# Multi-Dimensional Regression Models for Predicting the Wall Thickness Distribution of Corrugated Pipes

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Fundamentals

#### 2.1. Geometry and Modeling

- BS1 and BS3: symmetry
- BS2: free surface, constant normal forces (vacuum pressure) imposed, contact detection problem with the solid mold defined
- BS4: free surface, constant normal forces (inflation air pressure) imposed
- BS5: contact wall (contact between fluid parison and wall)
- BS6, BS7 and BS8: no contact walls and free of force

- The thickness of the extruded fluid parison, $S$, is constant in the initial state before the blow molding process.
- The temperature of the extruded fluid parison, $S$, is homogeneous and thus the viscosity remains constant over the entire cross section.
- The influence of the viscoelasticity, temperature, and pressures (air and vacuum) was neglected as they are insignificant for the final wall thickness distribution. However, they have an impact on the dynamics of the process, i.e., on how fast the molding process takes place.
- The extrusion speed of the parison is exactly equal to the speed of the mold blocks. Based on this assumption, it is allowed to neglect the dynamics of the process and reduce the geometry to a half model.

#### 2.2. Dimensional Analysis and Similitude

## 3. Numerical Simulation

^{3}. For operational conditions, a constant vacuum and inflation air pressure of 0.9 and 0.1 bar were applied, respectively, to the outer and inner surfaces of the fluid parison. This non-linear problem was solved numerically and iteratively by a very robust algebraic multi-frontal (AMF) direct solver based on the Gauss elimination method [22]. The final converged solution was obtained after performing the time-dependent calculation with the assigned parameters needed by the iterative scheme. Subsequently, the results are transformed back into a dimensionless representation. The blowing process over time is exemplarily shown in Figure 5 for various time steps. It can be seen that at first, the parison is inflated uniformly until it gets in contact with the mold. Then, the parison is further inflated into the mold, next getting into contact with the flanks and subsequently with the crest. The upper flank radius is shaped last, after an inflation time of approximately 1 s.

## 4. Design Study

#### 4.1. Screening Design

#### 4.1.1. Screening Design—Procedure

_{0}is rejected, and it can be concluded that the difference is significant. In other words, with a p-value < 0.05, the result is statistically significant, and with a p-value > 0.05, it is not [23].

#### 4.1.2. Screening Design—Results

#### 4.2. Parametric Design Study

#### Parametric Design Study—Results

## 5. Regression Analysis Using Heuristic Approaches

#### 5.1. Symbolic Regression—Modeling

#### 5.2. Symbolic Regression—Results

^{2}(Equation (14))—which describes how close the values estimated by our models are to the measured values, and is indicative of the response variation explained by a model, was utilized to determine the model quality.

^{2}value (=1) indicates that the predictions agree perfectly with the measured values, and a low R

^{2}value (=0) means that the predictions are poor and that the model is to be discarded [26].

^{2}was achieved after final model evaluation. Subsequently, the statistical accuracies of all derived models based on validation data set are given in Table 7. As indicated, the models given by Equations (9)–(13) achieved relatively small MAE values and MRE ≤ 1.632%, as well as the coefficient of determination R

^{2}> 0.996. The error analyses thus confirm that the derived models are statistically highly accurate.

## 6. Conclusions

## Author Contributions

## Funding

^{2}Future is funded within the Austrian COMET Program Competence Centers for Excellent Technologies under the auspices of the Austrian Federal Ministry for Climate Action, Environment, Energy, Mobility, Innovation and Technology, the Austrian Federal Ministry for Digital and Economic Affairs and of the Provinces of Upper Austria and Styria. COMET is managed by the Austrian Research Promotion Agency FFG.” Open Access funding is supported by Johannes Kepler Open Access Publishing Fund.

