# Hull Shape Design Optimization with Parameter Space and Model Reductions, and Self-Learning Mesh Morphing

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Shape and Grid Parameterization

#### How to Combine Different Shape Parametrization Strategies

## 3. The Mathematical Model for Incompressible Fluids

#### 3.1. The Full Order Model: Incompressible Rans

#### 3.2. The Reduced Order Model: POD-GPR

## 4. Optimization Procedure with Built-In Parameters Reduction

#### 4.1. Genetic Algorithm

#### 4.2. Active Subspaces

#### 4.3. Active Subspaces-Based Genetic Algorithm

## 5. Numerical Results

#### 5.1. Self-Learning Mesh Morphing Parameters

- $\mathit{x}$
**axis:**7 points layers located on sections 10, 12, 14, 16, 18, 20, 22; - $\mathit{y}$
**axis:**11 points layers that cover the whole hull beam, with the second and the second-to-last positioned on the lateral walls of the ship; - $\mathit{z}$
**axis:**7 points layers that cover the whole hull draft, aligning the 2nd and the 5th of them to the hull bottom and to the waterline, respectively.

#### 5.2. Reduced Order Model Construction

- at the inlet we set constant velocity, fixed flux condition for the pressure and a fixed profile for the VOF variable;
- at the outlet we set constant average velocity, zero-gradient condition for the pressure and variable height flow rate condition for VOF variable;
- at the bottom and lateral planes, we impose symmetric conditions for all the quantities;
- at the top plane, we set a pressure inlet outlet velocity condition for the velocity and nil pressure; VOF variable is fixed to 1 (air);
- at the hull surface, we impose no-slip condition for velocity, fixed flux condition for the pressure and zero-gradient condition for VOF variable.

#### 5.3. Optimization Procedure

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

AS | Active subspaces |

ASGA | Active subspaces genetic algorithm |

CAD | Computer-aided design |

CFD | Computational fluid dynamics |

FFD | Free form deformation |

FOM | Full order model |

GA | Genetic algorithm |

GPR | Gaussian process regression |

HPC | High performance computing |

PDE | Partial differential equation |

POD | Proper orthogonal decomposition |

POD-GPR | Proper orthogonal decomposition with Gaussian process regression |

RBF | Radial basis functions |

RANS | Reynolds averaged Navier–Stokes |

ROM | Reduced order method |

STL | Stereolithography tesselation language |

VOF | Volume of fluid |

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1. | Freely available at https://github.com/mathLab/ATHENA (accessed date 1 January 2020). |

2. | Imposed for y symmetry conservation. |

**Figure 1.**Illustration of the key steps of the proposed optimization pipeline with the methods and the softwares used.

**Figure 3.**A two dimensional sketch of the free form deformation (FFD) procedure applied to the surface of a container ship hull, including the three transformations $\psi $, $\widehat{T}(\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}},\mathit{\mu})$ and ${\psi}^{-1}$ composing the process.

**Figure 4.**A section view example illustrating the radial basis functions (RBF) morphing steps carried out to propagate the hull surface deformations to a volumetric mesh for ship hydrodynamics simulations.

**Figure 5.**Active subspaces-based genetic algorithm scheme. The main step of the classical genetic algorithm (GA) are depicted from top to bottom. The yellow boxes represent projections onto and from lower dimension active subspace. Thus, they are specific to ASGA.

**Figure 6.**The surface of the DTC hull. The highlighted sections divide the ship into 20 equispaced chunks at the free-surface level.

**Figure 7.**Visual examples of hull deformation with $\mathit{\mu}={[-0.2]}^{10}$ (on

**left**) and $\mathit{\mu}={\left[0.2\right]}^{10}$ (on

**right**). The red surface refers to the deformed ships, while the blue one is the original hull.

**Figure 8.**Values of the main mesh quality indicators as reported by checkMesh utility of OpenFOAM library, as a function of the index corresponding to each of the 200 volumetric meshes produced for the simulation campaign.

