstration example that shows the evolving process of the 2nd order deformation design. Fig. 13a is a randomly generated (6 × 3) frequency matrix. After evolving, the optimized frequency matrix (Fig. 13b) can be obtained, and the optimized structural PR distribution (Fig. 13c) can be calculated through inverse DCT.

Table 2 RMSE of deviation between FEA and experimental deformation and objective. Objective Deformation

Binary state

Continuous state (DCT)

FEA RMSE

Experimental RMSE

FEA RMSE

Experimental RMSE

1st order 2nd order “sine” wave

0.176 0.140 0.215

0.232 0.149 0.309

0.133 0.100 0.169

0.157 0.144 0.172

Fig. 15. 3D structure design: (a) shows the geometries and boundary conditions of the 3D beam, and (b) is the objective deformation (“vase” shape) on the x = y = 30 mm edge.

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Y. Han, W. Lu / Materials and Design 141 (2018) 384–394 Table 3 Parameters of the EA for the 3D beam design. Method Unit cell number Variable number Variable values Population size Mutation rate

Continuous (DCT encoding) 3×3×6 2×2×3 −1 to 1 100 0.2

Considering the fitness values and optimized PR distribution are similar for both encoding methods, only DCT encoded structures were modeled and tested. The results are demonstrated in Fig. 14. FEA results (Fig. 14a–c) and experimental results (Fig. 14d–f) all showed much better conformation with objectives, compared with the deformation of the binary state structures. From the FEA and experimental results of both binary and continuous structures, displacement at the tips of the upper edge unit cells (12 totally) were measured. The root mean square errors (RMSE) of the deviation between these 12 displacements and their objective values were calculated, and the results are listed in Table 2. Because manufacturing errors cannot be eliminated, all FEA results have lower RMSE compared with the experimental results. By comparing the deformation RMSE of binary and continuous state structures, we can see that continuous structures have much better performance. As a conclusion, DCT encoded continuous state structures not only have higher computational efficiency, but also better conformation accuracy.

Table 4 RMSE of deviation between FEA and experimental deformation and objective. Method of test RMSE

FEA result

Tensile test result

0.139

0.162

30 mm edge (the bold edge in Fig. 15a). To conduct the EA, several parameters were set as Table 3. After about 2.1 h of computation, the optimized PR distribution was obtained. By fitting 3D unit cells with corresponding PR (Fig. 5) into the distribution, the nonuniform cellular structure with optimized function of achieving a “vase” like shape can be generated. In Fig. 16, both FEA and deformation results are demonstrated. Considering the symmetry, only half of the tested structure is shown in the figure. The RMSE of the deviation between both results and objective deformation were calculated, and the results are listed in Table 4. Six displacements (midpoint of each unit cell's edge on the X = Y = 30 mm boundary) were compared with the objective curve, and the REM difference between FEA and tensile test is quite small. Same as the 2D structures, this 3D nonuniform cellular structure was also manufactured by a soft material FDM printer. Significantly, the cost of post-processing can be saved, because this particular structure can be printed without support structure. By comparing with the objective curve, both deformations showed good conformation. 6. Conclusions

5.2. 3D beam By applying a 3D linear static FEA algorithm in the “fitness evaluation” step, the proposed method can be easily introduced to solve a complex 3D structural design problem. Geometries of the structure are shown in Fig. 15a. The boundary conditions are set as follows: displacement of Z = 0 surface is constrained in Z direction; displacement of X = 0 surface is constrained in X direction; displacement of Y = 0 surface is constrained in Y direction; and a Z direction displacement of Dis_in = 20 mm is applied on the top (Z = 60 mm) surface. The design objective is to achieve a “vase” like shape (Fig. 15b) at the X = Y =

In this paper, an EA based method was proposed to design nonuniform PR cellular structures. By optimizing the distribution of different unit cells with different PR in the design domain, the result structure has the unique ability to obtain the desired deformation. Different from traditional topology optimization approaches, EA based optimization method can easily handle the structural PR and fulfill high-level objectives. The main challenge of applying EA to structural design is at the encoding stage. To cope with this difficulty, DCT encoding method was employed. Comparing the computational time of direct and DCT encoding during the evolving process of the 2D beam design, DCT encoding method can reduce the computational cost dramatically.

Fig. 16. Deformation of the optimized 3D cellular structure (half of the beam is shown): (a) is the FEA result of the beam (the plot shows the average Mises stress of each element in the structure, and the unit is Pa), and (b) is deformation result of the beam under stretch.