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Sub-Functions for the Regression model Predicting Ratio

Constant | Value | Constant | Value | Constant | Value | Constant | Value |
---|---|---|---|---|---|---|---|

${a}_{00}$ | 5.102 | ${a}_{10}$ | −7.786 × 10^{−2} | ${a}_{20}$ | 5.052 | ${a}_{30}$ | −2.894 × 10^{−1} |

${a}_{01}$ | −3.763 × 10^{−1} | ${a}_{11}$ | 5.775 | ${a}_{21}$ | 1.388 | ${a}_{31}$ | 6.817 |

${a}_{02}$ | −1.967 × 10^{−3} | ${a}_{12}$ | −7.779 | ${a}_{22}$ | 4.833 | ${a}_{32}$ | 1.197 |

${a}_{03}$ | −1.588 | ${a}_{13}$ | −1.202 × 10^{−2} | ${a}_{23}$ | 1.754 | ${a}_{33}$ | −6.157 × 10^{−1} |

${a}_{04}$ | −1.909 × 10^{−1} | ${a}_{14}$ | −1.103 | ${a}_{24}$ | −3.799 × 10^{−3} | ${a}_{34}$ | 6.009 |

${a}_{05}$ | 5.199 | ${a}_{15}$ | 6.869 × 10^{−1} | ${a}_{25}$ | 2.320 × 10^{−1} | ${a}_{35}$ | 8.257 × 10^{−1} |

${a}_{06}$ | 8.229 | ${a}_{16}$ | 3.796 | ${a}_{26}$ | −3.956 × 10^{−1} | ${a}_{36}$ | 1.114 |

${a}_{07}$ | −8.324 × 10^{−1} | ${a}_{17}$ | 1.562 × 10^{1} | ${a}_{27}$ | 4.595 | ${a}_{37}$ | 8.906 × 10^{−1} |

${a}_{08}$ | 7.659 | ${a}_{18}$ | 4.302 × 10^{−1} | ${a}_{28}$ | 1.890 | ${a}_{38}$ | 1.002 × 10^{−1} |

${a}_{09}$ | 5.134 | ${a}_{19}$ | −7.264 | ${a}_{29}$ | 5.028 |

#### Appendix A.2. Sub-Functions for the Regression Model Predicting ${\prod}_{TP\_C}$

Constant | Value | Constant | Value | Constant | Value | Constant | Value |
---|---|---|---|---|---|---|---|

${b}_{00}$ | 2.283 | ${b}_{07}$ | −1.010 × 10^{−1} | ${b}_{14}$ | 1.874 | ${b}_{21}$ | 8.775 × 10^{−2} |

${b}_{01}$ | 7.534 × 10^{−2} | ${b}_{08}$ | 5.358 × 10^{−4} | ${b}_{15}$ | 7.546 × 10^{−2} | ${b}_{22}$ | 4.176 × 10^{−2} |

${b}_{02}$ | 1.808 × 10^{−1} | ${b}_{09}$ | −0.112 | ${b}_{16}$ | 3.625 × 10^{−2} | ${b}_{23}$ | 1.293 × 10^{−4} |

${b}_{03}$ | 4.289 × 10^{−4} | ${b}_{10}$ | −1.602 × 10^{−1} | ${b}_{17}$ | −1.189 × 10 | ${b}_{24}$ | −3.388 × 10^{−2} |

${b}_{04}$ | −1.917 × 10^{3} | ${b}_{11}$ | −2.469 × 10^{−1} | ${b}_{18}$ | 2.060 | ${b}_{25}$ | −1.225 × 10^{−1} |

${b}_{05}$ | −8.858 × 10^{−1} | ${b}_{12}$ | 2.468 | ${b}_{19}$ | −1.189 | ${b}_{26}$ | 31.48 |

${b}_{06}$ | −8.320 × 10^{−1} | ${b}_{13}$ | −1.094 | ${b}_{20}$ | 1.762 × 10^{2} | ${b}_{27}$ | 4.664 × 10^{−3} |

#### Appendix A.3. Sub-Functions for the Regression Model Predicting ${\prod}_{TP\_V}$

Constant | Value | Constant | Value | Constant | Value | Constant | Value |
---|---|---|---|---|---|---|---|

${c}_{00}$ | 1.355 | ${c}_{09}$ | −2.382 × 10^{−1} | ${c}_{18}$ | −21.15 | ${c}_{27}$ | −10.48 |

${c}_{01}$ | −1.756 | ${c}_{10}$ | 11.28 | ${c}_{19}$ | 1.566 × 10^{2} | ${c}_{28}$ | −19.62 |

${c}_{02}$ | −7.893 × 10^{−2} | ${c}_{11}$ | 11.22 | ${c}_{20}$ | −1.540 × 10^{2} | ${c}_{29}$ | 3.222 |

${c}_{03}$ | 17.11 | ${c}_{12}$ | 1.925 | ${c}_{21}$ | 1.750 × 10^{−2} | ${c}_{30}$ | −2.579 × 10^{2} |

