Uncertainty quantification of inverse analysis for geomaterials using probabilistic programming

Uncertainty is an essentially challenging for safe construction and long-term stability of geotechnical engineering.The inverse analysis is commonly utilized to determine the physico-mechanical parameters.However,conventional inverse analysis cannot deal with uncertainty in geotechnical and geological systems.In this study,a framework was developed to evaluate and quantify uncertainty in inverse analysis based on the reduced-order model(ROM)and probabilistic programming.The ROM was utilized to capture the mechanical and deformation properties of surrounding rock mass in geomechanical problems.Probabilistic programming was employed to evaluate uncertainty during construction in geotechnical engineering.A circular tunnel was then used to illustrate the proposed framework using analytical and numerical solution.The results show that the geomechanical parameters and associated uncertainty can be properly obtained and the proposed framework can capture the mechanical behaviors under uncertainty.Then,a slope case was employed to demonstrate the performance of the developed framework.The results prove that the proposed framework provides a scientific,feasible,and effective tool to characterize the properties and physical mechanism of geomaterials under uncertainty in geotechnical engineering problems.

A logistic-Lasso-regression-based seismic fragility analysis method for electrical equipment considering structural and seismic parameter uncertainty

Damage to electrical equipment in an earthquake can lead to power outage of power systems.Seismic fragility analysis is a common method to assess the seismic reliability of electrical equipment.To further guarantee the efficiency of analysis,multi-source uncertainties including the structure itself and seismic excitation need to be considered.A method for seismic fragility analysis that reflects structural and seismic parameter uncertainty was developed in this study.The proposed method used a random sampling method based on Latin hypercube sampling(LHS)to account for the structure parameter uncertainty and the group structure characteristics of electrical equipment.Then,logistic Lasso regression(LLR)was used to find the seismic fragility surface based on double ground motion intensity measures(IM).The seismic fragility based on the finite element model of an±1000 kV main transformer(UHVMT)was analyzed using the proposed method.The results show that the seismic fragility function obtained by this method can be used to construct the relationship between the uncertainty parameters and the failure probability.The seismic fragility surface did not only provide the probabilities of seismic damage states under different IMs,but also had better stability than the fragility curve.Furthermore,the sensitivity analysis of the structural parameters revealed that the elastic module of the bushing and the height of the high-voltage bushing may have a greater influence.

Bayesian data analysis to quantify the uncertainty of intact rock strength

One of the main difficulties in the geotechnical design process lies in dealing with uncertainty. Uncertainty is associated with natural variation of properties, and the imprecision and unpredictability caused by insufficient information on parameters or models. Probabilistic methods are normally used to quantify uncertainty. However, the frequentist approach commonly used for this purpose has some drawbacks.First, it lacks a formal framework for incorporating knowledge not represented by data. Second, it has limitations in providing a proper measure of the confidence of parameters inferred from data. The Bayesian approach offers a better framework for treating uncertainty in geotechnical design. The advantages of the Bayesian approach for uncertainty quantification are highlighted in this paper with the Bayesian regression analysis of laboratory test data to infer the intact rock strength parameters σand mused in the Hoek-Brown strength criterion. Two case examples are used to illustrate different aspects of the Bayesian methodology and to contrast the approach with a frequentist approach represented by the nonlinear least squares(NLLS) method. The paper discusses the use of a Student’s t-distribution versus a normal distribution to handle outliers, the consideration of absolute versus relative residuals, and the comparison of quality of fitting results based on standard errors and Bayes factors. Uncertainty quantification with confidence and prediction intervals of the frequentist approach is compared with that based on scatter plots and bands of fitted envelopes of the Bayesian approach. Finally, the Bayesian method is extended to consider two improvements of the fitting analysis. The first is the case in which the Hoek-Brown parameter, a, is treated as a variable to improve the fitting in the triaxial region. The second is the incorporation of the uncertainty in the estimation of the direct tensile strength from Brazilian test results within the overall evaluation of the intact rock strength.

