Modelling Eddy Current Testing of Gaps in Carbon Fibre Structures Based on Spline Approximation

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

1.It is recommended to further clarify the advantages of the proposed method by quantitatively comparing it with existing approaches, and specifying in which metrics the performance is improved.

2.The defect configurations used in the experiments and simulations are not fully consistent; therefore, the discrepancy should be explicitly explained and, if possible, the conditions should be better aligned.

3.Please clarify how the Fourier-series parameters (coefficients and the number of terms) are obtained, and provide the corresponding identification/fitting procedure.

Author Response

Thank You for the thorough review, below you'll find our responses to the reasonable comments.

1. It is recommended to further clarify the advantages of the proposed method by quantitatively comparing it with existing approaches, and specifying in which metrics the performance is improved.

Response: Was quantitatively compared with correlation coefficient (Line 218).

2.The defect configurations used in the experiments and simulations are not fully consistent; therefore, the discrepancy should be explicitly explained and, if possible, the conditions should be better aligned.

Response: Has been adjusted, Gap size is now aligned with experiment. (Fig. 5, Fig. 14)

3. Please clarify how the Fourier-series parameters (coefficients and the number of terms) are obtained, and provide the corresponding identification/fitting procedure.

Response: Has been done for spline approximation (Line 100-123)

Reviewer 2 Report

Comments and Suggestions for Authors

Regarding the modeling of eddy current testing for "missing tow/gap" defects in carbon fiber fabric/laminates, the proposed approach uses Fourier series expansion to describe the spatial heterogeneity of conductivity within/between tows, combines virtual scanning and adaptive mesh refinement to reduce computational cost and numerical noise from re-meshing, and incorporates interlaminar interface resistance measured by the four-terminal method into the model to characterize interlayer current pathways. The overall concept is clear and the engineering significance is well-defined. However, the current work has shortcomings in parameter reproducibility, quantitative validation, and consistency with experimental defects, which need to be addressed before the reliability of the conclusions can be assessed.

1.What are the advantages of using the eddy current method compared to ultrasonic testing for inspecting carbon fibers?

2.The authors should clarify the novel contributions of this work compared to existing methods using "position-dependent conductivity functions/virtual scanning" (e.g., Hughes et al.). Specifically, what quantifiable improvements does the Fourier series expansion bring (e.g., in fitting error, sharpness of gap boundaries, signal error, computation time)?

3.The method for obtaining the coefficients in the Fourier series (Eq. 3) is not explained.

4.The paper uses a Fourier series order of N=70 without justification. Please provide a convergence analysis and explain the rationale for choosing N=70.

5.There is an inconsistency between the simulated and experimental gap sizes: the experimental gap is 10 mm, while the simulation only removes a tow of about 5 mm. The authors also note that this leads to a significantly wider experimental signal. It is recommended to implement a 10 mm gap in the simulation (by removing two tows or widening the low-conductivity region) or to supplement the experimental samples with a 5 mm gap to enable a stricter comparison.

6.The current validation relies mainly on visual comparison after normalization and contrast enhancement. It is recommended to supplement this with quantitative metrics and provide mean values ± standard deviation.

7.Table 1 provides interface resistances, but the measurement details are insufficient. Please supplement information on the four-terminal method: electrode dimensions and spacing, applied pressure, contact resistance handling, etc.

8.The authors should explain how the interface resistance is implemented in the Maxwell model.

9.Although the frequency range is 1–10 MHz, the results primarily show 4 MHz. It is recommended to include comparisons at multiple frequencies, present the variation of the gap signal with frequency, and explain why 4 MHz was chosen as the representative frequency.

10.The paper emphasizes that "virtual scanning + adaptive meshing" reduces the time and noise associated with re-meshing, but lacks quantitative evidence. It is recommended to supplement quantifiable parameters such as: number of mesh elements, convergence criteria, single solution time, total virtual scanning time.

Author Response

The overall concept is clear and the engineering significance is well-defined. However, the current work has shortcomings in parameter reproducibility, quantitative validation, and consistency with experimental defects, which need to be addressed before the reliability of the conclusions can be assessed.

Thank you for the review. We answer to the following comments in detail.

1.What are the advantages of using the eddy current method compared to ultrasonic testing for inspecting carbon fibers?

Response: Added to introduction (line 56f. ): “This paper analyses dry fabric samples to minimize the matrix's influence and focus on the fibres as eddy current inspection is especially beneficial early in CFRP production. It monitors dry fabrics where ultrasonic testing is limited by poor coupling.” 

