Your intelligence in this field is tremendous and you have been an excellent resource throughout my research. Basu, you have been my strongest advocate throughout my post graduate experience and it has been invaluable.
INTRODUCTION
- Motivation
- Summary
- Calibration, Selection, and Uncertainty Quantification for Permanent
- Surrogate Model Construction
- Uncertainty Propagation for Mechanistic Empirical Pavement Design
- Risk-Based Design for M-E Design of Flexible Pavement Design
- Organization of Dissertation
Accurate and efficient methods for reliability analysis are needed for uncertainty management in M-E pavement design. Through these four main objectives, this dissertation presents a comprehensive framework for managing uncertainty in flexible pavement design.
BACKGROUND
AASHTO MEPDG
- Terminal International Roughness Index (IRI) (Smoothness)
- Asphalt Concrete Layer Fatigue Cracking (Alligator Cracking)
- Permanent Deformation
The mechanical properties are calculated incrementally and accumulated over the design life of the pavement. Fatigue cracks in the AC layer of the pavement structure are estimated by the bottom-up (crocodile cracks) and surface-down (alongside cracks) fatigue stress models.
Accelerated Pavement Testing (APT) & WesTrack
In equation II.12, 𝜀𝑝 and 𝜀𝑟 are plastic and elastic deformation, respectively, and the values k and 𝛽 are regression coefficients and calibration factors. Traffic experiments were conducted on WesTrack using four triple trailer combinations (FIGURE II.2) that use driverless vehicle technology to ensure consistency of speed and driving performance as well as jobsite safety.
Methods for Surrogate Modeling
Surrogate modeling methods vary in computational complexity, which can affect the accuracy of the model compared to the true function. In addition, functions with many parameters can become complex, and the improvement in the computational speed of the model compared to the true function is reduced.
Analytical Reliability Methods for Pavement Design
The result is that the mean is calculated by evaluating the limit state function at the means of the random dependent variables. The reliability estimates resulting from different but equivalent expressions for the limit state function may be different using these methods.
Reliability Based Design Optimization
- Optimization Problem Definitions
- RBDO Problem Formulations
𝑃2 = min𝑑{max. Methods such as the direct FORM method can lead to inaccuracies if the approximation to the shape of the limit state is poor.
Discussion
EGRA evaluates the function on a small number of samples, constructs a surrogate model for the function, and solves an auxiliary optimization problem by finding the point of maximum expected feasibility. The final surrogate model can then be used to make reliability predictions for the true function. A single loop method improves the efficiency of the nested method by using a surrogate model that evaluates reliability across a domain rather than at individual candidate points.
The sequential formulation improves the one-loop process by intermittently improving the accuracy of the surrogate model to include verification in the iterative process.
CALIBRATION, SELECTION, AND UNCERTAINTY QUANTIFICATION
Introduction
A third theory presented in this dissertation assumes that permanent deformation is best described by a model that combines both shear and axial theories. Six permanent deformation predictive models are investigated in this dissertation to determine the most accurate predictive model for use in flexible pavement design. First, permanent deformation is a failure mode that affects driver safety and contributes significantly to design life maintenance cost.
The behavior of such binders can increase pavement life, which may be contrary to the behavior predicted by current permanent deformation models.
Background
The MEPDG predictive model for predicting permanent deformations of asphalt pavements is derived from empirical data obtained from LTPP data and linear elastic analysis of asphalt layers. The calibration procedure for the permanent deformation model in the asphalt layer is performed by minimizing the error between the actual and predicted capacity using Equation III.1, where a plastic elastic analysis program is used to determine the elastic deformation. The permanent strain in the asphalt concrete layer for these models is based on the assumption that shear strain governs the strain.
The WesTrack Level 1-B equation for permanent deformation (rdHMA), derived by the regression procedure described previously, is defined as the product of a regression coefficient (κ) and the permanent (inelastic) shear strain (γi).
Construction of Predictive Models
- MEPDG Rutting Models
- Shear Theory Model
- Weighted Models
Equation III.7 identifies the optimal calibration factors that minimize the sum of squared residuals for the first 12 months of the WesTrack experiment for all N test sections. The optimization is considered converged if the difference between the sums of squares of the residuals between iterations is less than 0.01. The combination of NCHRP and locally calibrated models shows that both models contribute information to rut prediction.
