Anisotropic design and optimization of conformal gradient lattice structures

Journal Pre-proof Anisotropic design and optimization of conformal gradient lattice structures Dawei Li, Wenhe Liao, Ning Dai, Yi Min Xie

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S0010-4485(19)30238-6 https://doi.org/10.1016/j.cad.2019.102787 JCAD 102787

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Computer-Aided Design

Received date : 17 May 2019 Revised date : 24 September 2019 Accepted date : 27 October 2019 Please cite this article as: D. Li, W. Liao, N. Dai et al., Anisotropic design and optimization of conformal gradient lattice structures. Computer-Aided Design (2019), doi: https://doi.org/10.1016/j.cad.2019.102787. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Journal Pre-proof Anisotropic Design and Optimization of Conformal Gradient Lattice Structures Dawei Li a*; Wenhe Liao a; Ning Dai a*; Yi Min Xie b College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China b. Centre for Innovative Structures and Materials, School of Engineering, RMIT University, GPO Box 2476, Melbourne 3001, Victoria, Australia Email: [email protected] (Dawei Li); [email protected] (Ning Dai)

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a.

Abstract:

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In this work, we present a novel anisotropic lattice structure design and multi-scale optimization method that can generate conformal gradient lattice structures (CGLS). The goal of optimization is to achieve gradient density, adaptive orientation and variable scale (or periodic) lattice structures with the highest mechanical stiffness. The asymptotic homogenization method is employed for the calculation of the mechanical properties of various lattice structures. And an equation of elastic tensor and relative density of the unit cell is established. The established function above is then considered in the numerical optimization schemes. In the post-processing, we propose a numerical projecting method based on Fourier transform, which can synthesize conformal gradient lattice structure without changing the size and shape of the unit cells. Besides, the algorithm allows us to minimize distortion and prevent defects in the final lattice and keep the lattice structures smooth and continuous. Finally, in comparison with different parameters and methods are performed to demonstrate the superiority of our proposed method. The results show that the optimized anisotropic conformal gradient lattice structures are much stiffer and exhibit better structural robustness and buckling resistance than the uniform and the directly mapped designs. Keywords:

1. Introduction

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Anisotropic Design; Multi-scale Optimization; Conformal Lattice; Synthesize Algorithm; Lattice Optimization

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Periodic lattice structures are widely used in lightweight aerospace components [1], armor-protected energy absorption [2], mold heat transfer [3][4], and biomedical implants design [5]. They are usually a regular lattice structure (along x, y, and z-direction in a 2D plane or 3D space) with a uniform or gradient density distribution. Typically, these lattice structures can be complete isotropic (e.g., circular structures) or orthotropic (e.g., hexagon, square and square-x mixed structures). For an orthotropic lattice structure, it has the maximum elastic modulus in a certain direction [6]. After hundreds of millions of years of evolution, the microstructures in nature exhibit multi-scale, adaptive anisotropic features that give them excellent physical properties. For example, the highly anisotropic wood material gives it better mechanical properties along the trunk direction [7]. The microstructure of human or bird bones mainly shows three main characteristics [8]. First, the internal cancellous bone is mainly distributed along the baseline of the principal stress (Fig. 1a). Second, from the outer cortical bone to the inner cancellous bone, the density changes from dense to loosen (Fig. 1b). Third, both cortical and cancellous bones exhibit macroscopic and microscopic multi-scale morphology (Fig. 1c). Inspired by gradient micro-structures in nature, we would improve the overall mechanical stiffness of the target object by optimizing the density, orientation and period distribution of the infilled lattice structures in this work. 1

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Fig. 1. Human bones [9] with adaptive (a) principal stress directions, (b) gradient densities and (c) multi-scale morphologies.

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We extend the homogenization-based topology optimization method into the variable density optimization of lattice structures. An equivalent coarse mesh is used to represent the lattice unit, and the density and orientation of the lattice structures are taken into account simultaneously in the optimization process. We propose a mesoscale lattice structure projecting synthesis algorithm based on Fourier transform. The ability to control the lattice structures to rotate, translate, scale, and minimize shape distortion as much as possible. We demonstrate the effectiveness of our proposed conformal gradient lattice structures optimization for designing lightweight and stiff infill structures for numerical analysis and additive manufacturing.

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Although the computing power is growing so that the topology optimization has been widely used. However, to achieve the above high-resolution lattice structures iterative optimization design still requires a high amount of computation, mainly consumed in finite element analysis. Theoretically, the best topology optimization results are composed of infinite fine-scale periodic lattice structures. However, it is also necessary to take additive manufacturability into account and to artificially control the minimum feature size of the mesoscale. The homogenization-based topology optimization technique allows the design space to be divided into coarse meshes for numerical calculation, which provides the basis for the optimization design of the lattice structures at the manufacturing scale. Therefore, to realize the optimized design of the lattice material distribution of the above-mentioned biomimetic features in Fig. 1, an optimization design algorithm of conformal gradient lattice structure (CGLS) is proposed. In summary, our key contributions are as follows:

The rest of this paper is organized as follows. After reviewing related work in Section 1.1, in Section 1.2 we present an overview workflow. In Section 2, we present our CGLS optimization technique in detail. Numerical analysis and physical experiments are presented in Section 3, before conclusions are drawn in Section 4. 1.1. Related works

1.1.1

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We consider our CGLS optimization method as a multi-scale structural topology optimization problem. Thus, this paper reviews the research status of the following contents: Effective properties of lattice structures

The effective mechanical properties of the lattice structure are the main factor that needs to be concerned to solve the multi-scale design and optimization problem. They are primarily determined by its topological shape and relative densities. The periodic lattice structure can be considered as a material when its length of the unit cell is less than the length of the component and is more than one order of magnitude lower [10]. There are several numerical analysis approaches for predicting the effective mechanical properties of lattice 2

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1.1.2

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materials, mainly including “Gibson-Ashby” model [11], equivalent strain energy approach [12], and asymptotic homogenization (AH) theory [13]. However, the Gibson-Ashby model is only suitable for lattice structure with relatively low density (＜0.3), and not for lattice structure with complex shape. Compared with other methods, the AH method is a useful tool for solving multi-scale optimization problems. It mainly includes solving the effective mechanical properties of the micro-scale unit lattice structures and optimizing the macro-scale mechanical response according to the homogenized media with the calculated effective properties. Additionally, the AH method has no limitation on the topological shape and relative density of the unit lattice. Therefore, it has been widely used to predict the mechanical properties of heterogeneous periodic structures [14] and multi-scale structure optimization with microstructures [3]. Multi-scale optimization for infinitesimal scale lattice materials

