Prediction Method for Excess Surface Temperature Peak Time Inclusion Defect Depth Based on Conjugate Gradient Algorithm

Abstract

This study addresses the challenge of accurately calculating the depth of inclusion defects in Glass Fiber-Reinforced Plastic (GFRP), which is commonly used in onshore wind turbine blades. To overcome this issue, we proposed a novel Excess Surface Temperature Peak Time (ESPT) estimation method that combines a conjugate gradient algorithm with a conventional analytical approach. This research employed the Inverse Heat Transfer Problem (IHTP) solution method to estimate the boundary conditions of an experimental sample subjected to pulse excitation. By drawing analogies with traditional depth detection methods, we analyzed specific physical models and determined the calculated thickness of the sample. The Excess Surface Temperature Peak Time characteristics were then used to estimate the defect depth, and the resulting estimates and relative errors were evaluated. Our results demonstrated that the proposed method achieved a relative error of less than 15% when calculating defect depth, confirming its effectiveness. This approach provides new insights and possibilities for improving defect depth estimation in GFRP materials, offering valuable contributions to the assessment and maintenance of wind turbine blade safety.

1. Introduction

The Inverse Heat Transfer Problem (IHTP) involves determining parameters within a heat transfer system that are either difficult or impossible to measure directly. These parameters are inferred from known and easily measurable inputs, such as system temperature. IHTP has been extensively applied in various engineering fields, including aerospace, healthcare, civil engineering, and metallurgical casting [1,2,3,4]. The solution methods for IHTP can be broadly categorized into non-iterative solutions, regularization methods, gradient methods, predictive control methods, and intelligent optimization algorithms. Each method has its own set of advantages and limitations, making the choice of method dependent on the specific application scenario.

Among these methods, the Conjugate Gradient Method (CGM) is the most commonly used gradient-based approach. It strikes a balance between calculation accuracy and computational efficiency, making it particularly popular for solving inverse heat conduction problems [5,6]. For instance, Lin et al. [7] developed an inversion algorithm based on the conjugate gradient method to estimate boundary conditions in unsteady laminar forced convection heat transfer scenarios. Similarly, Heng Yi et al. [8] introduced an iterative regularization strategy utilizing the conjugate gradient method to address the inverse problem in pool boiling experiments.

This study is grounded in the use of infrared thermal imaging to detect defects in wind power composite materials. The infrared thermal imaging method captures the thermal radiation emitted by an object to obtain surface temperature information, which is then used for analysis or problem-solving. However, other boundary conditions often cannot be directly measured, making this approach well-suited for solving inverse problems. Previous research has successfully combined infrared thermal imaging with the Conjugate Gradient Method (CGM) to estimate the boundary conditions of defects, particularly in scenarios involving the inner walls of cavities or pipes [9]. For instance, Shi Hongchen et al. [10] proposed an inverse heat transfer method for quantitatively identifying the liquid level inside a cylinder based on the temperature distribution on the cylinder’s outer surface. Similarly, Ching-Yu Yang et al. [11] developed a method for directly estimating the boundary conditions of a two-dimensional heat conduction inverse problem.

This paper aimed to optimize the physical parameters or boundary conditions in the mathematical model by using the Conjugate Gradient Method to invert the thermal diffusion efficiency and the energy received by the surface of the sample plate. However, since there is no established analytical formula for directly calculating the depth of inclusion defects, it is necessary to integrate the analytical methods and concepts of the direct problem to analyze and solve the specific physical model.

In the positive problem, most existing studies on defect depth rely on numerical and image analysis techniques [12,13,14,15,16]. Numerous methods have been developed for defect detection and depth assessment, with a common focus on the relationship between the temperature decay over time on the measured surface, such as in Active Pulse Thermography (APT) and Absolute Peak Slope Time (APST). For instance, Zhi Zeng et al. [17] proposed using APST to quantitatively measure defect depth. However, the experimental materials used in much of the current research are flat-bottom hole specimens, which are characterized by large defect thicknesses and simple shapes. These specimens do not adequately represent inclusion-type defects and are not reflective of real-world defects like bubbles, cracks, or adhesive voids. This highlights an urgent need to improve defect samples to better align with actual detection.

