# Optimisation of the Filament Winding Approach Using a Newly Developed In-House Uncertainty Model

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Background

^{2}, which is superior to that of wind (0.5 W/m

^{2}) and solar energy (0.1–0.3 W/m

^{2}) [1] Although several experiments have been conducted on the system, computational modelling of the device immersed in sea water is also required. The proposed float is made of a continuously wound glass-reinforced composite shell with stainless steel bolting plates integrated into the float walls to connect the float to a large barge on the seabed using 5 stainless steel cables, as shown in Figure 1. The barge hosts the energy harvesting devices that depend on the variation of the tension loads in the cables due to the constrained float movement. A sub-modelling technique is introduced with a layered (float) model and a solid (anchoring system) model to fully explore and assess the stress levels within each of the winding layers of the float. The integrity of the bonding between the cable connectors and the composite layers under normal operation will also be investigated, as it represents a major potential weakness in the integrity of the structure. In other words, the existence of high stress levels at the boundary between the anchoring system and the float under normal operation is one of the issues that might be encountered in the structure. In addition to locally altering the design in these zones to improve the structural response, it is necessary to perform further computations to assess if a delamination of the float layers is likely, taking into account the variable wave loading, i.e., the development of faults due to fatigue, which is beyond the scope of the current paper. The system suffers from various uncertainties, such as the number of plies in the top and bottom shells, the strength of the steel cable, the bond strength between the steel plates and the composite layers, the bond strength between the composite layers, and the fibre orientations of the composite material relative to the applied load. The current study will concentrate on exploring the effects of the fibre orientations and the number of plies on the overall performance of the structure.

#### 1.2. The Winding Angle

## 2. The Modelling Procedure and Assumptions

#### 2.1. Geodesic Path Principle

^{−1}, which is a part of a circular helix. This definition is useful in filament winding, since the fibres will follow the shortest path on the mandrel surfaces between any given two points. Logically, the geodesic arc is a stable path, as the fibres would have to stretch to deviate from the set path. On a cylinder, the geodesic arc is a helical path with a constant winding angle, α. This means that in order to wind the end points of a cylinder, the geodesic path cannot be followed completely. A change in the winding direction is required to generate a complete layer, where this deviation from the geodesic path ensures that the fibres stay in place through friction [7].

#### 2.2. Material and Boundary Conditions

#### 2.3. Multi-Level Optimisation Strategy

**Phase 1.**- The minimum number of plies for the top and bottom shells

**Phase 2.**- The optimum stacking sequence, i.e., fibre orientations

**Phase 3.**- The optimum combination of plies vs. stacking sequence

#### 2.4. Uncertainty and Robust Design Procedure

^{2}= 256 solutions). The reason for the 5° increments is that in real-world situations, filament winding processes are restricted to winding the fibres at angles of 0°, 5°, 10°, 15°, 20°,…….., 90°. These input values of the stacking sequence were manually input into the innovative model that was developed at Swansea University. Through this, ANSYS Workbench was linked via MATLAB utilising the developed and in-house written code. The code automatically operates ANSYS, runs the model, retrieves the results, and continues to obtain the remaining sets of results. This reduces the effort, time, and number of trials used to choose the optimum design parameters for any given design. The framework for robust design proposed and employed by the authors has recently been published elsewhere, although for carbon fibre composite materials bonded via aluminium joints [11]. For the optimum and robust design, the blind kriging approach [12], a typical meta-model process, was utilised in the analysis. This approach provided outstanding results with a minimal amount of variation. This facilitated the construction of the response surfaces for each stage in the analysis, from which the optimum and robust design parameters were determined. The strategy used to optimise the uncertain parameters and the robust design starts by assuming that a certain number of plies are oriented at 90° to provide the maximum possible strength while optimising the number of plies, which is the first phase. This reduces the risk of failure of the glass fibre material. The optimum number of plies will then be taken into the second phase to obtain the best angle combinations of the filament-wound fibres. At this stage, the number of plies is kept unchanged while optimising the fibre orientation so that the likelihood of failure is eliminated. The third phase optimises the number of plies and the fibre orientations in order to obtain the best performance, i.e., maximum strength and no failure. This exercise provides a procedure that can be applied to any engineering application that involves more than a single variable through the employment of 3D response surfaces that are able to define the safest operational region.

#### 2.5. Meta-Model Design Optimisation

#### 2.6. The Ordinary Kriging Model

_{1}(x), b

_{2}(x), …, b

_{p}(x), are the basis functions (i.e., the basis polynomials) and α

_{i}= (α

_{1}, α

_{2}, …, α

_{p}) denote the coefficients. In this approach, the regression function reproduces the general trend in the data (i.e., the largest variance), after which a Gaussian process is used to interpolate the residuals.

#### 2.7. The Blind Kriging Model

_{i}= (β

_{1}, β

_{2}, …, β

_{t}). The second term involves the parameters that are estimated based on the relevance of the candidate features. The estimation of the parameters, i.e., the least-squared solution, is a straightforward approach that can help to rank the features.