Uncertainty Analysis and Optimization of Quasi-Zero Stifness Air Suspension Based on Polynomial Chaos Method

To improve the vibration isolation performance of suspensions,various new structural forms of suspensions have been proposed.However,there is uncertainty in these new structure suspensions,so the deterministic research cannot refect the performance of the suspension under actual operating conditions.In this paper,a quasi-zero stifness isolator is used in automotive suspensions to form a new suspension−quasi-zero stifness air suspension(QZSAS).Due to the strong nonlinearity and structural complexity of quasi-zero stifness suspensions,changes in structural parameters may cause dramatic changes in suspension performance,so it is of practical importance to study the efect of structural parameter uncertainty on the suspension performance.In order to solve this problem,three suspension structural parameters d_(0),L_(0) and Pc_(0) are selected as random variables,and the polynomial chaos expansion(PCE)theory is used to solve the suspension performance parameters.The sensitivity of the performance parameters to diferent structural parameters was discussed and analyzed in the frequency domain.Furthermore,a multi-objective optimization of the structural parameters d_(0),L_(0) and Pc_(0) of QZSAS was performed with the mean and variance of the root-mean-square(RMS)acceleration values as the optimization objectives.The optimization results show that there is an improvement of about 8%−1_(0)%in the mean value and about 4_(0)%−55%in the standard deviation of acceleration(RMS)values.This paper verifes the feasibility of the PCE method for solving the uncertainty problem of complex nonlinear systems,which provide a reference for the future structural design and optimization of such suspension systems.

Generalized nth-Order Perturbation Method Based on Loop Subdivision Surface Boundary Element Method for Three-Dimensional Broadband Structural Acoustic Uncertainty Analysis

In this paper,a generalized nth-order perturbation method based on the isogeometric boundary element method is proposed for the uncertainty analysis of broadband structural acoustic scattering problems.The Burton-Miller method is employed to solve the problem of non-unique solutions that may be encountered in the external acoustic field,and the nth-order discretization formulation of the boundary integral equation is derived.In addition,the computation of loop subdivision surfaces and the subdivision rules are introduced.In order to confirm the effectiveness of the algorithm,the computed results are contrasted and analyzed with the results under Monte Carlo simulations(MCs)through several numerical examples.

The prediction of projectile-target intersection for moving tank based on adaptive robust constraint-following control and interval uncertainty analysis

To improve the hit probability of tank at high speed,a prediction method of projectile-target intersection based on adaptive robust constraint-following control and interval uncertainty analysis is proposed.The method proposed provides a novel way to predict the impact point of projectile for moving tank.First,bidirectional stability constraints and stability constraint-following error are constructed using the Udwadia-Kalaba theory,and an adaptive robust constraint-following controller is designed considering uncertainties.Second,the exterior ballistic ordinary differential equation with uncertainties is integrated into the controller,and the pointing control of stability system is extended to the impact-point control of projectile.Third,based on the interval uncertainty analysis method combining Chebyshev polynomial expansion and affine arithmetic,a prediction method of projectile-target intersection is proposed.Finally,the co-simulation experiment is performed by establishing the multi-body system dynamic model of tank and mathematical model of control system.The results demonstrate that the prediction method of projectile-target intersection based on uncertainty analysis can effectively decrease the uncertainties of system,improve the prediction accuracy,and increase the hit probability.The adaptive robust constraint-following control can effectively restrain the uncertainties caused by road excitation and model error.

