Using the Kriging Response Surface Method for the Estimation of Failure Values of Carbon-Fibre-Epoxy Subsea Composite Flowlines under the Influence of Stochastic Processes
Abstract
:1. Introduction
2. Preliminaries
2.1. Failure Criteria
2.1.1. Maximum Stress Failure Criterion
2.1.2. Tsai-Wu Failure Criterion
2.1.3. Hashin Failure Criterion
2.1.4. Calculation of Failure Values
2.2. Parameter Correlation
- 0.0 to 0.2—Slightly correlated, the relationship is almost negligible.
- 0.2 to 0.4—Lowly correlated, but the relationship is definite.
- 0.4 to 0.6—Moderately correlated, the relationship is substantial.
- 0.6 to 0.8—Highly correlated, the relationship is marked
- 0.8 to 1.0—Very highly correlated, the relationship is very dependable
2.3. Response Surface Methodology
3. Case Study of the Burst Design of a Subsea CFEC Flowline
3.1. General Properties
3.2. Nominal Load Values
3.3. Finite Element Model
3.3.1. Loads and Boundary Conditions
3.3.2. Mesh Refinement Study
3.4. Generation of Measured Values
3.4.1. Convergence Study on Population Size
3.4.2. Fitting Statistical Models to Measured Values
3.5. Calculating the Response Surface
3.5.1. Parametric Correlation to Identify Most Influential Input Parameters
3.5.2. Range of Input Parameters
4. Using Response Surfaces for Prediction of Failure Rates
4.1. Comparison of Statistical Moments
- Measured values, 500 samples. These were 500 sample points calculated directly from the finite element model. These 500 samples were considered the base case population.
- Lognormal distribution fitted to measured values: a Lognormal distribution fitted to ‘Measured values, 500 samples’.
- Response surface, 500 samples. These were 500 sample points calculated directly from the response surface.
- Lognormal distribution fitted to response surface values: a Lognormal distribution fitted to ‘Response surface, 500 samples’.
- The % differences were calculated with respect to ‘Measured values, 500 samples’ using Equation (16)
- There are negligible differences in the mean values.
- The values obtained from the response surface, when fitted to the Lognormal distribution, have smaller standard deviation values, which are about 9% smaller than the measured values.
- There are large differences in the skewness values. The response surface skewness values are about 10% smaller than the measured values. Fitting the response surface values to a Lognormal distribution makes the skewness values even smaller, i.e., about 22% smaller than the measured values. In addition, fitting the measured values to a Lognormal distribution gives smaller skewness values of about 14% smaller than the measured values.
- There are only slight differences of below 2% in the kurtosis values.
4.2. Comparison of Probability and Cumulative Distribution Plots
- The fitted distribution curves overlap the raw data, i.e., the measured and response surface sampled values.
- The response surface has a higher probability density at the most probable value; see area A in Figure 15.
- There are some differences in the tail regions in the cumulative probability distribution functions; see Figure 16. However, these differences are insignificant and lead to only negligible differences in the failure values calculated based on the upper tail region, as presented in Section 4.3.
4.3. Comparison of Failure Values
5. Optimising Response Surface Generation
5.1. Number of Input Parameters
5.1.1. Number of Parameters—Effect on Statistical Moments
- The mean values were not significantly affected when more than 10 input parameters are used where the differences were within 1%.
- The standard deviations were affected by the number of input parameters used. However, using the response surface would already result in a difference in the standard deviation of about 9%, as presented in Section 4.1. Using a smaller number of input parameters would increase this difference to about 20%.
- The skewness values were significantly affected by the number of input parameters used. However, using the response surface would already result in a difference in the standard deviation of about 14%, as presented in Section 4.1. Using a smaller number of input parameters would increase this difference to about 250%.
- The kurtosis values were not significantly affected when enough input parameters were used. However, the difference became as large as 60% when too few input parameters were used.