In summary, to more accurately determine defect depth in real-world detection scenarios, this paper sought to enhance the calculation process by integrating inverse problem algorithms, replacing traditional flat-bottom hole specimens with inclusion defect specimens, and extending conventional calculation methods and approaches. By innovatively constructing a physical model for a uniform sample, this study developed a defect depth calculation method that is better suited for practical applications.

2. Basic Principles of Infrared Thermal Imaging Defect Detection and Inverse Problem Application

2.1. Principle of Infrared Thermal Imaging Defect Detection

Common infrared thermal imaging excitation sources include pulse (PT), phase lock, ultrasound, and laser. Pulse excitation was chosen for this study due to its relatively larger action area, shortest action time, and higher tolerance for image noise.

The working principle of pulse excitation is as follows: an active pulse emits a short, high-energy light pulse onto the sample’s surface through a flash lamp. In a sample with defects, the local structure changes affect the heat conduction rate, causing some of the heat to “reflect” back to the surface. In contrast, defect-free areas maintain their original conduction state, resulting in a temperature difference on the surface of the sample.

Infrared thermal imaging experiments utilize this principle to detect defects based on surface image contrast or temperature differences. Excess surface temperature is a key parameter for defect detection, representing the temperature difference between defective and defect-free areas during the heat transfer process, as illustrated in Figure 1. This temperature difference is measured between points a and b.

Assume that the entire heat transfer process is governed solely by heat conduction, with convection and radiation being negligible. For a uniform plate, the surface temperature caused by the back wall at depth L is given by Equation (1) [18]. This equation represents the one-dimensional solutions of Fourier equations for Dirac functions in semi-infinite isotropic solids.

$$\Delta \mathrm{T}\left(0,t\right)={\displaystyle \frac{Q}{\sqrt{\pi \rho Cpkt}}}\left[1+2{\displaystyle \sum _{n=1}^{\infty }{R}^n\exp (-\frac{n^2{L}^2}{\alpha t})}\right]$$

(1)

ΔT is the surface temperature change, that is, the difference between any time t and the initial temperature at time 0, °C; ρ is the material density, kg/m³; C_p is the material specific heat capacity, kJ/(kg∙°C); k is the material thermal conductivity, W/(m∙°C); Q is the energy applied to the surface of the object, kJ; L is the thickness of the uniform sample plate, m; α is the thermal diffusivity, m²/s; R is the thermal reflection coefficient of the interface with air in actual materials, R ≤ 1.

In the formula, t^−1/2 represents time dependence, which is a common property of all solids with uniform structure, excluding areas containing subsurface defects.

Adisorn Sirikham et al. [19] proposed a new least squares analysis method (NLSF) by analyzing the above heat conduction model, improving and proving its effectiveness and accuracy. As shown below,

$$\tilde{\mathrm{T}}\left(t,\mathrm{B},W,R,t_s,s\right)={\displaystyle \frac{B}{\sqrt{t+t_s}}}\left[1+2{\displaystyle \sum _{n=1}^M{R}^n\exp (-\frac{n^{2}W}{t+t_s})}\right]-s(t+t_s)$$

(2)

where $B={\displaystyle \frac{Q}{\sqrt{\pi \rho C_{p}k}}}$; $W={\displaystyle \frac{L^2}{\alpha }}$; t_s is the sampling start time (the sampling start time does not represent the experiment start time); M is the number of iterations to replace ∞; s is the slope of the linear fit in the time period t_a and t_b; s is usually very small. For a defect-free plate with uniform thickness, this term can be ignored. t_a and t_b represent the Absolute Peak Slope Time [17], which refers to the characteristic time obtained in the absence of a reference curve. The optimized event-related function is derived twice to obtain (3).

$$t_a={\displaystyle \frac{L^2}{2\alpha }},t_b=3t_a$$

(3)

t_a represents the time when 3D heat conduction starts.

The improved formula describes the parameters more accurately and has higher calculation accuracy. According to this formula, we can also use any unknown quantity in the formula as the parameter to be calculated.