${c}_{04}$ | −11.62 | ${c}_{13}$ | 1.002 × 10^{−1} | ${c}_{22}$ | 3.369 × 10^{−1} | ${c}_{31}$ | 1.004 |

${c}_{05}$ | 17.19 | ${c}_{14}$ | −3.621 | ${c}_{23}$ | 10.13 | ${c}_{32}$ | 8.134 × 10^{−3} |

${c}_{06}$ | −11.62 | ${c}_{15}$ | 1.739 | ${c}_{24}$ | 17.24 | ||

${c}_{07}$ | −7.048 × 10^{−3} | ${c}_{16}$ | 2.974 × 10^{−3} | ${c}_{25}$ | 8.078 × 10^{2} | ||

${c}_{08}$ | −1.687 × 10^{−1} | ${c}_{17}$ | −2.246 | ${c}_{26}$ | −2.228 × 10^{−1} |

#### Appendix A.4. Sub-Functions for the Regression Model Predicting ${\prod}_{TP\_LF}$

Constant | Value | Constant | Value | Constant | Value |
---|---|---|---|---|---|

${e}_{00}$ | 4.114 × 10^{−1} | ${e}_{11}$ | 1.461 | ${e}_{22}$ | 2.473 |

${e}_{01}$ | 6.584 × 10^{−3} | ${e}_{12}$ | 1.219 | ${e}_{23}$ | 1.736 |

${e}_{02}$ | 3.588 × 10^{−2} | ${e}_{13}$ | 8.923 × 10^{−1} | ${e}_{24}$ | 7.159 × 10^{−1} |

${e}_{03}$ | 9.938 × 10^{−1} | ${e}_{14}$ | −1.334 × 10^{−1} | ${e}_{25}$ | −1.772 × 10^{−1} |

${e}_{04}$ | −2.344 | ${e}_{15}$ | −8.782 × 10^{−1} | ${e}_{26}$ | 7.645 × 10^{−1} |

${e}_{05}$ | −5.395 × 10^{−3} | ${e}_{16}$ | 2.443 × 10^{−3} | ${e}_{27}$ | −1.009 |

${e}_{06}$ | −1.518 × 10^{−2} | ${e}_{17}$ | 1.046 | ${e}_{28}$ | −1.245 |

${e}_{07}$ | 1.876 × 10^{−1} | ${e}_{18}$ | 1.730 | ${e}_{29}$ | −1.074 × 10^{−1} |

${e}_{08}$ | −3.506 | ${e}_{19}$ | −10.680 | ${e}_{30}$ | −1.016 |

${e}_{09}$ | −2.727 × 10^{−3} | ${e}_{20}$ | −1.563 × 10^{−1} | ${e}_{31}$ | 8.869 × 10^{−1} |

${e}_{10}$ | 8.860 × 10^{−1} | ${e}_{21}$ | 5.067 × 10^{−3} | ${e}_{32}$ | −1.991 × 10^{−4} |

#### Appendix A.5. Sub-Functions for the Regression Model Predicting ${\prod}_{TP\_UF}$

Constant | Value | Constant | Value | Constant | Value |
---|---|---|---|---|---|

${f}_{00}$ | −4.410 × 10^{−3} | ${f}_{11}$ | −1.41418 | ${f}_{22}$ | −1.925 |

${f}_{01}$ | −2.280 × 10^{−3} | ${f}_{12}$ | 2.249 | ${f}_{23}$ | −2.795 × 10^{−2} |

${f}_{02}$ | −3.333 | ${f}_{13}$ | 4.547 | ${f}_{24}$ | 1.276 |

${f}_{03}$ | 9.931 × 10^{−3} | ${f}_{14}$ | 1.510 × 10^{−2} | ${f}_{25}$ | 2.415 × 10^{−2} |

${f}_{04}$ | 1.222 | ${f}_{15}$ | −139.413 | ${f}_{26}$ | −1.762 |

${f}_{05}$ | 2.486 × 10^{−1} | ${f}_{16}$ | 17.694 | ${f}_{27}$ | −1.129 |

${f}_{06}$ | 62.361 | ${f}_{17}$ | 1.106 | ${f}_{28}$ | 1.041 |

${f}_{07}$ | −7.463 | ${f}_{18}$ | 1.913 × 10^{−1} | ${f}_{29}$ | 4.270 × 10^{−3} |

