# Variable Thickness Strain Pre-Extrapolation for the Inverse Finite Element Method

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Inverse Finite Element Method

#### 2.1. Numerical Strain Formulation

#### 2.2. Input Strain Formulation

#### 2.3. Euclidean Norms and Minimization

#### 2.4. Variable Thickness Pre-Extrapolation with the iFEM

Algorithm 1: Variable Thickness Pre-Extrapolation | |

1: | Given surface measurements, compute curvatures $\mathit{k}$ and membrane strains $\mathit{e}$ (Equation (3)). |

2: | Normalize the $\mathit{k}$ and $\mathit{e}$, computing the normalized curvatures $\mathit{\xi}$ and membrane strains $\mathit{\eta}$ (Equation (9)). |

3: | Pre-extrapolate/interpolate the normalized membrane strain components using any interpolation/extrapolation method. |

4: | $\mathrm{Unnormalize}\text{}\mathit{\xi}$and $\mathit{\eta}$ (Equation (12)) to recompute $\mathit{k}$ and $\mathit{e}$. |

5: | $\mathit{k}$ and $\mathit{e}$ to the iFEM input. |

## 3. Case Study: Composite Variable Thickness Plate

#### 3.1. Traction

#### 3.2. Out-of-Plane Tip Loading

#### 3.3. Uniform Pressure

#### 3.4. Shear Loading

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Farrar, C.R.; Park, G.; Allen, D.W.; Todd, M.D. Sensor network paradigms for structural health monitoring. Struct. Control Health Monit.
**2006**, 13, 210–225. [Google Scholar] [CrossRef] - Bellemare, A. Continuous-wave silica-based erbium-doped fibre lasers. Prog. Quantum Electron.
**2003**, 27, 211–266. [Google Scholar] [CrossRef] - Todd, M.D.; Johnson, G.A.; Althouse, B.L. A novel Bragg grating sensor interrogation system utilizing a scanning filter, a Mach-Zehnder interferometer and a 3×3 coupler. Meas. Sci. Technol.
**2001**, 12, 771–777. [Google Scholar] [CrossRef] - Mascarenas, D.L.; Todd, M.D.; Park, G.; Farrar, C.R. Development of an impedance-based wireless sensor node for structural health monitoring. Smart Mater. Struct.
**2007**, 16, 2137–2145. [Google Scholar] [CrossRef] - Cross, E.J.; Worden, K.; Chen, Q. Cointegration: A novel approach for the removal of environmental trends in structural health monitoring data. Proc. R. Soc. A
**2011**, 467, 2712–2732. [Google Scholar] [CrossRef] - Worden, K.; Farrar, C.R.; Haywood, J.; Todd, M. A review of nonlinear dynamics applications to structural health monitoring. Struct. Control Health Monit.
**2008**, 15, 540–567. [Google Scholar] [CrossRef] - Figueiredo, E.; Park, G.; Figueiras, J.; Farrar, C.; Worden, K. Structural Health Monitoring Algorithm Comparisons Using Standard Data Sets; Los Alamos National Lab.: Los Alamos, NM, USA, 2009. [Google Scholar] [CrossRef]
- Figueiredo, E.; Brownjohn, J. Three decades of statistical pattern recognition paradigm for SHM of bridges. Struct. Health Monit.
**2022**, 21, 3018–3054. [Google Scholar] [CrossRef] - Farrar, C.R.; Worden, K. Structural Health Monitoring: A Machine Learning Perspective, 1st ed.; Wiley: Chichester, UK, 2013; ISBN 978-1-119-99433-6. [Google Scholar]
- Yan, R.; Chen, X.; Mukhopadhyay, S.C. (Eds.) Structural Health Monitoring: An Advanced Signal Processing Perspective; Springer: Cham, Switzerland, 2017; ISBN 978-3-319-56126-4. [Google Scholar]
- Chadha, M.; Ramancha, M.K.; Vega, M.A.; Conte, J.P.; Todd, M.D. The modeling of risk perception in the use of structural health monitoring information for optimal maintenance decisions. Reliab. Eng. Syst. Saf.