Uncertainty and sensitivity analysis of in-vessel phenomena under severe accident mitigation strategy based on ISAA-SAUP program

The phenomenology involved in severe accidents in nuclear reactors is highly complex.Currently,integrated analysis programs used for severe accident analysis heavily rely on custom empirical parameters,which introduce considerable uncertainty.Therefore,in recent years,the field of severe accidents has shifted its focus toward applying uncertainty analysis methods to quantify uncertainty in safety assessment programs,known as“best estimate plus uncertainty(BEPU).”This approach aids in enhancing our comprehension of these programs and their further development and improvement.This study concentrates on a third-generation pressurized water reactor equipped with advanced active and passive mitigation strategies.Through an Integrated Severe Accident Analysis Program(ISAA),numerical modeling and uncertainty analysis were conducted on severe accidents resulting from large break loss of coolant accidents.Seventeen uncertainty parameters of the ISAA program were meticulously screened.Using Wilks'formula,the developed uncertainty program code,SAUP,was employed to carry out Latin hypercube sampling,while ISAA was employed to execute batch calculations.Statistical analysis was then conducted on two figures of merit,namely hydrogen generation and the release of fission products within the pressure vessel.Uncertainty calculations revealed that hydrogen production and the fraction of fission product released exhibited a normal distribution,ranging from 182.784 to 330.664 kg and from 15.6 to 84.3%,respectively.The ratio of hydrogen production to reactor thermal power fell within the range of 0.0578–0.105.A sensitivity analysis was performed for uncertain input parameters,revealing significant correlations between the failure temperature of the cladding oxide layer,maximum melt flow rate,size of the particulate debris,and porosity of the debris with both hydrogen generation and the release of fission products.

Machine learning based damage state identification:A novel perspective on fragility analysis for nuclear power plants considering structural uncertainties

Seismic fragility analysis(SFA)is known as an effective probabilistic-based approach used to evaluate seismic fragility.There are various sources of uncertainties associated with this approach.A nuclear power plant(NPP)system is an extremely important infrastructure and contains many structural uncertainties due to construction issues or structural deterioration during service.Simulation of structural uncertainties effects is a costly and time-consuming endeavor.A novel approach to SFA for the NPP considering structural uncertainties based on the damage state is proposed and examined.The results suggest that considering the structural uncertainties is essential in assessing the fragility of the NPP structure,and the impact of structural uncertainties tends to increase with the state of damage.Subsequently,machine learning(ML)is found to be superior in high-precision damage state identification of the NPP for reducing the time of nonlinear time-history analysis(NLTHA)and could be applied in the damage state-based SFA.Also,the impact of various sources of uncertainties is investigated through sensitivity analysis.The Sobol and Shapley additive explanations(SHAP)method can be complementary to each other and able to solve the problem of quantifying seismic and structural uncertainties simultaneously and the interaction effect of each parameter.

Uncertainty analysis of flow rate measurement for multiphase flow using CFD

The venturi meter has an advantage in its use,because it can measure flow without being much affected by the type of the measured fluid or flow conditions.Hence,it has excellent versatility and is being widely applied in many industries.The flow of a liquid containing air is a representative example of a multiphase flow and exhibits complex flow characteristics.In particular,the greater the gas volume fraction（GVF）,the more inhomogeneous the flow becomes.As a result,using a venturi meter to measure the rate of a flow that has a high GVF generates an error.In this study,the cause of the error occurred in measuring the flow rate for the multiphase flow when using the venturi meter for analysis by CFD.To ensure the reliability of this study,the accuracy of the multiphase flow models for numerical analysis was verified through comparison between the calculated results of numerical analysis and the experimental data.As a result,the Grace model,which is a multiphase flow model established by an experiment with water and air,was confirmed to have the highest reliability.Finally,the characteristics of the internal flow Held about the multiphase flow analysis result generated by applying the Grace model were analyzed to find the cause of the uncertainty occurring when measuring the flow rate of the multiphase flow using the venturi meter.A phase separation phenomenon occurred due to a density difference of water and air inside the venturi,and flow inhomogeneity happened according to the flow velocity difference of each phase.It was confirmed that this flow inhomogeneity increased as the GVF increased due to the uncertainty of the flow measurement.

