Variance-based Global Sensitivity Analysis Research Articles

An Improved Sobol Sensitivity Analysis Method

The Sobol method is a variance-based global sensitivity analysis method that evaluates single-input and multiple-input interaction effects by calculating the contribution of a single input to the output variance and the contribution of multiple inputs to the output variance. The Sobol method requires each input to obey a uniform distribution of U [0,1], but it is difficult to meet the requirements in practice. Taking the sum function as an example, this paper analyzes the inapplicability of the existing Sobol method when the input does not obey the uniform distribution U [0,1]. To solve the inapplicability of the Sobol method and broaden the application scope, an improved Sobol sensitivity analysis method is proposed. First, the effect of the joint probability density function not 1 on sensitivity calculation is studied; second, the input parameters are changed to uniform distribution U [0,1] through variable substitution; finally, a complete algorithm model is presented and logical sensitivity analysis results are obtained. Application verification shows that the improved Sobol method is more scientific, applicable and practical.

Multi-objective optimization for sustainable and economical polycarbonate production with reaction kinetics inference for real-world industrial process

Polycarbonate has a wide range of applications in electronic devices and medicine because of its remarkable properties such as transparency and mechanical strength. However, the polycarbonate production process using diphenyl carbonate leads to the inevitable generation of nitrogen oxide (NOx) from byproduct incineration. Therefore, it is essential to optimize the process conditions while simultaneously considering the production costs and NOx emissions. Herein, we propose a multi-objective optimization model for sustainable and economical polycarbonate production using a high-fidelity reaction model for real-world industrial processes. Additionally, a data-driven model was developed using industrial data to identify the operating conditions that minimize NOx emissions and maximize profitability. Four of the ten reaction parameters were selected through variance-based global sensitivity analysis, and the posterior distribution of each parameter was determined through Bayesian inference. In addition, NOx emissions were predicted using the data-driven model developed from industrial data. A process model with coefficients of determination above 0.7 and a data-driven model with R-squared value above 0.99 for pilot-scale experimental and industrial data were generated using our approach. The Pareto optimality, identified based on the developed models, suggested operating conditions that optimize both economic and environmental benefits. Consequently, NOx emissions could be reduced by 27.2%–27.4% and the unit costs of production could be reduced by 0.8%–1.6% compared with the base case.

Advancement in ice-jam flood risk management: Integrating dynamic adaptive behavior by an agent-based modeling in Fort McMurray, Canada

Ice-jam flooding is a prevalent extreme event that impacts flood hazard and vulnerability. We introduce a conceptual model framework for Dynamic Ice-jam Flood Risk Assessment (DIFRA). DIFRA integrates ice-jam flood hazard, ice-jam flood risk, and human adaptation. Using agent-based modeling, we captured top-down (artificial breakup) and bottom-up (flood-proofing) adaptive behavior. Our study in Fort McMurray, Canada, shows the complex interaction between micro-level behaviors and macro-level phenomena over time. Our variance-based global sensitivity analysis shows the role of dynamic adaptive behavior in ice-jam flood risk, where the artificial breakage by the government can lead to a regime shift and a decrease in the ice-jam flood risk. However, it can also decrease the number of newly adapted residents to flood-proofing and the role of residents in ice-jam flood risk. DIFRA offers a comprehensive approach to understanding and managing ice-jam flood risk, with potential applications to similar riverine communities in cold regions.

Identification of parameter importance for benzene transport in the unsaturated zone using global sensitivity analysis

