# A Method to Determine Core Design Problems and a Corresponding Solution Strategy

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## Abstract

**:**

## 1. Introduction

## 2. Establishment and Transformation of Hierarchical Representation Model of Design Problem

## 3. Method to Determine Core Problems Based on Analytic Network Process

#### 3.1. Construction of Network Layer and Control Layer for Design Problem

#### 3.2. Calculation of Importance of Root Causes and Determination of Core Problems

## 4. Product Design Process Based on Solving the Core Problem

## 5. Case Studies

#### 5.1. Importance Analysis of Root Causes of the Coolant Flow Distribution Device

#### 5.2. Solving the Root Causes Sequentially Based on Importance

#### 5.3. Conceptual Design Scheme Generation

#### 5.4. Simulation Analysis

- (1)
- Turbulent model: K-epsilon (2 eqn)-standard
- (2)
- Fluid material: water (liquid)
- (3)
- Wall: no slip
- (4)
- Inlet (velocity inlet): velocity magnitude: 5.084 m/s;turbulent intensity: 1.755%; hydraulic diameter: 1200 mm
- (5)
- Outlet (pressure outlet): gauge pressure: 0 pa
- (6)
- Convergence absolute criterion: ${10}^{-3}$
- (7)
- Number of meshes: 5.37 million

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 7.**Conceptual design scheme of coolant flow distribution device: (1) core support plate; (2) radial support key; (3) flow equalizing plate; (4) flow distribution cylinder; (5) supporting column; (6) energy-absorbing device; (7) pressure vessel bottom head; (I) inverted cone structure; (II) cap structure.

**Figure 11.**Velocity vectors of cross section of lower chamber. The velocity of the coolant around the upper surface of the core support plate is slightly greater than that in the middle. Although the velocity is not exactly the same on the upper surface, the velocity of the coolant in the middle is basically the same. What is more, the change of velocity consistency from the middle to the periphery is very small. This proves the effectiveness of the flow distribution effect of the design scheme. The direction of coolant flow in the lower chamber is stable, the variation degree of the flow direction is slow, and there is only a few of vortices. Structurally, the structure of the lower chamber is simplified by reducing the number of supporting columns and detachable connectors. Therefore, the design scheme has the characteristics of good flow distribution effect, a few of vortices, and simple structures.

**Table 1.**The ratio scale [32].

Ratio Scale | Meaning (If: Element 1 Is A; Element 2 Is B; The Third Party Element Is C) |
---|---|

1 | A and B have the same effect on C |

3 | The effect of A on C is slightly larger than the effect of B on C |

5 | The effect of A on C is medially larger than the effect of B on C |

7 | The effect of A on C is highly larger than the effect of B on C |

9 | The effect of A on C is extremely larger than the effect of B on C |

2,4,6,8 | Median value of the above adjacent scale |

Reciprocal of the above scale | The ratio of the effect of A on C to the effect of B on C is ${E}_{AB}^{C}$; The ratio of the effect of B on C to the effect of A on C is $1/{E}_{AB}^{C}$ |

**Table 2.**Judgment matrix (${A}_{m}^{{i}_{j}}$) of elements in cluster (m) based on element (${i}_{j}$ ).

${\mathit{i}}_{\mathit{j}}$ | ${\mathit{m}}_{1}$ | ${\mathit{m}}_{2}$ | $\dots $ | ${\mathit{m}}_{\mathit{n}}$ | Weight Vector ${\mathit{w}}_{\mathit{m}}^{\left({\mathit{i}}_{\mathit{j}}\right)}$ |
---|---|---|---|---|---|

${m}_{1}$ | ${e}_{11}$ | ${e}_{12}$ | $\dots $ | ${e}_{1n}$ | ${w}_{{m}_{1}}^{\left({i}_{j}\right)}$ |

${m}_{2}$ | ${e}_{21}$ | ${e}_{22}$ | $\dots $ | ${e}_{2n}$ | ${w}_{{m}_{2}}^{\left({i}_{j}\right)}$ |

$\vdots $ | $\vdots $ | $\vdots $ | $\ddots $ | $\vdots $ | $\vdots $ |

${m}_{n}$ | ${e}_{n1}$ | ${e}_{n2}$ | $\dots $ | ${e}_{nn}$ | ${w}_{{m}_{n}}^{\left({i}_{j}\right)}$ |

$\mathit{B}{\mathit{p}}_{1}$ | $\mathit{B}{\mathit{p}}_{1}$ | $\mathit{B}{\mathit{p}}_{2}$ | $\dots $ | $\mathit{B}{\mathit{p}}_{\mathit{N}}$ | Weight Vector |
---|---|---|---|---|---|

