# Optimum Electrode Configurations for Two-Probe, Four-Probe and Multi-Probe Schemes in Electrical Resistance Tomography for Delamination Identification in Carbon Fiber Reinforced Composites

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## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation and Solution

#### 2.1. Finite Element Model

^{6}S/m conductivity.

#### 2.2. Measuring Schemes

#### 2.3. Effective Independence Based Electrode Selection

#### 2.4. Electrode Combinations Reduction

#### 2.5. Inverse Identification Accuracy with Electrode Selection

#### 2.5.1. Two-Probe Schemes

#### 2.5.2. Four-Probe Schemes

#### 2.5.3. Multi-Probe Schemes

## 3. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CFRP | Carbon Fiber Reinforced Polymer |

ERT | Electrical Resistance Tomography |

EIT | Electrical Impedance Tomography |

EI | Effective Independence |

FIM | Fisher Information Matrix |

FEA | Finite Element Analysis |

FF | Full Factorial |

APDL | ANSYS Parametric Design Language |

GA | Genetic Algorithm |

DOE | Design of Experiments |

RE | Relative Error |

LHS | Latin Hypercube Sampling |

RMS | Root Mean Square |

RMSE | Root Mean Square Error |

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**Figure 1.**Schematic representation of a $[{0}_{4}/{90}_{4}{]}_{s}$ composite laminate under damage with 14 electrodes where electric current is injected in pair 1–2.

**Figure 2.**Scheme used for damage identification using two-probe schemes and two-sided electrodes. The matrix of observations A is obtained from the responses in order to apply effective independence. Responses are grouped by damage scenario in column vectors (${\nu}_{s}$) in matrix A: (

**a**) starting with the response for consecutive electrodes on the left edge at the top surface; (

**b**) following with increase of electrode distance and (

**c**) finishing with the electrode pair on the right edge at the bottom surface.

**Figure 3.**Scheme used for damage identification using four-probe schemes and two-sided electrodes. The matrix of observations A is obtained from the responses in order to apply effective independence. Responses are grouped by damage scenario in column vectors (${\nu}_{s}$) in matrix A: (

**a**) starting on the top surface with schemes for surface resistance; (

**b**) following with an increase of electrode distance between electric current electrodes and (

**c**) finishing with schemes used for oblique resistance.

**Figure 4.**Scheme used for damage identification using multi-probe schemes and two-sided electrodes. The matrix of observations A is obtained from the responses in order to apply effective independence. Responses are grouped by damage scenario in column vectors (${\nu}_{s}$) in matrix A: (

**a**) for an electrode pair used in current injection, voltages in all electrodes but the reference (e.g., electrode 2) will be associated with different responses; (

**b**) the last response is the thirteenth response associated with the electric current injection pair on the rightmost bottom surface.

**Figure 5.**Results of Effective Independence (EI) for two-probe schemes using (

**a**) one-sided electrodes on ${({[0/90]}_{4})}_{s}$ layup; (

**b**) one-sided electrodes on ${[{0}_{4}/{90}_{4}]}_{s}$ layup and (

**c**) two-sided electrodes on ${({[0/90]}_{4})}_{s}$ layup and (

**d**) two-sided electrodes on ${[{0}_{4}/{90}_{4}]}_{s}$ layup. An example of the EI independence before and after reduction is shown in (

**e**)

**Figure 6.**Results of Effective Independence on four-probe schemes on ${[{(0/90)}_{4}]}_{s}$ layup using (

**a**) one-sided electrodes and (

**b**) two-sided electrodes. Column sections show groups for surface resistance (${S}_{1},{S}_{2}$) and oblique resistance (${S}_{3},{S}_{4}$) measurements. Cells in dark blue indicate eliminated combination with low EI value.

**Figure 7.**Results of Effective Independence on four-probe schemes on ${[{0}_{4}/{90}_{4}]}_{s}$ layup using (

**a**) one-sided electrodes and (

**b**) two-sided electrodes. Column sections show groups for surface resistance (${S}_{1},{S}_{2}$) and oblique resistance (${S}_{3},{S}_{4}$) measurements. Cells in dark blue indicate eliminated combination with low EI value.

