# Lightweight Design of Variable-Stiffness Cylinders with Reduced Imperfection Sensitivity Enabled by Continuous Tow Shearing and Machine Learning

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## Abstract

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## 1. Introduction

- a new parameterization scheme for variable-stiffness cylindrical shells manufactured using continuous tow shearing (CTS), which result in a vast design space that is explored in order to attain designs with reduced imperfection sensitivity;
- the comparison of the performance of three classes of machine learning strategies with multiple kernel models, in a total of eighteen distinct instances;
- an algorithm to calculate the optimal design that meets the target levels of mass, buckling load, and post-buckling stiffness. The inverse problem is formulated within the scope of global optimization for mixed-integer variables.

## 2. Design Parameterization with Circumferentially Oriented Thickness Patterns

- n: the number of regions ${c}_{2}$;
- ${r}_{CTS}$: the CTS steering radius in the transition zone;
- ${{c}_{2}}_{ratio}$: the ratio between the design length ${c}_{2}$ with respect to ${{c}_{2}}_{max}$, which is calculated as per Equation (5);
- ${\theta}_{1}$ and ${\theta}_{2}$: the respective angles at regions ${c}_{1}$ and ${c}_{2}$.

- ${{n}_{x}}_{t}$: the number of nodes along x, along the length of each transition zone, given by t;
- ${n}_{y}$: the number of nodes along the circumferential direction, which is constant along the length. If ${n}_{y}$ is not given, then it is calculated assuming a desired aspect ratio of $\frac{{n}_{y}}{{n}_{x}}=1$;
- the maximum aspect ratio of the elements in the circumferential direction, with respect to the axial direction:$${\left(\right)}_{\frac{{n}_{y}}{{n}_{x}}}max$$If this value is exceeded, more nodes in the x direction are added to the regions ${c}_{1}$ and ${c}_{2}$ to keep$$\frac{{n}_{y}}{{n}_{x}}\le {\left(\right)}_{\frac{{n}_{y}}{{n}_{x}}}max$$Note that Equation (7) does not affect the transition regions with length t, which have their number of nodes along the length determined by ${{n}_{x}}_{t}$.

Algorithm 1: The modeling phase | |

/* Initial setup | */ |

$n\leftarrow $ number of regions ${c}_{2}$; | |

${r}_{CTS}\leftarrow $ steering radius; | |

${{c}_{2}}_{ratio}\leftarrow $ ratio in agreement with Equation (5); | |

${\theta}_{1}\leftarrow $ tow angle at ${c}_{1}$; | |

${\theta}_{2}\leftarrow $ tow angle at ${c}_{2}$; | |

/* Intermediary computation | */ |

$t\leftarrow $ length of the transition regions (Equation (2)); | |

${c}_{2}\leftarrow $ length (Equations (4) and (5)); | |

${c}_{1}\leftarrow $ length (Equation (3)); | |

Ensure $\frac{{n}_{y}}{{n}_{x}}\le {\left(\right)}_{\frac{{n}_{y}}{{n}_{x}}}max$; | |

Proceed with model evaluation; | |

/* Output | */ |

Return the number of nodes along the lengths ${c}_{1}$ and ${c}_{2}$. |

## 3. Linear Buckling Constraint

## 4. Post-Buckling Stiffness

- (1)
- $\xi $ is a scalar parameter;
- (2)
- ${\mathit{u}}_{\mathit{I}}$ is a first-order field, taken directly from one or a linear combination of multiple linear buckling modes. Vector ${\mathit{u}}_{\mathit{I}}$ is customarily re-scaled by dividing with the maximum normal displacement amplitude and multiplying by the plate or shell thickness;
- (3)
- ${\mathit{u}}_{\mathit{II}}$ is a second-order field that provides a correction to the first-order field;
- (4)
- the third-order field ${\mathit{u}}_{\mathit{III}}$, and higher fields, are assumed to have a negligible contribution;
- (5)
- ${a}_{I}$ and ${b}_{I}$ are, respectively, the first- and second-order load parameters to be determined.

## 5. Machine Learning Strategies for Meta-Modeling

#### 5.1. Design of Experiment

#### 5.2. Support Vector Machine

#### 5.3. Kriging Surrogate

#### 5.4. Random Forest

`tree`$\in \{50,100,200\}$) and the number of variables randomly sampled as candidates at each split (

`split`$\in \{3,5\}$). The performance of each model is evaluated against a set of ${n}_{test}=400$ designs not used in the training phase. The executions are performed thought the randomForest toolbox in R [71].

