# Multi-Defect Detection in Additively Manufactured Lattice Structures Using 3D Electrical Resistance Tomography

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory and Methods

#### 2.1. ERT Theory

_{l}is the location of the l

^{th}electrode, n is the normal direction from the boundary, z

_{l}is the contact impedance at the l

^{th}electrode, and I

_{l}and V

_{l}are the injected current and electric potential on the l

^{th}electrode, respectively. Here, Γ

_{1}is the boundary with electrodes, while Γ

_{2}refers to the remainder of the boundary. Equations (1)–(4) are numerically solved by the finite element (FE) method with a known σ distribution for simulated voltage responses V(σ) [22,28].

_{m}) and simulated voltages V(σ), starting with an assumed conductivity distribution σ. The objective function (g) is added with a Tikhonov regularization term as Equation (5) and solved with the Gauss-Newton iterative algorithm due to the ill-posed nature of the ERT inverse problem [28,31]:

^{i}

^{+1}in each (i+1)

^{th}iteration as:

^{i}, is the derivative of V(σ

^{i}), and ${\varphi}_{p}^{i}$ and ${\varphi}_{q}^{i}$ are the nodal electric potentials considering a current passing through the pair of current injection electrodes and voltage measurement electrodes, respectively [31,32,33]. The reconstruction process continues until the error ratio, which is defined as the norm of the difference between V(σ

^{i}) and V

_{m}normalized by the norm of V

_{m}, is not improving by 0.1% for the following iterations, then returns the final reconstructed conductivity distribution, σ

_{r}[22].

#### 2.2. Adjusted Absolute Imaging

_{undamaged}) and the voltages V(σ

_{ref}) calculated by the assumed homogenous model, before locally subtracting them from damaged state measurements (V

_{damaged}), as is shown in Equations (8) and (9). The undamaged state measurements usually could be easily obtained from itself or other qualified structures in a mass production [29,34]. The updated measurements (V′

_{damaged}) compensate for modeling inaccuracy and are directly used to reconstruct the absolute conductivity distribution (σ

_{rd}) of the target. This mechanism enhances reconstruction quality by transforming the inverse problem from a global to a local minimization process [34].

#### 2.3. Modification of the Sensitivity Map

_{max}) could be acquired in step (4) when damage is assigned on the boundary strut. During the calculation of the strut-wise normalization matrix (N) in step (5), diagonal components of the matrix N are normalized to compensate for relatively low responses in the central struts, and the non-diagonal components are used for suppressing artifacts to 0.1σ

_{k}, which would not affect the defect evaluation. With this normalization, the adjusted sensitivity map could be obtained as JN in step (6), and the change in voltage measurements corresponding to a single perturbation would be:

#### 2.4. Representative Strut Conductivity and Defect Quantification

_{s}. Here, σ

_{s}is the equivalent conductivity of a damaged strut, which is calculated using the electric potential drop between the two ends of the strut, V

_{ab}, while assuming its dimensions remain the same. V

_{ab}is affected by the size, shape, and amount of damage developed in the strut. With known damage features shown in Figure 3, σ

_{s}could be calculated as:

_{0}is the material conductivity in its undamaged state, L is the length of the strut, and A

_{r}is the residual area (i.e., the cross-sectional area where the defected region A

_{d}is subtracted from the undamaged cross-section A

_{0}) for a differential length, dl.

_{r}) in each strut could be used to examine damage severity. Here, σ

_{s}which corresponds to the damage could serve as a comparison parameter with respect to σ

_{r}solved by ERT with an invariant struts model. In this study, σ

_{r}was compared with σ

_{s}in both simulations and experiments to validate the quantitative defect detection capabilities of the proposed ERT method.

## 3. Simulation Details and Results

#### 3.1. 3D ERT Numerical Simulations

^{2}cross-section struts was constructed in Abaqus, as is shown in Figure 4; the cellular lattice structure was meshed using 9229 tetrahedral elements. Electrodes were defined at the 24 intersecting nodes along the boundaries. The conductivity of all the elements was assumed to be 1000 S/m, based on the resistance measurements of the CNT thin film coat used in the following experiments.

#### 3.2. Sensitivity Discussion

^{−13}to 10

^{−4}is plotted in Figure 5b, and 10

^{−7}(near the inflection point) was chosen.