**Figure 8.**Results of Effective Independence on multi-probe schemes ${[{(0/90)}_{4}]}_{s}$ layup using (

**a**) one-sided electrodes and (

**b**) two-sided electrodes.

**Figure 9.**Results of Effective Independence on multi-probe schemes on ${[{0}_{4}/{90}_{4}]}_{s}$ layup using (

**a**) one-sided electrodes and (

**b**) two-sided electrodes.

**Figure 10.**Failed damage cases for ${[{(0/90)}_{4}]}_{s}$ with two-sided electrodes and: (

**a**) two-probe scheme and full set of responses; (

**b**) two-probe scheme and reduced set of responses; (

**c**) four-probe scheme and full set of responses; (

**d**) four-probe scheme and reduced set of responses; (

**e**) multi-probe scheme and full set of responses; (

**f**) multi-probe scheme and reduced set of responses. Crack size is shown next to the case box.

**Figure 11.**Failed damage cases for ${[{0}_{4}/{90}_{4}]}_{s}$ with two-sided electrodes and: (

**a**) two-probe scheme and full set of responses; (

**b**) two-probe scheme and reduced set of responses; (

**c**) four-probe scheme and full set of responses; (

**d**) four-probe scheme and reduced set of responses; (

**e**) multi-probe scheme and full set of responses; (

**f**) multi-probe scheme and reduced set of responses. Crack size is shown next to the case box.

Scheme | Number of Measurements (Ns) | Total for One-Sided Electrodes (h = 7) | Total for Two-Sided Electrodes (h = 14) |
---|---|---|---|

Two-probe | ${C}_{2}^{h}$ | 21 | 91 |

Four-probe | $k\times {\sum}_{i=1}^{h-3}\left[{\sum}_{j=2}^{h-(i+1)}{C}_{2}^{j}\right]$ | 35; k = 1 | 140; k = 4 |

Multi-probe | $(h-1)\times {C}_{2}^{h}$ | 126 | 1183 |

**Table 2.**Comparison of inverse identification accuracy for different electrode schemes (for [(0/90)

_{4}]

_{s}and [0

_{4}/90

_{4}]

_{s}layups with full and reduced sets of measurements).

[(0/90)_{4}]_{s} Layup | [0_{4}/90_{4}]_{s} Layup | |||||
---|---|---|---|---|---|---|

Scheme, Setting | No. of Measurements Used | No. of Failed Cases (out of 40 Test Cases) | RMS of Inv. Id. Errors (%) | No. of Measurements Used | No. of Failed Cases (out of 40 Test Cases) | RMS of Inv. Id. Errors (%) |

2P, 07EF | 21 | 12 | 18.29 | 21 | 10 | 14.61 |

2P, 07ER | 12 | 11 | 13.77 | 13 | 15 | 20.69 |

2P, 14EF | 91 | 1 | 6.75 | 91 | 5 | 6.77 |

2P, 14ER | 34 | 1 | 5.17 | 32 | 5 | 7.83 |

4P, 07EF | 35 | 14 | 19.60 | 35 | 13 | 25.83 |

4P, 07ER | 13 | 17 | 22.54 | 1 | 39 | 60.19 |

4P, 14EF | 140 | 2 | 5.05 | 140 | 0 | 2.64 |

4P, 14ER | 55 | 3 | 5.18 | 21 | 2 | 4.18 |

MP, 07EF | 126 | 10 | 16.45 | 126 | 16 | 22.53 |

MP, 07ER | 7 | 12 | 16.52 | 6 | 16 | 28.37 |

MP, 14EF | 1,183 | 3 | 5.23 | 1,183 | 4 | 6.50 |

MP, 14ER | 53 | 1 | 5.09 | 14 | 5 | 6.71 |

Scheme 2P: Two-probe measurements | ||||||

Scheme 4P: Four-probe measurements | ||||||

Scheme MP: Multi-probe measurements | ||||||

Setting 07EF: Electrodes on one side, full set of electrode pair combinations | ||||||

Setting 07ER: Electrodes on one side, reduced set of electrode pair combinations | ||||||

Setting 14EF: Electrodes on both sides, full set of electrode pair combinations | ||||||

Setting 14ER: Electrodes on both sides, reduced set of electrode pair combinations |

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**MDPI and ACS Style**

Escalona-Galvis, L.W.; Diaz-Montiel, P.; Venkataraman, S. Optimum Electrode Configurations for Two-Probe, Four-Probe and Multi-Probe Schemes in Electrical Resistance Tomography for Delamination Identification in Carbon Fiber Reinforced Composites. *J. Compos. Sci.* **2018**, *2*, 29.
https://doi.org/10.3390/jcs2020029

**AMA Style**

Escalona-Galvis LW, Diaz-Montiel P, Venkataraman S. Optimum Electrode Configurations for Two-Probe, Four-Probe and Multi-Probe Schemes in Electrical Resistance Tomography for Delamination Identification in Carbon Fiber Reinforced Composites. *Journal of Composites Science*. 2018; 2(2):29.
https://doi.org/10.3390/jcs2020029

**Chicago/Turabian Style**

Escalona-Galvis, Luis Waldo, Paulina Diaz-Montiel, and Satchi Venkataraman. 2018. "Optimum Electrode Configurations for Two-Probe, Four-Probe and Multi-Probe Schemes in Electrical Resistance Tomography for Delamination Identification in Carbon Fiber Reinforced Composites" *Journal of Composites Science* 2, no. 2: 29.
https://doi.org/10.3390/jcs2020029