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IIT Hyderabad

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. © 2022 Elsevier B.V.

Department of Civil Engineering

Journal of Computational and Applied Mathematics

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

An adaptive Bayesian polynomial chaos expansion (BPCE) is developed in this paper for uncertainty quantification (UQ) and reliability analysis. The sparsity in the PCE model is developed using automatic relevance determination (ARD) and the PCE coefficients are computed using the variational Bayesian (VB) inference. Further, Sobol sequence is utilized to evaluate a response quantity sequentially. Finally, leave one out (LOO) error is used to obtain the adaptive BPCE model. UQ and reliability analysis are performed of some numerical examples by the adaptive BPCE model. It is found that the optimal number of model evaluations and the optimal PCE degree are suitably selected simultaneously for a problem by the adaptive BPCE model. A highly accurate result is predicted by the proposed approach using very few model evaluation. Further, highly sparse PCE models are obtained by the ARD approach for most of the numerical examples. Additionally, distribution parameters of the predicted response quantity are also obtained by the VB inference, which are used to compute the confidence interval of the predicted response quantities. © 2021, The Author(s), under exclusive licence to Springer-Verlag London Ltd. part of Springer Nature.

Engineering with Computers

Uncertainty quantification and reliability analysis by an adaptive sparse Bayesian inference based PCE model

Uncertainty quantification (UQ) of dynamical systems with a surrogate model is difficult as the surrogate model parameters often need to be computed at each time step. For that reason, a different kind of surrogate model is presented in this article by representing the stochastic response quantity using lower number of bases. Proper orthogonal decomposition (POD) is used to project the response quantity with low number of POD bases and the uncertainty is propagated using the Kriging model. The combined POD–Kriging model is used for UQ of four dynamical systems and all the results are compared with the Monte Carlo simulation results. The first three problems are long time integration problems and the time-dependent statistical moments are predicted well by the POD–Kriging model using low number of model evaluations. For the last example (a building subjected to Gaussian white noise base acceleration), the prediction accuracy of the POD–Kriging model is also high as compared to a similar surrogate model. © 2022 Elsevier B.V.

Journal of Computational Science

Uncertainty quantification of dynamical systems by a POD–Kriging surrogate model

Propagating uncertainty through a crash problem is very difficult due to non-linear and non-smooth behavior. The required number of model evaluations is often high, and therefore the computational cost is prohibitive. To deal with such problems, an adaptive meta-model is developed using a polynomial chaos expansion (PCE) and a proper orthogonal decomposition (POD). The adaptive meta-model is used for uncertainty quantification and for global sensitivity analysis of a crash box under impact loading. The time-dependent uncertain response quantities are expressed with the reduced POD modes. The predicted stochastic contact force and impactor velocity by the adaptive meta-model are quite close to the actual simulations. The time-dependent mean and standard deviation for all responses are predicted quite well with low number of model evaluations. Furthermore, it is found that the material property and the crash box thickness are the most influential parameters for the contact force, and the impactor mass is the most influential parameter for the total dissipated energy. © 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

World University root node

Structural and Multidisciplinary Optimization

Stochastic analysis of a crash box under impact loading by an adaptive POD-PCE model

Accurate computation of failure probability considering uncertain input parameters is very challenging within limited computational cost. An efficient surrogate model, referred to here as sparse variational Bayesian inference based polynomial chaos expansion (SVB-PCE), is formulated in this paper for reliability analysis. The sparsity in the polynomial basis terms is introduced by the automatic relevance determination (ARD) algorithm and the coefficients corresponding to the sparse polynomial bases are computed using the VB framework. The reliability analysis is performed on four typical numerical problems using the SVB-PCE model. The failure probability and the reliability index for all the examples are assessed accurately by the SVB-PCE model using fewer number of model evaluations as compared to the state-of-art methods. Further, the ARD enables to capture the most important terms in the polynomial bases which also reduces the computational cost in assessing the failure probability. © 2021 Elsevier Ltd