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This advantage makes the proposed design method more practical for real life applications. Re-entrant 2D and 3D structures were chosen as the unit cells to construct the structure, considering their PR can be easily changed by adjusting interior angle. With FEA, the relationship between unit cells' PR and interior angle can be obtained. Based on the relationship, the nonuniform cellular structure can be constructed by mapping different PR unit cells to the optimized distribution. To validate the proposed method, three different objective deformations, 1st order, 2nd order and “sine” wave, were applied to a 2D beam design. Both binary and continuous transition strategies were used to construct the optimized structure. By comparing, it showed that DCT encoded continuous structures have higher computational efficiency and better deformation performance. Consequently, DCT encoding and continuous transition strategy were chosen to carry out the 3D beam design. For all the 2D and 3D design cases, both numerical and experimental results have demonstrated that the proposed method has the ability to design high-level functional nonuniform cellular structures. Because DCT encoding eliminates the high frequency components of the PR distribution matrix, sharply changed deformation is hard to achieve through this method. To make up this disadvantage, finding new encoding method with better performance is one of the future work. Besides, deforming property is only one of the objectives in common product design. With different objective structural properties, the proposed method can also be applied to design high-level functional structures in different areas, such as soft robotics, 4D printing, flexible electronics etc.

References [1] R. Lakes, Foam structures with a negative Poisson's ratio, Science 235 (1987) 1038–1040 (80-.). [2] R.H. Baughman, D.S. Galvão, Crystalline networks with unusual predicted mechanical and thermal properties, Nature 365 (1993) 735–737, https://doi.org/10.1038/ 365735a0. [3] F. Steffens, S. Rana, R. Fangueiro, Development of novel auxetic textile structures using high performance fibres, Mater. Des. 106 (2016) 81–89, https://doi.org/10. 1016/j.matdes.2016.05.063. [4] D.Y. Fozdar, P. Soman, J.W. Lee, L.H. Han, S. Chen, Three-dimensional polymer constructs exhibiting a tunable negative Poisson's ratio, Adv. Funct. Mater. 21 (2011) 2712–2720, https://doi.org/10.1002/adfm.201002022. [5] C. Körner, Y. Liebold-Ribeiro, A systematic approach to identify cellular auxetic materials, Smart Mater. Struct. 24 (2015), 25013. https://doi.org/10.1088/0964-1726/ 24/2/025013. [6] A. Alderson, Kenneth E. Evans, Auxetic materials, function materials and structure from lateral thinking, Adv. Mater. 12 (2000) 617–628. [7] L. Guiducci, P. Fratzl, Y.J.M. Bréchet, J.W.C. Dunlop, Pressurized honeycombs as softactuators: a theoretical study, J. R. Soc. Interface 11 (2014), 20140458. https://doi. org/10.1098/rsif.2014.0458. [8] J.B. Choi, R.S. Lakes, Design of a fastener based on negative Poisson's ratio foam, Cell. Polym. 10 (1991) 205–212. [9] Y. Wang, L. Wang, Z.D. Ma, T. Wang, A negative Poisson's ratio suspension jounce bumper, Mater. Des. 103 (2016) 90–99, https://doi.org/10.1016/j.matdes.2016.04.041. [10] F. Scarpa, Auxetic materials for bioprostheses, IEEE Signal Process. Mag. 25 (2008)https://doi.org/10.1109/MSP.2008.926663. [11] S. Hou, T. Liu, Z. Zhang, X. Han, Q. Li, How does negative Poisson's ratio of foam filler affect crashworthiness? Mater. Des. 82 (2015) 247–259, https://doi.org/10.1016/j. matdes.2015.05.050. [12] R. Gatt, M. Vella Wood, A. Gatt, F. Zarb, C. Formosa, K.M. Azzopardi, A. Casha, T.P. Agius, P. Schembri-Wismayer, L. Attard, N. Chockalingam, J.N. Grima, Negative Poisson's ratios in tendons: an unexpected mechanical response, Acta Biomater. 24 (2015) 201–208, https://doi.org/10.1016/j.actbio.2015.06.018. [13] O. Sigmund, S. Torquato, I.A. Aksay, On the design of 1–3 piezocomposites using topology optimization, J. Mater. Res. 13 (1998) 1038–1048, https://doi.org/10.1557/ JMR.1998.0145. [14] J. Smardzewski, R. Kłos, B. Fabisiak, Design of small auxetic springs for furniture, Mater. Des. 51 (2013) 723–728, https://doi.org/10.1016/j.matdes.2013.04.075. [15] D. Park, J. LEE, A. ROMO, Poisson’s Ratio Material Distributions. In Emerging Experience in Past, Present and Future of Digital Architecture, Proceedings of the 20th International Conference of the Association for Computer-Aided Architectural Design Research in Asia 2015, pp. 725–744. [16] D. Prall, R.S. Lakes, Properties of a chiral honeycomb with a Poisson's ratio of — 1, Int. J. Mech. Sci. 39 (1997) 305–314, https://doi.org/10.1016/S0020-7403(96)00025-2. [17] Y. Jiang, Y. Li, 3D printed chiral cellular solids with amplified auxetic effects due to elevated internal rotation, Adv. Eng. Mater. (2016) 1–8, https://doi.org/10.1002/ adem.201600609.

