# Finite-Element-Mesh Based Method for Modeling and Optimization of Lattice Structures for Additive Manufacturing

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## Abstract

**:**

## 1. Introduction

## 2. Lattice Structure Configuration

#### 2.1. Finite Element Mesh Based Modeling Method

- Base mesh: Base mesh is the initial finite element mesh, which is obtained by meshing the given geometry.
- Base points: Base points are the nodes of the initial finite element mesh.
- Boundary bars: Boundary bars are the bar on element edge. The end points of boundary bars are base points.
- Derivative points: Derivative points are the extra nodes introduced to construct the additional bars. The positions of the derivative points are interpolated by the coordinates of base points.
- Derivative bars: Derivative bars are the bars made by connecting base points and derivative points.
- Primitive lattice cells: Primitive lattice cells are the unit cells constructed by boundary bars, as shown in Figure 1.
- Derivative lattice cells: Derivative lattice cells are the unit cell constructed by boundary bars and derivative bars, as shown in Figure 1.

#### 2.2. Modification

## 3. Problem Statement and Formulation

#### 3.1. A Brief Overview of MIST

- Case (a) if ${V}_{k}>{V}_{cons}\to \{\begin{array}{l}{t}_{{\mathrm{min}}_{k+1}}={t}_{k}\\ {t}_{{\mathrm{max}}_{k+1}}={t}_{{\mathrm{max}}_{k}}\end{array}$

- Case (b) if ${V}_{k}<{V}_{cons}\to \{\begin{array}{l}{t}_{{\mathrm{min}}_{k+1}}={t}_{{\mathrm{min}}_{k}}\\ {t}_{{\mathrm{max}}_{k+1}}={t}_{k}\end{array}$

_{k}is the total volume of the structure at kth iteration step. V

_{cons}is the volume constraint.

#### 3.2. Size Optimization Algorithm Based on MIST Method

_{i}is the cross-sectional area of bar number i, $f(\mathit{A})$ is objective function, Φ is the response function for MIST, F is load vector, K is overall stiffness matrix, U is displacement vector, A

_{l}is the lower bound of the cross-sectional area, A

_{u}is the upper bound of the cross-sectional area, and ${V}_{cons}$ is the volume constraint.

_{i}is the length of bar number i, F

_{i}is the axial force of bar number i, E is Young’s modulus, and A

_{i}is the cross-sectional area of bar number i.

## 4. Numerical Results

^{5}and v = 0.3, respectively. For the convenience of description, the design variable of cross section A is replaced by the circular radius r. The relationship between r and A is ${A}_{i}=\pi \cdot {{r}_{i}}^{2};{A}_{l}=\pi \cdot {{r}_{l}}^{2};{A}_{u}=\pi \cdot {{r}_{u}}^{2}$.

#### 4.1. Numerical Examples for 2-Dimensional Structures

#### 4.2. Numerical Examples for Three-Dimensional Structures

## 5. Experimental Validation

#### 5.1. Models for the Test

#### 5.2. Mechanical Test and Result

_{ini}= (2.79 ± 0.01) × 10

^{3}N/mm and S

_{opt}= (3.12 ± 0.01) × 10

^{3}N/mm for the initial and optimized hollow round platform, respectively. The measured stiffness values clearly demonstrate the superior stiffness of the optimized structure over the initial structure, with a significant improved global stiffness of 11.83%. Here, experiments are conducted only for comparing stiffness between initial and optimized lattice structures, which is in general a standard requirement for most engineering applications.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 17.**Final optimized structure of the hollow round platform: (

**a**) three-dimensional stereogram; and (

**b**) vertical view.

**Figure 18.**Analysis results: (

**a**) von Mises stress for initial structure; (

**b**) displacement for initial structure; (

**c**) von Mises stress for optimized hollow round platform; and (

**d**) displacement for optimized hollow round platform.

**Figure 21.**Final optimized structure of the horn structure: (

**a**) three-dimensional stereogram; and (

**b**) vertical view.

**Figure 23.**3D printed samples of: (

**a**) initial lattice hollow round platform; and (

**b**) optimized lattice hollow round platform.

Element Type | Triangle | Quadrangle | Tetrahedron | Hexahedron |
---|---|---|---|---|

Element and nodes | ||||

Primitive cell | ||||

Derivative cell | ||||

Number of derivative points | 1 | 1 | 1 | 1 |

Interpolation format | $\begin{array}{l}{x}_{Der}=\frac{1}{3}{\displaystyle \sum _{i=1}^{3}{x}_{i}}\\ {y}_{Der}=\frac{1}{3}{\displaystyle \sum _{i=1}^{3}{y}_{i}}\\ {z}_{Der}=\frac{1}{3}{\displaystyle \sum _{i=1}^{3}{z}_{i}}\end{array}$ | $\begin{array}{l}{x}_{Der}=\frac{1}{4}{\displaystyle \sum _{i=1}^{4}{x}_{i}}\\ {y}_{Der}=\frac{1}{4}{\displaystyle \sum _{i=1}^{4}{y}_{i}}\\ {z}_{Der}=\frac{1}{4}{\displaystyle \sum _{i=1}^{4}{z}_{i}}\end{array}$ | $\begin{array}{l}{x}_{Der}=\frac{1}{4}{\displaystyle \sum _{i=1}^{4}{x}_{i}}\\ {y}_{Der}=\frac{1}{4}{\displaystyle \sum _{i=1}^{4}{y}_{i}}\\ {z}_{Der}=\frac{1}{4}{\displaystyle \sum _{i=1}^{4}{z}_{i}}\end{array}$ | $\begin{array}{l}{x}_{Der}=\frac{1}{8}{\displaystyle \sum _{i=1}^{8}{x}_{i}}\\ {y}_{Der}=\frac{1}{8}{\displaystyle \sum _{i=1}^{8}{y}_{i}}\\ {z}_{Der}=\frac{1}{8}{\displaystyle \sum _{i=1}^{8}{z}_{i}}\end{array}$ |

${x}_{Der},\text{\hspace{0.17em}}{y}_{Der},\text{\hspace{0.17em}}{z}_{Der}$ are the Cartesian co-ordinates of derivative points ${x}_{i},\text{\hspace{0.17em}}{y}_{i},\text{\hspace{0.17em}}{z}_{i}$ are the Cartesian co-ordinates of node number i |

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**MDPI and ACS Style**

Chen, W.; Zheng, X.; Liu, S.
Finite-Element-Mesh Based Method for Modeling and Optimization of Lattice Structures for Additive Manufacturing. *Materials* **2018**, *11*, 2073.
https://doi.org/10.3390/ma11112073

**AMA Style**

Chen W, Zheng X, Liu S.
Finite-Element-Mesh Based Method for Modeling and Optimization of Lattice Structures for Additive Manufacturing. *Materials*. 2018; 11(11):2073.
https://doi.org/10.3390/ma11112073

**Chicago/Turabian Style**

Chen, Wenjiong, Xiaonan Zheng, and Shutian Liu.
2018. "Finite-Element-Mesh Based Method for Modeling and Optimization of Lattice Structures for Additive Manufacturing" *Materials* 11, no. 11: 2073.
https://doi.org/10.3390/ma11112073