#
Uncertainty Evaluation in the Design of Structural Health Monitoring Systems for Damage Detection^{†}

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Numerical Modeling of SHM Signals

^{3}. All of the defects were invariant in the z-direction, thus, the rivet hole and damage were uniform from the top of the plate to the bottom of the plate. Emergent cracks were symmetric and appeared on diametrically opposing sides of the rivet hole with equal lengths on both sides. Flaw size was defined as the combined lengths of the cracks from both sides of the hole. Finally, cracks were modeled as infinitely thin, straight notches, with no wave propagation through the crack faces.

- Built a model of the reduced domain near the rivet hole in ANSYS (ANSYS
^{®}Mechanical APDL) with a sensing circle and non-reflecting boundary conditions. - Evaluated the time-harmonic solutions to the wave scattering problem for frequencies of 2–1000 kHz at 2 kHz intervals, using the finite elements in ANSYS.
- Post-process FEM outputs generated complex scatter-cubes with wave-damage interaction coefficients (WDIC),
- Imported WDIC to Waveform Revealer 2-D (WFR-2D) in order to generate signal data analytically.

#### Finite Element Modeling

#### 2.2. Experimental Factors and Damage Indices

^{6}= 64 experimental design settings from which to recover signals for both the flaw state and baseline (signals that were the result of a plate with a rivet hole containing no crack). The signal output was measured as the vertical displacement (in nanometers) at the receiver location. An average analysis window size of 40 microseconds was used in order to compute damage indices. We chose to examine three common and simple damage indices, all of which were relative to the baseline signal, namely: a correlation coefficient, the sum of squared deviation, and the difference in peak amplitude. The damage index for the correlation coefficient (DIcc) (ρ) was given as follows:

_{min}, t

_{max}]. The Sum of squared deviations (sqdev) compares the baseline (f) and damage (g) signals and is computed for the discretized observed signal values as follows:

#### 2.3. Statistical Experimental Design and Analysis Methods

^{5}experimental configurations (all factor levels of Table 1, except for flaw size). Table 2 provides the settings for each of these 64 runs. These settings were specifically chosen so as to utilize an efficient design of the experiments, namely, the 2

^{k}factorial design (here k = 6 for six factors and 2 indicated that each factor was assessed at two different levels). One of the efficiencies of this design was in its ability to test the factor level combinations through a reduced number of experiments, namely, half or even one-fourth of the number of experiments. Therefore, the goal of this experiment was twofold. Firstly, we demonstrated the sensitivity for each of the three damage indices to the various SHM design settings, using the full factorial design (2

^{6}runs), and secondly, we conducted the analysis on a fractional factorial design using half of the number of runs (2

^{6−1}) and compared the findings. If the statistical design was efficient, we should have seen comparable results between the full design (2

^{6}experimental runs) and the design used that half of the number of runs (2

^{6−1}= 32 experimental runs). The factorial design that used half of the number of runs was called the half-fraction factorial design.

#### 2.3.1. Full Factorial Design

^{2}. Therefore, the observed factor effects were assumed to be normally distributed and the specific values that were observed were used to compute a (empirically) cumulative normal probability. This was achieved by ordering the observed values from smallest to largest and estimating the cumulative normal probability for the ith ordered effect as the (i − 0.5)/n quantile of the normal distribution, where n denoted the number of effects. Factors with small effects would lie along a straight line within the plot. Factors with large effects on the response would deviate substantially away from the other plotted factors. The use of the normal probability plot allowed us to use modeling assumptions to not only identify large, potentially significant effects, but to identify those effects that were essentially negligible (near zero effect), in order to use these small effects to estimate error. The use of the word “effect” was both colloquial and statistically specific. A factor’s effect was estimated by a linear combination of the observed responses from all of the experimental runs. In essence, for a particular factor, for example the transmitter angle, we would add the observed responses for all of the runs, where the transmitter angle was set to high, and subtract the observed responses for all of those runs in which the transmitter angle was set to low. This linear combination was then divided by (n2

^{k}

^{−1}) to get an estimate of the factor’s (average) effect. The standard DOE textbook explains these calculations and how to efficiently compute these effects for all factors and interactions [29].