393

[18] M. Konaković, K. Crane, B. Deng, S. Bouaziz, D. Piker, M. Pauly, Beyond developable: computational design and fabrication with auxetic materials, ACM Trans. Graph. 35 (2016), 89. https://doi.org/10.1145/2897824.2925944. [19] C. Lv, D. Krishnaraju, G. Konjevod, H. Yu, H. Jiang, Origami based mechanical metamaterials, Sci. Rep. 4 (2015), 5979. https://doi.org/10.1038/srep05979. [20] M.P. Bendsøe, O. Sigmund, Topology Optimization: Theory, Methods, and Applications, Second, Co, 2004. [21] U.D. Larsen, O. Sigmund, S. Bouwstra, Design and fabrication of compliant mecromechanisms and structures with negative Poisson's ratio, J. Microelectromech. Syst. 6 (1997) 365–371, https://doi.org/10.1109/MEMSYS. 1996.494009. [22] A. Clausen, F. Wang, J.S. Jensen, O. Sigmund, J.A. Lewis, Topology optimized architectures with programmable Poisson's ratio over large deformations, Adv. Mater. 27 (2015) 5523–5527, https://doi.org/10.1002/adma.201502485. [23] P. Vogiatzis, S. Chen, X. Wang, T. Li, L. Wang, Topology optimization of multimaterial negative Poisson's ratio metamaterials using a reconciled level set method, Comput. Des. 83 (2017) 15–32, https://doi.org/10.1016/j.cad.2016. 09.009. [24] K. Bertoldi, P.M. Reis, S. Willshaw, T. Mullin, Negative Poisson's ratio behavior induced by an elastic instability, Adv. Mater. 22 (2010) 361–366, https://doi.org/10. 1002/adma.200901956. [25] J.T.B. Overvelde, S. Shan, K. Bertoldi, Compaction through buckling in 2D periodic, soft and porous structures: effect of pore shape, Adv. Mater. 24 (2012) 2337–2342, https://doi.org/10.1002/adma.201104395. [26] J. Shim, S. Shan, A. Košmrlj, S.H. Kang, E.R. Chen, J.C. Weaver, K. Bertoldi, Harnessing instabilities for design of soft reconfigurable auxetic/chiral materials, Soft Matter 9 (2013) 8198, https://doi.org/10.1039/c3sm51148k. [27] S. Yuan, F. Shen, J. Bai, C. Kai Chua, J. Wei, K. Zhou, 3D soft auxetic lattice structures fabricated by selective laser sintering: TPU powder evaluation and process optimization, Mater. Des. 120 (2017) 317–327, https://doi.org/10.1016/j.matdes.2017.01. 098. [28] J. Hiller, H. Lipson, Automatic design and manufacture of soft robots, IEEE Trans. Robot. 28 (2012) 457–466, https://doi.org/10.1109/TRO.2011.2172702. [29] J.D. Hiller, H. Lipson, Multi material topological optimization of structures and mechanisms, 11th Annu. Conf. Genet. Evol. Comput 2009, pp. 1521–1528, https://doi.org/ 10.1145/1569901.1570105. [30] X. Jin, G. Li, M. Zhang, Design and optimization of nonuniform cellular, Structure 0 (2017) 1–14, https://doi.org/10.1177/0954406217704677. [31] D. Li, J. Ma, L. Dong, R.S. Lakes, A bi-material structure with Poisson's ratio tunable from positive to negative via temperature control, Mater. Lett. 181 (2016) 285–288, https://doi.org/10.1016/j.matlet.2016.06.054. [32] J.N. Grima, R. Caruana-Gauci, M.R. Dudek, K.W. Wojciechowski, R. Gatt, Smart metamaterials with tunable auxetic and other properties, Smart Mater. Struct. 22 (2013), 84016. https://doi.org/10.1088/0964-1726/22/8/084016. [33] D. Li, L. Dong, R.S. Lakes, A unit cell structure with tunable Poisson's ratio from positive to negative, Mater. Lett. 164 (2016) 456–459, https://doi.org/10.1016/j.matlet. 2015.11.037. [34] C.S. Ha, M.E. Plesha, R.S. Lakes, Chiral three-dimensional lattices with tunable Poisson’s ratio, Smart Mater. Struct. 