Reliability Analysis of Interface Oxidation for Thermal Barrier Coating Based on Proxy Model
Abstract
1. Introduction
2. Surrogate Model Based on Radial Basis Function
2.1. Construction of a Radial Basis Function Model
2.2. K-Fold Cross-Validation
3. RBF-MCS Method
3.1. Multiple Predictions Based on Different Shape Parameters and Multiple-Subset Initiation
3.2. Comparison of Predicted Variance Significance
3.3. Construction of the Learning Function
3.4. Failure Probability
3.5. Main Steps of Active Learning Process
- Step 1: Generating an MC population with a size of based on the probability distribution of the input variables.
- Step 2: Initializing the kernel function type, shape parameters c, subset number k, and termination number of the RBF model, and generating P initial sample points that form the initial DoE by using the Limit State Function (LSF) method.
- Step 3: Constructing the initial RBF model through initial DoE, and then obtaining the failure probability by MCS.
- Step 4: Predicting MC populations to obtain weighted coefficients , predicted mean , and predicted mean variance .
- Step 5: Selecting new sample points from MC population by using learning functions and updating DoE.
- Step 6: Reconstructing the RBF model based on Equation (1) by using the updated DoE, and predicting the failure probabilities of MC sampling points.
- Step 7: Executing stop criteria judgment. If satisfied, going to the next step of judgment, otherwise, returning to step 4 to continue iterating.
- Step 8: Calculating and determining if it is less than . If satisfied, ending iteration; otherwise, expanding by 10 times and returning to step 3.
3.6. Implementation Details
4. Example Verification
Results of MCS or References | Method | (%) | (%) | |
---|---|---|---|---|
MCS | 0.4324 | / | ||
Reference [24] | ARBFM-MCS | 60.1 | 0.4466 | 1.13 |
Reference [24] | CVRBF-MCS () | 65.4 | 0.4480 | 1.44 |
Reference [32] | RBF-GA | 33.3 | 0.4367 | 1.10 |
Reference [26] | MCRBF-MCS | 36.3 | 0.4479 | 1.43 |
RBF method | RBF+IM (, ) | 73.4 | 0.4473 | 3.44 |
RBF+G (, ) | 35.6 | 0.4382 | 1.36 | |
(, ) | 41.8 | 0.4383 | 1.02 | |
(, ) | 42.3 | 0.4374 | 0.96 | |
RBF+M (, ) | 58.9 | 0.4197 | 2.94 | |
(, ) | 64.2 | 0.4267 | 1.32 | |
(, ) | 38.2 | 0.4251 | 1.68 | |
(, ) | 43.3 | 0.4300 | 1.26 | |
(, ) | 40.8 | 0.4365 | 0.94 |
5. Reliability Assessment of TBC Interface Oxidation Failure
5.1. Finite Element Model and Boundary Conditions
Ceramic Layer | Oxide Layer | Bonding Layer | Substrate | |
---|---|---|---|---|
Young’s modulus E (GPa) | 35 | 375 | 120 | 160 |
Poisson’s ratio v | 0.1 | 0.25 | 0.32 | 0.33 |
density (kg/) | 5650 | 3978 | 8100 | 8100 |
tensile strength (GPa) | 0.2 | 0.38 | 0.5 | / |
Fracture energy (J/) | 61 | 49 | 47 | / |
Coefficient of thermal expansion () | 9.8 × | 9.2 × | 1.4 × | 1.5 × |
Critical energy release rate (N/mm) | 50 | 40 | 2700 | 2700 |
Thermal conductivity coefficient () | 1.16 | 4.4 | 14.5 | 26 |
5.2. Numerical Simulation Results
5.3. Construction of Failure Criteria
5.4. Reliability Calculation of TBC Interface Oxidation Failure Based on RBF-MCS Method
6. Conclusions
- When constructing multiple predictions with different numbers of DoE subsets , the number of subsets has a significant impact on the sequential sampling used for updating the RBF model, since the quantity of affects the local uncertainty. Simultaneously adjusting the coefficient also has an impact on the efficiency of the active learning function. When is small, the influence of local uncertainty is reduced, which also reduces the differences among the candidate sample points. Therefore, for the RBF-MCS method, the performance is the best when and in the example.
- The acceptable results under different subset numbers and adjustment coefficients can be obtained with the RBF-MCS method. The comparison with other methods in the literature can demonstrate the advantage of the proposed method in terms of accuracy.
- The accuracy and efficiency of reliability calculation for TBC oxidation failure problems can be improved via the proposed RBF-MCS method, and the results obtained are within the acceptable error range.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
TBC | Thermal Barrier Coating |
TGO | Thermally Grown Oxide |
MCS | Monte Carlo Simulation |
RBF | Radial Basis Function |
PRBF-MCS | Predicted Variance RBF-MCS Method |
DoE | Design of Experiments |
COV | Coefficient of Variation |
LSF | Limit State Function |
IM | Inverse Multi-quadratic |
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Kernel Function | |
---|---|
Gaussian Kernel function | |
Multi-quadratic Kernel function | |
Inverse Multi-quadratic Kernel function | |
Cubic Kernel function |
Material Properties | Values |
---|---|
Oxygen diffusion coefficient in bonding layer (/s) | 2 × |
Oxygen diffusion coefficient in oxide layer (/s) | 2 × |
Oxygen reaction rate ( ) | 1 × |
Molar concentration of oxygen M (mol ) | 1.11 × |
Random Variable | Distribution Type | Mean Value | Standard | |
---|---|---|---|---|
Young’s modulus | (GPa) | Normal distribution | 380 | 100 |
(GPa) | Normal distribution | 120 | 30 | |
Thermal expansion coefficient | (/K) | Normal distribution | 9.2 | 0.37 |
(/K) | Normal distribution | 9.8 | 0.39 | |
Fracture energy | (J/) | Weibull distribution | 60 | 6 |
(J/) | Weibull distribution | 50 | 5 | |
Temperature difference | (°C) | Normal distribution | 1050 | 5 |
Method | Failure Probability (%) | Efficiency | Accuracy (%) |
---|---|---|---|
Experiment [47] | 41.4 | / | / |
MC () | 42.9 | 5 × | / |
MC () | 35.8 | 5 × | / |
RBF-MC () | 43.7 | 143.3 | 1.92 |
RBF-MC () | 36.6 | 126.4 | 2.23 |
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Ma, J.; Wang, A.; Junker, P.; Alshawawreh, A.W.; Li, Q.; Xu, H.; Xue, R. Reliability Analysis of Interface Oxidation for Thermal Barrier Coating Based on Proxy Model. Modelling 2025, 6, 61. https://doi.org/10.3390/modelling6030061
Ma J, Wang A, Junker P, Alshawawreh AW, Li Q, Xu H, Xue R. Reliability Analysis of Interface Oxidation for Thermal Barrier Coating Based on Proxy Model. Modelling. 2025; 6(3):61. https://doi.org/10.3390/modelling6030061
Chicago/Turabian StyleMa, Juan, Anyi Wang, Philipp Junker, Anas W. Alshawawreh, Qingya Li, Haoqi Xu, and Runzhuo Xue. 2025. "Reliability Analysis of Interface Oxidation for Thermal Barrier Coating Based on Proxy Model" Modelling 6, no. 3: 61. https://doi.org/10.3390/modelling6030061
APA StyleMa, J., Wang, A., Junker, P., Alshawawreh, A. W., Li, Q., Xu, H., & Xue, R. (2025). Reliability Analysis of Interface Oxidation for Thermal Barrier Coating Based on Proxy Model. Modelling, 6(3), 61. https://doi.org/10.3390/modelling6030061