`tree = 200`and

`split = 5`provided the best mass predictor, based on the median error.

`tree = 200`and

`split = 3`provided the best ${P}_{critical}$ predictor.

## 6. The Inverse Problem

Algorithm 2: The training phase |

Algorithm 3: Obtaining the design as an inverse problem |

#### Imperfection-Sensitivity Analysis

- effective modeling strategy that explores the design space;
- uses popular machine learning methods without needing to adjust special parameters;

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 13.**Error dispersion of the ${P}_{critical}$ estimate using distinct Random Forest parameters.

**Figure 14.**Comparison of ${b}_{I}$ error dispersion considering the best SVM, Kriging, and Random Forest estimates.

**Figure 15.**Comparison of mass error dispersion considering the best SVM, Kriging, and Random Forest estimates.

**Figure 16.**Comparison of ${P}_{critical}$ error dispersion considering the best SVM, Kriging, and Random Forest estimates.

**Figure 22.**Optimum design ID = 12 from the Pareto front of Table 10, with $m=1.448\phantom{\rule{3.33333pt}{0ex}}\mathrm{kg}$. The contour shows the thickness distribution. The black and blue lines represent, respectively, the CTS tow paths of each layer.

**Figure 23.**Optimum design ID = 1 from the Pareto front of Table 10, with $m=1.510\phantom{\rule{3.33333pt}{0ex}}\mathrm{kg}$.

**Figure 24.**Optimum design ID = 13 from the Pareto front of Table 10, with $m=1.690\phantom{\rule{3.33333pt}{0ex}}\mathrm{kg}$.