$B{p}_{1}$ | ${e}_{11}$ | ${e}_{12}$ | $\dots $ | ${e}_{1n}$ | ${a}_{11}$ |

$B{p}_{2}$ | ${e}_{21}$ | ${e}_{22}$ | $\dots $ | ${e}_{2n}$ | ${a}_{12}$ |

$\vdots $ | $\vdots $ | $\vdots $ | $\ddots $ | $\vdots $ | $\vdots $ |

$B{p}_{N}$ | ${e}_{n1}$ | ${e}_{n2}$ | $\dots $ | ${e}_{nn}$ | ${a}_{1N}$ |

**Table 4.**Judgment matrix (${A}_{3}^{{1}_{2}}$) of elements in cluster ($B{p}_{3}$) based on element (${1}_{2}$).

${1}_{2}.$ | $\mathit{R}{\mathit{c}}_{1}^{3}$ | $\mathit{R}{\mathit{c}}_{2}^{3}$ | $\mathit{R}{\mathit{c}}_{3}^{3}$ | $\mathit{R}{\mathit{c}}_{4}^{3}$ | Weight Vector ${\mathit{w}}_{3}^{\left({1}_{2}\right)}$ |
---|---|---|---|---|---|

$R{c}_{1}^{3}$ | 1 | 6 | 3 | 1/2 | 0.33393 |

$R{c}_{2}^{3}$ | 1/6 | 1 | 1/2 | 1/6 | 0.06607 |

$R{c}_{3}^{3}$ | 1/3 | 2 | 1 | 1/3 | 0.13214 |

$R{c}_{4}^{3}$ | 2 | 6 | 3 | 1 | 0.46786 |

${\mathit{C}}_{\mathit{T}}$ | $\mathit{R}{\mathit{c}}_{1}^{1}$ | $\mathit{R}{\mathit{c}}_{2}^{1}$ | $\mathit{R}{\mathit{c}}_{1}^{2}$ | $\mathit{R}{\mathit{c}}_{2}^{2}$ | $\mathit{R}{\mathit{c}}_{3}^{2}$ | $\mathit{R}{\mathit{c}}_{1}^{3}$ | $\mathit{R}{\mathit{c}}_{2}^{3}$ | $\mathit{R}{\mathit{c}}_{3}^{3}$ | $\mathit{R}{\mathit{c}}_{4}^{3}$ |
---|---|---|---|---|---|---|---|---|---|

$R{c}_{1}^{1}$ | 0.25000 | 0.33333 | 0.16667 | 0.14286 | 0.12500 | 0.12500 | 0.50000 | 0.16667 | 0.14286 |

$R{c}_{2}^{1}$ | 0.75000 | 0.66667 | 0.83333 | 0.85714 | 0.87500 | 0.87500 | 0.50000 | 0.83333 | 0.85714 |

$R{c}_{1}^{2}$ | 1.00000 | 0.54545 | 0.59489 | 0.28571 | 0.63275 | 0.30915 | 1.00000 | 1.00000 | 0.28571 |

$R{c}_{2}^{2}$ | 0.00000 | 0.27273 | 0.27661 | 0.57143 | 0.19240 | 0.58126 | 0.00000 | 0.00000 | 0.57143 |

$R{c}_{3}^{2}$ | 0.00000 | 0.18182 | 0.12850 | 0.14286 | 0.17485 | 0.10959 | 0.00000 | 0.00000 | 0.14286 |

$R{c}_{1}^{3}$ | 0.00000 | 0.33393 | 0.33333 | 0.83333 | 0.33333 | 0.15385 | 0.12500 | 0.33333 | 0.54545 |

$R{c}_{2}^{3}$ | 0.00000 | 0.06607 | 0.00000 | 0.00000 | 0.00000 | 0.07692 | 0.25000 | 0.33333 | 0.18182 |

$R{c}_{3}^{3}$ | 0.00000 | 0.13214 | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 0.00000 |

$R{c}_{4}^{3}$ | 1.00000 | 0.46786 | 0.66667 | 0.16667 | 0.66667 | 0.76923 | 0.62500 | 0.33333 | 0.27273 |

${\mathit{C}}_{\mathit{T}}$ | $\mathit{B}{\mathit{p}}_{1}$ | $\mathit{B}{\mathit{p}}_{2}$ | $\mathit{B}{\mathit{p}}_{3}$ |
---|---|---|---|

$B{p}_{1}$ | 0.40000 | 0.26429 | 0.30915 |

$B{p}_{2}$ | 0.40000 | 0.64556 | 0.58126 |

$B{p}_{3}$ | 0.20000 | 0.09015 | 0.10959 |

${\mathit{C}}_{\mathit{T}}$ | $\mathit{R}{\mathit{c}}_{1}^{1}$ | $\mathit{R}{\mathit{c}}_{1}^{1}$ | $\mathit{R}{\mathit{c}}_{1}^{2}$ | $\mathit{R}{\mathit{c}}_{2}^{2}$ | $\mathit{R}{\mathit{c}}_{3}^{2}$ | $\mathit{R}{\mathit{c}}_{1}^{3}$ | $\mathit{R}{\mathit{c}}_{2}^{3}$ | $\mathit{R}{\mathit{c}}_{3}^{3}$ | $\mathit{R}{\mathit{c}}_{4}^{3}$ |
---|---|---|---|---|---|---|---|---|---|