Lattice structure optimization for Additive Manufacturing

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1.1.3

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Multi-scale design and optimization with microstructures have many pieces of research. For example, Bendsoe and Kikuchi [15][16] optimized the macroscopic performance by considering the optimal material topology for each relevant material point within the unit cell. The inverse homogenization-based structure optimization method was implemented by Sigmund [17], and the material layout within a single cell towards certain targets was searched. Liu et al. [18] proposed a penalization-based concurrent optimization with two-scale design variables, including solid isotropic material with penalization (SIMP) [19]in micro-scale and porous anisotropic material with penalization (PAMP) in macro-scale. Recently, Wang et al. [20] presented a multi-scale isogeometric topology optimization method for lattice materials and coupled to AH method. Zhang et al. [21] and Gao et al. [22] proposed a multi-scale concurrent topology optimization approach for porous structures with multi-domain microstructures based on ordered SIMP interpolation. Fu et al. [23] proposed a multiscale level set topology optimization method to design of shell-infill structures. Similarly, Wang et al. [24] designed the connectable graded micro-structures. Moreover, the microstructures are represented by a parametric level set method (PLSM), whose mechanical properties are predicted by AH method. As the above research methods without considering manufacturability, the AH-based multi-scale design and optimization approach consists of infinitesimal features.

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However, manufacturability is an important value manifestation for multi-scale optimization design to practical engineering applications. High-precision additive manufacturing (AM) techniques have made it possible to manufacture and apply multi-scale lattice structures with complex geometries [25]. Moreover, the lattice structure design and optimization have become an important component of DfAM (design for additive manufacturing) [26][27]. To obtain manufacturable designs from relaxed solutions, several multi-scale techniques have been proposed. The initial idea of the AM-oriented lattice structure design is to use a uniform lattice structure for lightweight [28]. Later, to make the infilling design reasonable, the stress-based density optimization method was appeased [29]. However, these methods are not the best design method. Recently, topology optimization (TO) technology has provided numerical design strategies for the optimal design of the lattice structure for AM [30]. Brackett et al. [31] earlier used the SIMP method but without the penalization to map the distribution of 2D lattice structures for AM. It is similarly based on SIMP direct mapping method, Panesar et al. [32] realized density interpolation mapping of various three-dimensional (3D) lattice structures. Wu et al. [33] used a self-supporting rhombic infill structures for AM which considering mechanical stiffness and static stability optimization. Besides, Wu [34] presented a novel continuous optimization method to obtain adaptive quadtree structures based on the SIMP. Liu et al. [35][36] proposed a sample-based two-scale structure synthesis approach which can control the anisotropic 3

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effective material properties of the structures. Moreover, a texture-guided generative structural design and optimization was presented by Hu et al. [37]. Recently, considering the effective mechanical properties of a lattice structure into macroscopic optimization has become a hot research topic. Jin et al. [38] proposed a 3D non-uniform lattice structures optimal design method based on SIMP, and the effective properties of the 3D lattice structures are predicted by the equivalent strain energy method. According to the same prediction method, Jiang et al. [39] given a lattice structure conformal mapping approach by using level set based structure optimization. Wu et al. [3] proposed a multi-scale thermomechanical cellular structures optimization method for porous injection molds. Considering the effective properties of a lattice structure

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On the other hand, the prediction of the effective mechanical properties of lattice structures based on the AH method is theoretically valid only when the scale of the lattice structure is much smaller than the macroscopic object scale. However, some studies in recent years have proved that this method can also be used to optimize the lattice structure distribution at a manufacturable scale when it meets a certain size [33-42]. Zhang et al. [40] and Cheng et al. [41][4][42] proposed the 2D and 3D functionally graded lattice structure topology optimization approaches for the design of AM components with stress constraints and heat conduction constraints. Besides, Li et al. [43][44] proposed a TPLS (triply periodic level surface) based functionally graded cellular structures (FGCS) optimization method. Towards anisotropic design, Liu et al. [45] and Zhu et al. [46], based on the MMC/MMV (moving morphable component/moving morphable void) topology optimization framework [47], proposed an orientation controllable lattice structure design method with an anisotropic mechanical performance. Groen et al. [48][49] presented a homogenization-based stiffness optimization and projection the 2D lattice structures with orthotropic infill.

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1.2. Overview

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In this work, a generative design and optimization technique for the generation of conformal gradient lattice structures is developed. Firstly, the AH method is used to predict the relationship between the elastic tensor of the lattices and the relative densities of the lattices, and to fit its curve equation in Section 2.1. Secondly, the tensor function fitted by the AH method is brought into the topology optimization process to obtain the material gradient distribution field. The field consists of three parts of sub-information: a density distribution field, a principal stress direction distribution field, and a scale distribution field in Section 2.2. Then, based on the information of the gradient field, the adaptive lattice filling structures are obtained with fine scale in Section 2.3. For example, suppose a design area with boundary conditions and constrains are shown in Fig. 2a. And a square lattice material is used as an optimized design material selected from the lattice library (Fig. 2b) for different filling design strategies with the same volume fraction 0.5. Fig. 2c shows the result of the traditional direct uniform lattice filling design, and Fig. 2d shows the gradient distribution field calculated by the topological optimization of the square lattice material constitutive equation. If only the density parameter is used as the design variable, the result is shown in Fig. 2e. In this paper, density, principal stress orientation, and periodic scale are used as optimization design parameters. The results are shown in Fig. 2f, and the stiffness is better than the direct density parameter as the design variable and uniform infilled lattice structures. Finally, multiple optimized design cases are analyzed and compared to existing methods in Section 3.

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Fig. 2. Overview of the proposed methodology to obtain conformal gradient lattice structures. (a) Design domain with loading and boundary conditions; (b) Lattice library; (c) Uniform lattice infill design (compliance: 275.9); (d) Density field optimized by coarse mesh (compliance: 87.6); (e) Density mapping with lattice materials directly (compliance: 142.1); (f) Projected fine-scale conformal gradient lattice structures (compliance: 89.3); (g) Variable periodic scale.

