Polynomial Chaos Expansion Model Research Articles

Probabilistic-based burst failure mechanism analysis and risk assessment of pipelines with random non-uniform corrosion defects, considering the interacting effects

Corrosion defects were primary causes for pipeline burst failures. Traditional methodologies simplified the realistic irregular-shaped defects as uniform ones, ignoring the effects of random morphologies on failure behaviours, which caused deviations in remaining strength pb estimation and reliability analysis. Addressing this issue, through coupling random field, non-linear finite element analysis and Monte-Carlo Simulation, an integrated methodology was developed to describe failure behaviours of pipelines with random defects. The failure mechanism and reliability analysis were performed. The pbs for random defects exhibited significant variabilities, of which the probability distribution patterns altered with different corrosion depths attributed to multiple failure modes. The main value of pb for random defects was lower than that of uniform defects, especially for deeper corrosion and interacting defects. At maximum, this reduction attained over 28%; The spatial variation gradient of remaining wall thickness dominated failure developments, based on which three failure modes were revealed; The wall thickness, corrosion degree and interacting effects were influential on failure probabilities Pfs. An average failure risk of Pf over 0.6 would be suffered for employing traditional methodologies. Polynomial chaos expansion models were fitted to estimate pb from corrosion morphologies, showing reasonable accuracies. Besides, the performance variations with different cases were discussed.

Uncertainty quantification in low-probability response estimation using sliced inverse regression and polynomial chaos expansion

For wave energy converters (WECs), wind turbines, etc., estimation of response extremes over a selected exposure time is important during design. Sources of uncertainty arising from background slowly-varying environmental conditions and from shorter time-scale fluctuations in ocean winds, turbulence, etc. must all be considered. Together, these different sources can comprise a high-dimensional vector of stochastic variables (often on the order of hundreds or thousands). To accurately propagate the influence of these uncertainty sources to model outputs, conventional surrogate model building approaches such as polynomial chaos expansion (PCE), stochastic collocation, low-rank tensor approximations, etc. must consider dimension reduction. We explore the use of sliced inverse regression (SIR) combined with polynomial chaos expansion. SIR first reduces the original high-dimensional problem to a low-dimensional one; then, an optimal polynomial PCE model is proposed and applied on “effective” components in the low-dimensional space. SIR-PCE can mitigate the curse of dimensionality. It is employed here in the prediction of the long-term extreme response of offshore structures; it is demonstrated using classical benchmark analytical functions as well as offshore applications including extreme waves and the response of a wave energy converter. Efficiency and accuracy gains over Monte Carlo simulation and other methods in literature are found.

Multi-dimensional population balance modeling of sonocrystallization of pyrazinamide with systematic estimation of kinetic parameters based on uncertainty and sensitivity analyses

Application of ultrasound has several beneficial effects on both crystallization process and product crystal properties. This work focuses on experimental study and population balance modeling of ultrasound-assisted unseeded batch cooling crystallization of an important polymorphic pharmaceutical. The δ-polymorph of pyrazinamide, a drug for mycobacterium tuberculosis, exhibits plate-like morphology, and such crystals are best described by atleast two dimensions: length and width. A multi-dimensional population balance model for batch cooling sonocrystallization of pyrazinamide from its 1,4-dioxane solution is developed and validated. A series of experiments are performed with different ultrasound amplitudes (silent, 10%, and 50%) to study the effect of ultrasound on nucleation rate, crystal size, and polymorphism. The ultrasound is applied till the point of nucleation and an enhanced nucleation rate is introduced in the model in the presence of ultrasound irradiation. The various kinetic parameters are estimated following a systematic methodology based on uncertainty quantification and sensitivity analysis using experimental observations related to solute concentration and crystal size distribution. The Polynomial Chaos Expansion (PCE) is used to quantify the uncertainties in model prediction due to kinetic parameter uncertainties and sensitivity analysis is performed using total order Sobol’s sensitivity indices derived from PCE model. The kinetic parameters are estimated in two steps. First, a global search technique is used to simultaneously estimate all the kinetic parameters. Next, the less sensitive parameters are kept fixed and the more significant parameters are fine-tuned using a gradient-based local optimizer. Out of total nine kinetic parameters, only four parameters appeared to be significant for both silent and sonicated cases. The high-resolution finite volume scheme is used to solve the multi-dimensional population balance model and the simulation results agree very well with experimental data related to both concentration and crystal size distributions. The results revealed that compared to silent cases, sonication increased the nucleation rate by 102 times and 104 times for 10% and 50% amplitude, respectively. The growth rate was also enhanced significantly with increase in ultrasonic amplitude. All the experiments yielded the desired δ-polymorph of pyrazinamide.