#### 2.3.2. Half-Fraction Factorial Design

^{6}factorial experiment for a second time using only half of the number of runs, that is, in only 2

^{6−1}= 2

^{5}number of runs. This design was called a half-fraction factorial design, since we were only using 32 of the 64 required runs to examine every high/low combination of all of the six factors. This meant that there would be confounding, in which two factors/interactions would mathematically produce the same exact effect when calculated from the observed responses. However, by deliberately picking 32 of the 64 experimental configurations to run, we accomplished what is called a resolution VI design, in which the factors of interest to us were estimated separately and not aliased (or confounded) together. A resolution VI design for our experiment meant that each main effect (A through F) was aliased with specific five-way interactions (which we deemed as negligible by the sparsity of effects principle); two-way interactions were aliased with four-way interactions (also likely negligible as a higher order interaction); and three-way interactions were aliased with each other. Therefore, if a main effect, such as transmitter angle (A), was significant, although it was aliased with the five-way interaction—between receiver angle, frequency, sensor size, distance to excitation and flaw size (BCDEF)—we attributed the effect to that of the transmitter angle (A) rather than the complex, five-way interaction (BCDEF). The alias structure for our resolution VI half-fraction factorial design is listed in Table 3. The statistical analysis included the same steps as for the full factorial analysis, although now the normal probability plots and resulting ANOVA were conducted on 32 runs instead of 64. For this portion of the analysis, the goal was to determine whether the main sources of variation in the SHM configuration could be identified with half of the required experimental runs. If this is achievable, this reduced design would result in a significant savings with respect to time and cost in the SHM validation.

## 3. Results

#### 3.1. SHM Configuration Factors Producing the Most Variation in the SHM Response (Damage Index)

^{2}value can be found in Table 4. For lnDIcc, the main effects of the transmitter angle (A), frequency (C), and distance to excitation (E) were all statistically significant (p-values <0.0001, <0.0001, and 0.0002, respectively). The R

^{2}value for the model that contained the transmitter angle (A), frequency (C), and distance to excitation (E) was 0.97 (model RMSE = 0.8125). This meant that 97% of the variability in lnDIcc could be explained by the transmitter angle, frequency, and distance to excitation, which was used in the SHM configuration. A smaller (9°) transmitter angle resulted in a higher damage index than a larger (27°) transmitter angle, based on the correlation coefficient. Similarly, a higher (550 kHz) frequency resulted in a higher DIcc, than a lower (200 kHz) frequency did, as did a shorter distance between the excitation and transmitter. Plots of these effects and their associated 95% confidence intervals are given in Figure 4 by the levels of each factor. Figure 4 demonstrates that altering the transmitter angle in the SHM configuration had the largest effect on DIcc, such that the larger DIcc values could be invoked by the use of smaller transmitter angles. In configurations where the transmitter angle was larger, only smaller values of DIcc were observed. However, a larger frequency and shorter distance between the excitation and transmitter increased the observed DIcc, even if a smaller transmitter angle was required because of the physical constraints of the application.

^{2}value for this model was 0.98, which indicated that 98% of the variability in the sum of squared deviation could be explained by the transmitter angle, frequency, interaction between transmitter angle and frequency, distance to excitation and the interaction between transmitter angle, and the distance to the excitation (model RMSE = 0.7903). Figure 5 shows plots of the effects on lnsqdev and their associated 95% confidence intervals, for both the transmitter angle/distance to excitation interaction and the transmitter angle/frequency interaction. At smaller transmitter angles, increasing the distance to the excitation had a smaller effect on lnsqdev than at larger transmitter angles, for which increasing the distance to excitation had a much larger effect, resulting in a smaller lnsqdev value (Figure 5a). In contrast, when the frequency was increased (from 200 kHz to 550 kHz) at the smaller transmitter angle (9°), there was a larger drop in lnsqdev than when the frequency was increased at the larger transmitter angle (27°), as shown in Figure 5b.

^{2}value out of all of the damage indices having been examined with only 78% of the variability in the difference in peak amplitude being explained by the combined levels of the transmitter angle, receiver angle, and the frequency used (model RMSE = 1.1312). Furthermore, lndiffPA was the only damage index examined that was sensitive to the receiver angle and was not sensitive to the distance to excitation. For none of these damage indices was the sensor size or flaw size a major contributor to the response.

#### 3.2. SHM Configuration Factors Identified through the Half-Fraction Factorial Design

^{2}value was 0.97 (RMSE = 0.8993).

^{2}= 0.97 (RMSE = 0.9992) with significant p-values for the transmitter angle/distance to excitation interaction (p-value = 0.0039), transmitter angle/frequency interaction (p-value = 0.0019).

^{2}= 0.6468, RMSE = 1.4624).

## 4. Discussion

^{2}value and the inconsistencies between the full factorial and half-fraction factorial statistical designs. When considering the use of a sensitive damage index, the difference in the peak amplitude remained less sensitive than the other indices that were considered.