Structural Safety

Structural reliability analysis by a Bayesian sparse polynomial chaos expansion

A proper orthogonal decomposition (POD)-based polynomial chaos expansion (PCE) is utilized in this article for the uncertainty quantification (UQ) of an impact dynamic oscillator. The time-dependent nonsmooth behavior and the uncertainties are decoupled using the POD approach. The uncertain response domain is reduced using the POD approach, and the dominant POD modes are utilized for the UQ of the response quantity. Furthermore, the PCE model is utilized for the propagation of the input uncertainties. Two different cases of impact oscillator are considered, namely, single impact and multiple impact. The contact between two bodies is modeled by Hertz's law. For both the cases, UQ is performed on the projectile displacement, projectile velocity, and contact force. A highly nonsmooth behavior is noticed for the contact force. For that reason, most number of POD modes are required to assess the UQ of contact force. All the results are compared with the Monte Carlo simulation (MCS) and time domain PCE results. Very good accuracies are observed for the PCE and the POD-PCE predicted results using much less number of model evaluations compared to MCS. As the PCE coefficients are dependent on time, the PCE model is computed at each time step. On the contrary, for the POD-PCE model, the PCE coefficients are computed for the number of POD modes only: it is much less than the PCE model. © 2020 by ASME.

Fulltext

Journal of Vibration and Acoustics, Transactions of the ASME

Uncertainty Quantification of Stochastic Impact Dynamic Oscillator Using a Proper Orthogonal Decomposition-Polynomial Chaos Expansion Technique

A novel approach, referred to as sparse Kriging-NARX (KNARX), is proposed for the uncertainty quantification of nonlinear stochastic dynamical systems. It combines the nonlinear autoregressive with exogenous (NARX) input model with the high fidelity surrogate model Kriging. The sparsity in the proposed approach is introduced in the NARX model by reducing the number of polynomial bases using the least-angle regression (LARS) algorithm. Sparse KNARX captures the nonlinearity of a problem by the NARX model, whereas the uncertain parameters are propagated using the Kriging surrogate model, and LARS makes the model efficient. The accuracy and the efficiency of the sparse KNARX was measured through uncertainty quantification applied to three nonlinear stochastic dynamical systems. The time-dependent mean and standard deviation were predicted for all the numerical examples. Instantaneous stochastic response characteristics and maximum absolute response were also predicted. All the results were compared with the full scale Monte Carlo simulation (MCS) results and a mean error was calculated for all the numerical problems to measure the accuracy. All the results had excellent agreement with the MCS results at a very limited computational cost. The efficiency of the sparse KNARX also was measured by the CPU time and the required number of surrogate model evaluations. In all instances, sparse KNARX outperformed other state-of-the-art methods, which justifies the applicability of this model for nonlinear stochastic dynamical systems. © 2020 American Society of Civil Engineers.

Journal of Engineering Mechanics

A Kriging-NARX Model for Uncertainty Quantification of Nonlinear Stochastic Dynamical Systems in Time Domain

This paper presents a detailed study on an E-plan-shaped (asymmetry about both plan axes) building under wind excitation (wind angles varied from 0° to 330° at an interval of 30°). This study is conducted by the wind-tunnel procedure and by utilizing the computational fluid dynamics (CFD) technique numerically. The numerical study is performed using the k-ϵ and shear stress transport (SST) k-ω models. The pressure contours and mean pressure coefficients of all the faces are computed by both approaches. The results show a good agreement between the experimentally and numerically computed results. The SST k-ω model performed better in comparison with the k-ϵ model for computing the mean pressure coefficients. The maximum positive and negative mean pressures of some faces of the building are found at skew wind angles. Additionally, a comparative study of the pressure variation with a symmetrical E-plan-shaped building (having the same cross-section area) is conducted in this paper. This study has shown that a drastic increment in the pressure variation is noticed for both the maximum positive and negative mean pressure coefficients due to a small aerodynamic modification. © 2020 American Society of Civil Engineers.

Journal of Structural Engineering (United States)

Experimental and Numerical Study of Wind-Pressure Distribution on Irregular-Plan-Shaped Building

A novel sparse polynomial chaos expansion (PCE) is proposed in this paper for global sensitivity analysis (GSA). The proposed model combines variational Bayesian (VB) inference and automatic relevance determination (ARD) with the PCE model. The VB inference is utilized to compute the PCE coefficients. The PCE coefficients are obtained through a simple optimization procedure in the VB framework. On the other hand, the curse of dimensionality issue of PCE model is tackled using the ARD which reduces the number of polynomial bases significantly. The applicability of the proposed approach is illustrated by performing GSA on five numerical examples. The results show that the proposed approach outperforms a similar state-of-art surrogate model in obtaining an accurate sensitivity index using limited number of model evaluations. For all the examples, the PCE models are highly sparse, which require very few polynomial bases to assess an accurate sensitivity index. © 2020 Elsevier Inc.