Abstract. One of the greatest threats to groundwater is contamination from fuel derivatives. Benzene, a highly mobile and toxic fuel derivative, can easily reach groundwater from fuel sources and lead to extensive groundwater contamination and drinking water disqualification. Modelling benzene transport in the unsaturated zone can quantify the risk for groundwater contamination and provide needed remediation strategies. Yet, characterization of the problem is often complicated, due to typical soil heterogeneity, numerous unknown site and solute parameters, and the difficulty of distinguishing important from non-important parameters. Thus, sensitivity analysis (SA) methods, such as global SA (GSA), are applied to reduce uncertainty and detect key parameters for groundwater contamination and remediation. Nevertheless, studies devoted to identifying the parameters that determine transport of fuel derivatives in the unsaturated zone are scarce. In this study, we performed GSA to assess benzene transport in the unsaturated zone. First, a simple GSA (Morris) screening method was used for a homogenous sandy vadose zone. Then, a more computationally demanding (Sobol) variance-based GSA was run on the most influential parameters. Finally, the Morris method was tested for a heterogeneous medium containing clay layers. To overcome model crashes during GSA, several methods were tested for imputation of missing data. The GSA results indicate that benzene degradation rate (λk) is the utmost influential parameter controlling benzene mobility, followed by aquifer depth (z). The adsorption coefficient (Kd) and the van Genuchten n parameter of the sandy soil (n1) were also highly influential. The study emphasizes the significance of λk and the presence of clay layers in predicting aquifer contamination. The study also indicates the importance of heterogenous media representation in the GSA. Though identical parameters control the transport in the different soil types, in the presence of both sand and clay, parameters directly affecting the solute concentration like λk and Kd have increased influence in clay, whereas n is more influential for sand comprising most of the profile. Overall, GSA is demonstrated here as an important tool for the analysis of transport models. The results also show that in higher dimensionality models, the radial basis function (RBF) is an efficient surrogate model for missing data imputation.

Global sensitivity analysis and optimization of multi-stage TEG based on random and fuzzy mixed input variables

In practical engineering, uncertainties in thermoelectric power generation arise from various factors such as material properties, geometric dimensions, assembly processes, and operational fluctuations of the thermoelectric components. Addressing subjective and objective uncertainties in the design parameters of multi-stage thermoelectric generators (TEGs), this study establishes a robust TEG model incorporating non-equilibrium thermodynamics. Fuzzy and stochastic theories are employed for uncertainty representation, and a variance-based global sensitivity analysis is conducted. The results indicate that the uncertainties in the design variables have varying effects on the output responses of the three-stage TEG. Among them, the temperature of the hot and cold sources has the most significant impact on the output responses（STHP: 0.5018, STLP:0.2199, STHη: 0.5436, STLη: 0.3336）, Parameters like Seebeck coefficient, thermal resistance, thermal conductivity, and operating current exhibit relatively minor effects. The coefficient of variation of the output response varies consistently with changes in the standard deviation or coefficient of variation of input variables, following a similar order as the sensitivity indicators. The variations in the distribution parameters of the key design variable, i.e., the temperature of the hot and cold sources, do not affect the sensitivity index rankings of the variables concerning the system output responses. As the distribution parameters of the hot and cold source temperatures gradually increase, the sensitivity indices of the design variables to the system output responses exhibit different trends, indicating a change in the degree of sensitivity to the variations in distribution parameters of the design variables. Optimization based on global sensitivity analysis enhances power output by 43.37 %, reaching a maximum of 11.37 W under constraints compared to the original 7.93 W. This research provides guidance for designing and optimizing thermoelectric power systems under mixed uncertainties.

Sensitivity analysis of multiple time-scale building energy using Bayesian adaptive spline surfaces

Sensitivity analysis has been widely used in building energy analysis to identify key variables influencing building energy and carbon emissions. However, most previous studies only focus on the single time scale energy use of buildings when conducting sensitivity analysis. The multiple time-scale sensitivity analysis would lead to increased complexity and challenges in building energy analysis. This paper proposes a new systematical procedure for global variance-based sensitivity analysis of multiple time-scale energy use of buildings. A case study of an office building is used to demonstrate the application of this approach using the Bayesian adaptive spline surface (BASS) models at three different time scales (annual, monthly, and daily). The results indicate that the procedure proposed here can provide fast and reliable sensitivity results for the multiple time-scale building energy assessment. The BASS learning models have the advantages of high predictive performance and the fast computation of sensitivity indicators in a closed form without Monte Carlo integrals. The application of principal component analysis can deal with the highly correlated multi-output energy use at a smaller time scale, which can further reduce computational costs. Moreover, the sensitivity analysis at the different time scales can provide new insights into energy characteristics due to the intrinsic variations of weather conditions. This systematical procedure can provide useful guidance for the multiple time-scale model calibration and multi-step ahead predictions of buildings. Moreover, the method proposed here can be extended to the sensitivity analysis of energy use in different time scales (sub-hourly, hourly, or daily) for other energy systems (such as PV, solar thermal, and wind).

