# L2-Sea hull optimization

## Overview

This optimization benchmark is based on the L2-Sea model. The goal is to optimize the ship hull's shape to minimize total resistance.

![L2-Sea-Model](https://raw.githubusercontent.com/UM-Bridge/benchmarks/main/models/l2-sea/l2sea_example.png "DTMB 5415 view of the wave elevation pattern and pressure field on the hull surface")

## Authors
- [Andrea Serani](mailto:andrea.serani@cnr.it)
- [Matteo Diez](mailto:matteo.diez@cnr.it)

## Run
```
docker run -it -p 4242:4242 linusseelinger/model-l2-sea
```

## Properties

Model | Description
---|---
forward | l2-sea

### forward
Mapping | Dimensions | Description
---|---|---
input | [14] | The 14 inputs are the $x$ design variables for the shape modification with $-1\leq x_i \leq 1$  (with parent hull has $x_i = 0$) for $i=1,\dots,14$\\.
output | [5] | The first output is the model scale total resistance ($R_\mathrm{T}$) in Newton, whereas the other four are geometrical constraints (negative to be satisfied), related to the beam, draft, and sonar dome dimensions.

Feature | Supported
---|---
Evaluate | True
Gradient | False
ApplyJacobian | False
ApplyHessian | False

Config | Type | Default | Description
---|---|---|---
fidelity | integer | 7 | Fidelity level for the total resistance evaluation associated to the numerical grid discretization. Fidelity goes from 1 to 7, where 1 is highest-fidelity level (finest grid) and 7 is the lowest-fidelity level (coarsest grid).

## Mount directories
Mount directory | Purpose
---|---
/output | ASCII files for visualization of pressure distribution along the hull `preXXXX.plt` and free-surface `intfrXXXX.plt` formatted for Tecplot and Paraview, where `XXXX` is the Froude number.

## Source code

[Model sources here.](https://github.com/UM-Bridge/benchmarks/tree/main/models/l2-sea)

## Description

The benchmark, developed within the activities of the NATO-AVT-331 Research Task Group on ''Goal-driven, multifidelity approaches for military vehicle system-level design'' [Beran et al. (2020)](https://doi.org/10.2514/6.2020-3158), pertains to the total resistance reduction of the DTMB 5415 in calm water at fixed speed, corresponding to a Froude number (Fr) equal to 0.28. The optimization problem reads
%
\begin{eqnarray}\label{eq:5415prob}
    \begin{array}{rll}
        \mathrm{minimize}      & \Delta R (\mathbf{x}) = \frac{R (\mathbf{x})}{R_0}-1  \qquad \mathrm{with} \qquad \mathbf{x}\in\mathbb{R}^N\\
        \mathrm{subject \,  to}& L_{\rm pp}(\mathbf{x}) = L_{\rm pp_0}\\
        \mathrm{and \, to}     & \nabla(\mathbf{x}) = \nabla_0 \\
        & |\Delta B(\mathbf{x})| \leq 0.05B_0 \\
        & |\Delta T(\mathbf{x})| \leq 0.05T_0 \\
        & V(\mathbf{x})\geq V_0\\
        & -1\leq x_i \leq 1 \qquad \mathrm{with} \qquad \forall i=1,\dots, N\\
    \end{array}
\end{eqnarray}
%
where $\mathbf{x}$ are the design variables, $L_{\rm pp}$ is the length between perpendiculars, $B$ is the overall beam, $T$ is the drought, and $V$ is the volume reserved for the sonar in the bow dome. Subscript ''0'' indicates parent (original) hull values. Equality and inequality constraints for the geometry deformations are taken from [Grigoropoulos et al. (2017)](https://www.researchgate.net/publication/316941318_Mission-based_hull_form_and_propeller_optimization_of_a_transom_stern_destroyer_for_best_performance_in_the_sea_environment).

The shape modifications $\tilde{\boldsymbol{\gamma}}(\boldsymbol{\xi},\mathbf{x})$ are produced directly on the Cartesian coordinates $\boldsymbol{\xi}$ of the computational body surface grid $\mathbf{g}$, as per

\begin{equation}\label{eq:reducedspace}
    \mathbf{g}(\boldsymbol{\xi},\mathbf{x})=\mathbf{g}_0(\boldsymbol{\xi}) + \boldsymbol{\gamma}(\boldsymbol{\xi},\mathbf{x})
\end{equation}

where $\mathbf{g}_0$ is the original geometry and $\boldsymbol{\gamma}$ is a shape modification vector obtained by a physics-informed design-space dimensionality reduction [Serani et al. (2019)](https://doi.org/10.2514/6.2019-2218)

\begin{equation}
    {\boldsymbol{\gamma}}(\boldsymbol{\xi},\mathbf{x}) = \sum_{k=1}^N x_k \boldsymbol{\psi}_k(\boldsymbol{\xi})
    \label{e:exp_gamma}
\end{equation}

with $\boldsymbol{\psi}$ a set of orthonormal functions, with $N=14$ the number of design variables ($\mathbf{x}$). It may be noted that the design variables and the associated shape modifications are organized in a hierarchical order, meaning that the first variables produce larger design modifications than the last ones [Serani et al. (2021)](https://link.springer.com/article/10.1007/s00366-021-01375-x).

The multifidelity levels are defined by the computational grid size. Specifically, the benchmark is defined with seven grid (fidelity) levels with a refinement ratio of 2$^{0.25}$