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2. Anisotropic Conformal Gradient Lattice Optimization 2.1. Prediction of Effective Elastic Tensor of Lattice Structures

Full-scale simulation of a micro-scale lattice structure by the finite element method (FEM) directly can be time-consuming. Therefore, it is necessary to analyze a representative volume element (RVE) [50] of a

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lattice material to obtain effective mechanical properties. In an elastic system, the macroscopic behavior of an anisotropic lattice structure can be characterized by the effective stress tensor σ ij and strain tensor  ij on a homogenized medium. The relationship between them can be combined by the homogenized elasticity tensors C cH .

σ ij = C c  ij

(1)

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where C

H c

H

is determined by the volume fraction and shape of the lattice structure, and it can be obtained

via AH method as follows. 2.1.1

Asymptotic homogenization method

To obtain the effective mechanical properties of the lattice structures, an AH-based framework is established to predict the homogenized elastic tensor C cH of the lattice structures at mesoscale. A simple schematic diagram of the procedure of AH approach [13] is described in Fig. 3. It is supposed that the body

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Ωε having a periodic lattice structure is subjected to the traction t at the traction boundary Γt, the displacement d at the displacement boundary Γd, and the body force f is distributed in the domain. Furthermore, by using the AH method, the lattice structure is considered to be a continuum real solid domain Ω with equivalent mechanical properties. Moreover, the computational cost is significantly reduced compared to the full-size simulation of the lattice structures that are explicitly represented.

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Fig. 3. Asymptotic homogenization (AH) concept of a lattice structure. The basic assumption of AH is that each field quantity depends on the macroscale x and the microscale y, and ε=y/x is the magnification factor of geometric features at the macroscale and microscale. It also assumes that field quantities, such as stress, strain, and displacement, varies smoothly at the macroscopic level and is periodic at the microscopic level. Besides, each mesoscale unit lattice follows the periodic boundary conditions (PBC) in the macrostructure. Therefore, the displacement field u can be expressed as a two-scale asymptotic expansion:

u  = u 0 ( x , y ) +  u1 ( x , y ) +  2 u 2 ( x , y ) +   

y = x / ,

= 1

(2)

For linear elasticity, u0 is the average displacement on the macroscopic scale, and u1, u2, …, etc. represent

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the perturbation displacement on the microscale. If only the first order terms of the asymptotic in Eq. (2) are considered, the effective elastic tensor C cH of a periodic lattice structure can be described in terms of elemental energy in an equivalent discrete form: 1 V

ne

  (I − B Ve

T

e

 e ) D e ( I − B e  e )dVe

(3)

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C cH =

e =1

where |V| is the unit lattice structure volume, the summation denotes the assembly of ne finite elements, Ve is the element e volume, I is the identity matrix, Be is the element strain-displacement matrix,  e denotes a matrix containing the element displacement vectors  ije obtained by the global execution unit test strain

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ij fields  , and De is the elastic tensor matrix of the element of the basic material of the object. Additionally,

the global stiffness matrix K of the unit lattice structure can be expressed as K =  e =1 k e , n

ke =



Ve

B Te D e B e dVe

(4)

where ke is element stiffness matrix. Besides, the load vector Fij which corresponds to macroscopic volumetric straining can be written as:

F =  e =1 f e , n

fe =



Ve

B Te D e  ij dVe

(5)

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ij For a two-dimensional (2D) case, three independent (six in 3D) unit strains  are chosen to be:

 = (1, 0, 0 ) ,  11

T

22

= ( 0,1, 0 )   T

12

= ( 0, 0,1 )

T

(6)

Therefore, the global displacement fields  ijc can be calculated with three load cases by the following function:

n n ij T ij T ij K  c = F    e =1  B e D e B e dVe   c =  e =1  B e D e  dV   Ve Ve

(7)

H Then, the homogenized constitutive matrix C ijkl can be obtained from the following equation:

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C ijkl =

1 V

n

(   ( e =1

0 ij ) e

Ve

( ij )

− e

)

T

(

0 ( kl )

De e

( kl )

− e

)dV

e

(8)

where  0e contains the three displacement fields corresponding to the solid unit strains in Eq. (6), and ( ij )

e

includes three columns corresponding to the three displacement fields which are produced by the

globally enforced unit strains. In this work, 2D lattice materials are taken as the objective infill structures; hence the homogenized elasticity tensor can be expressed using the following form:

2.1.2

C13H   C 23H  C 33H 

C12H C 22H

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C cH

 C11H  =  sym 

(9)

Effective mechanical properties of lattice structures by AH

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According to the "Gibson-Ashby" model [11], the relative elastic modulus of the lattice structure is a function of its relative density ρr=ρc/ρ* (ρc is the density of the lattice structure and ρ* is the density of the solid materials). Moreover, the function is known as “scaling law,” and referred as “elastic tensor scaling law” in this work. To capture the effective characteristics of each unit lattice topology, the properties are normalized as the base material regardless of the solid material. Moreover, each unit lattice has a very approximately uniform wall thickness. A MATLAB program is established to build, mesh and solve the 2D problem of the lattice material, which is referred the work of Andreassen [51]. Since the symmetry plane existing in the lattice structure determines the type of anisotropy of the lattice, and it can be generally divided into three symmetry of isotropic, orthogonal isotropic and orthotropic. Simultaneously, these symmetries also determine the number of independent effective elastic constants in

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H Eq. (9). As shown in Fig. 4, the elastic coefficients C ij are corresponding to different densities of

different lattice types calculated by the AH method (obtained by the Eq. (8)). To find the right elastic tensor scaling law fitting function for the computational results, different orders of polynomials and exponential functions are tried to be fitted. It is found that an exponential function not only ensures that the function

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monotonically increases in the interval of 0 to 1, but also has a small fitting error. Additionally, it provides the best combination of scaling rules between accuracy and compactness. Therefore, the equation of the elastic coefficients of a lattice structure with respect to its relative density can be described as H

C ij

(  r ) = a1e a   2

r

− a1

(10)

where C ijH = C ijH / C ijH * denotes the normalized effective elastic constant, C ijH represents the effective elastic constant of the lattice structure, while C ijH * is the elastic constant of the solid material component. And a1, a2 are the constant symmetric which are obtained after fitting to the homogenization calculation

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results. To make the fitting results accurate, each type of unit takes fifteen different volume fraction values between zero and one. Then, the fitted scaling laws of the selected lattice structures are plotted in Fig. 4a~b. It can be seen that the curve smoothly passes through the interpolation points and increases monotonically as ρ increases from 0 to 1. This means that the used homogenization model is valid over the entire density range.