Variance-based Global Sensitivity Analysis of Surface Runoff Parameters for Hydrological Modeling of a Real Peri-urban Ungauged Basin

Variance-based Global Sensitivity Analysis of Surface Runoff Parameters for Hydrological Modeling of a Real Peri-urban Ungauged Basin

EXTREME LEARNING MACHINES FOR VARIANCE-BASED GLOBAL SENSITIVITY ANALYSIS

Variance-based global sensitivity analysis (GSA) can provide a wealth of information when applied to complex models. A well-known Achilles' heel of this approach is its computational cost, which often renders it unfeasible in practice. An appealing alternative is to instead analyze the sensitivity of a surrogate model with the goal of lowering computational costs while maintaining sufficient accuracy. Should a surrogate be "simple" enough to be amenable to the analytical calculations of its Sobol' indices, the cost of GSA is essentially reduced to the construction of the surrogate.We propose a new class of sparse-weight extreme learning machines (ELMs), which, when considered as surrogates in the context of GSA, admit analytical formulas for their Sobol' indices and, unlike the standard ELMs, yield accurate approximations of these indices. The effectiveness of this approach is illustrated through both traditional benchmarks in the field and on a chemical reaction network.

Sensitivity analysis of fifth generation district heating and cooling coupled with borehole thermal energy storage with respect to cooling adoption

Fifth generation district heating and cooling (5GDHC) is a promising solution for decarbonising building thermal demands, as it allows for simultaneous heating and cooling through a bi-directional network. This paper aims to investigate the general performance of 5GDHC under uncertainty. Initially, an uncertainty analysis is performed on a 5GDHC model in a case study, revealing significant variation in economic, technical, and environmental performance due to uncertain parameters. Subsequently, variance-based global sensitivity analysis is used to rank the uncertain parameters based on their impact on the economic performance of 5GDHC. The results identify cooling adoption rate, borehole pricing, and discount rate as the most influential parameters. Lastly, factors mapping demonstrates that cooling adoption rate within 0.6-0.9, low borehole costs, low discount rates, and high gas prices will drive the switch to 5GDHC. Overall, this study emphasizes the general applicability of 5GDHC.

Estimating global identifiability using conditional mutual information in a Bayesian framework

A novel information-theoretic approach is proposed to assess the global practical identifiability of Bayesian statistical models. Based on the concept of conditional mutual information, an estimate of information gained for each model parameter is used to quantify the identifiability with practical considerations. No assumptions are made about the structure of the statistical model or the prior distribution while constructing the estimator. The estimator has the following notable advantages: first, no controlled experiment or data is required to conduct the practical identifiability analysis; second, unlike popular variance-based global sensitivity analysis methods, different forms of uncertainties, such as model-form, parameter, or measurement can be taken into account; third, the identifiability analysis is global, and therefore independent of a realization of the parameters. If an individual parameter has low identifiability, it can belong to an identifiable subset such that parameters within the subset have a functional relationship and thus have a combined effect on the statistical model. The practical identifiability framework is extended to highlight the dependencies between parameter pairs that emerge a posteriori to find identifiable parameter subsets. The applicability of the proposed approach is demonstrated using a linear Gaussian model and a non-linear methane-air reduced kinetics model. It is shown that by examining the information gained for each model parameter along with its dependencies with other parameters, a subset of parameters that can be estimated with high posterior certainty can be found.