25 (5) (2016), 054005. [35] X. Zhang, D. Yang, Mechanical properties of auxetic cellular material consisting of re-entrant hexagonal honeycombs, Materials (Basel) 9 (2016)https://doi.org/10. 3390/ma9110900. [36] Nonlinear Analysis of the Poisson's Ratio of Negative Poisson's Ratio Forms1994. [37] M.P. Bendsøe, N. Kikuchi, Generating optimal topologies in structural design using a homogenisation method, Comput. Methods Appl. Mech. Eng. 71 (1988) 197–224. [38] S. Chellappa, A.R. Diaz, M.P. Bendsøe, Layout optimization of structures with finitesized features using multiresolution analysis, Struct. Multidiscip. Optim. 26 (2004) 77–91, https://doi.org/10.1007/s00158-003-0306-7. [39] A. Radman, X. Huang, Y.M. Xie, Topology optimization of functionally graded cellular materials, J. Mater. Sci. 48 (2013) 1503–1510, https://doi.org/10.1007/s10853-0126905-1. [40] P. Zhang, J. Toman, Y. Yu, E. Biyikli, M. Kirca, M. Chmielus, A.C. To, Efficient designoptimization of variable-density hexagonal cellular structure by additive manufacturing: theory and validation, J. Manuf. Sci. Eng. 137 (2014), 21004. https://doi.org/10.1115/1.4028724. [41] S. Daynes, S. Feih, W.F. Lu, J. Wei, Optimisation of functionally graded lattice structures using isostatic lines, Mater. Des. 127 (2017) 215–223, https://doi.org/10. 1016/j.matdes.2017.04.082. [42] H. Lipson, J.B. Pollack, Automatic design and manufacture of robotic lifeforms, Nature 406 (2000) 974–978, https://doi.org/10.1038/35023115. [43] J.H. Holland, Adaptation in natural and artificial systems: an introductory analysis with applications to biology, control, and, Artif. Intell. (1975) 121–122. [44] P. Hajela, E. Lee, C.Y. Lin, Genetic algorithms in structural topology optimization, Topol. Des. Struct. 227 (1993) 117–133, https://doi.org/10.1007/978-94-011-1804-0_10. [45] R. Kicinger, T. Arciszewski, K. De Jong, Evolutionary computation and structural design: a survey of the state-of-the-art, Comput. Struct. 83 (2005) 1943–1978, https:// doi.org/10.1016/j.compstruc.2005.03.002. [46] M.J. Jakiela, C. Chapman, J. Duda, A. Adewuya, K. Saitou, Continuum structural topology design with genetic algorithms, Comput. Methods Appl. Mech. Eng. 186 (2000) 339–356, https://doi.org/10.1016/S0045-7825(99)00390-4. [47] C. Kane, M. Schoenauer, Topological optimum design using genetic algorithms, Control. Cybern. 25 (1996) 1059–1087. [48] I. Gibson, D. Rosen, B. Stucker, Additive Manufacturing Technologies: 3D Printing, Rapid Prototyping, and Direct Digital Manufacturing, 2nd 2015, 2015https://doi. org/10.1007/978-1-4939-2113-3.

394

Y. Han, W. Lu / Materials and Design 141 (2018) 384–394

[50] A. Ingrole, A. Hao, R. Liang, Design and modeling of auxetic and hybrid honeycomb structures for in-plane property enhancement, Mater. Des. 117 (2016) 72–83, https://doi.org/10.1016/j.matdes.2016.12.067. [51] X.-T. Wang, X.-W. Li, L. Ma, Interlocking assembled 3D auxetic cellular structures, Mater. Des. 99 (2016) 467–476, https://doi.org/10.1016/j.matdes.2016.03.088.

[52] K. Wang, Y.H. Chang, Y. Chen, C. Zhang, B. Wang, Designable dual-material auxetic metamaterials using three-dimensional printing, Mater. Des. 67 (2015) 159–164, https://doi.org/10.1016/j.matdes.2014.11.033. [53] L.D. Whitley, The GENITOR algorithm and selection pressure: why rank-based allocation of reproductive trials is best, ICGA 89 (1989) 116–123 (doi:10.1.1.18.8195).