**Figure 9.**ASGA runs. The reduction of the ${C}_{t}$ is to be intended with respect to the undeformed reference hull.

**Figure 10.**The sections (from 10 to 20) of the original ship in blue and of the optimized one in red.

**Figure 11.**Distribution of the shear stresses measured in Pascal over the undeformed hull: the ull order model (FOM) validation (

**top**) is compared to the reduced order model (ROM) approximation (

**middle**) and the absolute error is shown (

**bottom**).

**Figure 12.**Distribution of the shear stresses measured in Pascal over the optimal hull: the FOM validation (

**top**) is compared to the ROM approximation (

**middle**) and the absolute error is shown (

**bottom**).

**Figure 13.**Distribution of pressure measured in Pascal over the undeformed hull: the FOM validation (

**left**) is compared to the ROM approximation (

**center**) and the absolute error is shown (

**right**).

**Figure 14.**Distribution of the pressure measured in Pascal over the optimal hull: the FOM validation (

**left**) is compared to the ROM approximation (

**center**) and the absolute error is shown (

**right**).

**Figure 15.**Contours of free surface elevation field around the original hull (

**top**half) and optimal (

**bottom**half).

Quantity | Value |
---|---|

Length between perpendiculars ${L}_{pp}\phantom{\rule{3.33333pt}{0ex}}\left[\mathrm{m}\right]$ | $5.976$ |

Waterline breadth ${B}_{wl}\phantom{\rule{3.33333pt}{0ex}}\left[\mathrm{m}\right]$ | $0.859$ |

Draught midships ${T}_{m}\phantom{\rule{3.33333pt}{0ex}}\left[\mathrm{m}\right]$ | $0.244$ |

Volume displacement $V\phantom{\rule{3.33333pt}{0ex}}\left[{\mathrm{m}}^{3}\right]$ | $0.827$ |

Block coefficient ${C}_{B}$ | $0.661$ |

**Table 2.**FFD control points displacement. The indices refer to the relative position of the points within the lattice. The layers order, which starts from 0, is maintained consistent with the reference system. The intervals indicated by the—symbol are inclusive.

Lattice Points | Parameter | Displacement Direction | ||
---|---|---|---|---|

Index x | Index y | Index z | ||

2 | 0 | 2–4 | ${\mu}_{0}$ | x |

2 | 10 | 2–4 | ${\mu}_{0}$ | x |

3 | 0 | 2–4 | ${\mu}_{1}$ | x |

3 | 10 | 2–4 | ${\mu}_{1}$ | x |

4 | 0 | 2–4 | ${\mu}_{2}$ | x |

4 | 10 | 2–4 | ${\mu}_{2}$ | x |

4 | 2–4 | 2 | ${\mu}_{3}$ | y |

4 | 6–8 | 2 | $-{\mu}_{3}$ 2 | y |

4 | 2–4 | 3 | ${\mu}_{4}$ | y |

4 | 6–8 | 3 | $-{\mu}_{4}$ 2 | y |

4 | 2–4 | 4 | ${\mu}_{5}$ | y |

4 | 6–8 | 4 | $-{\mu}_{5}$ 2 | y |

3 | 2–4 | 2 | ${\mu}_{6}$ | y |

3 | 6–8 | 2 | $-{\mu}_{6}$ 2 | y |

5 | 2–4 | 3 | ${\mu}_{7}$ | y |

5 | 6–8 | 3 | $-{\mu}_{7}$ 2 | y |

4 | 0–1 | 2 | ${\mu}_{8}$ | z |

4 | 9–10 | 2 | ${\mu}_{8}$ | z |

5 | 0 | 3 | ${\mu}_{9}$ | z |

5 | 10 | 3 | ${\mu}_{9}$ | z |

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**MDPI and ACS Style**

Demo, N.; Tezzele, M.; Mola, A.; Rozza, G.
Hull Shape Design Optimization with Parameter Space and Model Reductions, and Self-Learning Mesh Morphing. *J. Mar. Sci. Eng.* **2021**, *9*, 185.
https://doi.org/10.3390/jmse9020185

**AMA Style**

Demo N, Tezzele M, Mola A, Rozza G.
Hull Shape Design Optimization with Parameter Space and Model Reductions, and Self-Learning Mesh Morphing. *Journal of Marine Science and Engineering*. 2021; 9(2):185.
https://doi.org/10.3390/jmse9020185

**Chicago/Turabian Style**

Demo, Nicola, Marco Tezzele, Andrea Mola, and Gianluigi Rozza.
2021. "Hull Shape Design Optimization with Parameter Space and Model Reductions, and Self-Learning Mesh Morphing" *Journal of Marine Science and Engineering* 9, no. 2: 185.
https://doi.org/10.3390/jmse9020185