5.1.2. Number of Parameters—Effect on Fitted Probability Distributions
- In general, the probability distribution functions are highly inaccurate when a small number of input parameters, i.e., 5, is used.
- As presented in Figure 19, the probability density functions fitted from the response surfaces do not generally fit well with the measured values. A similar observation was also previously reported in Figure 15 and in Section 4.2.
- As presented in Figure 20 and Figure 21, there are some differences in the cumulative probability distribution functions. These differences increase with decreasing number of input parameters. However, these do not significantly affect the upper tail regions and do not lead to large differences in the failure values calculated as presented in Section 5.1.3. Similar observations were also previously reported in Section 4.2.
5.1.3. Number of Parameters—Effect on Predicted Failure Values
5.2. Size of Response Surface
5.2.1. Size of Response Surface—Effect on Statistical Moments
- In general, the differences in statistical moments calculated increased with the size of the response surface used.
- The differences in the mean values increase with the size of the response surface used but were within 10% for ‘extremely large size’.
- The response surface size has limited or no influence on the standard deviation values. The differences in the standard deviation are within approximately 10%, which were like that of the ‘base case’, which had 8.5%.
- The skewness is strongly affected by the response surface size. For the ‘larger size’ and ‘extremely large size’, the differences were as much as 40%.
- The differences in the kurtosis values increased with the size of the response surface used but were within 10% for ‘extremely large size’.
5.2.2. Size of Response Surface—Effect on Fitted Probability Distributions
- In general, using ‘extremely large size’ results in significant differences in the probability distributions.
- As presented in Figure 24, the probability density functions fitted from the response surfaces do not generally fit well with the measured values. The ‘extremely large size’ probability distribution function is especially far away from that of the measured values.
- As shown in Figure 25, the cumulative probability distributions of ‘base case’ and ‘larger size’ were close to those of the measured values. Furthermore, as presented in Figure 26, these differences became more minor at the upper tail regions. As observed in the probability density functions, the ‘extremely large size’ cumulative probability function is especially far away from that of the measured values.
5.2.3. Size of Response Surface—Effect on Predicted Failure Values
5.3. Recommendations for Optimisation of Response Surface
- Reduce the input parameters selected to generate the response surface by only selecting the parameters with parametric correlation coefficients greater than +/−0.15.
- Consider using a larger response surface to maximise flexibility if the accuracy in the predicted failure values is not extremely important. A larger response surface would lead to some decreased accuracy in the results.
6. Conclusions
- In general, the response surface method produced predicted failure results close to those of the measured values. Most errors were minor unless too few input parameters are selected to generate the response surface and/or the size of the response surface was too large.
- The response surfaces do not accurately represent the skewness values in general; there was at least a 9% difference in the results. However, this is not of practical significance as it did not affect the prediction of failure values.
- In general, using more input parameters increases the accuracy of the response surface. However, it also increases the time required to generate the response surface, as the design of experiment will increase in size.
- It is recommended to select input parameters with correlation coefficients greater than +/−0.15, i.e., input parameters that are slightly correlated. In this present study, this results in about 10 input parameters.
- In general, a more extensive response surface leads to reduced accuracy. However, this enables greater flexibility in its utilisation as a more comprehensive range of input parameters is covered.