${f}_{08}$ | 1.443 × 10^{−1} | ${f}_{19}$ | −1.521 | ||

${f}_{09}$ | −1.890 × 10^{−2} | ${f}_{20}$ | 2.146 | ||

${f}_{10}$ | 1.327333 | ${f}_{21}$ | 1.616 |

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**Figure 1.**(

**a**) Schematic of a corrugated pipe production line and of the corrugator with (

**b**) the die head.

**Figure 2.**(

**a**) 3D Mold block geometry on which the 2D axisymmetric model is based; (

**b**) simulation boundary conditions.

**Figure 4.**(

**a**) Optimization of the simulation time, (

**b**) comparison study of wall thickness distribution and simulation time.

**Figure 5.**Simulation results of the blow-molding process showing that the parison is fully inflated after an inflation time of 1 s: (

**a**) wall thickness distribution, (

**b**) shear stress distribution.

**Figure 7.**Ratio of wall thicknesses at crest and valley as a function of dimensionless (

**a**) half profile width at valley ${\prod}_{{B}_{T}}$ and (

**b**) valley diameter ${\prod}_{{D}_{2}}$.

**Figure 8.**Ratio of wall thicknesses at crest and valley as a function of dimensionless (

**a**) initial thickness of fluid parison ${\prod}_{S};($

**b**) half profile width at crest ${\prod}_{{B}_{A}}$; (

**c**) crest diameter ${\prod}_{{D}_{1}}$; (

**d**) mold block inner radius ${\prod}_{{R}_{I}}$; and (

**e**) flank angle $\alpha $.

**Figure 9.**Ratio of wall thicknesses at crest and valley and dimensionless wall thickness as functions of (

**a**,

**b**) flank angle $\alpha $ and (

**c**,

**d**) dimensionless crest diameter ${\prod}_{{D}_{1}}$.

**Figure 10.**Ratio of wall thicknesses at crest and valley and dimensionless wall thickness as functions of dimensionless (

**a**,

**b**) half profile width at crest ${\prod}_{{B}_{A}}$ and (

**c**,

**d**) initial thickness of fluid parison ${\prod}_{S}$.

**Figure 11.**(

**a**) Ratio of wall thicknesses at crest and valley and (

**b**) dimensionless wall thickness as functions of dimensionless mold block inner radius ${\prod}_{{R}_{I}}$.

**Figure 13.**Comparisons of the estimated models for wall thickness distribution ratio and dimensionless wall thickness at several evaluation positions as functions of (

**a**,

**b**) dimensionless crest diameter ${\prod}_{{D}_{1}}$, (

**c**,

**d**) dimensionless half profile width at crest ${\prod}_{{B}_{A}},$ (

**e**,

**f**) dimensionless mold block inner radius ${\prod}_{{R}_{I}}$.

**Figure 14.**Comparisons of the estimated models for wall thickness distribution ratio and dimensionless wall thickness at several evaluation positions as functions of (

**a**,

**b**) dimensionless initial thickness of fluid parison ${\prod}_{S}$ and (

**c**,

**d**) flank angle $\alpha $.

**Figure 15.**Scatter plots of (

**a**) wall thickness ratio, (

**b**) dimensionless wall thickness at the crest, (

**c**) dimensionless wall thickness at the valley, (

**d**) dimensionless wall thickness at the lower flank, and (

**e**) dimensionless wall thickness at the upper flank. The dashed red lines indicate a relative error of ±5%.

Parameter | Mesh 1 | Mesh 2 | Mesh 3 | Mesh 4 | Mesh 5 | Mesh 6 |
---|---|---|---|---|---|---|

Edge Sizes [mm] | 0.01 | 0.025 | 0.05 | 0.075 | 0.1 | 0.15 |

CPU Time [s] | 5282 | 2300 | 1358 | 992 | 953 | 519 |

Wall thickness at crest [mm] | 0.3475 | 0.3470 | 0.3450 | 0.3447 | 0.3435 | 0.3417 |

Wall thickness at valley [mm] | 0.8997 | 0.8992 | 0.8992 | 0.8987 | 0.8985 | 0.8976 |

