# Wave Exciting Force Maximization of Truncated Cylinders in a Linear Array

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Problem’s Definition

## 3. Numerical Modeling

#### 3.1. Hydrodynamic Model

^{th}cylinder and Equation (14) expresses the zero velocity condition on the bottom of each cylinder.

^{th}cylinder and a global Cartesian coordinate system $\left(X,Y\right)$ fixed on the sea bottom, the incident wave potential ${\varphi}_{I}$ with respect to $\left({x}_{q},{y}_{q}\right)$ is expressed as:

^{th}cylinder is given by:

^{th}cylinder.

^{th}cylinder of the array and $\rho $ is the water density. Equations (24) and (25) result in robust analytical formulas. Full details of these formulas can be found in [10,25].

^{th}cylinder, with $j\ne q$.

#### 3.2. Genetic Algorithms Solver

#### 3.3. Optimization Numerical Process (ONP)

## 4. ONP Validation

## 5. Results and Discussions

#### 5.1. Examined Cases

^{®}Xeon

^{®}Silver 4110 CPU@ 2.1 GHz 2.1 GHz (2 processors).

#### 5.2. Optimum Solutions for EC1.1–EC1.8

^{th}cylinder, the successive increase of ${l}_{array}/b$ in the case of $\beta ={0}^{\mathrm{o}}$ leads to larger values of ${X}_{q}/b$, $q=1,\dots ,9$, especially for $q\ge 5$. For $\beta ={90}^{\mathrm{o}}$, this trend is only observed for the last three cylinders of the array, namely, for $q\ge 7$. Moreover, under the action of head waves a linear correlation between $q$ and ${X}_{q}/b,q=1,\dots ,\text{}9,$ for each examined ${l}_{array}/b$ value is observed. This is also shown in Figure 6, where ${X}_{q}/{l}_{array},q=1,\dots ,\text{}9$, is plotted as a function of$q$. By implementing linear fitting to the aforementioned data, a linear trend line along with its equation is obtained and it is additionally included in Figure 6. It is evident that the formula shown in Figure 6 does not correspond to a direct outcome of the solution of the examined optimization problem (Equations (2), (4)–(7)), but to an approximation of the relevant optimum solutions obtained for $\beta ={0}^{\mathrm{o}}$, by utilizing the present ONP. In the case of $\beta ={90}^{\mathrm{o}}$ (Figure 5b), a linear dependence of ${X}_{q}/b$ with $q$ exists only for EC1.3 and EC1.7 (${l}_{array}/b$ = 40 and 32, respectively) restricting the use of any possible fitting scheme of the data related to $\beta ={90}^{\mathrm{o}}$.

#### 5.3. Optimum Solutions for EC2.1–EC2.10

^{th}and $j$

^{th}is realized by placing one or more adjacent bodies at small distances from the $q$

^{th}and/or the $j$

^{th}cylinder. It is also worth noting that for both examined ${l}_{array}/b$ values, the largest maxima of ${F}_{Z}^{subtot}$ occur for EC2.9 and EC2.10 (Table 4), where maximization of the total heave exciting force applied on CYL4 and CYL6 of the array is implemented. For the rest of the examined pairs of cylinders, the ${F}_{Z}^{subtot}$ maxima successively decrease and the smallest values are observed for cases EC2.3 and EC2.4 (maximization of total heave exciting force applied on the outer cylinders, CYL1 and CYL9, of the array).

^{th}and $j$

^{th}cylinders, the consideration of a larger ${l}_{array}/b$ for maximizing the corresponding ${F}_{Z}^{subtot}$, leads to a more linear relationship between ${X}_{q}/b$ and q.

^{th}cylinder, the largest ${F}_{Z}^{\left(q\right)}$ value occurs when ${F}_{Z}^{subtot}$ is maximized for the pair of cylinders that includes the specific $q$

^{th}body. For the outer cylinders of the array ($q=1$ and $9$), the second largest value occurs for EC2.5 and EC2.6, where ${F}_{Z}^{subtot}$ is maximized for the pair of the two adjacent cylinders (i.e., $q=2$ and $8$). Same conclusions can be derived for CYL2 and CYL8, where the second largest value occurs for EC2.3 and EC2.4 (maximization of ${F}_{Z}^{subtot}$ for the pair of the two outer cylinders, i.e., $q=1$ and $9$). For the rest of the cylinders, a clear trend can not be observed. However, it is evident that the consideration of different pairs of cylinders for maximizing the corresponding ${F}_{Z}^{subtot}$ has a direct impact on the values of ${F}_{Z}^{\left(q\right)}$ for each $q$

^{th}cylinder of the array.