Application of polynomial chaos expansion in sensitivity analysis of towed cable parameters of the underwater towing system

The key to achieving the optimal design of towed cables, maintaining numerical simulation accuracy, and achieving precise control of the towed body lies in sensitivity analysis. However, the traditional global sensitivity analysis method presents challenges such as high calculation costs and low accuracy. To address these issues, this paper introduces polynomial chaos expansion (PCE) to quantitatively analyze the impact of uncertainties in physical and environmental parameters on the position and attitude of the towed cable. Latin hypercube sampling is employed to obtain sample sets of input parameters, and these samples are applied to the lumped mass method to calculate the end position coordinates of the towed cable, which serves as the output response. PCE is utilized to quantitatively compute the Sobol global sensitivity index of the towed cable parameters. The accuracy of the PCE model is verified, and the optimal degree of basis functions is selected using the bias-variance trade-off. The advantages of PCE are demonstrated by comparing it with the Monte Carlo and Morris methods. The results indicate that PCE accurately calculates the global sensitivity index of towed cable parameters even with a limited sample size. Under the condition of a fixed cable length, the position and attitude of the towed cable are sensitive to the current rate, liquid density, cable diameter, normal drag coefficient, and specific gravity. The feasibility and efficiency of PCE applied to the sensitivity analysis of towed cable parameters is verified, and recommendations for the engineering application of towed cables are summarized.

HPOSS: A hierarchical portfolio optimization stacking strategy to reduce the generalization error of ensembles of models.

Surrogate models are frequently used to replace costly engineering simulations. A single surrogate is frequently chosen based on previous experience or by fitting multiple surrogates and selecting one based on mean cross-validation errors. A novel stacking strategy will be presented in this paper. This new strategy results from reinterpreting the model selection process based on the generalization error. For the first time, this problem is proposed to be translated into a well-studied financial problem: portfolio management and optimization. In short, it is demonstrated that the individual residues calculated by leave-one-out procedures are samples from a given random variable ϵi, whose second non-central moment is the i-th model's generalization error. Thus, a stacking methodology based solely on evaluating the behavior of the linear combination of the random variables ϵi is proposed. At first, several surrogate models are calibrated. The Directed Bubble Hierarchical Tree (DBHT) clustering algorithm is then used to determine which models are worth stacking. The stacking weights can be calculated using any financial approach to the portfolio optimization problem. This alternative understanding of the problem enables practitioners to use established financial methodologies to calculate the models' weights, significantly improving the ensemble of models' out-of-sample performance. A study case is carried out to demonstrate the applicability of the new methodology. Overall, a total of 124 models were trained using a specific dataset: 40 Machine Learning models and 84 Polynomial Chaos Expansion models (which considered 3 types of base random variables, 7 least square algorithms for fitting the up to fourth order expansion's coefficients). Among those, 99 models could be fitted without convergence and other numerical issues. The DBHT algorithm with Pearson correlation distance and generalization error similarity was able to select a subgroup of 23 models from the 99 fitted ones, implying a reduction of about 77% in the total number of models, representing a good filtering scheme which still preserves diversity. Finally, it has been demonstrated that the weights obtained by building a Hierarchical Risk Parity (HPR) portfolio perform better for various input random variables, indicating better out-of-sample performance. In this way, an economic stacking strategy has demonstrated its worth in improving the out-of-sample capabilities of stacked models, which illustrates how the new understanding of model stacking methodologies may be useful.