Journal of Computational Physics

Global sensitivity analysis: A Bayesian learning based polynomial chaos approach

Propagating uncertainties through mechanical systems has been widely studied for the last thirty years. In particular metamodels based on polynomial chaos expansion (PCE) have been successfully developed in the context of both intrusive and non-intrusive methods. However, modelling random dynamical systems is much more challenging and requires increasing computational resources when the time integration becomes longer. Therefore separating the time aspect of the dynamical response and the random contributions is an appealing approach, which has been used in this paper. Thus, a non-intrusive method is proposed by associating a proper orthogonal decomposition (POD) and a PCE. The POD-PC model was applied on three examples. On two examples, the method was very efficient not only in calculating the first two statistical moments, but also in estimating the responses corresponding to several samples of the random parameters. As to the last example, the Kraichnan-Orszag three-mode problem, the model was able to estimate well the evolution of the first two statistical moments for the first part of the time duration, but a noticeable discrepancy occurred for the remaining part. It seems that the model is more appropriate to estimate the transient response of a random dynamical system than the steady-state response. © 2019 Elsevier Ltd

Mechanical Systems and Signal Processing

Random dynamical system in time domain: A POD-PC model

Uncertainty quantification of nonlinear stochastic dynamic problem is always a challenging task due to the complexity of the systems. In this paper, a hybrid surrogate modelling approach is proposed for the uncertainty quantification of nonlinear stochastic dynamical systems in the time domain. The proposed hybrid surrogate model is constructed using a nonlinear system identification tool, the Nonlinear AutoRegressive with eXogenous (NARX) input model, and the Kriging approach for uncertainty propagation. Further, to increase the computational efficiency, least angle regression (LARS) is utilized in the hybrid framework. The method is applied on a nonlinear stochastic dynamic oscillator to check its applicability. The time dependent mean and standard deviation are predicted using the proposed approach, and all the results are compared with the Monte Carlo simulation (MCS) results. A high-level accuracy is noticed using the proposed approach as compared to other state-of-the-art methods. This accuracy is achieved using a very limited number of model evaluations which is suggesting the efficiency of the proposed approach. Moreover, an excellent accuracy and efficiency is achieved using the proposed approach in predicting the probability density function (PDF) at several time instances for the nonlinear stochastic dynamic oscillator. © 2019 Proceedings of the 3rd International Conference on Uncertainty Quantification in Computational Sciences and Engineering, UNCECOMP 2019. All rights reserved.

Proceedings of the 3rd International Conference on Uncertainty Quantification in Computational Sciences and Engineering, UNCECOMP 2019

Uncertainty quantification of nonlinear stochastic dynamic problem using a kriging-narx surrogate model

Due to shortage of land and architectural aesthetics, sometimes the buildings are constructed as unconventional in plan. The wind force acts differently according to the plan shape of the building. So, it is of utter importance to study wind force or, more specifically wind pressure on an unconventional plan shaped tall building. To address this issue, this paper demonstrates a comprehensive study on mean pressure coefficient of „E‟ plan shaped tall building. This study has been carried out experimentally and numerically by wind tunnel test and computational fluid dynamics (CFD) simulation respectively. Mean wind pressures on all the faces of the building are predicted using wind tunnel test and CFD simulation varying wind incidence angles from 0° to 180° at an interval of 30°. The accuracy of the numerically predicted results are measured by comparing results predicted by CFD with experimental results and it seems to have a good agreement with wind tunnel results. Besides wind pressures, wind flow patterns are also obtained by CFD for all the wind incidence angles. These flow patterns predict the behavior of pressure variation on the different faces of the building. For better comparison of the results, pressure contours on all the faces are also predicted by both the methods. Finally, polynomial expressions as the sine and cosine function of wind angle are proposed for obtaining mean wind pressure coefficient on all the faces using Fourier series expansion. The accuracy of the fitted expansions are measured by sum square error, R2 value and root mean square error. Copyright © 2018 Techno-Press, Ltd.