## 6. Conclusions

^{th}cylinder of the array.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Geometry of the examined linear arrangement and definition of basic quantities: (

**a**) $X-Y$ plane and (

**b**)$Y-Z$ plane.

**Figure 2.**Flow chart of the developed optimization numerical process (ONP) in the present paper for calculating the optimum arrangement of cylinders in a linear array, in terms of maximizing the heave exciting force.

**Figure 4.**Effect of ${l}_{array}/b$ on the maximum non-dimensional values of ${F}_{Z}^{tot}$ (cases EC1.1–EC1.8): (

**a**) $\beta ={0}^{\mathrm{o}}$ and (

**b**) $\beta ={90}^{\mathrm{o}}$.

**Figure 5.**Effect of ${l}_{array}/b$ on the optimum ${X}_{q}/b$ values for EC1.1–EC1.8. (

**a**) $\beta ={0}^{\mathrm{o}}$ and (

**b**) $\beta ={90}^{\mathrm{o}}$.

**Figure 8.**Optimum values ${X}_{q}/b$ as a function of $q$ for EC2.1–EC2.10. (

**a**) ${l}_{array}/b=40$ and (

**b**) ${l}_{array}/b=60$.

**Figure 9.**Optimization cases EC2.1–EC2.10. (

**a**) Largest values of ${F}_{Z}^{\left(q\right)},q=1,\dots ,9$ , for each case; and (

**b**) ${F}_{Z}^{tot}$ as resulted from each optimum layout configuration.

**Figure 10.**${F}_{Z}^{\left(q\right)},q=1,\dots ,9$, for the optimum layout configurations of EC2.1–EC2.10. (

**a**) ${l}_{array}/b=40$ and (

**b**) ${l}_{array}/b=60$.

**Table 1.**Comparison of the optimum solutions between the present ONP and Kagemoto [11].

$\mathit{Q}$ | ONP | Kagemoto [11] | ||||
---|---|---|---|---|---|---|

${\mathit{X}}_{\mathbf{2}}\text{}\left(\mathbf{m}\right)$ | ${\mathit{X}}_{\mathbf{3}}\text{}\left(\mathbf{m}\right)$ | ${\mathit{F}}_{\mathit{X}}^{\mathit{t}\mathit{o}\mathit{t}}$ | ${\mathit{X}}_{\mathbf{2}}\text{}\left(\mathbf{m}\right)$ | ${\mathit{X}}_{\mathbf{3}}\text{}\left(\mathbf{m}\right)$ | ${\mathit{F}}_{\mathit{X}}^{\mathit{t}\mathit{o}\mathit{t}}$ | |

$3$ | $1.170$ | - | $0.162$ | $1.168$ | - | $0.16271$ |

$4$ | $1.153$ | $2.258$ | $0.002$ | $1.150$ | $2.251$ | $\approx 0$ |

Case | Objective Function | ${\mathit{l}}_{\mathit{o}\mathit{u}\mathit{t}}/\mathit{b}$ | ${\mathit{l}}_{\mathit{a}\mathit{r}\mathit{r}\mathit{a}\mathit{y}}/\mathit{b}$ | $\mathit{\beta}(\mathbf{\xb0})$ |
---|---|---|---|---|

EC1.1 | Maximization of ${F}_{Z}^{tot}$ with $Q=9$ in Equation (26) | $4$ | $40$ | $0$ |

EC1.2 | $6$ | $60$ | ||

EC1.3 | $4$ | $40$ | $90$ | |

EC1.4 | $6$ | $60$ | ||

EC1.5 | $0$ | $32$ | $0$ | |

EC1.6 | $48$ | |||

EC1.7 | $32$ | $90$ | ||

EC1.8 | $48$ | |||

EC2.1 | Maximization of ${F}_{Z}^{subtot}$ for $q=5$ and $j=0$ in Equation (27) | $4$ | $40$ | $90$ |

EC2.2 | $6$ | $60$ | ||

EC2.3 | Maximization of ${F}_{Z}^{subtot}$ for $q=1$ and $j=9$ in Equation (27) | $4$ | $40$ | |

EC2.4 | $6$ | $60$ | ||

EC2.5 | Maximization of ${F}_{Z}^{subtot}$ for $q=2$ and $j=8$ in Equation (27) | $4$ | $40$ | |

EC2.6 | $6$ | $60$ | ||

EC2.7 | Maximization of ${F}_{Z}^{subtot}$ for $q=3$ and $j=7$ in Equation (27) | $4$ | $40$ | |

EC2.8 | $6$ | $60$ | ||

EC2.9 | Maximization of ${F}_{Z}^{subtot}$ for $q=4$ and $j=6$ in Equation (27) | $4$ | $40$ | |

EC2.10 | $6$ | $60$ |

**Table 3.**Optimum values of ${X}_{q}/b,q=1,\dots ,\text{}9$ and corresponding maximum ${F}_{Z}^{tot}/\left(\rho g{b}^{2}A\right)$ values for EC1.1–EC1.8.