A discrete adjoint framework coupled with adaptive PCE for robust aerodynamic optimization of turbomachinery under flow uncertainty

Flow uncertainty is commonly encountered in turbomachinery. To mitigate the negative effects caused by the flow uncertainty, a framework coupled with adaptive polynomial chaos expansion (PCE) and discrete adjoint method for robust aerodynamic design optimization (RADO) of turbomachinery blades is developed. The optimization framework relies on gradient algorithms, which effectively avoids “dimensional curse” problem. Adaptive PCE method improves the efficiency of uncertainty quantification and only once PCE model need to be constructed in per optimization step, making the framework highly efficient. The PCE matrix and chain rule are employed to determine the gradient of the RADO objective function. This framework is employed for optimizing turbine blade, considering both inviscid and viscous flow. Monte Carlo simulation and finite difference PCE are used to validate the reliability of uncertainty quantification and gradient propagation methods. A comparative analysis of the optimization results obtained by both RADO and deterministic aerodynamic design optimization (DADO) is demonstrated. The higher robustness brought by RADO indicates the practicality of this framework in robust optimization of turbomachinery blades.

Analysis of Dynamic Mesh Stiffness and Dynamic Response of Helical Gear Based on Sparse Polynomial Chaos Expansion

This paper presents an efficient method for obtaining the dynamic mesh stiffness and dynamic response of a helical gear pair. Unlike the traditional dynamic model that utilizes a time-dependent sequence, the mesh stiffness using the presented method is updated according to the gear displacement vector at each sub-step of the numerical calculation. Three-dimensional loaded tooth contact analysis (3D LTCA) is used to determine the mesh stiffness, and a surrogate model based on sparse polynomial chaos expansion (SPCE) is proposed to improve the computational efficiency, which is achieved by reducing the number of coefficients in the polynomial chaos expansion (PCE) model though a quantum genetic algorithm. During the calculation, the gear displacement vector at each sub-step is converted into the changes in center distance, misalignment angle, and mesh force, which are then introduced into the SPCE model to update the mesh stiffness for subsequent calculations. The results suggest that the SPCE model exhibits high accuracy and can significantly improve the computational efficiency of the PCE model, making it suitable for dynamic calculations. Upon updating the mesh stiffness during the dynamic calculation, the mesh stiffness declines, the dynamic transmission error (DTE) increases, and the frequency components of the responses change significantly.

Propagation algorithm for hybrid uncertainty parameters based on polynomial chaos expansion

AbstractThis article presents an uncertainty analysis method for systems with hybrid stochastic and fuzzy uncertainty parameters based on polynomial chaos expansion (PCE). Parameters in the system are described by probability boxes, interval numbers, and fuzzy sets, respectively, based on the differences in their limited stochastic knowledge. First, this method transforms the uncertain parameters into standard normal distribution and interval variables through equal probability transformation or ‐cut operations. Second, the Legendre and Hermite polynomials are used as the PCE model's primary functions, and the expansion coefficients are calculated by the Galerkin projection method based on sparse grid numerical integration. Then, the system response bounds under the pre‐defined confidence level can be obtained using a genetic algorithm to solve the optimization problem constructed based on PCE models. Finally, the feasibility and effectiveness of the method are illustrated by taking the tank bi‐directional stabilized system and the double‐pendulum‐controlled system as examples. The numerical results show that the system response bounds obtained by the PCE model optimization algorithm are consistent with the Monte Carlo simulation. Still, the computational efficiency is much higher. The proposed method effectively combines fuzzy sets and probability boxes and dramatically simplifies the analysis process of uncertain systems. The method exhibits fine precision even in high‐dimensional uncertainty analysis problems.