Mixed uncertainty analysis of polycyclic aromatic hydrocarbon inhalation and risk assessment in ambient air of Beijing

This article presents the application of an integrated method that estimates the dispersion of polycyclic aromatic hydrocarbons （PAHs） in air, and assesses the human health risk associated with PAHs inhalation. An uncertainty analysis method consisting of three components were applied in this study, where the three components include a bootstrapping method for analyzing the whole process associated uncertainty, an inhalation rate （IR） representation for evaluating the total PAH inhalation risk for human health, and a normally distributed absorption fraction （AF） ranging from 0% to 100% to represent the absorption capability of PAHs in human body. Using this method, an integrated process was employed to assess the health risk of the residents in Beijing, China, from inhaling PAHs in the air. The results indicate that the ambient air PAHs in Beijing is an important contributor to human health impairment, although over 68% of residents seem to be safe from daily PAH carcinogenic inhalation. In general, the accumulated daily inhalation amount is relatively higher for male and children at 10 years old of age than for female and children at 6 years old. In 1997, about 1.73% cancer sufferers in Beijing were more or less related to ambient air PAHs inhalation. At 95% confidence interval, approximately 272-309 individual cancer incidences can be attributed to PAHs pollution in the air. The probability of greater than 500 cancer occurrence is 15.3%. While the inhalation of ambient air PAHs was shown to be an important factor responsible for higher cancer occurrence in Beijing, while the contribution might not be the most significant one.

Multi-parameters uncertainty analysis of logistic support process based on GERT

The uncertainty analysis is an effective sensitivity analysis method for system model analysis and optimization. However,the existing single-factor uncertainty analysis methods are not well used in the logistic support systems with multiple decision-making factors. The multiple transfer parameters graphical evaluation and review technique（MTP-GERT） is used to model the logistic support process in consideration of two important factors, support activity time and support activity resources, which are two primary causes for the logistic support process uncertainty. On this basis,a global sensitivity analysis（GSA） method based on covariance is designed to analyze the logistic support process uncertainty. The aircraft support process is selected as a case application which illustrates the validity of the proposed method to analyze the support process uncertainty, and some feasible recommendations are proposed for aircraft support decision making on carrier.

Measurement uncertainty in pharmaceutical analysis and its application

The measurement uncertainty provides complete information about an analytical result. This is very important because several decisions of compliance or non-compliance are based on analytical results in pharmaceutical industries. The aim of this work was to evaluate and discuss the estimation of uncertainty in pharmaceutical analysis. The uncertainty is a useful tool in the assessment of compliance or non-compliance of in-process and final pharmaceutical products as well as in the assessment of pharmaceutical equivalence and stability study of drug products.

Sensitivity and Uncertainty Analysis of Regional Marine Ecosystem Services Value

Marine ecosystem services are the benefits which people obtain from the marine ecosystem,including provisioning services,regulating services,cultural services and supporting services. The human species,while buffered against environmental changes by culture and technology,is fundamentally dependent on the flow of ecosystem services. Marine ecosystem services become increasingly valuable as the terrestrial resources become scarce. The value of marine ecosystem services is the monetary flow of ecosystem services on specific temporal and spatial scales,which often changes due to the variation of the goods prices,yields and the status of marine exploitation. Sensitivity analysis is to study the relationship between the value of marine ecosystem services and the main factors which affect it. Uncertainty analysis based on varying prices,yields and status of marine exploitation was carried out. Through uncertainty analysis,a more credible value range instead of a fixed value of marine ecosystem services was obtained in this study. Moreover,sensitivity analysis of the marine ecosystem services value revealed the relative importance of different factors.

Uncertainty analysis of runoff and sedimentation in a forested watershed using sequential uncertainty fitting method