Iterative hierarchical clustering algorithm for automated operational modal analysis

Recent developments in sensors and data processing made the structural health monitoring (SHM) sector expanding to big-data field, particularly when continuous long-term strategies are implemented. Nevertheless, main shortcomings are due to the identification and extraction of modal features. In fact, although machine learning methods have been implemented to automate modal identification processes, intense user interaction and time-consuming procedures are still required, limiting the extensive use of these techniques. In order to provide a fully automated procedure capable of identifying and extracting modal properties from covariance driven SSI analyses, an innovative and flexible algorithm for Iterative Hierarchical Clustering Analysis (IHCA) is proposed. To evaluate the stability and robustness of the IHCA method, a Variance-Based Global sensitivity Analysis (VBGA) was performed considering a numerical and experimental case study. The outcomes demonstrated that the IHCA is stable in clustering the physical structural modes and selecting the most representative modal features.

Effect of uncertainty of material parameters on stress triaxiality and Lode angle in finite elasto-plasticity—A variance-based global sensitivity analysis

This work establishes a computational framework for the quantification of the effect of uncertainty of material model parameters on extremal stress triaxiality and Lode angle values in plastically deformed devices, whereby stress triaxiality and Lode angle are accepted as key indicators for damage initiation in metal forming processes. Attention is paid to components, the material response of which can be represented as elasto-plastic with proportional hardening as a prototype model, whereby the finite element method is used as a simulation approach generally suitable for complex geometries and loading conditions. Uncertainty about material parameters is characterized resorting to probability theory. The effects of material parameter uncertainty on stress triaxiality and Lode angle are quantified by means of a variance-based global sensitivity analysis. Such sensitivity analysis is most useful for apportioning the variance of the stress triaxiality and Lode angle to the uncertainty on material properties. The practical implementation of this sensitivity analysis is carried out resorting to a Gaussian process regression, Bayesian probabilistic integration and active learning in order to decrease the associated numerical costs. An example illustrates the proposed framework, revealing that parameters governing plasticity affect stress triaxiality and Lode angle the most.

On quantifying uncertainty in lightning strike damage of composite laminates: A hybrid stochastic framework of coupled transient thermal-electrical simulations

Lightning strike damage can severely affect the thermo-mechanical performance of composite laminates. It is essential to quantify the effect of lightning strikes considering the inevitable influence of material and geometric uncertainties for ensuring the operational safety of aircraft. This paper presents an efficient support vector machine (SVM)-based surrogate approach coupled with computationally intensive transient thermal-electrical finite element simulations to quantify the uncertainty in lightning strike damage. The uncertainty in epoxy matrix thermal damage and electrical responses of unprotected carbon/epoxy composite laminates is probabilistically quantified considering the stochasticity in temperature-dependent multi-physical material properties and ply orientations. Further, the SVM models are exploited for variance-based global sensitivity analysis to investigate the input parameters' relative influence on the lightning strike-induced damage behavior. Due to the adoption of a coupled SVM-based simulation approach here, it has become possible to carry out a comprehensive uncertainty quantification leading to complete probabilistic descriptions of the electrical and lightning damage parameters despite the requirement of performing a large number of computationally intensive function evaluations. The results reveal that source-uncertainty of the unprotected laminates significantly influences the epoxy matrix decomposition, electrical current density and electric potential, wherein longitudinal electrical conductivity is most sensitive to stochastic variations followed by other electrical, thermal and geometric parameters.

Bayesian parameter identification in electrochemical model for lithium-ion batteries

Electrochemical models can characterize the internal behavior of cells and are powerful and effective tools for the design and management of batteries. This study proposes a comprehensive framework of Bayesian parameter identification to determine the parameter distributions in the electrochemical model and to estimate the global sensitivity of the parameters for lithium-ion batteries. Bayesian inference in parameter identification can simultaneously determine accurate parameter estimates and parameter identifiability, given specific voltage measurements. Among the several parameters in the pseudo-two-dimensional model, 15 parameters are selected to estimate the posterior distributions. The reconstructed voltage curves through the estimated parameter distributions are consistent with the reference voltages, with relative errors of less than 0.7% at various discharge rates. Changes in the parameter distributions and identifiability were investigated for different discharge rates through the estimated joint and marginal distributions of the parameters. Moreover, based on variance-based global sensitivity analysis, the identifiability of the electrochemical parameters according to the discharge rates is quantitatively analyzed. We demonstrate that the Bayesian parameter identification simultaneously obtains the parameter distributions and identifiability, considering the correlation between various parameters. The proposed framework can help to analyze the behaviors of batteries according to specific operating conditions and materials.

