# Design and Optimization of Lattice Structures: A Review

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- The manuscript especially distinguishes the differences between lattices and foams and honeycombs from aspects of structure and properties. Additionally, the manuscript divides lattice structures into cellular structures.
- (2)
- The manuscript reviews the definition and classification, properties and applications, and manufacturing methods of lattice structures. Especially, there are some shortcomings in the definition of lattice structures in some literature. Therefore, based on the understanding of lattice structures, the definition is given new content, which makes the definition more perfect.
- (3)
- The most excellent work is to reclassify the lattice structures, which is divided into uniform lattice structure and non-uniform lattice structure. The design and optimization methods of lattice structure are reviewed comprehensively over existing literature.
- (4)
- The manuscript also emphasizes the lattice structure of non-uniform Poisson’s ratio, which is rarely involved in other literature. The lattice structures with different Poisson’s ratios are discussed in detail.

## 2. Lattice Structures

#### 2.1. The Definition and Classification of Lattice Structure

#### 2.2. Properties and Applications of Lattice Structures

#### 2.2.1. Properties

^{3}. Mahshid et al. [88] fabricated structure samples including solid, hollow, and lattice structure, and compared the strength of structures by compression test. The deformation of samples after compression test is shown in Figure 4. The result indicated that though the strength of lattice structure was lower than that of solid structure, it can still meet the application requirements, and reduced the use of materials, and achieve lightweight design.

^{3}, and by the uniform lattice structure was 2.6 ± 0.2 MJ/m

^{3}. The total cumulative energy absorption per unit volume was higher in functionally graded lattice than in uniform lattice (as shown in Figure 6).

#### 2.2.2. Applications

#### 2.3. Manufacturing Methods of Lattice Structures

## 3. Design and Optimization of Uniform Lattice Structures

#### 3.1. Unit Cell Design Based on Geometric Wireframe

#### 3.2. Unit Cell Design Based on Mathematical Algorithm

_{p}:

_{G}:

_{D}:

_{IWP}:

_{S}:

_{FRD}:

_{x}, y = 2πY/L

_{y}, z = 2πZ/L

_{z}, and L

_{x}, L

_{y}and L

_{z}are the unit cell sizes in the X, Y, and Z directions.

^{3}, and satisfies the Equation (7):

^{3}R)

#### 3.3. Unit Cell Design Based on Topology Optimization

## 4. Design and Optimization of Non-Uniform Lattices

#### 4.1. Non-Uniform Lattice Structures Based on Functional Gradient Design

#### 4.2. Non-Uniform Lattice Structure Based on Structural Optimization

## 5. Summary and Overview

- (1)
- In the current researches of lattice structures, the design of cell topological shape is insufficient, and most of the literature are still studying the existing cell structures. The performance of cell structure is directly related to lattice structure, so it is necessary to design cell topology with special performance. Leonardi et al. [206] made a preliminary exploration of the potentialities of H-C and H-C+D lattice structures.
- (2)
- In the topology optimization of lattice structures, most researcher are still focused on the strut’s sizes. The single optimization is not conducive to the design of lattice structures. Only from the point of view of optimizing struts, the lattice structures cannot meet the specific requirements. Kazemi et al. [179] proposed the geometry projection method, to optimizing the multi-material lattice structures. This is a new method to optimize lattice structure, which enriches the method of lattice topology optimization.

- (1)
- Multi-material lattice structures. Because additive manufacturing technology adopts the manufacturing method of material layer printing, different materials can be used to manufacture lattice structure in order to meet the demand of using performance. For example, with the wide application of composite materials, a variety of composite materials can be mixed for designing lattice structures.
- (2)
- Multi-structure lattice structures. At present, the research of Poisson’s ratio structure has changed from uniform Poisson’s ratio to non-uniform Poisson’s ratio. From the former positive Poisson’s ratio structure to the present negative Poisson’s ratio structure, the design and optimization of multiple structures in lattice structures are bound to be the future research trend.
- (3)
- Multi-method to design the lattice structures. With the development of CAD technology and programming language, the design of lattice structures in the future will rely heavily on computer technology to develop lattice structure databases with different topology of unit cells. At the same time, equipped with detailed performance parameters using standards for various industries.
- (4)
- Micro-scale lattice structures. Because of different design methods and fabricating methods (such as additive manufacturing technology) of lattice structures, the scales of lattice structures are divided into macro size (>0.5 mm), micro size (0.1 mm–5 mm), micro size (micron level: 100 nm–100 um; nano level: <100 nm) [74,207]. Micro-nano scale lattice structures can be realized in the future, and will be a hot spot in research field.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A3.**Unit cells [144].