Id | Kernel | Min. | Median | Max. |
---|---|---|---|---|

1 | Laplace | 0.00000 | 0.00008 | 0.18026 |

2 | RBF | 0.00000 | 0.00012 | 0.18438 |

3 | Poly | 0.00000 | 0.00019 | 0.18768 |

4 | Vanilla | 0.00000 | 0.00019 | 0.18769 |

5 | Tanh | 0.00162 | 1.29576 | 6.39274 |

6 | Bessel | 0.00001 | 0.00341 | 0.18597 |

7 | ANOVA | 0.00506 | 0.61795 | 1.71422 |

Id | Kernel | Min. | Median | Max. |
---|---|---|---|---|

1 | Laplace | 0.00224 | 0.09454 | 14.09538 |

2 | RBF | 0.00020 | 0.12610 | 9.27851 |

3 | Poly | 0.00544 | 0.49763 | 529.40430 |

4 | Vanilla | 0.00628 | 0.49850 | 529.81813 |

5 | Tanh | 0.12038 | 41.58680 | 5921.04232 |

6 | Bessel | 0.00129 | 0.35197 | 339.96942 |

7 | ANOVA | 1.93005 | 28.31336 | 12,837.05073 |

Id | Kernel | Min. | Median | Max. |
---|---|---|---|---|

1 | Laplace | 0.00020 | 0.11691 | 46.73723 |

2 | RBF | 0.00024 | 0.14848 | 35.81538 |

3 | Poly | 0.00210 | 0.62256 | 166.42386 |

4 | Vanilla | 0.00242 | 0.62194 | 166.51978 |

5 | Tanh | 0.04686 | 52.67757 | 16,211.19438 |

6 | Bessel | 0.00086 | 0.33308 | 87.76449 |

7 | ANOVA | 0.17326 | 19.83168 | 5415.23160 |

**Table 4.**Error statistics of the ${b}_{I}$ meta-model using a distinct Kriging covariance kernel structure.

Id | Kernel | Min. | Median | Max. |
---|---|---|---|---|

1 | Gauss | 0.00000 | 0.00005 | 0.18798 |

2 | Matern ${5}_{2}$ | 0.00000 | 0.00010 | 0.18809 |

3 | Matern ${3}_{2}$ | 0.00000 | 0.00015 | 0.17966 |

4 | Exp | 0.00000 | 0.00033 | 0.18004 |

5 | Powexp | 0.00000 | 0.00080 | 0.18080 |

**Table 5.**Error statistics of the mass estimate using a distinct Kriging covariance kernel structure.

Id | Kernel | Min. | Median | Max. |
---|---|---|---|---|

1 | Gauss | 0.00004 | 0.01341 | 13.12694 |

2 | Matern ${5}_{2}$ | 0.00001 | 0.02327 | 3.54187 |

3 | Matern ${3}_{2}$ | 0.00010 | 0.02933 | 6.50619 |

4 | Exp | 0.00036 | 0.05714 | 20.58523 |

5 | Powexp | 0.00003 | 0.01734 | 22.41985 |

**Table 6.**Error statistics of the ${P}_{critical}$ estimate using a distinct Kriging covariance kernel structure.

Id | Kernel | Min. | Median | Max. |
---|---|---|---|---|

1 | Gauss | 0.00008 | 0.06522 | 43.00029 |

2 | Matern ${5}_{2}$ | 0.00016 | 0.05478 | 39.23206 |

3 | Matern ${3}_{2}$ | 0.00007 | 0.05416 | 36.73150 |

4 | Exp | 0.00011 | 0.05613 | 22.85848 |

5 | Powexp | 0.00042 | 0.05325 | 20.13985 |

**Table 7.**Error statistics of the ${b}_{I}$ meta-model using a Random Forest with distinct number of trees and splits.

Id | Tree/Split | Min. | Median | Max. |
---|---|---|---|---|

1 | 50/3 | 0.00000 | 0.00012 | 0.18004 |

2 | 100/3 | 0.00000 | 0.00013 | 0.18000 |

3 | 200/3 | 0.00000 | 0.00015 | 0.22901 |

4 | 50/5 | 0.00000 | 0.00005 | 0.31076 |

5 | 100/5 | 0.00000 | 0.00005 | 0.30622 |

6 | 200/5 | 0.00000 | 0.00007 | 0.27966 |

**Table 8.**Error statistics of the mass estimate using a Random Forest with distinct number of trees and splits.

Id | Tree/Split | Min. | Median | Max. |
---|---|---|---|---|

1 | 50/3 | 0.00001 | 0.09256 | 93.12868 |

2 | 100/3 | 0.00041 | 0.08894 | 81.82169 |

3 | 200/3 | 0.00026 | 0.08498 | 107.73466 |

4 | 50/5 | 0.00034 | 0.08707 | 77.93849 |

5 | 100/5 | 0.00003 | 0.08891 | 72.21142 |

6 | 200/5 | 0.00012 | 0.08465 | 94.27213 |

**Table 9.**Error statistics of the ${P}_{critical}$ estimate using a Random Forest with a distinct number of trees and splits.

Id | Tree/Split | Min. | Median | Max. |
---|---|---|---|---|

1 | 50/3 | 0.00018 | 0.07656 | 31.20040 |

2 | 100/3 | 0.00024 | 0.08051 | 16.20204 |

3 | 200/3 | 0.00004 | 0.07359 | 24.21144 |

4 | 50/5 | 0.00002 | 0.08778 | 23.46504 |

5 | 100/5 | 0.00007 | 0.08512 | 31.84951 |

6 | 200/5 | 0.00016 | 0.08478 | 20.88944 |

Id | ${\mathit{v}}_{1}$ | ${\mathit{v}}_{2}$ | ${\mathit{v}}_{3}$ | ${\mathit{v}}_{4}$ | ${\mathit{v}}_{5}$ | m | ${\mathit{P}}_{\mathit{critical}}$ | ${\mathit{b}}_{\mathit{I}}$ |
---|---|---|---|---|---|---|---|---|