#### 3.3. Assessment of Conductivity Reconstruction

_{C}and area error e

_{A}are calculated as:

_{s}and A

_{s}are the centroid and damage area of the real damaged strut, respectively, while C

_{r}and A

_{r}are the reconstructed damage centroid and damage area, respectively, which are defined by conductivity changes larger than one-fourth of the maximum conductivity change [37,38]. The undamaged strut’s length L

_{p}and area A

_{p}are included for normalization. In addition, the reconstructed error value, e

_{σ}, is defined to assess the difference between reconstructed conductivity (σ

_{r}) and the calculated strut representative conductivity (σ

_{s}) normalized by undamaged state conductivity σ

_{p}.

#### 3.4. Single-Defect Detection

_{C}and e

_{A}are all zeros as shown in Table 1. The improved reconstruction performance occurred because normalization compensates for the low central region sensitivity by imposing corresponding weighting factors that facilitated accurate conductivity reconstruction.

_{r}) within the damaged strut is consistent with the strut representative conductivity (σ

_{s}) for both imposed damage propagation scenarios. Their consistency expresses the significance of calculating strut representative conductivity and the capability of the ERT method with the normalized sensitivity map to return conductivity values corresponding to the damaged states. The reconstructed errors (e

_{σ}) of the damage cases with defect propagation along depth are shown in Table 2. The trends shown in Figure 7 demonstrate that the reconstructed values are more sensitive to damage propagated in the transverse direction (as opposed to the longitudinal direction), as suggested by Equation (10).

#### 3.5. Multi-Defect Detection

_{σ}of strut 1 (see Table 3) is approximately twice that of strut 2, because strut 2 is closer to the boundary electrodes. In contrast, Figure 8c,d show the reconstructed conductivity distribution in 3D visualization and with respect to finite elements, respectively, when using the normalized sensitivity map. In addition to significantly reducing conductivity reconstruction artifacts, normalization yielded uniform sensitivity throughout the cellular lattice structure. The reconstructed conductivity for both damaged struts approaches 0 S/m and can be clearly interpreted as breakages, as can be seen in Figure 8d, and both error values are 20 times lower than the case without normalization. Overall, these simulation results demonstrated improved spatial and quantitative accuracy when the normalized sensitivity map is incorporated with ERT.

#### 3.6. Defect Detection in Complex Lattice Structures

## 4. Experimental Details and Results

#### 4.1. 3D-Printed Lattice Structures

#### 4.2. 3D ERT Data Acquisition and Testing

#### 4.3. Single-Defect Detection

_{0}and one-fourth L, as illustrated in Figure 12a. The reconstructed image and value were then evaluated with the representative model and strut representative conductivity (σ

_{s}), which was calculated using Equations (9) and (10).

#### 4.4. Multi-Defect Detection

_{σ}in Table 7 clearly show that ERT with the normalized sensitivity map outperforms classical ERT.

## 5. Discussion

_{C}, e

_{A}, and e

_{σ}.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Disclaimer

## References

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**Figure 3.**Damage that was introduced in a strut is illustrated, and the strut representative (σ

_{s}) conductivity could be further calculated.

**Figure 4.**A 3 × 3 × 1 lattice structure model was created in Abaqus. Electrodes in the upper z-plane are marked in white.

**Figure 5.**(

**a**) The summed sensitivity map of the lattice structure was calculated. (

**b**) The L-curve was plotted, with λ ranging from 10

^{−13}to 10

^{−4}.

**Figure 6.**(

**a**) A lattice structure was imposed with damage. (

**b**) An ERT reconstruction was solved with the normalized sensitivity map. (

**c**) The reconstructed conductivity values of each element when solved without and (

**d**) with the normalized sensitivity map are plotted. (

**e**) The normalized errors are plotted with iterations.

**Figure 7.**Reconstructed conductivity in the strut σ

_{r}is consistent with representative strut conductivity σ

_{s}. Depth and length of the damage feature are varied by assigning 0 S/m to n finite elements.