A fully Bayesian sparse polynomial chaos expansion approach with joint priors on the coefficients and global selection of terms

Polynomial chaos expansion (PCE) is a versatile tool widely used in uncertainty quantification and machine learning, but its successful application depends strongly on the accuracy and reliability of the resulting PCE-based response surface. High accuracy typically requires high polynomial degrees, demanding many training points especially in high-dimensional problems through the curse of dimensionality. So-called sparse PCE concepts work with a much smaller selection of basis polynomials compared to conventional PCE approaches and can overcome the curse of dimensionality very efficiently, but have to pay specific attention to their strategies of choosing training points. Furthermore, the approximation error resembles an uncertainty that most existing PCE-based methods do not estimate. In this study, we develop and evaluate a fully Bayesian approach to establish the PCE representation via joint shrinkage priors and Markov chain Monte Carlo. The suggested Bayesian PCE model directly aims to solve the two challenges named above: achieving a sparse PCE representation and estimating uncertainty of the PCE itself. The embedded Bayesian regularizing via the joint shrinkage prior allows using higher polynomial degrees for given training points due to its ability to handle underdetermined situations, where the number of considered PCE coefficients could be much larger than the number of available training points. We also explore multiple variable selection methods to construct sparse PCE expansions based on the established Bayesian representations, while globally selecting the most meaningful orthonormal polynomials given the available training data. We demonstrate the advantages of our Bayesian PCE and the corresponding sparsity-inducing methods on several benchmarks.

A novel sparse polynomial chaos expansion technique with high adaptiveness for surrogate modelling

In this paper, a novel adaptive modelling framework for sparse polynomial chaos expansion is proposed, which can automatically determine adequate truncation degree and training sample set simultaneously. Moreover, the curse of dimensionality issue in polynomial chaos expansion, which generally arises in dealing with high input dimension or large truncation degree, can be alleviated to a large extent. In this framework, a new basis selection strategy, which leverages basis expansion, pruning and refinement, is pursued to adaptively select the polynomial terms of proper degree during the modelling process. Besides, an outstanding sequential sampling strategy is adopted to collect samples of high quality and in relatively small quantity for training polynomial chaos expansion model, and a sparse representation method, Bayesian compressive sensing, is employed for regression calculation. To reconcile the sequential sampling and adaptive basis selection in a consistent framework, a stability evaluation process which works in parallel with the sequential sampling process is performed. The performance of the proposed adaptive modelling framework is evaluated on two benchmark functions and a physical model through comparison with an existing adaptive polynomial chaos expansion modelling technique, a sequential sampling-only approach, and several basis adaptivity strategies. Results demonstrate that the proposed method has high adaptiveness in building surrogate models for various problems. It outperforms the existing adaptive polynomial chaos expansion modelling technique in terms of modelling precision and convergence rate, and it has similar performance with the sequential sampling-only method while lessening the burden in regression calculation and enabling adaptive determination of the truncation degree. Limitations of the proposed method are also summarized after comparing it with various basis adaptivity strategies.

Hybrid Framework for Safety Design of Human–Rail Vehicle Transportation System Using Stochastic Approach and Optimization

The human–rail vehicle transportation system safety design is complicated given that the complexity of the multilevel system with parameter uncertainties propagating from the vehicle structure (primary collision) to the interior human compartment (secondary collision). This study establishes a hybrid framework incorporating a stochastic approach and an integrated optimization strategy to improve train crashworthiness and reduce passenger crash injuries. The stochastic approach utilizes adaptive sparse polynomial chaos expansion models and variance-based sensitivity indices to evaluate the statistic characteristics of system responses and quantify the contribution ranking of uncertain parameters to response variations. The optimization strategy integrating the evolutionary algorithm and the multi-criteria decision making (MCDM) is proposed to solve the non-uniqueness of Pareto optimal solutions. In the optimization process, the modified DEMATEL–ANP method with interval type-2 fuzzy sets is developed to deal with vague linguistic judgments for the importance sequence of human injury responses. The q-rung orthopair trapezoidal fuzzy uncertain linguistic sets–TOPSIS method is established to address hesitant linguistic evaluations for the Pareto front and select the final optimal solution. Compared with the initial design, the driver Abbreviated Injury Scale (AIS) 3+ joint injury probability is reduced from 67.08% to 14.17% after optimization. Results prove that the proposed framework is a practical tool for improving the passive safety of railway industry.