- The response surface enables more rapid design optimisation process than using a brute force method; the computation of the failure values is instantaneous.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Nomenclature
Hoop stress | |
Longitudinal stress | |
Principle stress in x-direction | |
Principle stress in y-direction | |
Principle stress in z-direction | |
Shear stress in xy-plane | |
Shear stress in yz-plane | |
Shear stress in xz-plane | |
Compressive strength limit in x-direction | |
Compressive strength limit in y-direction | |
Compressive strength limit in z-direction | |
Tensile strength limit in x-direction | |
Tensile strength limit in y-direction | |
Tensile strength limit in z-direction | |
Shear strength limit in xy-plane | |
Shear strength limit in yz-plane | |
Shear strength limit in xz-plane | |
Spearman correlation coefficient | |
Covariance of the rank variable | |
Standard deviation of the rank variable | |
Standard deviation of the rank variable | |
Flow-line Diameter | |
Flowline wall thickness |
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Yield Strength (GPa) | Ultimate Tensile Strength (GPa) | Density (g/cm3) | Strength to Weight Ratio | |
---|---|---|---|---|
Epoxy carbon UD | - | 2.231 | 1.49 | 1.497 |
Steel, AISI 4130 | 0.95 | 1.11 | 7.85 | 0.141 |
Material Property | Symbol | Value | Unit |
---|---|---|---|
Elastic Modulus | E1, E2, E3 | 121,000, 8600, 8600 | MPa |
Shear Modulus | G12, G23, G13 | 4700, 3100, 4700 | MPa |
Poisson’s Ratio | ν12, ν23, ν13 | 0.27, 0.4, 0.27 | - |
Tensile Strength | σut1, σut2, σut3 | 2231, 29, 29 | MPa |
Compressive Strength | σuc1, σuc2, σuc3 | −1082, −100, −100 | MPa |
Shear Strength | τu12, τu23, τu13 | 60, 32, 60 | MPa |
Tsai-Wu Constants | F12, F23, F13 | −1, −1, −1 | - |
Maximum Stress | Tsai-Wu | Hashin | |
---|---|---|---|
Failure Criterion Value | 0.516 | 0.613 | 0.563 |
Failure Mode | σ2 exceeded | - | Matrix failure |
Element Size (mm) | No. of Elements | No. of Nodes |
---|---|---|
30 | 900 | 912 |
25 | 1040 | 1053 |
20 | 1536 | 1552 |
15 | 2803 | 2821 |
10 | 6430 | 6432 |
5 | 25,344 | 25,408 |
Parameter | Symbol | Unit | Mean | Standard Deviation |
---|---|---|---|---|
Elastic Modulus | E1 | MPa | 121,000 | 3630 |
E2, E3 | MPa | 8600 | 258 | |
Poisson’s Ratio | ν12, ν13 | - | 0.27 | 0.0081 |
ν23 | - | 0.4 | 0.012 | |
Shear Modulus | G13 | MPa | 4700 | 141 |
G23 | MPa | 3100 | 93 | |
Tensile Strength | σut1 | MPa | 2231 | 66.93 |
σut2, σut3 | MPa | 29 | 0.87 | |
Compressive Strength | σuc1 | MPa | −1082 | −32.46 |
σuc2, σuc3 | MPa | −100 | −3 | |
Shear Strength | τu12, τu13 | MPa | 60 | 1.8 |
τu23 | MPa | 32 | 0.96 | |
Internal pressure | P | MPa | −6.9 | −0.207 |
Axial Force | A | N | 20,000 | 600 |
Bending | B | N·m | 2000 | 60 |
Torsion | T | N·m | 2000 | 60 |
Tsai-Wu Constants | F12, F23, F13 | 0 | 0.3 | |
Diameter | D | mm | 125 | 8.75 |
Thickness | t | mm | 0.2 | 0.01 |
