# Surrogate-Based Optimization Using an Open-Source Framework: The Bulbous Bow Shape Optimization Case

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## Abstract

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## 1. Introduction

- How to efficiently and accurately simulate the physics involved.
- How to modify the geometry or mesh.
- How to search the optimal configuration.
- How to interface the applications involved in the optimization loop.

## 2. Description of the Optimization Framework and Optimization Strategy

## 3. Description of the CFD Model

#### 3.1. Reference Geometry and Modeling Assumptions

- All simulations were performed at model scale (1/20).
- No propeller nor appendages were modeled (bare hull).
- Only half of the hull was simulated (we used symmetry).
- A fixed trim condition of zero degrees was imposed.
- All the simulations were conducted in calm water conditions and no incoming waves.
- The thermophysical properties of the working fluids are constant (refer to Table 2).
- The simulations were conducted at the same Froude number as in the towing tank experiments.

#### 3.2. Computational Domain and Boundary Conditions

#### 3.3. Numerical Schemes

## 4. Calibration and Validation of the Solver

## 5. Results and Discussion

## 6. Conclusions and Future Perspectives

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Optimization loop. Notice that it only takes one single geometry. During the whole optimization loop, data are continuously collected.

**Figure 2.**Left: undeformed geometry. The green sphere represents the control point used to deform the geometry; this control point can move in the plane XZ. The surface region within the selection box is free to deform. Right: two deformed geometries.

**Figure 4.**Top: analytical solution. Bottom-left: surrogate model using a full-factorial experiment with 64 training points. Bottom-right: surrogate model using a space-filling experiment with 30 training points (Latin hypercube sampling, LHS).

**Figure 5.**Scatterplot matrix of the high-dimensional Rosenbrock function space exploration study ($d=6$). The Spearman correlation is shown in the upper triangular part of the matrix. In the diagonal of the matrix, the histograms showing the data distribution are displayed. In the lower triangular part of the matrix, the data distribution is shown using scatterplots. In the last row of the matrix plot, the response of the quantity of interest (QoI) in function of the design variables is illustrated, together with a quadratic regression model.

**Figure 7.**Sketch showing the characteristic lengths used to define the geometry parameters. Adapted from Reference [20].

**Figure 9.**Comparison of the normalized resistance as a function of the iteration number for the coarse and fine meshes. In the legend, LTS (local time-stepping) corresponds to the steady solution, and GTS (global time-stepping, i.e., a time-step identical for all control volumes) corresponds to the transient solutions.

**Figure 10.**Comparison of the normalized resistance as a function of the Froude number. The numbers next to the markers indicate the percentage error between the numerical results and the experimental data.

**Figure 11.**Sampling plan used in this study. The square represents the original geometry, whereas the circles represent the different geometry variations. Every point in the sampling plan represents a location where a high-fidelity computation was computed. All the simulations were conducted at a Froude number of 0.294.

**Figure 12.**Left: surrogate model constructed using Kriging interpolation. Right: surrogate model constructed using quadratic polynomial interpolation. In the image, the uniform spaced points represent the locations where the high-fidelity computations were computed. The square represents the starting geometry and the circles represent the new geometries. In both images, the color scale represents the resistance variation in reference to the original geometry (where negative values indicate resistance reduction, and positive values indicate resistance increment).

**Figure 13.**Left: the × symbol represents the global minimum, and the + symbol represents the local minimum. The square represents the starting geometry and the circles represent the new geometries. Right: the two squares represent two different starting points of the optimization algorithm. The yellow circles represent the path followed by the gradient-based algorithm when starting from the topmost position, whereas the green circles represent the path followed by the gradient-based algorithm when starting from the bottom-most position. In both images, the color scale represents the resistance variation in reference to the original geometry (where negative values indicate resistance reduction, and positive values indicate resistance increment).

**Figure 14.**Improved surrogate using infilling. The square represents the starting geometry and the circles represent the new geometries. The triangles represent the infill points used. The color scale represents the resistance variation in reference to the original geometry (where negative values indicate resistance reduction, and positive values indicate resistance increment).

**Figure 15.**Bulbous bow shapes. Left: original shape. Middle: shape obtained at global minimum. Right: shape obtained at local minimum.