#### 2.8. The Co-Kriging Model

## 3. Results and Discussion

#### 3.1. Phase 1 Results (Number of Plies of Top and Bottom Shells)

#### 3.2. Phase 2 Results (The Fibre Stacking Sequence)

#### 3.3. Phase 3 Results (Number of Plies vs. the Stacking Sequence)

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Replication of Results

## References

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**Figure 4.**The stainless steel support with the top and bottom plies of the filament-wound composite.

**Figure 6.**Phase 1: Optimising the number of plies: (

**a**) the blind kriging response surface; (

**b**) the contour plot; (

**c**) the variance plot.

**Figure 7.**Phase 2: Optimising the ply orientation: (

**a**) the blind kriging response surface; (

**b**) the contour plot; (

**c**) the variance plot. Note: All angles are in degrees (°).

**Figure 8.**A simple structural analysis explaining the optimum +60°/−60° combination of the optimum stacking sequence. Note: All angles are in degrees (°).

**Figure 9.**The combined solution for the thickness of the float (the top and bottom shells) at the optimum filament winding angle of ±60°.

Material Properties | Value |
---|---|

Young’s Modulus X direction | 28,000 MPa |

Young’s Modulus Y direction | 6900 MPa |

Young’s Modulus Z direction | 6900 MPa |

Poisson’s Ratio XY | 0.3 |

Poisson’s Ratio YZ | 0.19 |

Poisson’s Ratio XZ | 0.19 |

Shear Modulus XY | 11,100 MPa |

Shear Modulus YZ | 2980 MPa |

Shear Modulus XZ | 2980 MPa |

Tensile Strength X direction | 1100 MPa |

Tensile Strength Y direction | 35 MPa |

Tensile Strength Z direction | 35 MPa |

**Table 2.**A summary of the inverse reverse factor (IRF) values obtained for the shell optimization *.

Top Plies # | Bottom Plies # | IRF Values |
---|---|---|

30 | 31 | 0.6160867 |

26 | 30 | 0.6853152 |

22 | 31 | 0.6944406 |

28 | 29 | 0.6949207 |

29 | 28 | 0.7232109 |

24 | 29 | 0.7396711 |

20 | 30 | 0.7493208 |

18 | 30 | 0.790386 |

31 | 26 | 0.7919314 |

25 | 27 | 0.8047401 |

21 | 28 | 0.8076068 |

27 | 26 | 0.8297545 |

14 | 31 | 0.844326 |

30 | 25 | 0.8538816 |

23 | 26 | 0.8634863 |

16 | 29 | 0.8730014 |

19 | 27 | 0.8788239 |

22 | 25 | 0.914228 |

17 | 27 | 0.9276962 |

26 | 24 | 0.9404642 |

28 | 23 | 0.984838 |

12 | 29 | 0.9925366 |

13 | 28 | 0.9995041 |

**Table 3.**A summary of the IRF values obtained for the stacking sequence optimisation* (the first column of angles is α, whereas the second column is β).

Angle 1 | Angle 2 | IRF Values |
---|---|---|

60 | −60 | 0.936464153 |

65 | −60 | 0.936740126 |

55 | −60 | 0.937034556 |

75 | −60 | 0.937040329 |

70 | −60 | 0.937175776 |

80 | −60 | 0.937716612 |

85 | −60 | 0.938415852 |

50 | −60 | 0.940729225 |

70 | −65 | 0.982900049 |

55 | −65 | 0.982931364 |

60 | −65 | 0.983015988 |

65 | −65 | 0.983230577 |

75 | −65 | 0.983428191 |

50 | −65 | 0.983641761 |

80 | −65 | 0.984033312 |

85 | −65 | 0.984664912 |

50 | −55 | 0.999807636 |

55 | −66 | 0.999854478 |

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**MDPI and ACS Style**

Aldoumani, N.; Giannetti, C.; Abdallah, Z.; Belblidia, F.; Khodaparast, H.H.; Friswell, M.I.; Sienz, J.
Optimisation of the Filament Winding Approach Using a Newly Developed In-House Uncertainty Model. *Eng* **2020**, *1*, 122-136.
https://doi.org/10.3390/eng1020008

**AMA Style**

Aldoumani N, Giannetti C, Abdallah Z, Belblidia F, Khodaparast HH, Friswell MI, Sienz J.
Optimisation of the Filament Winding Approach Using a Newly Developed In-House Uncertainty Model. *Eng*. 2020; 1(2):122-136.
https://doi.org/10.3390/eng1020008

**Chicago/Turabian Style**

Aldoumani, Nada, Cinzia Giannetti, Zakaria Abdallah, Fawzi Belblidia, Hamed Haddad Khodaparast, Michael I. Friswell, and Johann Sienz.
2020. "Optimisation of the Filament Winding Approach Using a Newly Developed In-House Uncertainty Model" *Eng* 1, no. 2: 122-136.
https://doi.org/10.3390/eng1020008