**2023**, 229, 108845. [Google Scholar] [CrossRef] - Zeng, J.; Wu, Z.; Todd, M.D.; Hu, Z. Bayes risk-based mission planning of Unmanned Aerial Vehicles for autonomous damage inspection. Mech. Syst. Signal Process.
**2023**, 187, 109958. [Google Scholar] [CrossRef] - Cross, E.J.; Gibson, S.J.; Jones, M.R.; Pitchforth, D.J.; Zhang, S.; Rogers, T.J. Physics-Informed Machine Learning for Structural Health Monitoring. In Structural Health Monitoring Based on Data Science Techniques; Cury, A., Ribeiro, D., Ubertini, F., Todd, M.D., Eds.; Springer International Publishing: Cham, Switzerland, 2022; Volume 21, pp. 347–367. ISBN 978-3-030-81715-2. [Google Scholar] [CrossRef]
- Najera-Flores, D.A.; Todd, M.D. Ensemble of Numerics-Informed Neural Networks with Embedded Hamiltonian Constraints for Modeling Nonlinear Structural Dynamics. In Nonlinear Structures & Systems; Brake, M.R.W., Renson, L., Kuether, R.J., Tiso, P., Eds.; Springer International Publishing: Cham, Switzerland, 2023; Volume 1, pp. 27–30. ISBN 978-3-031-04085-6. [Google Scholar] [CrossRef]
- Tessler, A.; Spangler, J. A Variational Principle for Reconstruction of Elastic Deformations in Shear Deformable Plates and Shells; National Aeronautics and Space Administration, Langley Research Center: Hampton, VA, USA, 2003. [Google Scholar]
- Tessler, A.; Spangler, J. Inverse FEM for Full-Field Reconstruction of Elastic Deformations in Shear Deformable Plates and Shells; DEStech Publications, Inc.: Lancaster, PA, USA, 2004. [Google Scholar]
- Gherlone, M.; Cerracchio, P.; Mattone, M. Shape sensing methods: Review and experimental comparison on a wing-shaped plate. Prog. Aerosp. Sci.
**2018**, 99, 14–26. [Google Scholar] [CrossRef] - Gherlone, M.; Cerracchio, P.; Mattone, M.; Di Sciuva, M.; Tessler, A. An inverse finite element method for beam shape sensing: Theoretical framework and experimental validation. Smart Mater. Struct.
**2014**, 23, 045027. [Google Scholar] [CrossRef] - Kefal, A.; Oterkus, E. Displacement and stress monitoring of a Panamax containership using inverse finite element method. Ocean Eng.
**2016**, 119, 16–29. [Google Scholar] [CrossRef] - Kefal, A.; Mayang, J.B.; Oterkus, E.; Yildiz, M. Three dimensional shape and stress monitoring of bulk carriers based on iFEM methodology. Ocean Eng.
**2018**, 147, 256–267. [Google Scholar] [CrossRef] - Li, M.; Kefal, A.; Oterkus, E.; Oterkus, S. Structural health monitoring of an offshore wind turbine tower using iFEM methodology. Ocean Eng.
**2020**, 204, 107291. [Google Scholar] [CrossRef] - Li, M.; Kefal, A.; Cerik, B.C.; Oterkus, E. Dent damage identification in stiffened cylindrical structures using inverse Finite Element Method. Ocean Eng.
**2020**, 198, 106944. [Google Scholar] [CrossRef] - Oboe, D.; Colombo, L.; Sbarufatti, C.; Giglio, M. Shape Sensing of a Complex Aeronautical Structure with Inverse Finite Element Method. Sensors
**2021**, 21, 1388. [Google Scholar] [CrossRef] - Colombo, L.; Sbarufatti, C.; Giglio, M. Definition of a load adaptive baseline by inverse finite element method for structural damage identification. Mech. Syst. Signal Process.
**2019**, 120, 584–607. [Google Scholar] [CrossRef] - Oboe, D.; Poloni, D.; Sbarufatti, C.; Giglio, M. Crack Size Estimation with an Inverse Finite Element Model. In European Workshop on Structural Health Monitoring; Rizzo, P., Milazzo, A., Eds.; Springer International Publishing: Cham, Switzerland, 2023; Volume 253, pp. 443–453. ISBN 978-3-031-07253-6. [Google Scholar] [CrossRef]
- Kefal, A.; Diyaroglu, C.; Yildiz, M.; Oterkus, E. Coupling of peridynamics and inverse finite element method for shape sensing and crack propagation monitoring of plate structures. Comput. Methods Appl. Mech. Eng.
**2022**, 391, 114520. [Google Scholar] [CrossRef] - Kefal, A.; Tessler, A. Delamination Damage Identification in Composite Shell Structures Based on Inverse Finite Element Method and Refined Zigzag Theory; CRC Press: Boca Raton, FL, USA, 2021. [Google Scholar]
- Roy, R.; Gherlone, M.; Surace, C.; Tessler, A. Full-Field Strain Reconstruction Using Uniaxial Strain Measurements: Application to Damage Detection. Appl. Sci.