**Table 1.**Outcome of the optimization study of the high-dimensional Rosenbrock function. In the table, observations refer to the number of experiments used to construct the surrogate. Note that the same starting point ($\mathbf{x}=0$) was used for all the design variables in the gradient-based optimization studies. In the table, DV stands for design variable, HF stands for high-fidelity simulations, SBO stands for surrogate-based optimization, MFD stands for method of feasible directions, QN stands for quasi-Newton BFGS method, and DR stands for division of rectangles derivative-free method.

- | HF-MFD | HF-QN | SBO-MFD | SBO-QN | SBO-DR |
---|---|---|---|---|---|

DV-1 | 0.999 | 0.999 | 0.958 | 0.958 | 0.993 |

DV-2 | 0.999 | 0.999 | 0.992 | 0.994 | 1.007 |

DV-3 | 0.998 | 0.999 | 1.004 | 0.999 | 0.998 |

DV-4 | 0.997 | 0.998 | 1.115 | 1.091 | 1.037 |

DV-5 | 0.995 | 0.996 | 1.208 | 1.189 | 1.058 |

DV-6 | 0.989 | 0.992 | 1.317 | 1.251 | 1.108 |

QoI | 0.00003 | 0.00002 | $-48$ | $-48$ | $-44$ |

Function Evaluations | 1238 | 420 | - | - | - |

Observations | - | - | 500 | 500 | 500 |

Phase | Density $\mathit{\rho}\phantom{\rule{0.277778em}{0ex}}[\frac{\mathit{kg}}{{\mathit{m}}^{3}}]$ | Kinematic Viscosity $\mathit{\nu}\phantom{\rule{0.277778em}{0ex}}[\frac{{\mathit{m}}^{2}}{\mathit{s}}]$ |
---|---|---|

Water | $998.3$ | $1.02\times {10}^{-6}$ |

Air | $1.2$ | $1.48\times {10}^{-5}$ |

Mesh | Number of cells | Average ${\mathit{y}}^{+}$ |
---|---|---|

Coarse | ≈ 800000 | ≈ 80 |

Fine | ≈ 2800000 | ≈ 7 |

**Table 4.**Percentage error (with respect to the experimental data) and CPU time of the benchmark cases. All the simulations were run in parallel with four processors. In all cases, the reported CPU time was measured at 8000 iterations.

Benchmark Case | Percentage Error | CPU Time (seconds) |
---|---|---|

Steady - Coarse mesh | 1.8% | ≈ 12000 |

Steady - Fine mesh | 1.1% | ≈ 20000 |

Unsteady - Coarse mesh | 6.8% | ≈ 27000 |

Unsteady - Fine mesh | 4.3% | ≈ 35000 |

**Table 5.**Resistance reduction and percentage error (with respect to the high-fidelity simulations) of five cases not included in the original sampling plan.

${\mathit{C}}_{\mathit{L}\mathit{P}\mathit{R}}$ | ${\mathit{C}}_{\mathit{Z}\mathit{B}}$ | Resistance Reduction | Percentage Error | Note |
---|---|---|---|---|

0.119 | 0.660 | ≈ 5% | ≈ 7% | - |

0.131 | 0.566 | ≈ 5% | ≈ 7% | - |

0.131 | 0.755 | ≈ 5% | ≈ 6% | - |

0.109947 | 0.762845 | ≈ 7% | ≈ 2% | Global minima |

0.14 | 0.515651 | ≈ 6% | ≈ 2% | Local minima |

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## Share and Cite

**MDPI and ACS Style**

Guerrero, J.; Cominetti, A.; Pralits, J.; Villa, D.
Surrogate-Based Optimization Using an Open-Source Framework: The Bulbous Bow Shape Optimization Case. *Math. Comput. Appl.* **2018**, *23*, 60.
https://doi.org/10.3390/mca23040060

**AMA Style**

Guerrero J, Cominetti A, Pralits J, Villa D.
Surrogate-Based Optimization Using an Open-Source Framework: The Bulbous Bow Shape Optimization Case. *Mathematical and Computational Applications*. 2018; 23(4):60.
https://doi.org/10.3390/mca23040060

**Chicago/Turabian Style**

Guerrero, Joel, Alberto Cominetti, Jan Pralits, and Diego Villa.
2018. "Surrogate-Based Optimization Using an Open-Source Framework: The Bulbous Bow Shape Optimization Case" *Mathematical and Computational Applications* 23, no. 4: 60.
https://doi.org/10.3390/mca23040060