Case | ${\mathit{X}}_{1}/\mathit{b}$ | ${\mathit{X}}_{2}/\mathit{b}$ | ${\mathit{X}}_{3}/\mathit{b}$ | ${\mathit{X}}_{4}/\mathit{b}$ | ${\mathit{X}}_{5}/\mathit{b}$ | ${\mathit{X}}_{6}/\mathit{b}$ | ${\mathit{X}}_{7}/\mathit{b}$ | ${\mathit{X}}_{8}/\mathit{b}$ | ${\mathit{X}}_{9}/\mathit{b}$ | ${\mathit{F}}_{\mathit{Z}}^{\mathit{t}\mathit{o}\mathit{t}}/\left(\mathit{\rho}\mathit{g}{\mathit{b}}^{2}\mathit{A}\right)$ |
---|---|---|---|---|---|---|---|---|---|---|

EC1.1 | 0.10 | 8.70 | 10.90 | 18.20 | 20.70 | 27.40 | 30.10 | 36.30 | 39.50 | 6.841 |

EC1.2 | 1.10 | 9.70 | 18.40 | 21.30 | 30.70 | 40.40 | 47.30 | 49.90 | 59.90 | 7.137 |

EC1.3 | 0.50 | 3.20 | 12.70 | 15.60 | 22.90 | 26.00 | 33.60 | 36.00 | 38.10 | 10.212 |

EC1.4 | 0.00 | 2.80 | 5.30 | 46.70 | 50.00 | 52.70 | 55.00 | 57.40 | 60.00 | 10.017 |

EC1.5 | 0.00 | 2.20 | 9.30 | 11.80 | 19.60 | 21.80 | 27.50 | 30.00 | 32.00 | 6.075 |

EC1.6 | 0.00 | 7.10 | 9.70 | 19.10 | 26.50 | 29.00 | 35.40 | 38.30 | 48.00 | 6.968 |

EC1.7 | 0.00 | 2.90 | 5.70 | 18.90 | 21.90 | 24.40 | 26.80 | 29.10 | 32.00 | 10.035 |

EC1.8 | 0.00 | 2.20 | 4.70 | 10.90 | 13.40 | 15.80 | 43.50 | 45.90 | 48.00 | 10.090 |

**Table 4.**Optimum values of ${X}_{q}/b,q=1,\dots ,\text{}9$ and the corresponding maximum ${F}_{Z}^{subtot}$/$\left(\rho g{b}^{2}A\right)$ values for EC2.1–EC2.10.

Case | ${\mathit{X}}_{1}/\mathit{b}$ | ${\mathit{X}}_{2}/\mathit{b}$ | ${\mathit{X}}_{3}/\mathit{b}$ | ${\mathit{X}}_{4}/\mathit{b}$ | ${\mathit{X}}_{5}/\mathit{b}$ | ${\mathit{X}}_{6}/\mathit{b}$ | ${\mathit{X}}_{7}/\mathit{b}$ | ${\mathit{X}}_{8}/\mathit{b}$ | ${\mathit{X}}_{9}/\mathit{b}$ | ${\mathit{F}}_{\mathit{Z}}^{\mathit{s}\mathit{u}\mathit{b}\mathit{t}\mathit{o}\mathit{t}}/\left(\mathit{\rho}\mathit{g}{\mathit{b}}^{2}\mathit{A}\right)$ |
---|---|---|---|---|---|---|---|---|---|---|