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Fig. 4. Normalized elastic constants as a function of the relative density of the square (a) and (b) square-x.

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Furthermore, to verify the proposed homogenized model is available, the FEA simulation is used to compare with the homogenization method. An 8×8 lattice structure of different densities is selected for finite element analysis in the y direction, and the elastic modulus values in this direction are calculated by using Hooke’s law. Furthermore, the elastic modulus value of the proposed homogenization method in the y direction can be derived from the formula provided in the reference [52]. The result is shown in Fig. 5, and it can be seen that the calculated results of the homogenization method match well with the results of the FEA simulation.

Fig. 5. Comparison of elastic modulus values predicted by the homogenized model and those obtained from FEA: (a) square; (b) square-x. The mechanical properties of incompletely isotropic lattice structures are highest in some directions, and we call this direction the optimal direction of material properties. For a 2D lattice unit structure, its elastic tensor is usually calculated in its plane of symmetry. If the unit structure is rotated by an angle θ along with 8

Journal Pre-proof the original coordinate system (as shown in Fig. 6), its new elastic tensor calculation satisfies the following formula: C cH (  ,  ) = R −1 ( ) C cH (  ) R − T ( )

(11)

where R is the well-known rotation matrix, and the matrix is: sin 2  cos  2

(12)

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sin  cos 

2 sin  cos    −2 sin  cos   cos 2  − sin 2  

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 cos 2   R =  sin 2   − sin  cos  

Fig. 6. The isotropic polar plot of a unit lattice structure. 2.1.3

Size effect of lattice structures

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Since the equivalent elastic tensor calculated by the homogenization method is independent of the size of the lattice unit, it depends only on the configuration and density of the unit cell. Moreover, the AH method is effective only when the unit cell is sufficiently small in size relative to the macrostructure formed. Therefore, this method cannot reflect the size effect of the lattice structures. However, when considering the minimum manufacturing dimensions that existing AM techniques can meet, we need to find the optimal unit cell size that meets the homogenization theory. Here, the size effect of the lattice structure on the accuracy of homogenization is discussed by a numerical FEA simulation.

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As shown in Fig. 7a, a square design domain of size 20×20 mm is filled with the lattice structures of different sizes with the same density (0.3). It is assumed that the Young’s modulus for the solid material that constructs the lattice materials is 2.15GPa and the Poisson’s ratio is 0.3. Moreover, the loading conditions and fixed boundary conditions are shown in Fig. 7a. The FEA simulation process is performed by using OptiStruct 2017 software, and the constitutive model in Eq. (10) is used for homogenization simulation. Fig. 7b shows the maximum deformation of the upper boundary relative to the number of unit lattices in one direction. It can be seen from the results that when the number of unit lattices in one direction is larger than eight, the maximum total deformation gradually converges to the homogenized result. Therefore, eight-unit lattices along one direction of the design component is considered to be the minimum number in this work.

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Fig. 7. Size effect of lattice structures. (a) A square domain with boundary and loading conditions; (b) Maximum of the total deformation associated with unit cell number along one direction. 2.2. Multiscale Optimization Problem Problem statement

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2.2.1

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Topology optimization is to solve the optimal distribution of materials in the design domain according to given constraints and load conditions. Usually, to make the optimization results have physical meaning, the intermediate density value should be removed as much as possible by penalizing the density function (e.g., SIMP-based method [19]), or the solid area to be retained can be selected according to the stress value of the discrete unit (e.g., ESO/BESO-based method [53]). However, such artificial penalization leads to an optimization structure that is not better than the intermediate density without penalization. If there is a way to manufacture these intermediate density values, then there would be no need to penalize them. In this work, a novel approach for optimizing lattice material distribution is proposed. First, the overall lattice structure distribution is solved by a macroscopic structural optimization problem. Then, the density, orientation, and scale of the microscopic lattice structures are controlled and optimized. The goal is to achieve optimum stiffness of the object when filling with a lattice material.

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The objective of a minimum compliance problem is to find the material density distribution to minimize structural deformation under specified boundary and loading conditions. Accordingly, the mathematic expression of the lattice structure optimization problem can be described as find : ρ,  min J infill ( ρ, ) =

1 2

U K ( ρ, ) U T

s .t . C cH = C cH ( ρ, ) K ( ρ, ) U = F



m i =1

(13)

v ei  ei  V

0   min   ei   max  1 10

Journal Pre-proof where structural compliance J infill ( ρ, ) is the objective function, ρ is the vector of the element relative densities in the design area, U is the displacement vector, K ( ρ, ) is the global stiffness matrix, and F is the external load vector.  ei is the relative density of element e, while vei is its corresponding element volume. The design variable is range from ρmin to ρmax. The first constraint is the effective elastic tensor scaling law which is established in Section 2.1.2. The second constraint is the Hooke’s law equilibrium equation. The third constraint limits the total design volume to V. The final constraint is the minimum and criteria (OC) algorithm in this work. 2.2.2

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maximum design variables for the relative density. The optimization problem is solved by the optimality

Projecting information for the fine-scale lattice structures

In this work, the fine-scale lattice structure will be generated according to the field information decomposed from the above optimization results. And the field information mainly includes three variables of the density field, orientation field, and period field. The density  ij of each unit cell can be obtained directly from the optimized material gradient density distribution. The orientation  ij of each unit cell can T

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be calculated from the stress tensor σ ij =  ijx  ijy  ijxy  . Where  ijx and  ijy are the stress in the x and y directions, respectively, and  ijxy is the shear stress. Then, the orientation  ij can be calculated by the following equation:

 2 ijxy tan −1  x  − y 2 ij  ij

1

   

(14)

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 ij =

However, functionally graded microstructure materials in nature tend to have smaller features in its dense regions, while in its loose density regions have relative larger feature sizes. In addition to the characteristics of the human cancellous bone and dense bone in Fig. 1, the microstructure of bamboo also has a certain gradient scale features along the inner wall to the outer wall. Inspired by nature, here we indirectly establish

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a new design variable which is related to the density field  ij . In this work, this new design variable is called gradient period field ij . And it is defined here by dividing the density field  ij into n uniform sub-intervals.

 ij =  x1 , x 2 ,..., x n 

(15)

Then, the density interval of each subinterval xi can be expressed as:

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 x =   ,   , x =   , 2   ,..., x =  ( n − 1)  ,   2 n max   1  min       = (  max −  min ) n   s.t .