The Soil and Water Assessment Tool （SWAT） was implemented in a small forested watershed of the Soan River Basin innorthern Pakistan through application of the sequential uncertainty fitting （SUFI-2） method to investigate the associateduncertainty in runoff and sediment load estimation. The model was calibrated for a 10-year period （1991–2000） with aninitial 4-year warm-up period （1987–1990）, and was validated for the subsequent 10-year period （2001–2010）. Themodel evaluation indices R2 （the coefficient of determination）, NS （the Nash-Sutcliffe efficiency）, and PBIAS （percentbias） for stream flows simulation indicated that there was a good agreement between the measured and simulated flows.To assess the uncertainty in the model outputs, p-factor （a 95% prediction uncertainty, 95PPU） and r-factors （averagewideness width of the 95PPU band divided by the standard deviation of the observed values） were taken into account.The 95PPU band bracketed 72% of the observed data during the calibration and 67% during the validation. The r-factorwas 0.81 during the calibration and 0.68 during the validation. For monthly sediment yield, the model evaluation coefficients（R2 and NS） for the calibration were computed as 0.81 and 0.79, respectively; for validation, they were 0.78and 0.74, respectively. Meanwhile, the 95PPU covered more than 60% of the observed sediment data during calibrationand validation. Moreover, improved model prediction and parameter estimation were observed with the increasednumber of iterations. However, the model performance became worse after the fourth iterations due to an unreasonableparameter estimation. Overall results indicated the applicability of the SWAT model with moderate levels of uncertaintyduring the calibration and high levels during the validation. Thus, this calibrated SWAT model can be used for assessmentof water balance components, climate change studies, and land use management practices.

Uncertainty analysis of seawater intrusion forecasting

In order to describe the importance of uncertainty analysis in seawater intrusion forecasting and identify the main factors that might cause great differences in prediction results, we analyzed the influence of sea level rise, tidal effect, the seasonal variance of influx, and the annual variance of the pumping rate, as well as combinations of different parameters. The results show that the most important factors that might cause great differences in seawater intrusion distance are the variance of pumping rate and combinations of different parameters. The influence of sea level rise can be neglected in a short-time simulation （ten years, for instance）. Retardation of seawater intrusion caused by tidal effects is obviously important in aquifers near the coastline, but the influence decreases with distance away from the coastline and depth away from the seabed. The intrusion distance can reach a dynamic equilibrium with the application of the sine function for seasonal effects of influx. As a conclusion, we suggest that uncertainty analysis should be considered in seawater intrusion forecasting, if possible.

Application of SWAT Model to the Olifants Basin: Calibration, Validation and Uncertainty Analysis

The application of the Soil and Water Assessment Tool (SWAT) to the Olifants Basin in South Africa was the focus of our study with emphasis on calibration, validation and uncertainty analysis. The Basin was discretized into 23 sub-basins and 226 Hydrologic Response Units (HRUs) using 3 arc second (90 m × 90 m) pixel resolution SRTM DEM with stream gauge B7H015 as the Basin outlet. Observed stream flow data at B7H015 were used for model calibration (1988-2001) and validation (2002-2013) using the split sample approach. Relative global sensitivity analysis using SUFI-2 algorithm was used to determine sensitive parameters to stream flow for calibration of the model. Performance efficiency of the Olifants SWAT model was assessed using Nash-Sutcliffe (NSE), coefficient of determination (R2), Percent Bias (PBIAS) and Root Mean Square Error-Observation Standard deviation Ratio (RSR). Sensitivity analysis revealed in decreasing order of significance, runoff curve number (CN2), alpha bank factor (ALPHA_BNK), soil evaporation compensation factor (ESCO), soil available water capacity (SOIL_AWC, mm H2O/mm soil), groundwater delay (GW_ DELAY, days) and groundwater “revap” coefficient (GW_REVAP) to be the most sensitive parameters to stream flow. Analysis of the model during the calibration period gave the following statistics;NSE = 0.88;R2 = 0.89;PBIAS = -11.49%;RSR = 0.34. On the other hand, statistics during the validation period were NSE = 0.67;R2 = 0.79;PBIAS = -20.69%;RSR = 0.57. The observed statistics indicate the applicability of the SWAT model in simulating the hydrology of the Olifants Basin and therefore can be used as a Decision Support Tool (DST) by water managers and other relevant decisions making bodies to influence policy directions on the management of watershed processes especially water resources.

