# An Open-Source Processing Pipeline for Quad-Dominant Mesh Generation for Class-Compliant Ship Structural Analysis

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## Abstract

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## 1. Introduction

- M1
- Automatic/batch meshing. It is assumed that the geometry description has been corrected for connectivity errors before the algorithm is applied.
- M2
- Mapped or interactive meshing. In this approach, there is more human involvement than in the batch meshing but less than in fully manually generated mesh. It is also assumed that the geometry description has been corrected for connectivity errors before the processing pipeline is applied.
- M3
- Manual meshing where the meshing is intertwined with geometric modeling by the use of special graphical commands which correct the modeling errors by refining/idealizing the geometry based on a human decision.

`qmorph`algorithm in [5,21,22]. According to ([5], Section 5.3) this approach is available in several commercial meshers, including in Ansys, to automatically generate quad dominant meshes.

## 2. Materials and Methods

#### 2.1. Automatic Geometry Refinement Using Boolean Operations

#### 2.2. Automatic Mesh Generation Based on Virtual Stiffeners—Controlling the Local Mesh Orientation

#### 2.3. Controlling the Sizing Field of Web Surfaces

Algorithm 1: Mesh element size algorithm |

$\mathrm{factor}=\left[\phantom{\rule{3.33333pt}{0ex}}\right]$for S in web surfaces doDivide S in 4 new surfaces along the shorter side Calculate the size s of the shorter side of S Append s to the array factor. end forSet $\overline{\mathrm{factor}}=\mathrm{average}\left(\mathrm{factor}\right)$ Set $\overline{\mathrm{factor}}$ as the minimal mesh size factor in Gmsh Set maximal mesh size factor in Gmsh as $1.5\ast \left(\overline{\mathrm{factor}}\right)$ |

## 3. Results

#### 3.1. Effects of Geometry Refinement—We Did Not Idealize

#### 3.2. Mesh Refinement and the Virtual Stiffener-Controlled Refinement Localization

#### 3.3. Performance of the Algorithm

- (a)
- geometric preprocessing performed using Boolean in Open CASCADE augmented by the virtual stiffener algorithm;
- (b)
- marching front triangular mesh generation in ${L}^{\infty}$ norm;
- (c)
- recombination algorithm which produces a quadrilateral dominated mesh using the perfect matching algorithm.

#### 3.4. Performance of the Indicators

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

FE | finite element |

FEA | finite element analysis |

CAE | Computer aided engineering |

SICN | signed inverse condition number |

web surface | a parallelogram with one dimension much smaller than the other |

regular surface | quadrilateral defined by four co-planar corners |

warped surface | generalized quadrilateral defined by four not co-planar corners |

cs-conforming element | an element confirming to classification society’s rules |

$\overline{\eta}$ | an average of the set $\left\{\eta \right(q)\phantom{\rule{3.33333pt}{0ex}}:\phantom{\rule{3.33333pt}{0ex}}q\phantom{\rule{4.pt}{0ex}}\mathrm{a}\phantom{\rule{4.pt}{0ex}}\mathrm{quadrilateral}\phantom{\rule{4.pt}{0ex}}\mathrm{element}\phantom{\rule{4.pt}{0ex}}\mathrm{in}\phantom{\rule{4.pt}{0ex}}\mathrm{the}\phantom{\rule{4.pt}{0ex}}\mathrm{mesh}\}$. |

$\lambda $ | the percentage of cs-conforming elements in the mesh |

$\xi $ | the percentage of regular quadrilaterals in the mesh |

$\tau $ | the percentage of triangles in the mesh |

## Appendix A. Requirements on the Mesh

#### Appendix A.1. Coarse Mesh Requirements

- Rod (truss) element—line element with axial stiffness only and constant cross-sectional area along the length of the element;
- Beam element—line element with axial, torsional and bi-directional shear and bending stiffness and with constant properties along the length of the element;
- Shell (planar) surface element with constant thickness—with in-plane stiffness (membrane) and out-of-plane bending stiffness (plate).

- Two node line elements and four or three node shell elements are sufficient for hull structure representation—mesh descriptions given in this CG are based on the assumption that these elements are used in the FE models (however, higher order elements may also be used);
- Quadrilateral shell elements with inner angles below 45° or above 135° between edges should be avoided;
- Quadrilateral shell elements with high aspect ratio as well as distorted elements should be avoided—aspect ratio is to be kept close to 1 but should not exceed 3 for four node elements or 5 for eight node elements;
- The use of triangular shell elements is to be kept to a minimum.