^{k}design was in the inference between the two settings of each parameter. Since we only used two such settings, we assumed a linear trend across values of each of the various parameters (see Figure 5 for instance). This assumption might not have held in all design settings. However, there were other factorial designs, for instance 3

^{k}factorial designs, which considered a general high, medium, and low level for each factor. If it was thought that the response might have been curvilinear as the value of the parameter increased/decreased, then the 3

^{k}design could be used to model this curvilinear trend from low to high. With such designs, the trends in the parameter values could be examined while still reducing the number of runs that were required (e.g., a half-fraction for the 3

^{k}design was possible). Such a reduction in the number of experiments was especially important considering that a 2

^{6}design was 64 experimental runs, but a full 3

^{6}design was 729 experimental runs.

^{2}= 0.78). However, we utilized the normal probability plots in the spirit of the sparsity of effects principle and in order to establish appropriate degrees of freedom. As such, from only the half-fraction results, we concluded that if the damage index lndiffPA was of interest, further work would be needed in order to identify additional sources of variation—A result that might have also been concluded from the full factorial design, given the lower comparative R

^{2}value. In fact, as we conducted the comparison of the damage indices to determine which were the most sensitive and robust for the SHM system, our conclusion was to work with either the damage index, based upon the correlation coefficient, or the sum of the squared deviations, as lndiffPA was much less predictive. This was a conclusion that was determined by the half-fraction factorial design. Furthermore, an R

^{2}of about 97% was achievable with the correlation coefficient damage index and no interactions were present, demonstrating that this damage index was more malleable and robust for this SHM system.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Numerical simulation modeling the experimental configuration. R-PWAS—receiving piezoelectric wafer active sensor; T-PWAS—transmitting piezoelectric wafer active sensor.

**Figure 2.**Computational domain of the rivet hole for the finite element method (FEM): (

**a**) overall FEM model; (

**b**) zoomed in at the damage region (there are two nodes disbanded along the crack); and (

**c**) boundary conditions applied along the loading line, where the stress mode shapes were converted into the nodal force that was applied at each node.

**Figure 3.**Normal probability plots from the full factorial experiment for each damage index: (

**a**) normal probability plot for the natural log of the correlation coefficient damage index (lnDIcc); (

**b**) normal probability plot for the natural log of the sum of squared deviations (lnsqdev); and (

**c**) normal probability plot for the natural log of the difference in peak amplitude (lndiffPA). A—transmitter angle; B—receiver angle; C—frequency; D—sensor size; E—excitation distance; F—flaw size.

**Figure 4.**Estimated responses and associated 95% confidence intervals for the correlation coefficient damage index (DIcc) at each level of transmitter angle, frequency, and distance between excitation and transmitter.

**Figure 5.**Estimated responses and associated 95% confidence intervals for the natural log of the sum of squared deviations (lnsqdev) at the following: (

**a**) the interaction of transmitter angle with distance to excitation; (

**b**) the interaction of transmitter angle with frequency; (

**c**) for the natural log of the difference in peak amplitude (lndiffPA) at the interaction of transmitter angle and frequency fixed at a receiver angle of 5 degrees; and (

**d**) a receiver angle of 20 degrees.

**Figure 6.**Normal probability plots from the half-fraction factorial experiment for each damage index: (

**a**) normal probability plot for the natural log of the correlation coefficient damage index (lnDIcc); (

**b**) normal probability plot for the natural log of the sum of squared deviations (lnsqdev); and (

**c**) normal probability plot for the natural log of the difference in peak amplitude (lndiffPA).

Label | Factor | Generic Level | Specific Values |
---|---|---|---|

A | Transmitter Angle, θ | Low | 9° |

High | 27° | ||

B | Receiver Angle, φ | Low | 5° |

High | 20° | ||

C | Frequency | Low | 200 kHz |

High | 550 kHz | ||

D | Sensor Size | Low | 5 mm |

High | 7 mm | ||

E | Distance between Excitation and Damage Site | Low | 150 mm |

High | 250 mm | ||

F | Flaw Size | Low | 0.64 mm |

High | 3.20 mm |

Run | A | B | C | D | E | F |
---|---|---|---|---|---|---|

Transmitter Angle | Receiver Angle | Frequency | Sensor Size | Distance to Excitation | Flaw Size | |

