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The assessment of casualty risks associated with re-entry necessitates a comprehensive analysis of trajectories and the examination of pertinent safety-related quantities such as the ground impact area and ground reaching velocity. In practical scenarios, the presence of uncertainty in input conditions introduces variability in safety-related quantities. Consequently, employing stochastic re-entry trajectory analysis becomes crucial in overcoming the limitations of conventional deterministic analyses. Conducting sensitivity assessments during the break-up phase is imperative to gain more insights into how to manage the variability of safety-related measures. Therefore, this paper conducted a surrogate-based global sensitivity analysis and employed explainability machine learning techniques to unveil the complexities of the relationship between input uncertainty conditions and three key measures: ground-reaching velocity, falling range, and falling time, with the object of interest being the Apollo-type capsule. A three-step polynomial chaos expansion-based strategy was devised to efficiently approximate the discontinuous relationships. The results show that the relationship is characterized by severe discontinuity that separates two modes: low- and high-ground reaching velocity caused by the presence of two distinct trim points, with precautionary measures that should be taken to prevent the occurrence of the latter. From this set of procedures, three key input conditions that significantly affect the safety-related measures were identified, namely, the altitude, pitch rate, and path angle of the capsule during the breakup. Subsequently, explainability techniques were utilized to give suggestions on how to control the input variability and avoid the high ground reaching velocity mode, aiming to achieve more dependable predictions for the three safety parameters mentioned earlier. • Performed surrogate-based sensitivity analysis for stochastic re-entry trajectory. • Utilized explainability machine learning techniques to decipher relationships. • Identified key input conditions influencing safety measures for Apollo-type capsules. • Contributed insights to minimize hazardous re-entry modes and mitigate risks. • Uncovered complex discontinuity in response surface with explainable surrogate models. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the capability of the universal Kriging (UK) model for single-objective global optimization applied within an efficient global optimization (EGO) framework. We implemented this combined UK-EGO framework and studied four variants of the UK methods, that is, a UK with a first-order polynomial, a UK with a second-order polynomial, a blind Kriging (BK) implementation from the ooDACE toolbox, and a polynomial-chaos Kriging (PCK) implementation. The UK-EGO framework with automatic trend function selection derived from the BK and PCK models works by building a UK surrogate model and then performing optimizations via expected improvement criteria on the Kriging model with the lowest leave-one-out cross-validation error. Next, we studied and compared the UK-EGO variants and standard EGO using five synthetic test functions and one aerodynamic problem. Our results show that the proper choice for the trend function through automatic feature selection can improve the optimization performance of UK-EGO relative to EGO. From our results, we found that PCK-EGO was the best variant, as it had more robust performance as compared to the rest of the UK-EGO schemes; however, total-order expansion should be used to generate the candidate trend function set for high-dimensional problems. Note that, for some test functions, the UK with predetermined polynomial trend functions performed better than that of BK and PCK, indicating that the use of automatic trend function selection does not always lead to the best quality solutions. We also found that although some variants of UK are not as globally accurate as the ordinary Kriging (OK), they can still identify better-optimized solutions due to the addition of the trend function, which helps the optimizer locate the global optimum. [ABSTRACT FROM AUTHOR]

The presence of uncertainties is inevitable in engineering design and analysis, where failure in understanding their effects might lead to the structural or functional failure of the systems. The role of global sensitivity analysis in this aspect is to quantify and rank the effects of input random variables and their combinations to the variance of the random output. In problems where the use of expensive computer simulations is required, metamodels are widely used to speed up the process of global sensitivity analysis. In this paper, a multi-fidelity framework for global sensitivity analysis using polynomial chaos expansion (PCE) is presented. The goal is to accelerate the computation of Sobol sensitivity indices when the deterministic simulation is expensive and simulations with multiple levels of fidelity are available. This is especially useful in cases where a partial differential equation solver computer code is utilized to solve engineering problems. The multi-fidelity PCE is constructed by combining the low-fidelity and correction PCE. Following this step, the Sobol indices are computed using this combined PCE. The PCE coefficients for both low-fidelity and correction PCE are computed with spectral projection technique and sparse grid integration. In order to demonstrate the capability of the proposed method for sensitivity analysis, several simulations are conducted. On the aerodynamic example, the multi-fidelity approach is able to obtain an accurate value of Sobol indices with 36.66% computational cost compared to the standard single-fidelity PCE for a nearly similar accuracy. [ABSTRACT FROM AUTHOR]

Surrogate models are indispensable tools in uncertainty quantification and global sensitivity analysis. Polynomial chaos expansion (PCE) is one of the most widely used surrogate models, thanks to its faster convergence rate compared to Monte Carlo simulation. In some cases, especially for complex problems, analyzing the complexity of the random input–output relationship (e.g., nonlinearity and interactions between input variables) may reveal additional information and useful insight. To that end, this paper introduces the use of Shapley additive explanations (SHAP) to help the explanation of a PCE model. Originating from game theory and machine learning, SHAP computes the contribution of the input variables to the single prediction level. SHAP enables visual inspection of the nonlinearity and interaction between variables from a PCE model. In addition, as an alternative to Sobol indices, SHAP also quantifies the relative importance of the inputs to the output. This paper introduces a procedure to calculate SHAP values from a PCE model without explicitly building multiple PCE models. A fast and exact algorithm that enables the calculation of SHAP for high-dimensional problems is presented. The usefulness of SHAP with PCE is demonstrated on several algebraic and non-algebraic problems. • An explainable AI technique is implemented within polynomial chaos expansion (PCE). • The Shapley Additive Explanations (SHAP) enhances the explainability of PCE. • A method to analytically calculate SHAP for PCE is presented. • SHAP allows visualization of nonlinearity and interactions from the PCE model. • The method is demonstrated on three nonlinear uncertainty quantification problems. [ABSTRACT FROM AUTHOR]

In this paper we present a multi-fidelity (MF) extension of non-intrusive polynomial chaos based on regression (point collocation) for uncertainty quantification purposes. The proposed method uses the principle of a global correction function from a previous similar method that uses spectral projection to estimate the coefficients. Due to its usage of regression to estimate the coefficients, the present method offers high flexibility in the sampling and generation of the polynomial basis. The method takes advantage of a nested sampling plan to create the samples for the low-fidelity (LF) and correction expansions where the high-fidelity (HF) samples are a subset of the LF ones. To build the polynomial basis, a total order or hyperbolic truncation strategy is used with a highly flexible combination of the LF and correction polynomial expansions. The method is demonstrated on some artificial test problems and aerodynamic problems of the Euler flow around an airfoil and common three-dimensional research models. In order to derive the strategies for successful MF approximation, the effect of the correlation and the errors between the LF and HF functions is also studied. The results show that high correlation and moderately low errors are important to improve the MF approximation’s accuracy. On a common research model problem, the MF approach with partially-converged simulations as the LF samples can successfully reduce the computational cost to about 40% for similar accuracy compared to an approach using a single HF expansion. [ABSTRACT FROM AUTHOR]

This corrigendum corrects equations (18), (21) and (24) from Palar, Tsuchiya, and Parks [Comput. Methods Appl. Mech. Engrg. 305 (2016) 579–606]. These errors do not change the results, figures, discussions and main conclusions of the paper. [ABSTRACT FROM AUTHOR]