**Figure A8.**Octet-truss cell [151].

**Figure A10.**Rectangular prism: (

**a**) C01; (

**b**) C02; (

**c**) C03; (

**d**) C04; hexagonal structures: (

**e**) H01; (

**f**) H02; (

**g**) H03; (

**h**) H04 [153].

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**Figure 2.**Classification of lattice structures: (

**a**) random lattice structures; (

**b**) periodic lattice structures; (

**c**) periodic lattice structures [23].

**Figure 3.**Gradient lattice structures [76].

**Figure 6.**Cumulative energy absorption per unit volume vs. compressive strain curves of uniform and graded structures [90].

**Figure 7.**FEA simulations to estimate displacement of six lattice structures [92]: (

**a**) model 1; (

**b**) model 2; (

**c**) model 3; (

**d**) model 4; (

**e**) model 5; (

**f**) model 6.

**Figure 8.**Apparent thermal conductivity against actual volume fraction [98].

**Figure 11.**Extrusion and electro-discharge machining method [122].

**Figure 12.**3D unit cell structure library [157]: tetrahedron, octahedron, cube, vector (

**top row**, from left to right); icosahedron, dodecahedron, tetrakaidecahedron, triacontahedron (

**bottom row**, from left to right).

**Figure 13.**3D texture mapping method [157].

**Figure 17.**Topology optimization of uniform lattice structure [183]: (

**a**) unit cell; (

**b**) lattice structure.

**Figure 18.**Gradient lattice structure generated by top-down Voronoi-tessellation method [186].

**Figure 19.**The gradient lattice structure topologies [190]: (

**a**) gradient discrete; (

**b**) gradient increasing; (

**c**) gradient decreasing.

**Figure 20.**Non-uniform lattice structure based on SGM method [27].

**Figure 21.**Non-uniform lattice structure based on SMS method [191].

**Figure 22.**Non-uniform lattice structure based on HOC method [205].

Type of Cellular Structures | Applications |
---|---|

Foams | Energy absorbers [1,2,8,9], Filters [9,29], Silencers [9], Flame arresters [9], Heaters and heat exchangers [9,29], Eelectro-chemical devices [9,29]. |

Honeycombs | Energy absorbers [1,2,15,30,31,32,33], Biomedical implants [30,31,32], Filters [34,35], Sensors [34,35], Actuators [34,35], Vibration absorber or damper [36,37,38,39]. |

Lattices | Energy absorbers [1,2,40,41,42], Heaters and heat exchangers [23,27,43,44,45,46], Engine hood [47], Biomedical implants [48,49,50,51,52,53,54,55], Wings [56], Gas turbine engine fan blades [57], Vibration absorber [58,59,60,61], Robotic system [62], Spacecraft and aircraft structures (i.e., fuselage, rib) [63,64,65]. |

AM Process | Material | Lattice Structures | Reference |
---|---|---|---|

SLM | Ti-6Al-4V | [130] | |

[131] | |||

[132] | |||

[133] | |||

[134] | |||

[135] | |||

AlSi10Mg | [136] | ||

[137] | |||

| [138] | ||

CoCr alloy | [139] | ||

316L stainless steel | [140] | ||

| [141] | ||

SLS | Polyamide powder | [142] | |

Nylon | [143] | ||

Paracetamol | [144] | ||

SLA | Polymer | [145] | |

[146] | |||

[147] | |||

FDM | Acrylonitrile Butadiene Styrene (ABS) | [148] | |

[149] | |||

PLA filaments | [150] | ||

EBM | Ti-6Al-4V | [151] | |

[152] | |||

[153] |

Condition | Interpretation |
---|---|

f(x, y, z) = 0 | On surface |

f(x, y, z) < 0 | Inside |

f(x, y, z) > 0 | Outside |

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**MDPI and ACS Style**

Pan, C.; Han, Y.; Lu, J. Design and Optimization of Lattice Structures: A Review. *Appl. Sci.* **2020**, *10*, 6374.
https://doi.org/10.3390/app10186374

**AMA Style**

Pan C, Han Y, Lu J. Design and Optimization of Lattice Structures: A Review. *Applied Sciences*. 2020; 10(18):6374.
https://doi.org/10.3390/app10186374

**Chicago/Turabian Style**

Pan, Chen, Yafeng Han, and Jiping Lu. 2020. "Design and Optimization of Lattice Structures: A Review" *Applied Sciences* 10, no. 18: 6374.
https://doi.org/10.3390/app10186374