1 | L | L | L | L | L | L |

2 | H | L | L | L | L | L |

3 | L | H | L | L | L | L |

4 | H | H | L | L | L | L |

... | ... | ... | ... | ... | ... | ... |

63 | H | H | H | H | H | L |

64 | H | H | H | H | H | H |

Factors for the Half-Fraction Factorial Design (and Associated Aliased Factor) | |||||
---|---|---|---|---|---|

A = BCDEF | AB = CDEF | BD = ACEF | DE = ABCF | ABC = DEF | ACF = BDE |

B = ACDEF | AC = BDEF | BE = ACDF | DF = ABCE | ABD = CEF | ADE = BCF |

C = ABDEF | AD = BCEF | BF = ACDE | EF = ABCD | ABE = CDF | ADF = BCE |

D = ABDEF | AE = BCDF | CD = ABEF | ABF = CDE | AEF = BCD | |

E = ABCDF | AF = BCDE | CE = ABDF | ACD = BEF | ||

F = ABCDE | BC = ADEF | CF = ABDE | ACE = BDF |

Factor | lnDIcc | lnsqdev | lndiffPA | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Sums of Squares | df | p-Value | Sums of Squares | df | p-Value | Sums of Squares | df | p-Value | |||

A | 1166.46 | 1 | <0.0001 | 1518.10 | 1 | <0.0001 | 164.57 | 1 | <0.0001 | ||

B | - | - | - | - | - | - | 12.18 | 1 | 0.0030 | ||

C | 21.81 | 1 | <0.0001 | 38.05 | 1 | <0.0001 | 3.22 | 1 | 0.1180 | ||

E | 10.68 | 1 | 0.0002 | 54.86 | 1 | <0.0001 | - | - | - | ||

AE | - | - | - | 10.85 | 1 | <0.0001 | - | - | - | ||

AC | - | - | - | 41.97 | 1 | <0.0001 | 23.58 | 1 | <0.0001 | ||

AB | - | - | - | - | - | - | 24.96 | 1 | <0.0001 | ||

BC | - | - | - | - | - | - | 0.14 | 1 | 0.7440 | ||

ABC | - | - | - | - | - | - | 24.01 | 1 | <0.0001 | ||

Error | 39.61 | 60 | 36.23 | 58 | 71.66 | 56 | |||||

Total | 1238.56 | 63 | 1700.07 | 63 | 324.25 | 63 | |||||

R^{2} | 0.9680 | 0.9787 | 0.7790 | ||||||||

RMSE | 0.8125 | 0.7903 | 1.1312 |

Factor | lnDIcc | lnsqdev | lndiffPA | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Sums of Squares | df | p-Value | Sums of Squares | df | p-Value | Sums of Squares | df | p-Value | |||

A | 608.68 | 1 | <0.0001 | 834.34 | 1 | <0.0001 | 117.51 | 1 | <0.0001 | ||

C | 14.80 | 1 | 0.0002 | 15.86 | 1 | 0.0016 | - | - | - | ||

E | 7.00 | 1 | 0.0065 | 23.45 | 1 | <0.0001 | - | - | - | ||

AE | - | - | - | 5.96 | 1 | 0.0039 | - | - | - | ||

AC | - | - | - | 6.37 | 1 | 0.0019 | - | - | - | ||

Error | 22.64 | 28 | 25.96 | 26 | 64.16 | 30 | |||||

Total | 653.12 | 31 | 911.93 | 31 | 181.67 | 31 | |||||

R^{2} | 0.9653 | 0.9736 | 0.6468 | ||||||||

RMSE | 0.8993 | 0.9992 | 1.4642 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Schubert Kabban, C.; Uber, R.; Lin, K.; Lin, B.; Bhuiyan, M.Y.; Giurgiutiu, V.
Uncertainty Evaluation in the Design of Structural Health Monitoring Systems for Damage Detection^{†}. *Aerospace* **2018**, *5*, 45.
https://doi.org/10.3390/aerospace5020045

**AMA Style**

Schubert Kabban C, Uber R, Lin K, Lin B, Bhuiyan MY, Giurgiutiu V.
Uncertainty Evaluation in the Design of Structural Health Monitoring Systems for Damage Detection^{†}. *Aerospace*. 2018; 5(2):45.
https://doi.org/10.3390/aerospace5020045

**Chicago/Turabian Style**

Schubert Kabban, Christine, Richard Uber, Kevin Lin, Bin Lin, Md Yeasin Bhuiyan, and Victor Giurgiutiu.
2018. "Uncertainty Evaluation in the Design of Structural Health Monitoring Systems for Damage Detection^{†}" *Aerospace* 5, no. 2: 45.
https://doi.org/10.3390/aerospace5020045