Bayesian Model Updating for Structural Dynamic Applications Combing Differential Evolution Adaptive Metropolis and Kriging Model

The Bayesian model updating approach has attracted much attention by providing the most probable values (MPVs) of physical parameters and their uncertainties. However, the Bayesian approach has challenges in high-dimensional problems and requires high computational costs in large-scale engineering structures dealing with structural dynamics. This study proposes a new Bayesian updating framework using the Differential Evolution Adaptive Metropolis (DREAM) algorithm to enhance the Bayesian approach’s performance and computational efficiency. In addition, two time-saving strategies are proposed. Firstly, variance-based global sensitivity analysis is used to eliminate insignificant parameters to model responses and reduce model dimensionality. Secondly, a fast-running kriging model is employed as a surrogate of the time-consuming finite-element (FE) model to alleviate the computational burden. DREAM essentially is a multichain sampling method that runs different paths to seek all possible solutions and accurately approximate the posterior probability distribution function in the Bayesian approach. The proposed updating framework was demonstrated using one numerical example and a real-world cable-stayed pedestrian bridge. The results showed that the proposed method enables rationally identifying structural parameters and recovering dynamic responses. Compared with the traditional Bayesian approach without a surrogate model, the computational cost is orders of magnitude lower.

Sparse-grids uncertainty quantification of part-scale additive manufacturing processes

The present paper aims at applying uncertainty quantification methodologies to process simulations of powder bed fusion of metal. In particular, for a part-scale thermomechanical model of an Inconel 625 super-alloy beam, we study the uncertainties of three process parameters, namely the activation temperature, the powder convection coefficient and the gas convection coefficient. First, we perform a variance-based global sensitivity analysis to study how each uncertain parameter contributes to the variability of the beam displacements. The results allow us to conclude that the gas convection coefficient has little impact and can therefore be fixed to a constant value for subsequent studies. Then, we conduct an inverse uncertainty quantification analysis, based on a Bayesian approach on synthetic displacement data, to quantify the uncertainties of the two remaining parameters, namely the activation temperature and the powder convection coefficient. Finally, we use the results of the inverse uncertainty quantification analysis to perform a data-informed forward uncertainty quantification analysis of the residual strains. Crucially, we make use of surrogate models based on sparse grids to keep to a minimum the computational burden of every step of the uncertainty quantification analysis. The proposed uncertainty quantification workflow allows us to substantially ease the typical trial-and-error approach used to calibrate power bed fusion part-scale models, and to greatly reduce uncertainties on the numerical prediction of the residual strains. In particular, we demonstrate the possibility of using displacement measurements to obtain a data-informed probability density function of the residual strains, a quantity much more complex to measure than displacements.

Deterministic and probabilistic-based model updating of aging steel bridges

Numerical modeling is a very useful tool in different fields of bridge engineering, such as load-carrying capacity assessment or structural health monitoring. Developing a reliable computational model that accurately represents the actual bridge mechanical behavior entails advanced FEM-based modeling complemented by a comprehensive experimental campaign that provides the necessary supporting information and allows validating simulation outcomes. This paper proposes a unified approach aimed at the experimental characterization and FE model updating of aging steel bridges. It first involves the realization of an extensive experimental campaign aimed at the bridge's geometrical, material, and dynamic behavior characterization. Then, a model calibration framework is developed, where deterministic (optimization) and probabilistic (Bayesian inference) approaches are employed, and techniques such as global variance-based sensitivity analysis and Kriging-based surrogate modeling are further implemented in order to enhance the identification process and reduce the overall computational burden. The methodology has been validated in a historical riveted steel bridge in O Barqueiro, north of Galicia, Spain. The results show a good agreement in the identified model parameter values and a noticeable correlation between numerical and experimental modal properties, with an average relative error in frequencies of 0.34% and 0.44% for the deterministic and probabilistic approaches and an average MAC (Modal Assurance Criterion) ratio of 0.96.

