# Multi-Objective Design for Critical Supporting Parameters of Vacuum-Insulated Glazing with a Case Study

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Construction and Basic Assumptions

- (1)
- Elastomer assumption: when the load on the glass does not exceed the limit, it exhibits elastic properties;
- (2)
- Rigid supporting: the elastic modulus of supporting pillars is much larger than that of glass pane, and the glass deformation at the contact position is much larger than that of the supporting pillars. It can be assumed that the supporting pillars are regarded as rigid supports without deformation;
- (3)
- Negligible edge displacement: the edge is designed as a fixed constraint after sealing, so that the edge has negligible displacement.

## 3. Mechanical Model Establishment

#### 3.1. Stress Analysis

- (1)
- The bending stress of the glass pane;
- (2)
- The contact stress in the inner surface of the glass pane at the contact part with supporting pillars;
- (3)
- The stress of the edge sealing part of the VIG.

#### 3.2. Bending Stress of the Glass Substrate

_{0}from atmospheric pressure and supporting force F from the supporting pillar, the maximum superimposed stress occurs on the upper surface of the glass at the supporting point:

_{1,max}is the maximum superimposed stress (MPa); q

_{0}is atmospheric pressure (MPa); a is the pillar separation of the square unit (mm); h is thickness of the glass substrate (mm); ν is the Poisson ratio; r is the supporting pillar radius (mm); and β

_{1}and β

_{2}are structural coefficients.

_{2,max}is the maximum bending stress (MPa) and α

_{1}is the structural coefficient.

#### 3.3. Contact Stress between the Glass Substrate and Supporting Pillars

_{3,max}is the maximum contact tensile stress (MPa).

#### 3.4. Stress in the Edge Sealing Part of VIG

_{4}) was located in the outer edge. The inner edge of the edge sealing part was selected as the origin, and the force moment at this point was deduced. Each supporting pillar and atmospheric pressure contribute half of the force moment to the long and short sides.

_{4}is the maximum line tension of the edge sealing part (MPa) and m is the edge sealing width of VIG (mm).

_{l}= M

_{1}− M

_{2}),

_{4}) is mainly affected by the supporting force from the outermost supporting pillar (F

_{1}) and the structural parameters (m and ε). Obviously, the smaller the F

_{1}, the greater the f

_{4}. Therefore, the limit condition was considered, that is, when F

_{1}= 0, f

_{4}is the maximal value.

#### 3.5. Deformation of VIG

_{1}is the maximum bending deformation of the glass substrate (mm) and E is the elastic modulus of the glass substrate (MPa).

_{2}is the compression deformation (mm).

## 4. Multi-Objective Optimization Design

^{*}is the set of positive integers; g

_{j}(X) is the inequality constraint function; and h

_{k}(X) is the equality constraint function.

#### 4.1. Objective Function

_{2}= r

_{min}= [f

_{1}(X), f

_{2}(X)].

#### 4.2. Variable Analysis

#### 4.3. Objective Function

_{1,max}≤ [σ

_{1}], f

_{2,max}≤ [σ

_{1}]

_{1}] is the strength of the glass substrate under permanent stress, MPa.

_{3,max}) should not exceed the local strength:

_{3,max}≤ [σ

_{2}]

_{2}] is the local strength of the glass substrate, MPa.

_{4,max}≤ [σ

_{3}], ε ≤ a

_{3}] is the strength of the edge sealing area, MPa.

_{1}+ 2w

_{2}≤ h

_{2}

_{2}is the height of the supporting pillar (mm).

#### 4.4. Solution for Objective Function

## 5. Case Simulation and Verification

#### 5.1. Experimental Design

^{−3}density, a 72 GPa elastic modulus and a 0.24 Poisson ratio, consistent with Table 2. The supporting pillar was made of stainless steel with a density of 7800 kg m

^{−3}, an elastic modulus of 200 GPa and a Poisson ratio of 0.3.

#### 5.2. Simulation Method

#### 5.3. Results and Analysis

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Simplified square unit and its force diagram. Where a is the pillar separation of the square unit, q

_{0}is the atmospheric pressure, r is the supporting pillar radius and F is the support force from the supporting pillar.

**Figure 3.**Schematic diagram of the force unit, the width of force unit is b, and supporting pillars are located at the midpoint in the width direction.

**Table 1.**Coefficients when rectangular flat-plate functioned by uniform load [16].

a/b | 1.0 | 1.2 | 1.4 | 1.6 | 1.8 | 2.0 |

β_{1} | 0.1386 | 0.1794 | 0.2094 | 0.2286 | 0.2406 | 0.2472 |

β_{2} | −0.238 | −0.078 | 0.011 | 0.053 | 0.068 | 0.067 |

**Table 2.**Structure parameters of VIG from reference [30].

Structure Parameters | Abbreviations | Value | Unit |
---|---|---|---|

VIG size | L × L | 500 × 500 | mm × mm |

Glass substrate thickness | h | 5 | mm |

Vacuum gap/pillar height | h_{2} | 0.3 | mm |

Edge sealing width | m | 10 | mm |

Atmosphere pressure | q_{0} | 0.1 | MPa |

Constant coefficient | α_{1} | 0.8719 | |

β_{1} | 0.1386 | ||

β_{2} | −0.238 | ||

Glass density | 2500 | kg m^{−3} | |

Glass elastic modulus | 72 | GPa | |

Glass Poisson ratio | ν | 0.24 |

Treatments | Supporting Pillar Radius (mm) | Supporting Pillar Spacing (mm) |
---|---|---|

R3a63 | 0.3 | 63 |

R3a50 “^{1}” | 0.3 | 50 |

R2a63 | 0.2 | 63 |

^{1}Supporting pillar radius and spacing were consistent with reference [30].

**Table 4.**The supporting stress and deformation of the glass substrate under different supporting pillar radii and spacing distances. R3a63, with a 0.3 mm supporting pillar radius and 63 mm spacing distance; R3a50, with a 0.3 mm supporting pillar radius and 50 mm spacing distance; R2a63, with a 0.2 mm supporting pillar radius and 63 mm spacing distance.

Treatments | Strain Distribution and Maximum Deformation | Stress Distribution and Maximum Stress |
---|---|---|

R3a63 r = 0.3 mm a = 63 mm | ω _{max} = 18.413 μm | σ _{max} = 88.379 MPa |

R3a50 r = 0.3 mm a = 50 mm | ω _{max} = 8.2336 μm | σ _{max} = 74.674 MPa |

R2a63 r = 0.2 mm a = 63 mm | ω _{max} = 19.768 μm | σ _{max} = 88.549 MPa |

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**MDPI and ACS Style**

Zhang, Y.; Yuan, W.; Han, L.; Zhang, R.; Xi, X. Multi-Objective Design for Critical Supporting Parameters of Vacuum-Insulated Glazing with a Case Study. *Appl. Sci.* **2022**, *12*, 7504.
https://doi.org/10.3390/app12157504

**AMA Style**

Zhang Y, Yuan W, Han L, Zhang R, Xi X. Multi-Objective Design for Critical Supporting Parameters of Vacuum-Insulated Glazing with a Case Study. *Applied Sciences*. 2022; 12(15):7504.
https://doi.org/10.3390/app12157504

**Chicago/Turabian Style**

Zhang, Yifu, Wei Yuan, Lianjie Han, Ruihong Zhang, and Xiaobo Xi. 2022. "Multi-Objective Design for Critical Supporting Parameters of Vacuum-Insulated Glazing with a Case Study" *Applied Sciences* 12, no. 15: 7504.
https://doi.org/10.3390/app12157504