**2021**, 11, 1681. [Google Scholar] [CrossRef] - Poloni, D.; Oboe, D.; Sbarufatti, C.; Giglio, M. Gaussian Process Strain Pre-extrapolation and Uncertainty Estimation for Inverse Finite Elements. In European Workshop on Structural Health Monitoring; Rizzo, P., Milazzo, A., Eds.; Springer International Publishing: Cham, Switzerland, 2023; Volume 254, pp. 308–317. ISBN 978-3-031-07257-4. [Google Scholar] [CrossRef]
- Tessler, A.; Riggs, H.R.; Macy, S.C. A variational method for finite element stress recovery and error estimation. Comput. Methods Appl. Mech. Eng.
**1994**, 111, 369–382. [Google Scholar] [CrossRef] - Tessler, A.; Riggs, H.R.; Freese, C.E.; Cook, G.M. An improved variational method for finite element stress recovery and a posteriori error estimation. Comput. Methods Appl. Mech. Eng.
**1998**, 155, 15–30. [Google Scholar] [CrossRef] - Riggs, H.R.; Tessler, A.; Chu, H. C1-Continuous stress recovery in finite element analysis. Comput. Methods Appl. Mech. Eng.
**1997**, 143, 299–316. [Google Scholar] [CrossRef] - Oboe, D.; Colombo, L.; Sbarufatti, C.; Giglio, M. Comparison of strain pre-extrapolation techniques for shape and strain sensing by iFEM of a composite plate subjected to compression buckling. Compos. Struct.
**2021**, 262, 113587. [Google Scholar] [CrossRef] - Kefal, A.; Tabrizi, I.E.; Yildiz, M.; Tessler, A. A smoothed iFEM approach for efficient shape-sensing applications: Numerical and experimental validation on composite structures. Mech. Syst. Signal Process.
**2021**, 152, 107486. [Google Scholar] [CrossRef] - Poloni, D.; Oboe, D.; Sbarufatti, C.; Giglio, M. Towards a stochastic inverse Finite Element Method: A Gaussian Process strain extrapolation. Mech. Syst. Signal Process.
**2023**, 189, 110056. [Google Scholar] [CrossRef] - Roy, R.; Tessler, A.; Surace, C.; Gherlone, M. Efficient shape sensing of plate structures using the inverse Finite Element Method aided by strain pre-extrapolation. Thin-Walled Struct.
**2022**, 180, 109798. [Google Scholar] [CrossRef] - Roy, R.; Esposito, M.; Surace, C.; Gherlone, M.; Tessler, A. Shape Sensing of Stiffened Plates Using Inverse FEM Aided by Virtual Strain Measurements. In European Workshop on Structural Health Monitoring; Rizzo, P., Milazzo, A., Eds.; Springer International Publishing: Cham, Switzerland, 2023; Volume 253, pp. 454–463. ISBN 978-3-031-07253-6. [Google Scholar] [CrossRef]
- Oboe, D.; Sbarufatti, C.; Giglio, M. Physics-based strain pre-extrapolation technique for inverse Finite Element Method. Mech. Syst. Signal Process.
**2022**, 177, 109167. [Google Scholar] [CrossRef] - Kefal, A.; Oterkus, E.; Tessler, A.; Spangler, J.L. A quadrilateral inverse-shell element with drilling degrees of freedom for shape sensing and structural health monitoring. Eng. Sci. Technol. Int. J.
**2016**, 19, 1299–1313. [Google Scholar] [CrossRef] - Tessler, A.; Hughes, T.J.R. An improved treatment of transverse shear in the mindlin-type four-node quadrilateral element. Comput. Methods Appl. Mech. Eng.
**1983**, 39, 311–335. [Google Scholar] [CrossRef] - Cook, R.D. Four-node ‘flat’ shell element: Drilling degrees of freedom, membrane-bending coupling, warped geometry, and behavior. Comput. Struct.
**1994**, 50, 549–555. [Google Scholar] [CrossRef] - Kefal, A.; Tessler, A.; Oterkus, E. An enhanced inverse finite element method for displacement and stress monitoring of multilayered composite and sandwich structures. Compos. Struct.
**2017**, 179, 514–540. [Google Scholar] [CrossRef] - Jones, R.M. Mechanics of Composite Materials, 2nd ed.; CRC Press: Boca Raton, FL, USA, 2018; ISBN 978-1-315-27298-6. [Google Scholar] [CrossRef]
- Tsai, S.W. Double–Double: New Family of Composite Laminates. AIAA J.
**2021**, 59, 4293–4305. [Google Scholar] [CrossRef] - Singh, K.K.; Singh, N.K.; Jha, R. Analysis of symmetric and asymmetric glass fiber reinforced plastic laminates subjected to low-velocity impact. J. Compos. Mater.
**2016**, 50, 1853–1863. [Google Scholar] [CrossRef]