(16)

where  represents the average density interval. Next, different scales of lattice unit cell can be filled in different density intervals according to the already divided density intervals. Assuming that the lattice scale in the density interval x1 is 1 =  (i.e.,  max =  ), and the lattice scale of the n-th gradient interval is defined as: n =  / n

(17) 11

Journal Pre-proof Thus, the minimum scale of the lattice is  min =  / n . Thus, a global period field can be obtained according to the above definition of the lattice gradient scale. And, n is defined herein as a scale gradient or a periodic gradient. The following is indicated by the symbol p gradient . And the periodic gradient p gradient is defined as the value of the maximum cell size  max to the minimum cell size  min , that is, p gradient =  max /  min . As shown in Fig. 8a, the above optimization process is performed according to given

boundary conditions and load conditions. Then, the gradient density field  ij distribution of the coarse

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mesh can be obtained as shown in Fig. 8b, and the orientation field  ij distribution is shown in Fig. 8c, and Figs. 8d is the gradient period field ij , respectively. Later, the above fields (  ij ,  ij , ij ) can be projected to fine-scale lattice structures by using a corresponding mapping function which will be described c in the Section 2.3. Furthermore, the compliance of the optimized coarse mesh is referred as J infill , and the

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f compliance of the projected fine-scale lattice structure is referred as J infill in this work.

Fig. 8. Multi-scale field information obtained from the optimized coarse mesh. (a) Design domain with dimensions 30×30 mm and boundary conditions; (b) Density field with discrete coarse mesh 30×30 c =126.8 ); (c) Orientation field with each unit cells; (d) Period field with each interval areas. ( J infill

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2.3. Synthesize the Conformal Gradient Lattice Structures In this section, a compelling conformal synthesis approach for generating density gradient, orientation adaptive and period variable lattice structures is presented. The technique is a numerical method that allows us to manipulate any of the attributes of a periodic lattice structure. This technique is used to functionally grade the lattice structure without changing the size and shape of the unit lattices, keeping the lattice structure smooth and continuous. Moreover, this method can minimize distortion and prevents defects in the final lattice. Since it is not a coordinate transformation, it is easier to generate complex and arbitrary spatial gradient structures. The specific method is as follows. Synthesizing a density gradient lattice structure

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2.3.1

The periodic structure consists of repeated geometries, which are called unit cells. Each unit cell corresponds to one structural period, and these units are repeatedly arranged in space to construct one-dimensional, two-dimensional and three-dimensional periodic structures. Such periodicity can be seen as a periodic solid "wave" passing through space. In theory, almost any type of periodic lattice structures can be expressed via a complex Fourier series expansion [54]. For example, consider a 2D periodic structure in the x-y plane. This 2D structure is infinite in ±x and ±y with the unit cell dimensions λx and λy. Let us denote this by a periodic lattice function f lattice ( x , y ) that 12

Journal Pre-proof represents a field reaction of the 2D periodic structure. Moreover, the form of the function can be described as f lattice ( x + n  x , y + m  y ) = f lattice ( x , y )

n , m = 1, 2, 3...

(18)

Thus, the Fourier series expansion form of the 2D periodic lattice wave function f lattice ( x , y ) is

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  2 n   2 m   f lattice ( x , y ) =   a n , m exp  j  x+ y     n = − m = − y     x

(19)

where a n , m is the complex amplitude [55], and it can be calculated using a fast 2D Fourier transform r

(FFT), a n , m = FFT2D  f lattice ( x , y )  , and a wave vector function  n , m ( x , y ) can be defined from Eq. (19). r

 n,m ( x, y ) =

2 n

x

xˆ +

2 m

y

yˆ

(20)

where xˆ , yˆ are unit vectors in the x and y directions. For practical implementation, the expansion

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function is complex since the wave vector functions are complex. Thus, the original unit lattice cell is reconstructed by taking the real part of the summation in Eq. (19), and it can be expressed as E r r r r f lattice ( r ) = Re   a e exp  j e ( r ) gr    e =1 

(21)

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r In this equation, a finite set of series expansion is given (E=N×M, e=1,2,3…P), and r is position vector in the x-y plane. For the 3D lattice structure, the term in the z-direction can be expanded in the above equations.

For the 2D uniform lattice structure, to reconstruct the solid “wave” lattice structures, we can simplify the Eq. (21) to be a cosine wave function.

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r r r r 1 1 f lattice ( r ) = cos  ( r )gr  + 2 2

(22) r

To get a clear “solid-void” design, a Heaviside step function is employed, and a threshold function  ( r ) is established to control the fill fraction. Thus, the final solid lattice structure can be obtained by the following equation.

1 r  f lattice-solid ( r ) =   0

r r f lattice ( r )   ( r ) r r , f lattice ( r )   ( r )

r

 (r ) =

r 1 cos  ( r )  + 2 2

1

(23)

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As shown in Fig. 9, the density field information in Fig. 8b is projected into fine-scale lattice structures (the resolution is 1800×1800) by using square and square-x unit cell, respectively. So according to Eq. (22), the wave field can be generated is shown in Fig. 9a and 9c, and the color gradient of each sub-interval is represented as a level-set wave field from 0 to 1. Then, the target fill fraction results are obtained through Eq. (23), as shown in Fig. 9b and 9d. From the results, it can be seen that a very smooth lattice structure can be obtained. Further, from the value of the obtained compliance, it is understood that the stiffness of the infilled lattice structure obtained by direct density mapping is smaller than the optimized coarse mesh. Therefore, we need to do further orientation and period scale optimization, so that the obtained stiffness is consistent with the coarse mesh calculation results. 13

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2.3.2

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Fig. 9. Synthesizing density gradient lattice structures. (a) and (c) are wave fields for different unit cells; (b) and (d) are the corresponding fine-scale lattice structures. The compliance of (b) is 158.3, and (d) is 163.2. Synthesizing a conformal gradient lattice structure