Parameter | Unit | Value | ||
---|---|---|---|---|

${\prod}_{{R}_{I}}$ | - | 0.835 | 0.879 | 0.932 |

${\prod}_{{D}_{1}}$ | - | 0.005 | 0.049 | 0.097 |

${\prod}_{{D}_{2}}$ | - | 0.002 | 0.026 | 0.052 |

${\prod}_{{B}_{A}}$ | - | 0.007 | 0.028 | 0.049 |

${\prod}_{{B}_{T}}$ | - | 0.001 | 0.004 | 0.015 |

${\prod}_{S}$ | - | 0.009 | 0.011 | 0.030 |

$\alpha $ | 1 | 7 | 17.5 |

Term | Estimated Regression Coefficient | p-Value |
---|---|---|

${\prod}_{{R}_{I}}$ | 0.0401725 | 6.25418 × 10^{−5} |

${\prod}_{{D}_{1}}$ | 0.0332355 | 0.000293 |

${\prod}_{{D}_{2}}$ | −0.013532 | 0.186159 |

${\prod}_{{B}_{A}}$ | 0.0775136 | 0.000227 |

${\prod}_{{B}_{T}}$ | −0.040225 | 0.226997 |

${\prod}_{S}$ | 0.0526789 | 0.020054 |

$\alpha $ | 0.0092651 | 0.012899 |

Parameter | Unit | Value | ||||
---|---|---|---|---|---|---|

${\prod}_{{R}_{I}}$ | - | 0.835 | 0.857 | 0.879 | 0.920 | 0.932 |

${\prod}_{{D}_{1}}$ | - | 0.005 | 0.024 | 0.049 | 0.073 | 0.097 |

${\prod}_{{B}_{A}}$ | - | 0.007 | 0.018 | 0.028 | 0.038 | 0.049 |

${\prod}_{S}$ | - | 0.009 | 0.011 | 0.017 | 0.024 | 0.030 |

$\alpha $ | 1 | 3.5 | 7 | 10.5 | 17.5 |

Parameter | Value |
---|---|

Population size | 100 |

Selected parents | 200 |

Crossover probability | 90% |

Mutation probability | 25% |

Maximum tree depth | 30 |

Maximum tree length | 100 |

Fitness function | Pearson R^{2} |

Maximum generations | 75 |

Maximum selection pressure | 100 |

Operators | +, −, ∗, / |

Power functions (square) |

Parameter | Unit | Value | |||
---|---|---|---|---|---|

${\prod}_{{R}_{I}}$ | - | 0.846 | 0.868 | 0.899 | 0.926 |

${\prod}_{{D}_{1}}$ | - | 0.015 | 0.036 | 0.061 | 0.085 |

${\prod}_{{B}_{A}}$ | - | 0.012 | 0.023 | 0.033 | 0.043 |

${\prod}_{S}$ | - | 0.010 | 0.014 | 0.021 | 0.027 |

$\alpha $ | ° | 2.25 | 5.25 | 8.75 | 14 |

Model | R^{2} (−) | MAE (−) | MRE (%) |
---|---|---|---|

Ratio | 0.99942 | 3.101 × 10^{−3} | 0.720 |

${\prod}_{TP\_Crest}$ | 0.99903 | 6.255 × 10^{−5} | 1.232 |

${\prod}_{TP\_Valley}$ | 0.99985 | 2.550 × 10^{−5} | 0.194 |

${\prod}_{TP\_LF}$ | 0.99649 | 1.058 × 10^{−4} | 1.073 |

${\prod}_{TP\_UF}$ | 0.99828 | 9.235 × 10^{−5} | 1.632 |

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## Share and Cite

**MDPI and ACS Style**

Albrecht, H.; Roland, W.; Fiebig, C.; Berger-Weber, G.R.
Multi-Dimensional Regression Models for Predicting the Wall Thickness Distribution of Corrugated Pipes. *Polymers* **2022**, *14*, 3455.
https://doi.org/10.3390/polym14173455

**AMA Style**

Albrecht H, Roland W, Fiebig C, Berger-Weber GR.
Multi-Dimensional Regression Models for Predicting the Wall Thickness Distribution of Corrugated Pipes. *Polymers*. 2022; 14(17):3455.
https://doi.org/10.3390/polym14173455

**Chicago/Turabian Style**

Albrecht, Hanny, Wolfgang Roland, Christian Fiebig, and Gerald Roman Berger-Weber.
2022. "Multi-Dimensional Regression Models for Predicting the Wall Thickness Distribution of Corrugated Pipes" *Polymers* 14, no. 17: 3455.
https://doi.org/10.3390/polym14173455