**Figure 1.**Local (x, y, z) and global (X, Y, Z) reference systems for the iQS4 element; numbers 1 to 4 represent the local node labels.

**Figure 4.**(

**a**) iFEM mesh and (

**b**) direct FEM mesh: Triaxial strain sensors are placed on both the bottom and top sides at each blue-filled circle.

**Figure 7.**Traction: (

**a**) membrane strain ${e}_{XX}$ and (

**b**) normalized membrane strain ${\eta}_{XX}$, slice at Y = 250 mm.

**Figure 11.**(

**a**) out-of-plane tip loading: curvature ${\kappa}_{XX}$ and (

**b**) normalized curvature ${\xi}_{XX}$, slice at Y = 250 mm.

**Figure 15.**Uniform pressure: (

**a**) curvature ${\kappa}_{XX}$ and (

**b**) normalized curvature ${\xi}_{XX}$, slice at Y = 250 mm.

**Figure 18.**Shear loading: (

**a**) membrane strain ${e}_{XY}$ and (

**b**) normalized membrane strain ${\eta}_{XY}$, slice at Y = 250 mm.

**Table 1.**Fiberglass-epoxy lamina material properties [45] for the direct FEM.

${\mathit{E}}_{11}$$\mathbf{\left[}\mathbf{M}\mathbf{P}\mathbf{a}\mathbf{\right]}$ | ${\mathit{E}}_{22}\mathbf{\left[}\mathbf{M}\mathbf{P}\mathbf{a}\mathbf{\right]}$ | ${\mathit{\nu}}_{12}$ | ${\mathit{G}}_{12}\mathbf{\left[}\mathbf{M}\mathbf{P}\mathbf{a}\mathbf{\right]}$ | ${\mathit{G}}_{13}\mathbf{\left[}\mathbf{M}\mathbf{P}\mathbf{a}\mathbf{\right]}$ | ${\mathit{G}}_{23}\mathbf{\left[}\mathbf{M}\mathbf{P}\mathbf{a}\mathbf{\right]}$ | Thickness [mm] |
---|---|---|---|---|---|---|