#### Appendix A.2. Fine Mesh Zone

- A uniform quadratic mesh is to be used with a smooth transition leading up to the fine mesh zone;
- Finite element size it to be limited to 50 mm × 50 mm;
- The extent of the fine mesh zone is not to be less than 10 elements in all directions from the area under investigation;
- The use of extreme aspect ratio (greater than 3) and distorted elements (corner angles below 60° and greater than 120°) are to be avoided;
- The use of triangular elements is to be avoided;
- All structural parts within an extent of at least 500 mm in all directions leading up to the high stress area are to be modeled explicitly with shell elements;
- Stiffeners within the zone are to be modeled using shell elements;
- Stiffeners outside the zone may be modeled using beam elements;
- The transition between shell elements and beam elements is to be modeled so that the overall stiffener deflection is retained;
- Openings—the first two layers of elements around the opening are to be modeled with mesh size no greater than 50 mm × 50 mm;
- Face plates—of openings, primary supporting members and associated brackets are to be modeled with at least two elements across their width on either side.

## Notes

1 | https://www.maestromarine.com/wp-content/uploads/2015/09/MAESTRO-Manual.pdf (accessed on 23 December 2021) |

2 | https://www.rhino3d.com/ (accessed on 23 December 2021) |

3 | https://www.napa.fi/software-and-services/ship-design/structural-design/ (accessed on 23 December 2021) |

4 | https://structures.aero/wp-content/uploads/2016/10/Femap_MeshModelGeneration_SPLM_Sherman.pdf (accessed on 23 December 2021) |

5 | https://altairuniversity.com/wp-content/uploads/2014/02/HM_Automeshingintro.pdf, slide 22. (accessed on 23 December 2021) |

6 | https://github.com/tpaviot/oce/ (accessed on 23 December 2021) |

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**Figure 1.**Examples of dictionaries of elements representing a geometry of a section of ships’ structure.

**Figure 2.**Inserting an opening and a virtual stiffener in an angled girder. The resolution of the height of the web surface needs to be adapted to control the local quality of the mesh.

**Figure 3.**Overlapping girders, before and after Boolean operations. Shaded areas on the left Figure indicate the overlap.

**Figure 6.**Visualization of the location of the elements which do not meet the desired restrictions in the dictionary representing the original geometry.

**Figure 7.**A zoomin on a description of the surfaces of two girders which are unintentionally overlapping.

**Figure 8.**Section of a deck and a bulkhead. A fully automatically generated mesh using pyREMAKEmsh. It has 11,909 elements and the following quality measures $\overline{\eta}=0.93$, $\xi =80$, $\tau =4$, and $\lambda =97$.

**Figure 9.**Interactively generated mesh on the hand refined geometry using NAPA-Steel. The mesh has more than 18,481 elements, it has $\overline{\eta}=0.99$, $\xi =95$, $\tau =1$ and $\lambda =99$. This is a result of interactive incremental meshing and geometry refinement.

**Figure 12.**Performance benchmarking on four geometries. The y-axis is labeled by two numbers. The first number indicates the number of triangles generated. The second number indicates the number of elements obtained after the run of the recombination algorithm.

**Figure 15.**Statistics for the refined mesh of the structure. The number of regular quadrilaterals has increased, but not by much. This indicates that the marching front algorithm correctly detected the challenging parts of the geometry, even at the level of a coarser mesh.

**Table 1.**Quality measures for the studied geometries. Algorithm: Full recombination algorithm with smoothing.

Geometry Dictionary | $\overline{\mathit{\eta}}$ | $\mathit{\xi}$ | $\mathit{\tau}$ | $\mathit{\lambda}$ | Average SICN | Number of Elements |
---|---|---|---|---|---|---|

${G}_{1}$ | $0.93$ | 80 | 4 | 97 | $0.94$ | 11,909 |

${G}_{2}$ | $0.91$ | 67 | 10 | 89 | $0.88$ | 6549 |

${G}_{3}$ | $0.94$ | 49 | 6 | 95 | $0.93$ | 128,093 |

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**MDPI and ACS Style**

Grubišić, L.; Lacmanović, D.; Palaversa, M.; Prebeg, P.; Tambača, J. An Open-Source Processing Pipeline for Quad-Dominant Mesh Generation for Class-Compliant Ship Structural Analysis. *J. Mar. Sci. Eng.* **2022**, *10*, 209.
https://doi.org/10.3390/jmse10020209

**AMA Style**

Grubišić L, Lacmanović D, Palaversa M, Prebeg P, Tambača J. An Open-Source Processing Pipeline for Quad-Dominant Mesh Generation for Class-Compliant Ship Structural Analysis. *Journal of Marine Science and Engineering*. 2022; 10(2):209.
https://doi.org/10.3390/jmse10020209

**Chicago/Turabian Style**

Grubišić, Luka, Domagoj Lacmanović, Marin Palaversa, Pero Prebeg, and Josip Tambača. 2022. "An Open-Source Processing Pipeline for Quad-Dominant Mesh Generation for Class-Compliant Ship Structural Analysis" *Journal of Marine Science and Engineering* 10, no. 2: 209.
https://doi.org/10.3390/jmse10020209