# Substructure-Based Topology Optimization for Symmetric Hierarchical Lattice Structures

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. The Static Condensation of Substructure

## 3. Substructure-Based Symmetric Microstructure Model

^{2}attributed. Here, we assume that a condensed stiffness matrix at any density can be expressed as follows:

## 4. Symmetric Structures Design Method

#### 4.1. The Topology Optimization Model

#### 4.2. Sensitivity Analysis

^{−3}.

## 5. Numerical Example

#### 5.1. Clamped Ends Beam

#### 5.2. The Cantilever Design

#### 5.3. D Simply Supported Cube Design

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

SIMP | Solid isotropic material with penalization |

BESO | Bi-directional evolutionary structural optimization |

MOR | Model Order Reduction |

FEA | Finite element analysis |

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**Figure 4.**Optimized results for Case A lattices of ${N}_{x}\times {N}_{y}=20\times 4$ discretization resolutions.

**Figure 5.**Optimized results for Case B lattice of ${N}_{x}\times {N}_{y}=20\times 4$ discretization resolutions.

**Figure 6.**Optimized results for case A lattice of ${N}_{x}\times {N}_{y}=30\times 6$ discretization resolutions.

**Figure 7.**Optimized results for Case B lattice of ${N}_{x}\times {N}_{y}=30\times 6$ discretization resolutions.

**Figure 8.**Optimized results for Case A lattice of ${N}_{x}\times {N}_{y}=60\times 12$ discretization resolutions.

**Figure 9.**Optimized results for Case B lattice of ${N}_{x}\times {N}_{y}=60\times 12$ discretization resolutions.

**Figure 15.**Axonometric and zoom in view of design results for case B with a different penalty factor.

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**MDPI and ACS Style**

Wu, Z.; Xiao, R.
Substructure-Based Topology Optimization for Symmetric Hierarchical Lattice Structures. *Symmetry* **2020**, *12*, 678.
https://doi.org/10.3390/sym12040678

**AMA Style**

Wu Z, Xiao R.
Substructure-Based Topology Optimization for Symmetric Hierarchical Lattice Structures. *Symmetry*. 2020; 12(4):678.
https://doi.org/10.3390/sym12040678

**Chicago/Turabian Style**

Wu, Zijun, and Renbin Xiao.
2020. "Substructure-Based Topology Optimization for Symmetric Hierarchical Lattice Structures" *Symmetry* 12, no. 4: 678.
https://doi.org/10.3390/sym12040678