2.2. Principle of ESPT Defect Depth Calculation

Statistics show that most samples used in defect-depth research are flat-bottom hole samples, as illustrated in Figure 2a. The limitation of this approach is that it assumes that the material of the entire sample plate is uniform, and only the local thickness needs to be measured to determine the defect depth. In practical applications, however, defects come in various types, with many being inclusion defects that are hidden within the plate. These inclusion defects cannot be detected using traditional methods, as shown in Figure 2b.

If we treat the defective area as a single unit and assume that the material is uniform throughout, the inclusion defect sample can be modeled as two uniform sample plates spliced together. Under the same boundary conditions, the calculated thickness in the defective area changes due to altered physical parameters, which can be divided into two cases, as shown in Figure 2c,d. The ESPT defect depth calculation method can be analogous to that used for traditional flat-bottom hole defects. If the thicknesses of the two areas are denoted as d₁ and d, then, theoretically, their difference represents the defect depth h that we aimed to calculate.

Substitute the inverse related parameters into Equation (2), and use the solution of the direct problem to subtract the calculated thickness of the defective area from that of the defective area, and estimate the calculated depth of the defect, as shown below:

$$\mathrm{h}=abs\left(d-d_1\right),$$

(4)

where d and d₁ are the calculated thickness of the area shown in Figure 2, and h is the calculated depth of the defect.

2.3. Application of the Conjugate Gradient Method

The inverse problem of heat transfer involves determining parameters that are difficult or impossible to measure directly within a heat transfer system based on known quantities and easily measurable inputs such as heat source power and temperature. In practical applications, the limited data from tests, such as the surface temperature, often align with the characteristics of IHTP. To address incomplete defect information and enhance detection and prediction accuracy, this paper employed the traditional conjugate gradient algorithm to determine various defect characteristics. Specifically, the study uses physical property parameters from defect-free regions to invert the pulse energy received by the sample and subsequently uses this calculated value, Q, to determine the thermal diffusion efficiency, α, of the defect.

2.3.1. Optimization Objective Function of the Inverse Problem

Given a heat transfer model with unknown parameters, the objective function is inverted using the experimentally measured surface temperature. The optimization objective function of this problem is the following:

$$S={{\displaystyle \sum _{k=1}^G\left[T_k^{cal}\left(R\right)-T_k^{mea}\right]}}^2,$$

(5)

where k is the number of identification parameters; T_cal and T_mea are the measured temperature and the calculated temperature, respectively. When the optimization target S is less than a given sufficiently small positive number ε, the iteration ends, that is,

$$S\le \epsilon$$

(6)

2.3.2. Key Parameters

This paper adopted the traditional conjugate gradient method [7] to estimate the target parameters by minimizing the objective function. The iterative formula for identifying the parameters is the following:

$$R^{n+1}=R^n+{\alpha }^n{d}^n,$$

(7)

where α is the search step size, and d is the search direction, as shown below:

$$d^n={\left({\displaystyle \frac{\partial S}{\partial R}}\right)}^n+{\beta }^n{d}^{n-1}$$

(8)

The conjugation coefficient β is the following:

$${\beta }^n={\displaystyle \frac{\nabla S(R^n)\nabla S^T(R^n)}{\nabla S(R^{n-1})\nabla S^T(R^{n-1})}},$$

(9)

where $\nabla S(R^n)$ is the first-order partial derivative of the optimization objective function with respect to the identification parameter. The search step length is the following:

$${\alpha }^{\mathrm{n}}={\displaystyle \frac{{\displaystyle \sum _{k=1}^M\left[T_k^{cal}\left(R^n\right)-T_k^{mea}\right]\nabla T_k{d}^n}}{{\displaystyle \sum _{k=1}^M{\left[\nabla T_k{d}^n\right]}^2}}}$$

(10)

In the formula, $\nabla T_k=(\partial T_k^{cal}/\partial r_1,\cdots ,\partial T_k^{cal}/\partial r_{m+1})$ is the sensitivity coefficient [20], that is, the first derivative of temperature with respect to the identification parameter.