Statistical Model | Exponential | Lognormal | Normal | Weibull |
---|---|---|---|---|
R2 Value | 0.960 | 0.999 | 0.998 | 0.987 |
Parameter Group | Parameters | Level of Correlation |
---|---|---|
Loads | P, T | Moderate |
Geometry | t | Moderate |
Material properties | E2, ν12, ν23, ν13, G23, G13, σut1, σut3, τu23, D, F12, F13, A, B | Slight |
E1, σut2, τu12 | Low |
Parameters | Symbol | Unit | Lower Limit | Upper Limit |
---|---|---|---|---|
Elastic Modulus | E1 | MPa | 108,900 | 133,100 |
E2 | MPa | 7740 | 9460 | |
Poisson’s Ratio | ν12, ν13 | 0.243 | 0.297 | |
ν23 | 0.36 | 0.44 | ||
Shear Modulus | G13 | MPa | 4230 | 5170 |
G23 | MPa | 2790 | 3410 | |
Tensile Strength | σut1 | MPa | 2007.9 | 2454.1 |
σut2, σut3 | MPa | 26.1 | 31.9 | |
Shear Strength | τu12 | MPa | 54 | 66 |
τu23 | MPa | 28.8 | 35.2 | |
Internal pressure | P | MPa | 6.21 | 7.59 |
Axial Force | A | N | 18,000 | 22,000 |
Bending | B | Nm | 1800 | 2200 |
Torsion | T | Nm | 1800 | 2200 |
Tsai-Wu Constants | F12, F13 | −1 | 1 | |
Diameter | D | mm | 100 | 150 |
Thickness | t | mm | 0.18 | 0.22 |
Source | Mean | Standard Deviation | Skewness | Kurtosis |
---|---|---|---|---|
Measured values, 500 samples | 0.591 | 0.039 | 0.232 | 3.027 |
Lognormal distribution fitted to measured values |
0.591 (0.0%) |
0.039 (0.1%) |
0.199 (−14.1%) |
3.071 (1.4%) |
Response surface, 500 samples |
0.592 (0.3%) |
0.036 (−8.5%) |
0.209 (−9.8%) |
2.980 (−1.6%) |
Lognormal distribution fitted to response surface values |
0.592 (0.3%) |
0.036 (−8.6%) |
0.182 (−21.7%) |
3.059 (1.0%) |
Source | Failure Criterion | Failure Rate = 1 in 104 | Failure Rate = 1 in 105 | Failure Rate = 1 in 106 |
---|---|---|---|---|
Lognormal distribution fitted to measured values | Maximum Stress | 0.64 | 0.67 | 0.69 |
Tsai-Wu | 0.75 | 0.78 | 0.81 | |
Hashin | 0.76 | 0.79 | 0.81 | |
Lognormal distribution fitted to response surface values | Maximum Stress | 0.64 | 0.66 | 0.68 |
Tsai-Wu | 0.74 | 0.77 | 0.79 | |
Hashin | 0.75 | 0.78 | 0.80 |
Number of Input Parameters | 5 | 10 | 15 | 20 |
---|---|---|---|---|
Size of Design of Experiment, Number of Samples | 27 | 149 | 287 | 551 |
Minimum Parametric Correlation Value | 0.271 | 0.131 | 0.079 | 0.046 |
Source | Failure Criterion | Failure Rate = 1 in 104 | Failure Rate = 1 in 105 | Failure Rate = 1 in 106 | % Diff, Failure Rate = 1 in 104 | % Diff, Failure Rate = 1 in 105 | % Diff, Failure Rate = 1 in 106 |
---|---|---|---|---|---|---|---|
Lognormal distribution fitted to measured values | Maximum Stress | 0.64 | 0.67 | 0.69 | - | - | - |
Tsai-Wu | 0.75 | 0.78 | 0.81 | - | - | - | |
Hashin | 0.76 | 0.79 | 0.81 | - | - | - | |
Lognormal distribution fitted to response surface values (20 input parameters) | Maximum Stress | 0.64 | 0.66 | 0.68 | 0.0 | −1.5 | −1.4 |
Tsai-Wu | 0.74 | 0.77 | 0.79 | −1.3 | −1.3 | −2.5 | |
Hashin | 0.75 | 0.78 | 0.80 | −1.3 | −1.3 | −1.2 | |