EC2.1 | 1.80 | 3.90 | 10.90 | 12.90 | 15.40 | 18.10 | 27.80 | 36.70 | 39.00 | 1.5854 |

EC2.2 | 4.20 | 6.20 | 14.70 | 24.70 | 27.10 | 29.60 | 39.10 | 49.00 | 59.00 | 1.6066 |

EC2.3 | 1.40 | 4.70 | 13.40 | 15.40 | 23.10 | 25.30 | 32.30 | 34.30 | 37.20 | 2.5921 |

EC2.4 | 3.00 | 6.30 | 15.90 | 25.20 | 34.10 | 36.30 | 44.70 | 54.20 | 57.50 | 2.5742 |

EC2.5 | 0.00 | 2.20 | 5.00 | 14.10 | 21.70 | 23.90 | 32.10 | 34.90 | 37.20 | 2.7583 |

EC2.6 | 5.80 | 8.40 | 11.20 | 20.90 | 31.10 | 40.80 | 50.80 | 53.60 | 56.30 | 2.7670 |

EC2.7 | 0.60 | 2.60 | 5.40 | 8.60 | 10.80 | 31.30 | 34.60 | 37.50 | 39.50 | 2.7262 |

EC2.8 | 2.90 | 11.90 | 14.30 | 17.10 | 25.90 | 42.20 | 44.80 | 47.20 | 56.40 | 2.8065 |

EC2.9 | 1.50 | 3.50 | 11.70 | 14.50 | 22.40 | 25.30 | 27.80 | 36.20 | 38.20 | 2.8033 |

EC2.10 | 1.90 | 11.80 | 21.00 | 23.70 | 31.10 | 33.80 | 36.20 | 45.50 | 55.40 | 2.8618 |

**Table 5.**${F}_{Z}^{\left(q\right)}$ and ${F}_{Z}^{tot}$ for the optimum solutions of EC2.1–EC2.10 (values given are non-dimensional in terms of $\rho g{b}^{2}A$).

Case | ${\mathit{F}}_{\mathit{Z}}^{\left(1\right)}$ | ${\mathit{F}}_{\mathit{Z}}^{\left(2\right)}$ | ${\mathit{F}}_{\mathit{Z}}^{\left(3\right)}$ | ${\mathit{F}}_{\mathit{Z}}^{\left(4\right)}$ | ${\mathit{F}}_{\mathit{Z}}^{\left(5\right)}$ | ${\mathit{F}}_{\mathit{Z}}^{\left(6\right)}$ | ${\mathit{F}}_{\mathit{Z}}^{\left(7\right)}$ | ${\mathit{F}}_{\mathit{Z}}^{\left(8\right)}$ | ${\mathit{F}}_{\mathit{Z}}^{\left(9\right)}$ | ${\mathit{F}}_{\mathit{Z}}^{\mathit{t}\mathit{o}\mathit{t}}$ |
---|---|---|---|---|---|---|---|---|---|---|

EC2.1 | 0.8226 | 0.9252 | 0.6471 | 1.0907 | 1.5854 | 0.9715 | 0.8162 | 0.9043 | 0.8791 | 8.6421 |

EC2.2 | 0.8574 | 0.9521 | 0.8326 | 1.0353 | 1.6066 | 1.0241 | 0.9176 | 0.8459 | 0.8345 | 8.9062 |

EC2.3 | 1.2923 | 0.9728 | 0.9957 | 0.7696 | 0.8527 | 0.9442 | 0.7182 | 1.2133 | 1.2998 | 9.0586 |

EC2.4 | 1.2844 | 0.9606 | 0.7900 | 0.7714 | 0.8406 | 0.9169 | 0.7725 | 0.9369 | 1.2897 | 8.5631 |

EC2.5 | 1.1283 | 1.3630 | 0.9451 | 0.8343 | 0.9146 | 1.0193 | 0.9563 | 1.3954 | 1.1155 | 9.6716 |

EC2.6 | 1.0812 | 1.3888 | 0.9813 | 0.8426 | 0.8016 | 0.8232 | 0.9905 | 1.3782 | 1.0650 | 9.3523 |

EC2.7 | 0.8450 | 0.9402 | 1.3724 | 1.0471 | 0.7847 | 0.9590 | 1.3538 | 0.9844 | 0.8380 | 9.1245 |

EC2.8 | 1.0085 | 1.0164 | 1.4672 | 0.9892 | 0.9476 | 1.0915 | 1.3393 | 0.9905 | 0.9495 | 9.7998 |

EC2.9 | 0.9191 | 0.9416 | 0.9061 | 1.2613 | 1.0189 | 1.5420 | 0.9170 | 1.0359 | 0.8775 | 9.4195 |

EC2.10 | 0.9050 | 0.9024 | 0.9690 | 1.2821 | 0.9843 | 1.5796 | 1.0082 | 0.9658 | 0.9381 | 9.5347 |

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**MDPI and ACS Style**

Michailides, C.; Loukogeorgaki, E.; Chatjigeorgiou, I.K.
Wave Exciting Force Maximization of Truncated Cylinders in a Linear Array. *Energies* **2020**, *13*, 2400.
https://doi.org/10.3390/en13092400

**AMA Style**

Michailides C, Loukogeorgaki E, Chatjigeorgiou IK.
Wave Exciting Force Maximization of Truncated Cylinders in a Linear Array. *Energies*. 2020; 13(9):2400.
https://doi.org/10.3390/en13092400

**Chicago/Turabian Style**

Michailides, Constantine, Eva Loukogeorgaki, and Ioannis K. Chatjigeorgiou.
2020. "Wave Exciting Force Maximization of Truncated Cylinders in a Linear Array" *Energies* 13, no. 9: 2400.
https://doi.org/10.3390/en13092400