The Eq. (22) does not hold when the orientation and period of the lattice structures are conformal varying. A unit cell cannot be square if the spatial variation in angles has to be satisfied. Here, we use a r

r

r

r

mapping function  , which maps the density  ( r ) , orientation  ( r ) and periodic p ( r ) = 2 /  ( r ) domain  p onto a periodic set in ¡

2

that describes the composite. Using this mapping function, we

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can reformulate Eq. (22) such that the lattice wave function can be described as r r 1 1 ,p f lattice ( r ) = cos  ( r )  + 2 2

(24)

Additionally, the wave vector function that adds spatial orientation and periodic variables is rewritten as r r r r r   p ( r ) = p ( r ) R  ( r )  gr     r  cos  ( r )    r    R  ( r )  =  r    sin  r ( )     

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



(25) r

The challenge is thus to find a suitable mapping vector function  ( r ) , such that each unit lattice in r

domain  p corresponds to the correct composite shape. Instead, the mapping vector function  ( r ) is r

r

r

r

r

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related to  p ( r ) through a gradient operation   ( r ) =   p ( r ) . For such an overdetermined system of r

linear equations, it usually has no exact solution, so the goal is to find the  ( r ) that best fit the equations. Moreover, the minimizing least-squares method is used to solve this problem (Eq. (26)). r r r r 2 1 arg min S (  ( r ) ) =    ( r ) −   p ( r ) d  r 2  (r )

(26)

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where S is the objective function, and this minimization problem has a unique solution and is given by solving the normal equation. Therefore, the final lattice structure equation with spatial density, orientation, and periodic variation is  r 1  , , p f lattice-solid (r ) =   0

r r ,p f lattice (r )    (r ) r r , ,p f lattice (r )    (r )

r

  (r ) =

1 2

r 1 cos  ( r )  + 2

(27)

Therefore, as long as the type of the unit lattice structure, the density field, the orientation field, and period field are provided. Then, the corresponding target lattice structure can be obtained according to the above technique. Fig. 10 shows the ability of the proposed synthesizing algorithm for the design domain in Fig. 8, and the density field, orientation field and period field are considered simultaneously. Here, the 14

Journal Pre-proof maximum unit size is set to 2 mm, and the minimum unit size is 2 mm, 1 mm, and 0.5 mm, respectively. Then, the corresponding wave fields are as shown in Figs. 10a~c. Moreover, according to the definition, the periodic gradient p gradient is 1, 2, 4. Then, the fine-scale lattice structures are synthesized as shown in Fig. 10d~f, respectively. As can be seen from the results of the compliance, they are significantly better than the results of Fig. 9b and d, and they are basically consistent with the calculation results of the coarse grid in Fig. 8b. Besides, the change in the periodic gradient also have a certain effect on the stiffness of the

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structure and will be discussed further in the experimental section.

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Fig. 10. Synthesizing conformal gradient lattice structures. (a) (b) and (c) are wave fields for different periodic gradient; (d) (e) and (f) are the corresponding fine-scale lattice structures. The compliance of (d) is 130.2, (e) is 129.9, and (f) is 127.1.

3. Numerical Analysis and Results

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In this section, to examine the effectiveness of the above-proposed method herein, different design cases with various parameters are analyzed, and comparison with existing lattice structure design methods are conducted. The entire structural design and optimization framework is implemented by MATLAB programming. The Young’s modulus for the solid material is 1.0, and the Poisson’s ratio is 0.3. Besides, all examples are run on a laptop computer with CPU Intel i7-6700HQ of 3.2 GHz, RAM of 16 GB, and software environment MATLAB 2018b. 3.1. Concurrent Design of a Cantilever Beam 3.1.1

Cantilever beam

The first numerical example is the typical cantilever beam design problem, which is used here to verify the effectiveness of the proposed method compared to conventional design approaches. The design domain with dimension and boundary conditions is illustrated in Fig. 11a. The volume fraction constraint is set to 15

Journal Pre-proof 0.5, and the density interval is 0.1 to 0.9. Firstly, the design domain is meshed into 80×40 coarse discretization uniform grids to solve the topology optimization problem based on homogenized mechanical properties of the lattice materials. Fig. 11b shows the density distribution under a coarse grid and from which mapping field information can be obtained. Then, the size scale of the x-type unit cell is determined (4mm), the periodic gradient is set to 1, then the optimization result is projected onto a fine-scale mesh composed of 1600 × 800 elements (Fig. 11c). As a result, the compliance of the optimized coarse mesh c f =61.28 , and the compliance of fine-scale projected lattice structure J infill =64.11 . Besides, topology J infill

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the optimal orientation of the unit cell in Fig. 11c is mainly distributed along the largest principal stress direction, and the density of the gradient change also satisfies the macroscopic distribution characteristics of the material. Another notable feature is that the material is distributed throughout the design domain and the lattices are continuous smooth transition in the density interval.

Fig. 11e~f are other design strategies, uniform lattice filling, classical topology optimization (SIMP), and direct lattice mapping without considering lattice mechanical properties (same resolution as Fig. 11c). The target volume fraction of these methods are also set to 0.5. The magnitudes of the compliance values for these results are compared separately. Compared to the SIMP method, the stiffness is slightly lower when

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the material is distributed throughout the design domain. However, compared with the uniform lattice structure filling and the mapping method without considering the material properties, the optimized results of our CGLS method have obvious advantages. In the case of considering only the magnitude of the stiffness value, it is true that direct topology optimization has an optimal solution. All numerical calculation results are recorded in Table. 1, and it can be seen that the CGLS method of this work consumes very little

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time, almost one tenth of the SIMP method, but the stiffness is basically the same. However, from the aspect of structural damage resistance and buckling resistance, our optimization method has advantages and

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will be further discussed in the following sections.

Fig. 11. Cantilever Beam. (a) Design domain with boundary conditions and dimensions 80×40 mm. (b) Density field with discrete coarse gird 80×40 (compliance: 61.28). (c) Projected conformal gradient fine-scale lattice structures (compliance: 64.11). (d) Uniform lattice structures (compliance: 207.74). (e) Structure optimized by classical topology optimization (compliance: 58.32). (f) Directly density mapping results (compliance: 79.87).