This is expected because the locally calibrated model is a subset of the parameter-based model.
Model Validation and Comparison
The Bayes factor is calculated as a ratio of the probability of observing the validation data, conditional on the null and alternative hypotheses, as shown in Eq. The null hypothesis states that the behavior of the experimental data is well represented by the predictive model in the numerator. The alternative does not support the model in the numerator as a good predictor of the experimental data compared to the model.
The means and standard deviations of the normal PDFs are assumed to be functions of the models under investigation.
Model Validation and Comparison Results
The locally calibrated model has a lower mean squared error and a higher coefficient of determination than the national (MEPDG) model. The Bayes factor for the locally calibrated model indicates a decisive degree of support for the model over the national model. The Bayes factor for the parameter calibrated model indicates decisive support of this model over the national and local models.
The weights associated with each weighted model (shown in TABLE III.2), in conjunction with the metrics presented in FIGURE III.3, lead to the conclusion that the axial strain model and the shear load model contribute significantly to the prediction accuracy.
Conclusion
The results presented here indicate that a model combining both theories reduces model shape errors and improves accuracy in predictions of permanent deformation. The models presented indicate that calibrating the MEPDG model to incorporate local factors, site-specific factors, and mixing parameters is a critical step in accurate pavement predictions. Presented here is a procedure to incorporate these parameters through the calibration factors already present in the MEPDG software, resulting in a model that begins to capture the behavior of the coating with respect to both mechanistic theories.
A well-trained surrogate model can accurately simulate the MEPDG design equations and improves the computational time required for single design evaluations.
SURROGATE MODEL INITIALIZATION: VARIABLE SELECTION
- Introduction
- Surrogate Model Construction: Initialization
- Quantity of Training Point Parameters (N D )
- Location of Training Points for Evaluation of Selection Processes
- Selection Process Methods
- ANOVA
- Correlation Matrix
- Gaussian Process Model Length-Scale Factors
- Selection Process Comparison
- Results
- Sensitivity Analysis: Quantity of Training Point Parameters
- Method for Selection of Training Point Parameters
- Conclusion
The number of parameters for each training point refers to the dimensions of the model's input. Construction of the surrogate model also requires training values provided as a matrix of dimensions [NTP x NY], where each training point input has a paired training value (output) for each prediction model (Y). The construction of a surrogate model requires selection of the number of parameters for each training point (ND).
The construction of a well-trained surrogate model requires an appropriate selection of the amount of training point dimensions (ND) in an efficient manner.
UNCERTAINTY PROPAGATION WITH SURROGATE MODELS
Introduction
Understanding the importance of the impact of uncertainty due to these different sources is critical in validating predictive models of MEPDG distress and for reliable pavement design. In this chapter, quantification of important sources of uncertainty is performed, a method for uncertainty propagation is demonstrated, sensitivity of predicted performance models to sources of uncertainty is included, and a reliability analysis is demonstrated and discussed. Sensitivity analysis evaluates the sources of uncertainty and their impact on the accuracy of the predicted behavior.
The sensitivity analysis represents the contribution to the overall variance in the predicted values from each source of uncertainty.
Sources of Uncertainty
- Input Parameter Variability
- GP Model Predictive Uncertainty
- MEPDG Predictive Uncertainty
The standard deviation can be calculated from the standard deviation of the residuals between the true and surrogate functions. The standard deviation of the prediction model is obtained by regression analysis between measured and predicted values used in calibrating the MEPDG prediction functions. The distribution of uncertainty is treated as a normal distribution with a mean of zero, in accordance with the method developed in the MEPDG.
The standard deviation of the MEPDG is calculated as the ratio of the margin of safety to the inverse of the standard normal cumulative distribution function (CDF) evaluated by the probability of failure calculated by the MEPDG.