Uncertainty Analysis for Ship-Bank Interaction Tests in A Circulating Water Channel

This paper presents a systematic model test program to assess the uncertainty of the ship-bank interaction forces,using the planar motion mechanism(PMM)system in a circulating water channel(CWC).Therefore,the uncertainties due to ship-bank distance and water depth are considered,and they are calculated via the partial differentials of the regression formulae based on the test data.The general part of the uncertainty analysis(UA)is performed according to the ITTC recommended procedure 7.5-02-06.04,while the uncertainty of speed is identified as the bias limit due to the flow velocity maldistribution in the CWC.In each example test for the UA of ship-bank interaction forces,12 repeated measurements were conducted.Results from the UA show that the contribution of water depth error and flow velocity maldistribution to the total uncertainty is noticeable,and the paper explains how they increase with the change of the test conditions.The present study will be useful in understanding the uncertainty regarding the ship-bank interaction force measurement in a CWC.

Determination and stability analysis of ultimate open-pit slope under geomechanical uncertainty

In open-pit mines,pit slope as one of the important parameters affects the mine economy and total minable reserve,and it is also affected by different uncertainties which arising from many sources.One of the most critical sources of uncertainty effects on the pit slope design is rock mass geomechanical properties.By comparing the probability of failure resulted from deterministic procedure and probabilistic one,this paper investigated the effects of aforesaid uncertainties on open-pit slope stability in metal mines.In this way,to reduce the effect of variance,it implemented Latin Hypercube Sampling(LHS)technique.Furthermore,a hypothesis test was exerted to compare the effects on two cases in Middle East.Subsequently,the investigation approved high influence of geomechanical uncertainties on overall pit steepness and stability in both iron and copper mines,though on the first case the effects were just over.

Reliability Analysis for Complex Systems Based on Dynamic Evidential Network Considering Epistemic Uncertainty

Fault tolerant technology has greatly improved the reliability of modern systems on one hand and makes their failure mechanisms more complex on the other.The characteristics of dynamics of failure,diversity of distribution and epistemic uncertainty always exist in these systems,which increase the challenges in the reliability assessment of these systems significantly.This paper presents a novel reliability analysis framework for complex systems within which the failure rates of components are expressed in interval numbers.Specifically,it uses a dynamic fault tree(DFT)to model the dynamic fault behaviors and copes with the epistemic uncertainty using Dempster-Shafer(D-S)theory and interval numbers.Furthermore,an approach is presented to convert a DFT into a dynamic evidential network(DEN)to calculate the reliability parameters.Additionally,a sorting method based on the possibility degree is proposed to rank the importance of components represented by interval numbers in order to obtain the most critical components,which can be used to provide the guidance for system design,maintenance planning and fault diagnosis.Finally,a numerical example is provided to illustrate the availability and efficiency of the proposed method.

An Uncertainty Analysis Method for Artillery Dynamics with Hybrid Stochastic and Interval Parameters

This paper proposes a non-intrusive uncertainty analysis method for artillery dynamics involving hybrid uncertainty using polynomial chaos expansion(PCE).The uncertainty parameters with sufficient information are regarded as stochastic variables,whereas the interval variables are used to treat the uncertainty parameters with limited stochastic knowledge.In this method,the PCE model is constructed through the Galerkin projection method,in which the sparse grid strategy is used to generate the integral points and the corresponding integral weights.Through the sampling in PCE,the original dynamic systems with hybrid stochastic and interval parameters can be transformed into deterministic dynamic systems,without changing their expressions.The yielded PCE model is utilized as a computationally efficient,surrogate model,and the supremum and infimum of the dynamic responses over all time iteration steps can be easily approximated through Monte Carlo simulation and percentile difference.A numerical example and an artillery exterior ballistic dynamics model are used to illustrate the feasibility and efficiency of this approach.The numerical results indicate that the dynamic response bounds obtained by the PCE approach almost match the results of the direct Monte Carlo simulation,but the computational efficiency of the PCE approach is much higher than direct Monte Carlo simulation.Moreover,the proposed method also exhibits fine precision even in high-dimensional uncertainty analysis problems.