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Journal Pre-proof Table. 1 Performance and computational cost of the different optimization strategies for the cantilever beam Compliance (J)

Volume fraction

Time (s)

Resolution

Coarse mesh

61.28

0.5

56.7

80×40

CGLS

64.11

0.49

59.1

1600 × 800

Unform

207.74

0.5

1.2

1600 × 800

SIMP

58.32

0.5

576.8

400 × 200

GLS

79.97

0.49

57.4

1600 × 800

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3.1.2

Method

Periodical gradient

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In this paper, the density distribution of the lattice material and the optimal directional field are obtained directly by the optimization solution, but the size scale of the periodic unit is indirectly given by the density field. Therefore, it is necessary to discuss the effect of the size scale of the unit on the overall stiffness. Fig. 12 shows the projected results with different unit sizes. In all examples, the same boundary conditions as illustrated in Fig. 11a are applied. Among them, the unit cell sizes of Fig. 12a~c are the synthesized results with 6 mm, 4 mm, and 2 mm, respectively. And their periodic gradients are all 1 according to the definition of the periodic gradient. The results show that the intermediate scale (Fig. 12b) unit projected result has better stiffness than others (Fig. 12a and c). Since the cell size in the high-density region is too big that the integrity of the lattice cannot be preserved in Fig. 12a, thus the overall stiffness is reduced. On the other hand, the cell size in the lower density region is small that the unit structure has low stiffness, then the overall stiffness is relatively low in Fig. 12b. Therefore, it is necessary for different densities to correspond to a reasonable size scale lattice structure design. In Fig. 12e, the size of the minimum density corresponded unit cell is set to 4mm, then change the maximum density corresponded unit cell size. The compliance values for the different periodic gradients are then calculated and plotted as shown in Fig. 12f. As a result, it is understood that as the periodic gradient becomes larger, the compliance value of the structure gradually converges.

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Journal Pre-proof Fig. 12. (a) (b) and (c) are the projected fine-scale lattice structures with different unit cell size (compliance is 74.39, 64.11, 82.12 respectively). (d) is the period field with each interval areas. (e) is the projected fine-scale lattice structures with gradient period is 2 (compliance is 62.14). (f) is the compliance values for the different periodic gradients. 3.2. Concurrent Design of Bracket

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A second numerical example is a bracket object as shown in Fig. 13, which is referred from [34]. It is illustrated that the proposed anisotropic lattice structure optimization method works as well on the design domain with curve and holes. The boundary conditions and the dimension of the bracket design problem are illustrated in Fig. 13a. It contains curve boundaries so that unoptimized lattice structures are not conformal. The target volume fraction constraint is set 0.5, and the density interval is 0.1 to 0.9. Firstly, according to the bounding box of the bracket, the design domain meshes into 60×40 coarse discretization uniform grids. An optimization process is then performed to obtain the mapping information field of the material. Fig. 13b illustrates the calculated coarse mesh density field. Fig. 13c is a uniform lattice structure with the unit cell size is 2mm. Fig. 13d~f illustrates the optimized result are projected onto a fine-scale mesh composed of 1200 × 800 elements. And the size scale of the maximum density corresponded unit is set to 1, the size of the minimum density corresponded unit is set to 2. Numerical analysis suggests that the gradient lattice structures not only conformal with the curve boundaries, but the stiffness is about 3.5 stiffer than the uniform pattern.

Fig. 13. Anisotropic optimization on a curved design domain (a), and (b) is the coarse mesh optimized density field. The gradient lattice structures (d)~(f) is 3.5 times stiffer than the uniform pattern (c).

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3.3. Concurrent Design of a Wing Rib

Although the stiffness of the lattice structure is not as good as that of direct topology optimization, its structural robustness is an important reason for filling with a lattice structure. An advantage of multi-scale lattice structures is that their structures are subject to damage tolerance. For example, even if the part is partially broken, the lattice makes the whole still stiff. Moreover, local failure is easily generated in the actual application process, such as accidental collision, or corrosion of the structure itself, fatigue failure, and manufacturing error. In this section, according to the work of Wu [56], a simple local damage model is used to verify different optimization strategies, assuming that a fixed-size square damage region can be 18

Journal Pre-proof placed anywhere in the design area. The wing rib is an important part of the aircraft wing structure and is an essential component for the rigidity and safety of the entire aircraft. If the structure is subjected to local damage by an indeterminate load, it is important to maintain excellent stiffness.

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As shown in Fig. 14a, a simplified wing rib is presented here to verify the robustness of the optimized lattice structure. The design domain has three circular voids to represent inspection holes which is for the passage of electrical cables through the wing, and a square void representing the presence of a spar. The structure is fixed from the left and right edges, and a compression load is applied on both the upper and lower edge of the rib. A 160×40 coarse mesh is used for optimization, and volume fraction constraint is set to 0.45. The finally fine-scale mesh is set to 4800×1200 elements. The results of different optimization strategies are shown in Fig. 14b~d, which are truss structure, SIMP-based topology optimization, and our CGLS method. To simulate the effect of damage on the stiffness of the structure, the square damaged area (movable deficiency) gradually moves in the vertical direction in each case. Then, the compliance of each case corresponding to different damage locations is calculated and compared (Fig. 14e). First, the direct topology-optimized structure has the greatest stiffness when there is no damage, but its stiffness drops rapidly when it is broken. However, the structural stiffness of the adaptive lattice filling method in this work is less affected by the damage. For the uniform truss structure, although it also has minor stiffness fluctuations, its compliance is approximately three times greater than the proposed method herein. Therefore, it can be proved that the proposed method has better structural robustness.

Fig. 14. Wing rib optimization problem. (a) schematic diagram of a wing rib; (b) truss structure; (c) the classical topology optimization method (SIMP); (d) our CGLS method. (e) the compliance value curve as the damage area moves.