Lognormal distribution fitted to response surface values (15 input parameters) | Maximum Stress | 0.64 | 0.67 | 0.69 | −1.6 | −3.0 | −1.4 |
Tsai-Wu | 0.75 | 0.78 | 0.81 | −2.7 | −2.6 | −3.7 | |
Hashin | 0.76 | 0.79 | 0.81 | −2.6 | −2.5 | −2.5 | |
Lognormal distribution fitted to response surface values (10 input parameters) | Maximum Stress | 0.64 | 0.66 | 0.68 | 0.0 | −1.5 | −1.4 |
Tsai-Wu | 0.74 | 0.77 | 0.79 | −4.0 | −5.1 | −6.2 | |
Hashin | 0.75 | 0.78 | 0.8 | −1.3 | −1.3 | −1.2 | |
Lognormal distribution fitted to response surface values (5 input parameters) | Maximum Stress | 0.63 | 0.65 | 0.68 | 15.6 | 14.9 | 15.9 |
Tsai-Wu | 0.73 | 0.76 | 0.78 | 5.3 | 5.1 | 3.7 | |
Hashin | 0.74 | 0.77 | 0.79 | 2.6 | 2.5 | 2.5 |
Diameter (mm) | Thickness (mm) | |
---|---|---|
Base Case | 100–150 | 0.18–0.22 |
Larger Size | 75–175 | 0.14–0.26 |
Extremely Larger Size | 50–200 | 0.10–0.30 |
Source | Failure Criterion | Failure Rate = 1 in 104 | Failure Rate = 1 in 105 | Failure Rate = 1 in 106 | % Diff, Failure Rate = 1 in 104 | % Diff, Failure Rate = 1 in 105 | % Diff, Failure Rate = 1 in 106 |
---|---|---|---|---|---|---|---|
Lognormal distribution fitted to measured values | Maximum Stress | 0.64 | 0.67 | 0.69 | - | - | - |
Tsai-Wu | 0.75 | 0.78 | 0.81 | - | - | - | |
Hashin | 0.76 | 0.79 | 0.81 | - | - | - | |
Lognormal distribution fitted to base case response surface values | Maximum Stress | 0.64 | 0.66 | 0.68 | 0.0 | −1.5 | −1.4 |
Tsai-Wu | 0.74 | 0.77 | 0.79 | −1.3 | −1.3 | −2.5 | |
Hashin | 0.75 | 0.78 | 0.80 | −1.3 | −1.3 | −1.2 | |
Lognormal distribution fitted to larger size response surface values | Maximum Stress | 0.63 | 0.65 | 0.67 | −1.8 | −1.9 | −2.1 |
Tsai-Wu | 0.75 | 0.78 | 0.81 | −0.1 | −0.1 | −0.3 | |
Hashin | 0.75 | 0.77 | 0.80 | −1.2 | −1.4 | −1.5 | |
Lognormal distribution fitted to extremely large response surface values | Maximum Stress | 0.63 | 0.66 | 0.67 | −1.4 | −1.8 | −2.1 |
Tsai-Wu | 0.80 | 0.83 | 0.85 | 5.7 | 5.6 | 5.6 | |
Hashin | 0.76 | 0.79 | 0.81 | 0.6 | 0.4 | 0.2 |
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Xing, Y.; Xu, W.; Buratti, V. Using the Kriging Response Surface Method for the Estimation of Failure Values of Carbon-Fibre-Epoxy Subsea Composite Flowlines under the Influence of Stochastic Processes. Designs 2022, 6, 1. https://doi.org/10.3390/designs6010001
Xing Y, Xu W, Buratti V. Using the Kriging Response Surface Method for the Estimation of Failure Values of Carbon-Fibre-Epoxy Subsea Composite Flowlines under the Influence of Stochastic Processes. Designs. 2022; 6(1):1. https://doi.org/10.3390/designs6010001
Chicago/Turabian StyleXing, Yihan, Wenxin Xu, and Valentina Buratti. 2022. "Using the Kriging Response Surface Method for the Estimation of Failure Values of Carbon-Fibre-Epoxy Subsea Composite Flowlines under the Influence of Stochastic Processes" Designs 6, no. 1: 1. https://doi.org/10.3390/designs6010001