2.The authors should clarify the novel contributions of this work compared to existing methods using "position-dependent conductivity functions/virtual scanning" (e.g., Hughes et al.). Specifically, what quantifiable improvements does the Fourier series expansion bring (e.g., in fitting error, sharpness of gap boundaries, signal error, computation time)?

Response: Was quantitatively compared with correlation coefficient (line 218ff., fig. 16)

3.The method for obtaining the coefficients in the Fourier series (Eq. 3) is not explained.

Response: Fourier series has been changed to spline approximation and explanation of coefficients in line 100-120 added

4.The paper uses a Fourier series order of N=70 without justification. Please provide a convergence analysis and explain the rationale for choosing N=70. 

Response: Fourier series has been changed to spline approximation and explanation of coefficients in line 100-120 added

5.There is an inconsistency between the simulated and experimental gap sizes: the experimental gap is 10 mm, while the simulation only removes a tow of about 5 mm. The authors also note that this leads to a significantly wider experimental signal. It is recommended to implement a 10 mm gap in the simulation (by removing two tows or widening the low-conductivity region) or to supplement the experimental samples with a 5 mm gap to enable a stricter comparison.

Response: Simulation has been changed to 10 mm gap (fig. 5, fig. 14)

6.The current validation relies mainly on visual comparison after normalization and contrast enhancement. It is recommended to supplement this with quantitative metrics and provide mean values ± standard deviation. 

Response: Was quantitatively compared with correlation coefficient (line 218ff., fig. 16)

7.Table 1 provides interface resistances, but the measurement details are insufficient. Please supplement information on the four-terminal method: electrode dimensions and spacing, applied pressure, contact resistance handling, etc.

Response: The electrode strips are each 12 mm wide. The distance between the middle electrodes is 115 mm. A total weight of 1 kg was used to apply reproducible pressure. The resistance was measured with a Keithley DAQ6510 digital multimeter, and a four-terminal method was used to minimize errors from lead and terminal resistances. (Line 164-168)

8.The authors should explain how the interface resistance is implemented in the Maxwell model.

Response: The interface resistivity is simulated by assigning a resistive sheet boundary to the contacting surface of both layers. https://ansyshelp.ansys.com/public/Views/Secured/Electronics/v252/en/Subsystems/Maxwell/Maxwell.htm#Maxwell/AssigningaResitiveSheetBoundaryfortheEddyCurrentSolver.htm?Highlight=resistive%20sheet (line 188)

9.Although the frequency range is 1–10 MHz, the results primarily show 4 MHz. It is recommended to include comparisons at multiple frequencies, present the variation of the gap signal with frequency, and explain why 4 MHz was chosen as the representative frequency.

Response: Experiments have only been done at 4,7,8,12 MHz and simulation was conducted again after first review. Due to short amount of time, it could only be repeated for 4 MHz which showed the best experimental results (Table 2)

10.The paper emphasizes that "virtual scanning + adaptive meshing" reduces the time and noise associated with re-meshing but lacks quantitative evidence. It is recommended to supplement quantifiable parameters such as: number of mesh elements, convergence criteria, single solution time, total virtual scanning time.

Response: Has been removed as virtual scanning advantages have already been stated/proven in Ref.14 (Yi, et. al) (calculation time can be reduced from 80 h to 15 h with Ryzen 5900 with 12 cores + better signal-to noise ratio)

Reviewer 3 Report

Comments and Suggestions for Authors

In the reviewed manuscript, in contrast to Ref. 12, the authors are proposing an alternative approximation of the conductivity spatial distribution function in the form of Fourier series expansion. Its application is illustrated by numerical simulations in ANSYS and supplemented by certain experimental tests. Although the paper content might be treated as being related to the Scope of the Sensors Journal, substantial revisions should be introduced into the text of the manuscript to make it suitable for publishing. Two main issues are as follows:

• Without preliminary thorough reading of Ref. 12, it is very difficult to get into the content of the current manuscript. With this respect, it could be suggested to provide a brief mathematical description of the considered problem which might allow to clarify the meaning of Eq. (1)-(3) in a more thorough way.
• The novelty and even more, the practical need, of the introduced Fourier series expansion which is assumed by the authors as the main novelty of their work is not clarified at all. Since FEM software is finally used for the numerical simulation, it supports the introduction of any spatial distribution of the electrical conductivity function (see, e.g., comments prior to Eq. 7 in Ref. 12). With this respect, when simulating the gap occurring due to the absence of fiber yarns, any stepwise function could be easily introduced to the FEM software. At the same time, it is even possible to address in a more precise way experimental electric conductivity shown in Fig. 3 (a) employing, e.g., any spline-based approximation of the latter. Moreover, if changing the period of the sine function, the behavior analogous the one which is exhibited by the solid blue curve in Fig. 3 (b) could be easily achieved. Since the authors claim that they are proposing "a novel methodology", one could expect that it should be very carefully validated numerically and confirmed experimentally. Moreover, any benefit of the proposed approach over existing solutions must be illustrated. However, nothing of this is present in the current version of the manuscript.