Structure damage identification in dams using sparse polynomial chaos expansion combined with hybrid K-means clustering optimizer and genetic algorithm

Structural damage identification plays a crucial role in structural health monitoring. In this study, a novelty method for structural damage identification is developed, which employs an advanced surrogate modelling technique to drive a new hybrid optimization strategy, namely a combination of K-means clustering optimizer and genetic algorithm (HKOGA). The core of this method is using the reliable sparse polynomial chaos expansion model as a cost-effective substitute for the computationally expensive structural finite element models, thus greatly improving the efficiency of the optimization strategy in finding the optimal value of the objective function. To evaluate the performance of this hybrid optimization strategy, seven optimization algorithms are selected and compared with it for 23 classical benchmark functions, and the comparative results show that the HKOGA has the best performance. Taking a small-scaled laboratory dam as an example, the efficiency and reliability of the proposed method to cope with the problems concerning finite element model updating and structural damage identification are explored. Two important findings are as follows. (i) For finite element model updating, compared to the conventional method based on iterative optimization, the proposed method improves computational efficiency by a factor of 59 while maintaining computational accuracy. (ii) For structural damage identification, leaving aside the huge leap in computational efficiency, the HKOGA has a faster convergence rate, stronger robustness, and higher accuracy than its sub-algorithm K-means clustering optimizer (KO). The results show that this method can be severed as an extremely efficient and potential tool to identify damage in large and complex structures.

Conjunction Time and Collision Probability Calculation Based on Bayesian Optimization

Collision probability calculation is critical to space situational awareness or space traffic management. To determine the collision probability, the conjunction time needs to be determined first, but both collision probability and conjunction time usually exist large uncertainties. In this paper, a continuous representation of the polynomial chaos expansion (PCE)-based surrogate model is used to describe the uncertainty of space objects along with time. This PCE model facilitates the propagation of sampling points representing orbital uncertainty at any time. A machine learning-based Bayesian optimization method is employed to determine the conjunction time and then the collision probability. A data-driven Gaussian process regression model is built to approximate the Hausdorff distance, which is a distance metric between two space objects with uncertainty and is used as an objective function for Bayesian optimization. The proposed PCE model and Bayesian optimization do not require the Gaussian assumption and provide a general framework to calculate the conjunction time and collision probability. Two numerical experiments are used to show the effectiveness of the proposed algorithm and its close performance to the traditional Monte Carlo sampling-based method.

On the use of sparse Bayesian learning-based polynomial chaos expansion for global reliability sensitivity analysis

Global reliability sensitivity analysis determines the effects of input uncertain parameters on the failure probability of a system. Usually, the global reliability sensitivity analysis can be performed by the conventional Monte Carlo simulation (MCS) approach. However, the MCS approach requires a large number of model evaluations which limits MCS to apply for realistic problems. For that reason, a sparse polynomial chaos expansion (PCE) model is used in the present work based on a variational Bayesian (VB) inference. More specifically, the PCE coefficients are computed by the VB inference and the important terms in the PCE basis are selected by an automatic relevance determination (ARD) approach. Therefore, the VB inference is fully connected with the ARD approach. Global reliability sensitivity analysis is performed for some numerical examples using the sparse PCE model and all the results are compared with the MCS and the least angle regression (LARS)-based PCE model predicted results. The 95% confidence interval is also obtained by the VB approach to measure the prediction uncertainty. It is found that a very good result is obtained with the sparse PCE model using much less number of model evaluations as compared to the MCS approach. The accuracy of obtaining the PCE coefficients is higher by the VB inference than the LARS approach. Further, the required number of terms is small for the VB-PCE model and therefore, the number of PCE coefficients is also small.