**Figure 8.**(

**a**) A lattice structure with two damaged struts. (

**b**) The reconstructed conductivity values of each element when solved without the normalized sensitivity map. (

**c**) The reconstructed 3D conductivity distribution and (

**d**) the conductivity values for each element, when solved using the normalized sensitivity map.

**Figure 9.**3D conductivity distribution reconstructions of (

**a**) 3 × 3 × 3 and (

**b**) 4 × 4 × 1 (with diagonal struts) lattice structures successfully identified the broken strut in each structure.

**Figure 10.**(

**a**) A 3 × 3 × 1 lattice structure was spray-coated with a conductive, nanocomposite thin film. (

**b**) ERT measurements were obtained using a customized data acquisition system.

**Figure 11.**(

**a**) Simulated voltages are compared with experimentally measured voltages. (

**b**) The first etch (damage) was introduced in the lattice. (

**c**) The reconstructed conductivity values of each element when solved with the normalized sensitivity map are plotted. (

**d**) The corresponding 3D conductivity distribution successfully confirmed damage detection in strut 1.

**Figure 12.**(

**a**) The actual experimental damage (Case #1) was compared to what was modeled. (

**b**) Representative strut conductivity σ

_{s}and reconstructed conductivity in the strut σ

_{r}change in tandem as damage increased in severity, both along its length and depth (cross-section).

**Figure 13.**(

**a**) The reconstructed conductivity values of each element when solved with the normalized sensitivity map are plotted. (

**b**) The corresponding 3D conductivity distribution of the lattice structure successfully identified breaks in strut 1 and strut 2.

Strut-Based Evaluation | e_{C} | e_{A} |
---|---|---|

Without normalized sensitivity map | 0.0030 | 0.1314 |

With normalized sensitivity map | 0 | 0 |

Damage Scale | e_{σ} | Damage Scale | e_{σ} |
---|---|---|---|

1 | 0.0008 | 7 | 0.0192 |

2 | 0.0026 | 8 | 0.0198 |

3 | 0.0052 | 9 | 0.0198 |

4 | 0.0060 | 10 | 0.0251 |

5 | 0.0100 | 11 | 0.0209 |

6 | 0.0105 | 12 | 0.0137 |

Strut-Based Evaluation | e_{C} | e_{A} | e_{σ} of Strut 1 | e_{σ} of Strut 2 |
---|---|---|---|---|

Without normalized sensitivity map | 0.0146 | 0.1528 | 0.0752 | 0.0329 |

With normalized sensitivity map | 0 | 0 | 0.0032 | 0.0017 |

Single-Defect | Multi-Defect | |||||||
---|---|---|---|---|---|---|---|---|

Case | #1 | #2 | #3 | #4 | #5 | #6 | #7 | #8 |

Number of damaged struts | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 |

Number of damaged faces | 1 | 1 | 1 | 1 | 2 | 3 | 4 | 8 |

Total damaged length | L/4 | L/2 | 3L/4 | L | L | L | L | 2L |

Strut-Based Evaluation | e_{C} | e_{A} |
---|---|---|

Without normalized sensitivity map | 0.0696 | 0.1551 |

With normalized sensitivity map | 0 | 0 |

Damage Case | #1 | #2 | #3 | #4 | #5 | #6 | #7 |
---|---|---|---|---|---|---|---|

e_{σ} | 0.0053 | 0.0199 | 0.0257 | 0.0353 | 0.0134 | 0.0042 | 0.0244 |

Strut-Based Evaluation | e_{C} | e_{A} | e_{σ} of Strut 1 | e_{σ} of Strut 1 |
---|---|---|---|---|

Without normalized sensitivity map | 0.0287 | 0.2233 | 0.0653 | 0.0804 |

With normalized sensitivity map | 0 | 0 | 0.0056 | 0.0030 |

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**MDPI and ACS Style**

Shu, Y.; Mukherjee, S.; Chang, T.; Gilmore, A.; Tringe, J.W.; Stobbe, D.M.; Loh, K.J.
Multi-Defect Detection in Additively Manufactured Lattice Structures Using 3D Electrical Resistance Tomography. *Sensors* **2022**, *22*, 9167.
https://doi.org/10.3390/s22239167

**AMA Style**

Shu Y, Mukherjee S, Chang T, Gilmore A, Tringe JW, Stobbe DM, Loh KJ.
Multi-Defect Detection in Additively Manufactured Lattice Structures Using 3D Electrical Resistance Tomography. *Sensors*. 2022; 22(23):9167.
https://doi.org/10.3390/s22239167

**Chicago/Turabian Style**

Shu, Yening, Saptarshi Mukherjee, Tammy Chang, Abigail Gilmore, Joseph W. Tringe, David M. Stobbe, and Kenneth J. Loh.
2022. "Multi-Defect Detection in Additively Manufactured Lattice Structures Using 3D Electrical Resistance Tomography" *Sensors* 22, no. 23: 9167.
https://doi.org/10.3390/s22239167