$R{c}_{1}^{1}$ | 0.10000 | 0.13333 | 0.04405 | 0.03776 | 0.03304 | 0.03864 | 0.15458 | 0.05153 | 0.04416 |

$R{c}_{2}^{1}$ | 0.30000 | 0.26667 | 0.22024 | 0.22653 | 0.23125 | 0.27051 | 0.15458 | 0.25763 | 0.26499 |

$R{c}_{1}^{2}$ | 0.40000 | 0.21818 | 0.38404 | 0.18445 | 0.40848 | 0.17970 | 0.58126 | 0.58126 | 0.16608 |

$R{c}_{2}^{2}$ | 0.00000 | 0.10909 | 0.17857 | 0.36889 | 0.12420 | 0.33787 | 0.00000 | 0.00000 | 0.33215 |

$R{c}_{3}^{2}$ | 0.00000 | 0.07273 | 0.08296 | 0.09222 | 0.11288 | 0.06370 | 0.00000 | 0.00000 | 0.08304 |

$R{c}_{1}^{3}$ | 0.00000 | 0.06679 | 0.03005 | 0.07513 | 0.03005 | 0.01686 | 0.01370 | 0.03653 | 0.05977 |

$R{c}_{2}^{3}$ | 0.00000 | 0.01321 | 0.00000 | 0.00000 | 0.00000 | 0.00843 | 0.02740 | 0.03653 | 0.01992 |

$R{c}_{3}^{3}$ | 0.00000 | 0.02643 | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 0.00000 |

$R{c}_{4}^{3}$ | 0.20000 | 0.09357 | 0.06010 | 0.01503 | 0.06010 | 0.08430 | 0.06849 | 0.03653 | 0.02989 |

${\mathit{C}}_{\mathit{T}}$ | $\mathit{R}{\mathit{c}}_{1}^{1}$ | $\mathit{R}{\mathit{c}}_{2}^{1}$ | $\mathit{R}{\mathit{c}}_{1}^{2}$ | $\mathit{R}{\mathit{c}}_{2}^{2}$ | $\mathit{R}{\mathit{c}}_{3}^{2}$ | $\mathit{R}{\mathit{c}}_{1}^{3}$ | $\mathit{R}{\mathit{c}}_{2}^{3}$ | $\mathit{R}{\mathit{c}}_{3}^{3}$ | $\mathit{R}{\mathit{c}}_{4}^{3}$ |
---|---|---|---|---|---|---|---|---|---|

$R{c}_{1}^{1}$ | 0.06797 | 0.06797 | 0.06797 | 0.06797 | 0.06797 | 0.06797 | 0.06797 | 0.06797 | 0.06797 |

$R{c}_{2}^{1}$ | 0.24441 | 0.24441 | 0.24441 | 0.24441 | 0.24441 | 0.24441 | 0.24441 | 0.24441 | 0.24441 |

$R{c}_{1}^{2}$ | 0.28464 | 0.28464 | 0.28464 | 0.28464 | 0.28464 | 0.28464 | 0.28464 | 0.28464 | 0.28464 |

$R{c}_{2}^{2}$ | 0.19897 | 0.19897 | 0.19897 | 0.19897 | 0.19897 | 0.19897 | 0.19897 | 0.19897 | 0.19897 |

$R{c}_{3}^{2}$ | 0.07708 | 0.07708 | 0.07708 | 0.07708 | 0.07708 | 0.07708 | 0.07708 | 0.07708 | 0.07708 |

$R{c}_{1}^{3}$ | 0.04730 | 0.04730 | 0.04730 | 0.04730 | 0.04730 | 0.04730 | 0.04730 | 0.04730 | 0.04730 |

$R{c}_{2}^{3}$ | 0.00536 | 0.00536 | 0.00536 | 0.00536 | 0.00536 | 0.00536 | 0.00536 | 0.00536 | 0.00536 |

$R{c}_{3}^{3}$ | 0.00646 | 0.00646 | 0.00646 | 0.00646 | 0.00646 | 0.00646 | 0.00646 | 0.00646 | 0.00646 |

$R{c}_{4}^{3}$ | 0.06781 | 0.06781 | 0.06781 | 0.06781 | 0.06781 | 0.06781 | 0.06781 | 0.06781 | 0.06781 |

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**MDPI and ACS Style**

Xie, Y.; Li, W.; Luo, Y.; Li, Y.; Li, S.
A Method to Determine Core Design Problems and a Corresponding Solution Strategy. *Symmetry* **2019**, *11*, 576.
https://doi.org/10.3390/sym11040576

**AMA Style**

Xie Y, Li W, Luo Y, Li Y, Li S.
A Method to Determine Core Design Problems and a Corresponding Solution Strategy. *Symmetry*. 2019; 11(4):576.
https://doi.org/10.3390/sym11040576

**Chicago/Turabian Style**

Xie, Yuanming, Wenqiang Li, Yin Luo, Yan Li, and Song Li.
2019. "A Method to Determine Core Design Problems and a Corresponding Solution Strategy" *Symmetry* 11, no. 4: 576.
https://doi.org/10.3390/sym11040576