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Journal Pre-proof 3.4. Buckling Resistance for Optimized MBB Beams

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In this section, a three-point bending experiment is conducted to compare the mechanical behavior of buckling resistance of optimized MBB (Messerschmitt-Bölkow-Blohm) beams with our proposed method, uniform lattice structure and classical SIMP method. The design domain with a geometric dimension of 180 ×30 mm and boundary conditions is illustrated in Fig. 15a. The volume fraction constraint is set to 0.4, and the density interval is 0.1 to 0.9. The design domain is then discretized into coarse meshes (180×30) for homogenized lattice material based topology optimization. Moreover, the optimized result is projected onto a fine-scale mesh with 1800×300 elements. The maximum cell size is set to 4mm, and the periodic gradients synthesized by the CGLS method are set as 1, 2, and 3, respectively. For comparison, Fig. 15b shows the uniform lattice structures with a cell size of 4 mm and a density of 0.4, and Fig. 15c shows the traditional SIMP optimized results with the same density. Fig. 15d is the coarse mesh density field distribution from the numerical analysis of our CGLS method. And Fig. 15e is the lattice optimization that only considers density field synthesis. Fig. 15f~h are the conformal lattice materials synthesized by the CGLS algorithm with periodic gradients of 1, 2 and 3. The numerical results of different design strategies are shown in Table. 2. It can be seen from the results that the stiffness of the conformal lattice structure synthesized by the CGLS method is very close to the stiffness of the coarse mesh. And the CGLS method with a periodic gradient of 3 is slightly lower than the conventional SIMP method. Additionally, from the comparison of numerical results, the CGLS method is about 5 times more rigid than the Uniform method. In terms of efficiency, the CGLS method is significantly faster than the SIMP method. Moreover, it can be seen from the numerical results that the stiffness of the periodic gradient of 3 is the best.

Fig. 15. MBB beams optimization problem. (a) Design domain with boundary conditions; (b) Uniform lattice structures; (c) Classical topology optimization (SIMP); (d) Coarse mesh from CGLS method; (e) Synthesized results only considering density field (GLS); (f)~(h) CGLS method with periodic gradients of 1, 2 and 3.

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Journal Pre-proof Table. 2 Performance and computational cost of the different optimization strategies for the MBB beams Compliance (J)

Volume fraction

Time (s)

Resolution

Uniform

675

0.4

2.2

1800×300

SIMP

129

0.4

125.2

180×30

Coarse Mesh

116

0.4

9.4

180×30

GLS

207

0.39

12.4

1800×300

CGLS-1

148

0.42

12.1

1800×300

CGLS-2

136

0.38

19.8

1800×300

CGLS-3

133

0.39

22.3

1800×300

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Method

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Next, the 2D SIMP, Uniform, GLS, SGLS-1, SGLS-2, and SGLS-3 optimized results are thickened to 10 mm in the z-axis direction, respectively. Then, three-dimensional solid geometric models can be obtained (Fig. 16a). Moreover, the Form 2 (Formlabs Inc., USA) desktop 3D printer is used to produce the samples by SLA technology, and the manufactured results are shown in Fig. 16b. Where the elastic modulus of the base material is E = 2.5 GPa, Poisson ratio v = 0.33. Then, according to Fig. 16c, the optimization results of different design strategies were tested by three-point bending experiments, and the loading speed was set to 10 mm/min. The "load-displacement" curve is plotted according to the experimental results in Fig. 16d. Consistent with the expected results, the SIMP method has the best structural stiffness in the early stage of loading. The stiffness of the CGLS method is slightly lower than that of the SIMP method. The stiffness of the GLS method and the Uniform method are weak. Moreover, the Uniform method is significantly less rigid than all other design results. However, with the gradual increase of the load, the SIMP optimized solid structure yields quickly. In contrast, the CGLS method of this paper, although slightly buckling in the early stage, can maintain good carrying capacity with the increase of load. And the CGLS-3 model with a periodic gradient of 3 performs best. Additionally, although Uniform lattice also has good resistance to yield, its stiffness is relatively weak. In summary, it can be seen that the CGLS algorithm proposed in this work can not only make the overall structure retain better stiffness, but also its yield resistance performance is better than other design strategies.

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Fig. 16. (a) Geometric models for different design strategies; (b) additive manufactured results; (c) three-point bending experiments; (d) "load-displacement" curves for different design strategies

4. Conclusions

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In this work, a conformal anisotropic design and optimization framework is developed to optimize the components with lattice structures for additive manufacturing. The optimized lattice structure has three notable features. First, the density of the distributed lattice structure is gradually changing which adaptive the stiffness of the component. Second, the orientation of each unit lattice structure is along the principal stress direction, which results in the maximum mechanical properties of the lattice structures. Third, the size-scale of the unit lattice structure in different density regions are different, which in turn helps improve the overall stiffness. The mechanical properties of the corresponding lattice materials are calculated by the homogenization method and integrated into the topology optimization process. For the lattice structure projecting from coarse mesh to fine-scale mesh, a synthesizing framework based on Fourier transform is presented. Finally, several structures are numerically simulated and discussed from different perspectives. It is found that, in contrast to classical uniform lattice design method and single-scale topology optimization with solid material, not only is the stiffness of the structure optimized, but also the optimized design result has robustness structure for damage tolerant and buckling resistance ability. In future work, we will expand our technique to the study of particular functionalities such as thermal management, energy absorption, and multi-physics coupling optimization problem. Acknowledgment

This work is supported by the funding of the National Natural Science Foundation of China (No. 51775273), National Defense Pre-Research Foundation of China (No. 6141B07090119, 61409230305), National Defence Basic Scientific Research Program of China (No. JCKY2018605C010), Frontiers of 22

Journal Pre-proof Science and Technology Program of China (No. 1816312ZT00406301). References

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Graphic abstract:

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Highlights

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An anisotropic design and optimization method is proposed for conformal gradient lattice structures. The optimized lattice structures exhibit conformal gradient density, adaptive orientation, variable period. These anisotropic conformal lattice structures are much stiffer than uniform and directly mapped designs.

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Conflict of interest statement

We wish to draw the attention of the Editor to the following facts which may be considered as potential conflicts of interest and to significant financial contributions to this work.

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We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled, “Anisotropic Design and Optimization of Conformal Gradient Lattice Structures”.

We confirm that the manuscript has been read and approved by all named authors and that there are no other persons who satisfied the criteria for authorship but are not listed. We further confirm that the order of authors listed in the manuscript has been approved by all of us.

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We confirm that we have given due consideration to the protection of intellectual property associated with this work and that there are no impediments to publication, including the timing of publication, with respect to intellectual property. In so doing we confirm that we have followed the regulations of our institutions

Author list:

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concerning intellectual property.

Dawei Li (First author), [email protected]

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Wenhe Liao, [email protected]

Ning Dai (Corresponding author), [email protected]

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Yi Min Xie, [email protected]

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