Uncertainty design and optimization of a hybrid rocket motor with mixed random-interval uncertainties

Uncertainties, which exist widely in the design of aerospace systems, have a great impact on system response and may lead to design failures. Both the probability and non-probability uncertainties exist in real engineering cases. Thus, it is imperative to incorporate the mixed uncertainties into the aerospace systems design and optimization. Present study proposed an efficient design optimization method considering the random and interval uncertainties to conduct the uncertainty analysis and optimize the design of a hybrid rocket motor with mixed uncertainties. A polynomial chaos expansion model was selected to deal with the random uncertainties, and an interval analysis method based on the first order Taylor expansion was adopted to calculate the response bounds of interval uncertainties. A simple structure was firstly used to verify the efficiency and accuracy of the proposed method. The random-interval analysis method was very efficient in searching the optimal results, showing considerable computational cost savings. Subsequently, a design model of hybrid rocket motor was introduced and verified by a long-time firing test. In the optimization of the upper-stage hybrid rocket motor, the comparison between uncertainty analysis and traditional methods showed that the design optimization considering the uncertainty in design variables was able to achieve more robust results. Compared with the deterministic scheme, the uncertainty scheme increased the total engine mass by nearly 18%, but the probabilities of all constraints considering parameter fluctuations remained above 0.9. Capable of solving complex problems with mixed uncertainties, this method shows its potential in aerospace engineering applications.

Consistency regularization-based deep polynomial chaos neural network method for reliability analysis

Polynomial chaos expansion (PCE) is a powerful method for building a surrogate model that can be applied to assist reliability analysis. Generally, a PCE model with a higher expansion order is required to obtain an accurate surrogate model for some complex non-linear stochastic systems. However, the high-order PCE increases the number of labeled data necessary for solving the expansion coefficients. To alleviate this problem, this paper proposes a consistency regularization-based deep polynomial chaos neural network (Deep PCNN) method, including the low-order adaptive PCE model (the auxiliary model) and the high-order polynomial chaos neural network (the main model). The expansion coefficients of the main model are parameterized into the learnable weights of the polynomial chaos neural network, realizing iterative learning of expansion coefficients to obtain more accurate high-order PCE models. By unlabeled data, the auxiliary model uses the proposed consistency regularization loss function to assist in training the main model. The consistency regularization-based Deep PCNN method can precisely solve the expansion coefficients with fewer labeled data without reducing the high-order PCE model accuracy compared to the existing PCE methods. Two numerical examples validate the effectiveness of the consistency regularization-based Deep PCNN method, which is then applied to satellite reliability analysis.

3D Numerical Modeling and Quantification of Oblique Wave Forces on Coastal Bridge Superstructures

Simply supported bridges comprise the majority of bridge systems in coastal communities and are susceptible to severe damage from extreme waves induced by storms or tsunamis. However, the effects of oblique wave impacts have been less investigated due to the lack of appropriate numerical models. To address this issue, this study investigates the effects of wave incident angles on coastal bridge superstructures by developing an advanced computational fluid dynamics (CFD) model. Different wave scenarios, including wave height, relative clearance, incident angle, and wavelength are tested. It is found that the maximum wave forces in the horizontal and longitudinal directions could reach 1901 and 862 kN under extreme conditions, respectively, destroying bearing connections. Three surrogate models, i.e., the Gaussian Kriging surrogate model, the Artificial Neural Network (ANN), and the Polynomial Chaos Expansion (PCE), are established by correlating the wave parameters with the maximum wave forces. Through comparisons among the three surrogate models, it is found that the 3-order PCE model has better performance in predicting loads in vertical and horizontal directions, while the ANN model is more suitable for results in the longitudinal direction. This study contributes to the optimized design of coastal bridges and also offers an opportunity for future studies to investigate hazard damage-mitigation measures.