26,000 | 26,000 | 0.1 | 3800 | 2800 | 2800 | 0.25 |

${\mathit{\eta}}_{\mathit{X}\mathit{X}}$ | ${\mathit{\eta}}_{\mathit{Y}\mathit{Y}}$ | ${\mathit{\eta}}_{\mathit{X}\mathit{Y}}$ | ${\mathit{\xi}}_{\mathit{X}\mathit{X}}$ | ${\mathit{\xi}}_{\mathit{Y}\mathit{Y}}$ | ${\mathit{\xi}}_{\mathit{X}\mathit{Y}}$ | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Direction | $X$ | $Y$ | $X$ | $Y$ | $X$ | $Y$ | $X$ | $Y$ | $X$ | $Y$ | $X$ | $Y$ |

Polynomial degree | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

${\mathit{\eta}}_{\mathit{X}\mathit{X}}$ | ${\mathit{\eta}}_{\mathit{Y}\mathit{Y}}$ | ${\mathit{\eta}}_{\mathit{X}\mathit{Y}}$ | ${\mathit{\xi}}_{\mathit{X}\mathit{X}}$ | ${\mathit{\xi}}_{\mathit{Y}\mathit{Y}}$ | ${\mathit{\xi}}_{\mathit{X}\mathit{Y}}$ | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Direction | $X$ | $Y$ | $X$ | $Y$ | $X$ | $X$ | $Y$ | $X$ | $Y$ | $X$ | $Y$ | |

Polynomial degree | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

${\mathit{\eta}}_{\mathit{X}\mathit{X}}$ | ${\mathit{\eta}}_{\mathit{Y}\mathit{Y}}$ | ${\mathit{\eta}}_{\mathit{X}\mathit{Y}}$ | ${\mathit{\xi}}_{\mathit{X}\mathit{X}}$ | ${\mathit{\xi}}_{\mathit{Y}\mathit{Y}}$ | ${\mathit{\xi}}_{\mathit{X}\mathit{Y}}$ | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Direction | $X$ | $Y$ | $X$ | $Y$ | $X$ | $Y$ | $X$ | $Y$ | $X$ | $Y$ | $X$ | $Y$ |

Polynomial degree | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 |

${\mathit{\eta}}_{\mathit{X}\mathit{X}}$ | ${\mathit{\eta}}_{\mathit{Y}\mathit{Y}}$ | ${\mathit{\eta}}_{\mathit{X}\mathit{Y}}$ | ${\mathit{\xi}}_{\mathit{X}\mathit{X}}$ | ${\mathit{\xi}}_{\mathit{Y}\mathit{Y}}$ | ${\mathit{\xi}}_{\mathit{X}\mathit{Y}}$ | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Direction | $X$ | $Y$ | $X$ | $Y$ | $X$ | $Y$ | $X$ | $Y$ | $X$ | $Y$ | $X$ | $Y$ |

Polynomial degree | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 |

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**MDPI and ACS Style**

Poloni, D.; Oboe, D.; Sbarufatti, C.; Giglio, M.
Variable Thickness Strain Pre-Extrapolation for the Inverse Finite Element Method. *Sensors* **2023**, *23*, 1733.
https://doi.org/10.3390/s23031733

**AMA Style**

Poloni D, Oboe D, Sbarufatti C, Giglio M.
Variable Thickness Strain Pre-Extrapolation for the Inverse Finite Element Method. *Sensors*. 2023; 23(3):1733.
https://doi.org/10.3390/s23031733

**Chicago/Turabian Style**

Poloni, Dario, Daniele Oboe, Claudio Sbarufatti, and Marco Giglio.
2023. "Variable Thickness Strain Pre-Extrapolation for the Inverse Finite Element Method" *Sensors* 23, no. 3: 1733.
https://doi.org/10.3390/s23031733