Reliability-based structural assessment of historical masonry arch bridges: The case study of Cernadela bridge

Nowadays, several historical masonry arch bridges present a deficient state of conservation due to degradation processes induced by natural or human actions. Usually, these constructions have significant economic, cultural, and heritage value. Therefore, they shall be thoroughly assessed to verify their structural integrity and safety condition. For this purpose, reliability-based structural assessments are typically performed. However, the associated outcomes (i.e., reliability index and probability of failure) highly rely on the accuracy of the structural parameters uncertainty quantification. This work presents a study regarding the influence of the scattering of the arches' thickness dimensions in the load-carrying capacity assessment of the Cernadela Bridge, a historical stone bridge located in Galicia, Spain. The study first involved a comprehensive experimental campaign to characterize the outer and inner bridge geometry through geomatic techniques, i.e., terrestrial laser scanning and ground penetrating radar. Subsequently, a limit analysis model was developed, considering only the arches' outer (visible) data. From this initial structural assessment, a reliability index of 2.38 was obtained. The influence of the uncertain structural parameters, both geometric features and material properties, in the collapse load was investigated through a global variance-based sensitivity analysis (i.e., Sobol' indices) complemented by a surrogate modeling strategy based on the Kriging approach. Finally, the measured inner geometry of the arches was introduced in the computational model through Bayesian inference procedures. Thus, two new structural assessments were performed: first, by considering the updated distributions of all arches thicknesses, and second, by considering only the updated distributions of the non-influential ones. Reliability indexes of 1.51 and 2.33 were derived, thus highlighting the importance of a proper uncertainty quantification process and the relevance of the sensitivity analysis outcomes to identify the critical parameters on the bridge mechanical response.

Efficient global sensitivity analysis for high-dimensional outputs combining data-driven probability models and dimensionality reduction

• GSA for applications involving complex numerical models and high-dimensional outputs is examined • Two different, recently developed, GSA formulations are combined • Principal component analysis (PCA) is firstly adopted as a dimensionality reduction technique. • Relevant latent output GSA statistics are then calculated by extending probability model-based GSA. • Extension pertains to estimation of necessary covariance statistics This paper examines the efficient variance-based global sensitivity analysis (GSA), quantified by estimating first-/higher-order and total-effect Sobol’ indices, for applications involving complex numerical models and high-dimensional outputs. Two different, recently developed, techniques are combined to address the associated challenges. Principal component analysis (PCA) is first considered as a dimensionality reduction technique. The GSA for the original output vector is then formulated by calculating variance and covariance statistics for the low-dimensional latent output space identified by PCA. These statistics are efficiently approximated by extending recent work on data-driven, probability model-based GSA (PM-GSA). The extension, constituting the main novel contribution of this work, pertains to the estimation of covariance statistics beyond the variance statistics examined in the original PM-GSA formulation. Specifically, a Gaussian mixture model (GMM) is developed to approximate the joint probability density function between some subset of the input vector, and each latent output, or each pair of latent outputs. The GMM is then utilized to estimate the aforementioned statistics. Results across two natural hazards engineering examples show that the dimensionality reduction and transformation of output space established through PCA do not impact the overall accuracy of the PM-GSA, and that the proposed implementation accommodates highly-efficient GSA estimates.

A mechanism motion error sensitivity analysis method based on principal component analysis and artificial neural network

Present sensitivity analysis of motion error usually focuses on the trajectory deviation of the mechanism, which inevitably introduces an intractable time dependent problem. For efficiently and accurately measuring the motion error of the planar mechanism with dimension and clearance uncertainties by global sensitivity analysis (GSA), a novel method is proposed in this work. By applying the principal component analysis (PCA), the motion error is transformed into new vector output and cleverly avoids the time dependent problem. To ensure the accuracy of PCA in the case of small samples, the Bootstrap method is introduced. Based on the PCA results, the artificial neural network (ANN) surrogate model is established between the input variables and the vector output. Then the classical variance-based GSA method is applied to obtain the variable importance ranking for different PCs, and the synthesized GSA indices are introduced. Four representative examples are studied to demonstrate the versatility and effectiveness of the proposed method.