Wind and Structures, An International Journal

Investigation of mean wind pressures on ‘E’ plan shaped tall building

Surrogate models are widely used to predict response function of any system and in quantifying uncertainty associated with the response function. It is required to have response quantities at some preselected sample points to construct a surrogate model which can be processed in two way. Either the surrogate model is constructed using one shot experimental design techniques, or, the sample points can be generated in a sequential manner so that optimum sample points for a specific problem can be obtained. This paper addresses a comprehensive comparisons between these two types of sampling techniques for the construction of more accurate surrogate models. Two most popular one shot sampling strategies: Latin hypercube sampling and Sobol sequence, and four different type sequential experimental designs (SED) namely, Monte Carlo intersite projected (MCIP), Monte Carlo intersite projected threshold (MCIPT), optimizer projected (OP) and LOLA-Voronoi (LV) method are taken for the present study. Two most widely used surrogate models, namely polynomial chaos expansion and Kriging are used to check the applicability of the experimental design techniques. Three different types of numerical problems are solved using the two above-mentioned surrogate models considering all the experimental design techniques independently. Further, all the results are compared with the standard Monte Carlo simulation (MCS). Overall study depicts that SED performs well in predicting the response functions more accurately with an acceptable number of sample points even for high-dimensional problems which maintains the balance between accuracy and efficiency. More specifically, MCIPT and LV method based Kriging outperforms other combinations. © 2017, CIMNE, Barcelona, Spain.

Archives of Computational Methods in Engineering

A Critical Appraisal of Design of Experiments for Uncertainty Quantification

This paper presents detailed study of force and pressure coefficients of an unsymmetrical ’E’ plan shaped tall building under wind excitation. The building is unsymmetrical about both direction (along and across wind direction) in plan. Experimental and numerical study is carried out by wind tunnel test in boundary layer wind tunnel and Computational Fluid Dynamics (CFD) technique using ANSYS CFX software respectively to study the above model under wind excitation. Two different types of models were used for force and pressure measurements in wind tunnel. Rigid model was used at a model scale of 1:300 for both the cases. Force measurement model was tested under three different wind speeds of 6, 8, 10 m/s whether pressure measurement model was tested under wind speed of 10 m/s in wind tunnel. Wind flow pattern around the building model is studied by CFD technique for both wind flow direction which predicts about the nature of pressure distribution on different surfaces. Some dynamic nature of wind flow is observed at the rear side in case of across wind effects. Maximum positive force and pressure coefficients noticed in case of along wind effects as the surface area is maximum and maximum wind energy dissipates on frontal surface. Same mean pressure occurred on similar faces in case of along wind effects. Pressure contours are symmetrical about vertical axis on frontal faces for both directions of flow and pressure fluctuation noticed at the rear side face due to dynamic effects of flow pattern in case of across wind effect. A comparative study is carried out between experimental and numerical study and results found from these two methods have good agreement to each other. © Springer India 2015.

Advances in Structural Engineering: Dynamics, Volume Two

Along and across wind effects on irregular plan shaped tall building

The present study demonstrates the pressure distribution on various faces of 'E' plan shaped tall buildings under wind excitation. Experimental and analytical studies were carried out using wind angles varying from 0° to 180° with an interval of 30°. The experimental study was conducted by open circuit boundary layer wind tunnel; whereas the analytical study was conducted with Computational Fluid Dynamics (CFD) technique using ANSYS CFX software package using k-ε turbulence model. A rigid model (made of perspex sheet) was used for wind tunnel test with a model scale of 1:300. Mean pressure coefficients of all the faces are found for all wind incidence angles and pressure contours are plotted on all the surfaces for 0° wind angles. Mean pressure coefficients are also calculated by CFD technique and the results have a good agreement with experimental results. Also, pressure contours on all the faces for a 0° wind angle are plotted and the contours are almost similar to those of experimental investigation. The flow pattern around the building model is also shown to understand the variation of pressures on different faces for a particular wind angle. © 2014 JUST. All Rights Reserved.