$${\displaystyle \frac{\partial T_k}{\partial r_l}}=2{\displaystyle \sum _{k=1}^M\left[T_k^{cal}\left(R^n\right)-T_k^{mea}\right]}{\displaystyle \frac{\partial T_k^{cal}\left(R^n\right)}{\partial r_l}},(l=1,2,\cdots ,M+1)$$

(11)

2.3.3. Solution Process of Inverse Problem

The general steps for solving the inverse problem using the conjugate gradient method are as follows:

• Initialize Parameters: Choose an initial guess R₀ for the parameter to be inverted, along with other necessary parameters, and set a tolerance ε to determine calculation accuracy;
• Check Convergence Criteria: Verify whether the optimization objective function satisfies the iteration conditions;
• Compute Necessary Quantities: Calculate the sensitivity coefficient, search direction, search step size, conjugate coefficient, and other relevant values;
• Update and Iterate: Update R_k, check for convergence, and decide whether to continue the iteration based on the convergence criteria.

3. Experimental Equipment and Test Specimens

3.1. Experimental Equipment

The infrared thermal imaging experimental device and process are illustrated in Figure 3. The detection steps are as follows:

• The pulse power supply controls the excitation source, providing energy that causes the excitation source to emit heat waves toward the sample plate;
• The infrared thermal imager converts the detected thermal radiation into temperature information;
• The computer integrates and organizes this information, producing a thermal sequence diagram.

The infrared thermal imager model is Fast-M200, the sampling frequency is 25 Hz, the pulse power supply model is TRIA6000S which is invented and manufactured by Telops inc. It headquartered in 100-2600 avenue St-Jean-Baptiste, Québec QC G2E 6J5, Canada. And the pulse duration is about 4 ms (Figure 3).

3.2. Experimental Specimens

The test specimens in this experiment are Glass Fiber-Reinforced Plastic (GFRP). Unlike the previous flat-bottom hole specimens, this study employed inclusion defects. Specifically, while laying the fiber cloth, an insulation film with physical properties similar to air was incorporated to better simulate common inclusion bubbles and glue deficiencies. The details of the defect number, depth, and area arrangement are shown in Figure 4. Due to the large size of the specimen, energy calculations are typically based on measurements taken at two similar points on the board surface. The sample board measures 500 mm × 500 mm × 17 mm, with the defect areas consistent in column-wise arrangement but varying in depth. These defects are located in the 2nd, 4th, 6th, 8th, and 10th layers, each approximately 0.77 mm thick.

4. Results Display and Analysis

4.1. Characteristics of Excess Temperature Peak Time

As shown in Equation (1), because uniform solid materials exhibit time-dependent behavior, the calculated energy and thermal diffusivity for the entire time period are represented using time-related series. Consequently, the calculated thickness is also time-dependent, and the appropriate calculation depth must be selected based on time. This means that the depth should be chosen according to the time series data.

Figure 5 presents several thermal sequence diagrams from the defect detection process, illustrating defect recognition before the pulse and during the heat dissipation phase. The defect images appear according to defect depth, with the area effect being relatively minor. The excess surface temperature is typically used to indicate defect identification; thus, the order of defect appearance can also be inferred from the excess surface temperature. A higher excess temperature generally correlates with better defect recognition, indirectly suggesting a relationship between excess temperature and defect depth. Therefore, by analyzing the qualitative relationship between defect depth and excess temperature, the best selection time for determining the calculation depth can be established.

Based on the analysis of excess surface temperature obtained from the experiment, it is observed that for defects of the same area, the time at which the excess temperature reaches its peak value shifts as the depth increases. This is illustrated in Figure 6.

A preliminary analysis of the time range revealed a linear correlation between defect depth and the Excess Surface Temperature Peak Time (ESPT). As explained in Formula (1) on the Detection Principle, defects are detectable because varying internal structures result in different thermal conductivity rates. Consequently, the energy from the pulse on the surface of the sample plate does not uniformly conduct to the interior. Additionally, some heat is “reflected” at the defect, creating a temperature difference on the surface.