1 | 0.1199 | 3 | 0.4123 | 48.41 | 7.60 | 1.510 | 4885 | −0.004 |

2 | 0.1487 | 4 | 0.4377 | 47.44 | 41.43 | 1.707 | 8024 | 0.038 |

3 | 0.0953 | 5 | 0.5615 | 49.68 | 51.61 | 1.910 | 11,344 | −0.003 |

4 | 0.1545 | 5 | 0.6250 | 54.75 | 56.16 | 2.139 | 14,253 | −0.002 |

5 | 0.1500 | 5 | 0.5299 | 59.09 | 57.71 | 2.303 | 17,020 | −0.006 |

6 | 0.1524 | 4 | 0.5205 | 60.55 | 62.40 | 2.534 | 20,467 | −0.002 |

7 | 0.1520 | 4 | 0.4729 | 65.19 | 62.33 | 2.745 | 23,818 | 0.031 |

8 | 0.1520 | 4 | 0.5200 | 67.06 | 63.55 | 2.896 | 25,779 | 0.035 |

9 | 0.0885 | 8 | 0.4050 | 67.56 | 67.00 | 3.133 | 31,053 | 0.342 |

10 | 0.1582 | 4 | 0.5127 | 66.79 | 70.41 | 3.338 | 34,298 | 0.095 |

11 | 0.1217 | 4 | 0.4316 | 68.98 | 70.37 | 3.465 | 37,080 | 0.309 |

12 | 0.1405 | 2 | 0.6810 | 50.10 | 12.63 | 1.448 | 4758 | 0.095 |

13 | 0.1450 | 2 | 0.7022 | 48.38 | 42.34 | 1.690 | 8151 | 0.033 |

14 | 0.1575 | 4 | 0.5750 | 49.95 | 51.61 | 1.915 | 11,375 | −0.003 |

15 | 0.1520 | 5 | 0.5999 | 54.75 | 57.01 | 2.166 | 14,739 | −0.002 |

16 | 0.1520 | 5 | 0.6650 | 59.09 | 57.76 | 2.293 | 16,918 | −0.007 |

17 | 0.1525 | 6 | 0.6654 | 63.68 | 57.98 | 2.435 | 18,751 | −0.005 |

18 | 0.1805 | 4 | 0.4348 | 66.48 | 61.05 | 2.788 | 23,240 | −0.017 |

19 | 0.1617 | 4 | 0.4436 | 64.22 | 68.53 | 3.009 | 27,982 | 0.029 |

20 | 0.1043 | 6 | 0.4595 | 65.79 | 70.51 | 3.254 | 31,232 | 0.005 |

21 | 0.1632 | 4 | 0.4677 | 69.07 | 68.50 | 3.340 | 34,555 | 0.350 |

22 | 0.1624 | 4 | 0.4705 | 70.86 | 68.51 | 3.500 | 36,835 | 0.317 |

**Table 11.**Imperfection sensitivity of Case ${0}^{\circ}$, Case $\pm {45}^{\circ}$, and the design optimum ID = 12 from Table 10.

Pristine | BL $\left(\mathit{N}\right)$ | BL $\left(\mathit{N}\right)$ | KDF | KDF | |
---|---|---|---|---|---|

Buckling | with a | with a | with a | with a | |

Load $\left(\mathit{N}\right)$ | SPLI of $0.1\phantom{\rule{4pt}{0ex}}\mathit{N}$ | SPLI of $1.0\phantom{\rule{4pt}{0ex}}\mathit{N}$ | SPLI of $0.1\phantom{\rule{4pt}{0ex}}\mathit{N}$ | SPLI of $1.0\phantom{\rule{4pt}{0ex}}\mathit{N}$ | |

Case ${0}^{\circ}$ | 3017 | 1747 | 225 | 57.9% | 7.4% |

Case $\pm {45}^{\circ}$ | 4261 | 3402 | 2040 | 79.8% | 47.9% |

Optimum Id = 12 | 3017 | 2511 | 1831 | 83.2% | 60.7% |

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**MDPI and ACS Style**

dos Santos, R.R.; Castro, S.G.P.
Lightweight Design of Variable-Stiffness Cylinders with Reduced Imperfection Sensitivity Enabled by Continuous Tow Shearing and Machine Learning. *Materials* **2022**, *15*, 4117.
https://doi.org/10.3390/ma15124117

**AMA Style**

dos Santos RR, Castro SGP.
Lightweight Design of Variable-Stiffness Cylinders with Reduced Imperfection Sensitivity Enabled by Continuous Tow Shearing and Machine Learning. *Materials*. 2022; 15(12):4117.
https://doi.org/10.3390/ma15124117

**Chicago/Turabian Style**

dos Santos, Rogério R., and Saullo G. P. Castro.
2022. "Lightweight Design of Variable-Stiffness Cylinders with Reduced Imperfection Sensitivity Enabled by Continuous Tow Shearing and Machine Learning" *Materials* 15, no. 12: 4117.
https://doi.org/10.3390/ma15124117