Some other comments and questions regarding the paper content:

• As seen in Fig. 5 when using Fourier series expansion some typical oscillations are observed due to the stepwise nature of the approximated function. How is such behavior related to the real-case conductivity observed in experimental specimens with such gaps?
• How is the sketch of the meshed domain shown in Fig. 10 related to the experimental setup from Fig. 8?
• “The CF domain contains several fiber bundles in 0° and 90° orientations”: what does the term “several” mean here?
• If the authors are wishing to provide experimental verification of their model, either 5 mm gap should be introduced in the experiments or 10 mm gap should be simulated in the model.
• Visible light horizontal patterns are present in experimental results (Fig. 13, a) while being absent in the simulations. What is the reason of such behavior?
• It is not clear how the authors are comparing numerical data with experimental results in Fig. 15 while different sizes of gaps were considered in experiments and simulations and are trying to derive any conclusions based on such comparison.
• Line 28: "... the electrical behaviour of a coil is still poorly understood..." - Could the authors support this statement with any references?
• Line 58: When simulating ultrasonic-based NDT with air-coupled transducers, air which is surrounding the inspected object should be considered and meshed as well.
• Line 61: What does the term "networking effort" mean here?
• How is s_N(x) mentioned in Eq. (3) related to the Eq. (1)?
• Lines 100-101: What kind of behavior is mentioned here?
• What does each gray circle mean in Figure 4?
• Lines 127-128: The authors claim that eddy-current-based technique is a novel approach suitable for the investigation of CFRP structures. However, they mention that near the specimen edges no scanning is performed. At the same time such areas are also of the specific interest in the sense of NDT. Therefore, a certain contradiction could be observed here.
• Line 129: What kind of tape was used? To what extent is it essential for the experiment? What would happen if it is removed?
• Subsection 4.2: What is the manufacture of both the Tx-Rx-coild and the preamplifier which are employed in the experiments?
• Line 180: For which specific sample were these measurements and simulations performed?
• Lines 193-195: How do the authors judge that exactly these reasons are conditioning the aforementioned spatial variation?
• Lines 204-205: There are no data provided in the paper which could support this statement.

Author Response

Thank you for the thorough review and for the great idea to use a spline-based approach. You'll find our responses to the reasonable comments below.

In the reviewed manuscript, in contrast to Ref. 14, the authors are proposing an alternative approximation of the conductivity spatial distribution function in the form of Fourier series expansion. Its application is illustrated by numerical simulations in ANSYS and supplemented by certain experimental tests. Although the paper content might be treated as being related to the Scope of the Sensors Journal, substantial revisions should be introduced into the text of the manuscript to make it suitable for publishing. Two main issues are as follows:

Without preliminary thorough reading of Ref. 14, it is very difficult to get into the content of the current manuscript. With this respect, it could be suggested to provide a brief mathematical description of the considered problem which might allow to clarify the meaning of Eq. (1)-(3) in a more thorough way.

Response: More thorough description of the equations used in Ref. 14 have been added to subsection 2.2. (line 77-93)

The novelty and even more, the practical need, of the introduced Fourier series expansion which is assumed by the authors as the main novelty of their work is not clarified at all. Since FEM software is finally used for the numerical simulation, it supports the introduction of any spatial distribution of the electrical conductivity function (see, e.g., comments prior to Eq. 7 in Ref. 12). With this respect, when simulating the gap occurring due to the absence of fiber yarns, any stepwise function could be easily introduced to the FEM software. At the same time, it is even possible to address in a more precise way experimental electric conductivity shown in Fig. 3 (a) employing, e.g., any spline-based approximation of the latter. Moreover, if changing the period of the sine function, the behavior analogous the one which is exhibited by the solid blue curve in Fig. 3 (b) could be easily achieved. Since the authors claim that they are proposing "a novel methodology", one could expect that it should be very carefully validated numerically and confirmed experimentally. Moreover, any benefit of the proposed approach over existing solutions must be illustrated. However, nothing of this is present in the current version of the manuscript.