During heat conduction, before the surface heat is fully conducted to the defect, heat accumulation occurs, causing the excess temperature to rise gradually. As heat diffusion in the X and Y directions starts to dominate over heat conduction in the Z direction, the surface temperature distribution becomes more uniform, and the excess temperature slowly decreases until it approaches zero. Since the time required for heat to reach different depths varies, the peak time of the excess temperature also changes accordingly.

Taking Figure 6a as an example, for defects with the same area, the defect at position No. 11 has a depth of approximately 0.77 mm, while the defect at position No. 12 has a depth of approximately 2.31 mm, which is about twice as deep. Correspondingly, the ESPT for these defects also differs by approximately the same factor. Figure 6a–d shows the ESPT comparison for defects in columns 1–4. It is evident that the ESPT for each column of defects exhibits a linear relationship with depth. Therefore, we can estimate the defect depth based on ESPT.

Further analysis revealed that the accurate defect depth can be determined by calculating the difference between the thickness of the defect area with the defect and without the defect at the ESPT. This is expressed as follows:

$$\mathrm{h}=abs\left[d\left(ES\mathrm{PT}\right)-d_1\left(ESPT\right)\right]$$

(12)

Given the large volume of basic data and the complexity of data fluctuations, errors are inevitable when relying solely on the maximum value from the experimental data. Therefore, this experiment selected a uniform number of samples before and after the ESPT, removed any outliers, and used the average value as a substitute.

4.2. Estimated Depth and Its Relative Error

By solving the calculated depth, comparing it with the actual depth, and listing the corresponding errors, it can be observed that the relative error obtained by this method remains within 15%. As shown in

Table 1. Calculated defect depth and its relative error (mm).

Defect Number	No. 11	No. 12	No. 13	No. 21	No. 22	No. 23
Actual Depth	0.77	2.31	3.85	0.77	2.31	3.85
Estimated Depth	0.80	2.16	3.29	0.78	2.23	3.94
Relative error	0.044	0.066	0.145	0.016	0.033	0.024
percentage	4.36%	6.59%	14.53%	1.64%	3.25%	2.43%
Defect Number	No. 31	No. 32	No. 33	No. 41	No. 42
Actual Depth	0.77	2.31	3.85	0.77	2.31
Estimated Depth	0.76	2.08	3.55	0.794	2.150
Relative error	0.016	0.100	0.079	0.031	0.069
percentage	1.62%	9.96%	7.86%	3%	7%

. This indicates that the calculated depth is very close to the actual depth, demonstrating that the method performs well both practically and theoretically.

By plotting the diagonal error diagram of the relative errors for all defects, we obtained Figure 7. The distribution trend of the errors indicates that the calculation error increases with defect depth. This trend can be attributed to several factors: as depth increases, the impact of reflected heat at the defect becomes more pronounced; deeper defects result in a larger ESPT, leading to 3D heat conduction and diffusion over time, which affects the surface temperature; and smaller defect areas are more prone to sampling errors, further increasing the calculation error.

5. Conclusions

This paper employed the traditional conjugate gradient method to invert the defect boundary conditions and estimated defect depth using the variation in the one-dimensional solution of the Fourier equation with a Dirac function. The estimated results were compared to the actual defect depths. The findings demonstrate that this method offers notable advantages for calculating inclusion defects. The conclusions are summarized as follows.

• From the picture, we can see that there is a corresponding qualitative relationship between the defect depth and ESPT; that is, calculating the depth of a certain defect can roughly estimate the depth of other defects.
• It is worth noting that our results demonstrate that the proposed method achieves a relative error of less than 15% when calculating defect depth, confirming its effectiveness.
• The test results indicate that predicting the depth of deep defects is challenging due to difficulties in determining the defect’s location. To accurately estimate the depth, significant effort must be devoted to enhancing the excitation source to improve defect localization and measurement.
• Under this excitation condition, the detection of deeper defects is also affected by 3D heat conduction and sampling location, so the error needs further discussion.
• Additionally, the analysis confirms that the inverse problem algorithm is effective for determining defect depth. It provides high accuracy in reconstructing defect boundary conditions and accurately locates the depth of defects, thus serving as a valuable reference for further investigation into unknown defect information.