Uncertainty Propagation and Sensitivity Analysis
- Uncertainty Propagation Method
- Numerical Experiment
The same 170 design input sets were used in the training and calibration of the GP model. This confirms that the GP model's uncertainty is negligible compared to that of the MEPDG uncertainty. These results indicate that the uncertainty of the input parameters and GP is less significant compared to the predictive uncertainty in the MEPDG.
The importance of input uncertainty is in changing the expected outcome.
Conclusion
These results demonstrate the accuracy and validity of using surrogate models for pavement reliability analysis while harnessing the predictive power of the MEPDG. By combining the effects of the three sources of uncertainty, this chapter presented a unified approach to uncertainty analysis. By using surrogate models, the barrier of the computational cost of the MEPDG was eliminated for MCS-based reliability analysis.
The computational cost of the M-E design process can be offset by the efficiency of analytical methods, which typically require a small number of function evaluations.
ANALYTICAL RELIABILITY METHODS FOR MECHANISTIC-EMPIRICAL
- Introduction
- Distress Models for M-E Pavement Design
- Distributions of the Random Variables
- Numerical Results
- Implementation with the AASHTO MEPDG
- AASHTO MEPDG Prediction Equations
- Conclusion
A large number of simulation points must be evaluated to obtain a true representation of the performance function. Investigation of these analytical reliability methods requires an understanding of the disturbance models used in flexible pavement design. The AMV analysis is also evaluated with the surrogate model to compare the accuracy of the AMV method with the MCS analysis.
The direction cosines (𝛼𝑖) in equation VI.15 are evaluated based on the average of the input parameters.
LRFD AND CORRECTION FACTORS FOR ROUTINE RELIABILITY
- Introduction
- Inverse FORM
- Calculation of Load and Resistance Factors
- Distributions of the Random Variables
- GP Model Construction and Verification
- Calculation and Discussion of Load and Resistance Factors
- Correction Factors for AMV
- Conclusion
Once the u* point is determined, it is transformed into the x space to find the design values of the random variables. In calculating the load and resistance factors and design displacements, the random variable probability distributions are assumed to be normal and independent with means and standard deviations as shown in TABLE VII.1. The mean of the R-squared values for each distress state is high (with two near unity), indicating that the GP prediction is closely correlated with the MEPDG.
The results show that the choice of verification points will significantly affect the level of support for the model.
RISK-BASED DESIGN OPTIMIZATION METHOD UTILIZING M-E DESIGN
- Introduction
- Selection of a Surrogate Model Type
- Surrogate Model Construction: Adaptive Training Point Selection Process
- Quantity of Training Points (N TP )
- Verification of the Surrogate Model
- Results
- GP Construction and Verification
- RBDO Solution
- Conclusion
The improvement of the GP through the exploration process is determined by the variance of a set of randomly selected verification points in the domain space. Predictions for GP can be compared with results from MEPDG to determine the validity of the surrogate model at different points from the training data (i.e. validation points). Although visual inspection of the GP variance plots implies a potential minimum number of training points, verification of the GP against the actual function is also necessary.
The means of the random variables used in the GP model are presented in TABLE VIII.2.
CONCLUDING REMARKS
Specifically, it was found that a GP model accurately mimics M-E pavement design models and minimizes computational cost. A training point selection framework using a correlation matrix between input parameters and predicted performance was found to be an effective method for selecting the set of training point parameters for the GP model. Analytical reliability methods have also been shown to provide accurate reliability estimates in a computationally efficient manner.
Analytical reliability methods, especially FORM and AMV, have proven to be powerful reliability methods that allow for accurate and computationally efficient estimates.
FUTURE WORK
- Pay Factors and Performance Related Specifications
- Genetic Algorithms for GP Parameter Selection Process
- Additional MEPDG Distress Models & Various Pavement Structures
- Optimization Routine Improvement for Model Calibration
The procedures presented in this dissertation can be applied to any of the disturbance models. Future work also includes sensitivity analysis of the slope terms included in the parameter-calibrated model to investigate the influence of shear-based mixture properties. A quadratic model can also be constructed and included in sensitivity analysis to examine the impact of interaction terms and higher order terms.
Implementation of any of the aforementioned recommendations can improve the accuracy of the prediction of permanent deformation performance in flexible pavement structures.