A novel robust aerodynamic optimization technique coupled with adjoint solvers and polynomial chaos expansion

Uncertainty is common in the life cycle of an aircraft, and Robust Aerodynamic Optimization (RAO) that considers uncertainty is important in aircraft design. To avoid the curse of dimensionality in surrogate-based optimization, this study proposes an adjoint RAO technique called “R-Opt”. Polynomial Chaos Expansion (PCE) is coupled with the R-Opt technique to quantify uncertainty in the responses of the target (including its mean and standard deviation). Only one process of PCE model construction is required in each iteration, and the gradients of uncertainty can be inferred via chain rules. The proposed method is more efficient than prevalent methods, and avoids the problem of a disagreement over the best PCE basis from among a number of PCE models (especially in case of sparse PCE). It also supports the application of sparse PCE. Two benchmark tests and two airfoil cases were used to verify R-Opt, and the optimal solutions were deemed to be robust. It improved the mean aerodynamic performance and reduced the standard deviation of the target.

An adaptive dimension-reduction method-based sparse polynomial chaos expansion via sparse Bayesian learning and Bayesian model averaging

Polynomial chaos expansion (PCE) is a popular surrogate method for stochastic uncertainty quantification (UQ). Nevertheless, when dealing with high-dimensional problems, the well-known curse of dimensionality and overfitting pose great challenges to the PCE. This study proposes an adaptive sparse learning method based on the automatic relevance determination (ARD) and the Bayesian model averaging (BMA) to solve this problem. Firstly, we construct the sparse PCE model in the framework of sparse Bayesian learning. Secondly, the dimension-reduction method (DRM) is used to reduce the size of candidate PCE bases. Different from the common univariate and bivariate DRMs, this study considers the high-dimensional components in an adaptive way. Thirdly, we propose a novel ARD method based on the analytical B-LASSO to prune the candidate PCE bases. The pruned PCE is regarded as an initial model, and then the BMA is used to refine the high-fidelity sparse PCE. Compared with the direct use of the noninformation prior, the intervention of the pruned PCE model can enhance the rationality of the prior, and improve the accuracy of the posterior estimation. We illustrate the validity of the proposed method from different angles using five classical numerical examples and solve one practical engineering problem. The results indicate that the proposed method is a good choice for UQ under limited design samples.

A stochastic approach to estimate intraocular pressure and dynamic corneal responses of the cornea

IntraOcular Pressure (IOP) is one of the most informative factors for monitoring the eye-health. This is usually measured by tonometers. However, the outputs of the tonometers depend on the physical and geometrical properties of the cornea. Therefore, the common practice is to develop a numerical model to generate some correction factors. The main challenge here is the accuracy and efficiency of a numerical model in predicting the IOP and Dynamic Corneal Response (DCR) of each patient. This study addresses this issue by developing a two-step surrogate model based on adaptive sparse Polynomial Chaos Expansion (PCE) for fast and accurate prediction of the IOP. In this regard, first, an FE model of the cornea has been developed to predict the DCR parameters. This FE model has been replaced with a PCE-based surrogate model to speed up the simulation step. The uncertainties in the geometry and material model of the cornea have been propagated through the surrogate model to estimate the distributions of the DCR parameters. In the second step, the combination of DCR parameters and the input parameters provide a proper parameter space for developing an efficient data-driven PCE model to predict the IOP. Moreover, sensitivity analysis by using PCE-based Sobol indices has been performed. The results demonstrate the accuracy and efficiency of the proposed method in predicting the IOP. Sensitivity analysis revealed that IOP measurement was influenced mostly by deflection amplitude and applanation time. The analysis indicates the importance of the interactions between the parameters.