Response: Fourier series approximation has been replaced with spline approximation (line 100-120)

Some other comments and questions regarding the paper content:

As seen in Fig. 5 when using Fourier series expansion some typical oscillations are observed due to the stepwise nature of the approximated function. How is such behavior related to the real-case conductivity observed in experimental specimens with such gaps?

Response: Fourier series approximation has been replaced with spline approximation (line 100-120)

How is the sketch of the meshed domain shown in Fig. 10 related to the experimental setup from Fig. 8?

Response: Only the coil and the material are simulated as the scanning robot is not part of the simulation. (Coil was marked in Fig. 10)

“The CF domain contains several fiber bundles in 0° and 90° orientations”: what does the term “several” mean here?

Response: 10 (Was adjusted in line 182)

If the authors are wishing to provide experimental verification of their model, either 5 mm gap should be introduced in the experiments or 10 mm gap should be simulated in the model.

Response: 10 mm gap has been simulated now (Fig. 5, Fig. 14)

Visible light horizontal patterns are present in experimental results (Fig. 13, a) while being absent in the simulations. What is the reason of such behavior?

Response: they are present in the simulation but in relation to the gap they are a lot smaller, which is why they appear differently than in the experiment. These are the spaces between fibre bundles with lower conductivity. The simulation underestimates the conductivity diminution between the fibres.

It is not clear how the authors are comparing numerical data with experimental results in Fig. 15 while different sizes of gaps were considered in experiments and simulations and are trying to derive any conclusions based on such comparison. 

Response: 10 mm gap has been simulated now (Fig. 5, Fig. 14)

Line 28: "... the electrical behaviour of a coil is still poorly understood..." - Could the authors support this statement with any references?

Response: Was changed to: “Due to the complex physical effects underlying the eddy current method especially at high frequencies and the electrically complex properties of CF structures \citep{Wang2023, Bui2016, Horie2024}, the interaction between CF materials and electromagnetic fields in terms of eddy current density distribution and the resulting effects on the electrical behaviour of a coil is still subject to current research \citep{Xu.2023, Hughes2018}.” (added references) (line 25-29)

Line 58: When simulating ultrasonic-based NDT with air-coupled transducers, air which is surrounding the inspected object should be considered and meshed as well. 

Response: Ultrasonic testing does not involve electromagnetic simulation and is not comparable.

Line 61: What does the term "networking effort" mean here?

Response: Was changed to "computing time" (line 64)

How is s_N(x) mentioned in Eq. (3) related to the Eq. (1)?

Response: Is explained better now by using f(x) as the fiber volume fraction or conductivity function transverse to the fibres. (eq. 2)

Lines 100-101: What kind of behavior is mentioned here?

Response: Fibre conductivity distribution has been explained more in detail (line 100-120)

What does each gray circle mean in Figure 4?

Response: Confusing circles have been removed

Lines 127-128: The authors claim that eddy-current-based technique is a novel approach suitable for the investigation of CFRP structures. However, they mention that near the specimen edges no scanning is performed. At the same time such areas are also of the specific interest in the sense of NDT. Therefore, a certain contradiction could be observed here. This is only about the edge effect in the experiment, to establish comparability with the simulation, not generally about the fact that the edge cannot be inspected.

Response:  From inspection in NCF production lines the authors know that the edges of the fabric are cut off at the end and not used.

Line 129: What kind of tape was used? To what extent is it essential for the experiment? What would happen if it is removed?

Response: The tape is a 0.08 mm teflon tape that ensures that the sensor is sliding smoothly over the material ensuring a constant liftoff, while the sensor is spring-mounted and can adapt to minor height deviations. (line 145)

Subsection 4.2: What is the manufacture of both the Tx-Rx-coil and the preamplifier which are employed in the experiments?

Response: The parameters of the coil can be found in table 2

Line 180: For which specific sample were these measurements and simulations performed?

Response:  An exemplary sample is shown in fig 7.

Lines 193-195: How do the authors judge that exactly these reasons are conditioning the aforementioned spatial variation?

Response: Theoretical considerations of current paths in carbon fibres lead to this assumption (Line 100-120) Sources: https://doi.org/10.1016/j.compstruct.2023.116948, https://doi.org/10.1016/j.compositesa.2024.108232

Lines 204-205: There are no data provided in the paper which could support this statement.

Response: Has been removed as virtual scanning was already stated by Ref. 14 (calculation time can be reduced from 80 h to 15 h with Ryzen 5900 with 12 cores + better signal-to noise ratio)

Author Response File: Author Response.pdf

Round 2

Reviewer 3 Report

Comments and Suggestions for Authors

The authors have adequately responded to all the comments. The paper could be accepted.