Uncertainty Propagation of Structural Computation for Fatigue Assessment

Offshore wind substations are subjected to uncertain loads from waves,wind and currents.Sea states are composed of irregular waves which statistics are usually characterized.Irregular loads may induce fatigue failure of some structural components of the structures.By combining fatigue damage computed through numerical simulations for each sea state endured by the structure,it is possible to assess fatigue failure of the structure over the whole deployment duration.Yet,the influence of the discretization error on the fatigue damage is rarely addressed.It is possible to estimate the discretization error on the quantity of interest computed at the structural detail suspected to fail.However,the relation between this local quantity of interest and the fatigue damage is complex.In this paper,a method that allows propagating error bounds towards fatigue damage is proposed.While increasing computational burden,computing discretization error bounds is a useful output of finite element analysis.It can be utilized to either validate mesh choice or guide remeshing in case where potential error on the fatigue damage is too large.This method is applied to an offshore wind substation developped by Chantiers de l’Atlantique using two discretization error estimators in a single sea state.

Measurement of neutron capture cross section on^(71)Ga at 2.15 and 3.19 MeV and uncertainty propagation and covariance analysis

The cross section values of the^(71)Ga(n,γ)^(72)Ga reaction are measured,which are 9.14±0.81 mb and 5.74±0.50 mb at 2.15 and 3.19 MeV,respectively.The detailed uncertainty propagation and covariance analysis are also given.The^(7)Li(p,n)^(7)Be reaction was used to generate the neutrons,and the neutron flux was normalized using the^(115)In(n,n′)^(115)In^(m)monitor reaction.The measured cross section data are compared with the data available in the EXFOR database,the data obtained using nuclear reaction model codes EMPIRE-3.2 and TALYS-1.95,and also the evaluated nuclear data from ENDF/B-VIII.0 and JEFF-3.1/A.The comparison shows that our result at 3.19 MeV is in good agreement with those of EMPIRE-3.2 and JEFF-3.1/A.Since there are no other measurements available at3.19 MeV,our data could not be compared with literature data at 3.19 MeV,but they are consistent with the cross section values available at 2.98±0.26 and 3.0±0.1 MeV.Our result at 2.15 MeV is slightly higher than the literature value available in EXFOR,evaluated value,and theoretically predicted result.

Analytical configuration uncertainty propagation of geocentric interferometric detection constellation

Long-term configuration stability is essential for an interferometric detection constellation(IDC),which is closely related to initial uncertainty.Therefore,it is vital to evaluate the uncertainty and characterize the configuration stability.In this study,an analytical method was developed for the configuration uncertainty propagation of a geocentric triangular IDC.The angular momentum and the argument latitude were found to be significantly affected by the initial uncertainty and were selected as the core variables.By averaging the perturbation in one revolution,an analytical solution was proposed for propagating the core orbital elements in one revolution.Subsequently,the analytical solution of the orbit elements during the mission period is obtained by multiplying the solutions in iterative revolutions.The relationship between the selected orbital elements and the configuration stability parameters was established using an analytical solution.The effects of the initial uncertainty in different directions on the configuration and stable domains were studied.Simulations show that the developed method is highly efficient and accurate in predicting the configuration stability.The relative error with respect to the Monte Carlo simulations was less than 3%with a time consumption of 0.1%.The proposed method can potentially be useful for constellation design and stability analysis.

Acoustic propagation uncertainty in internal wave environments using an ocean-acoustic joint model

An ocean-acoustic joint model is developed for research of acoustic propagation uncertainty in internal wave environments.The internal waves are numerically produced by tidal forcing over a continental slope using an ocean model.Three parameters(i.e.,internal wave,source depth,and water depth)contribute to the dynamic waveguide environments,and result in stochastic sound fields.The sensitivity of the transmission loss(TL)to environment parameters,statistical characteristics of the TL variation,and the associated physical mechanisms are investigated by the Sobol sensitivity analysis method,the Monte Carlo sampling,and the coupled normal mode theory,respectively.The results show that the TL is most sensitive to the source depth in the near field,resulted from the initial amplitudes of higher-order modes;while in middle and far fields,the internal waves are responsible for more than 80%of the total acoustic propagation contribution.In addition,the standard deviation of the TL in the near field and the shallow layer is smaller than those in the middle and far fields and the deep layer.

Erratum:Measurement of(n,α)and(n,2n)reaction cross sections at a neutron energy of 14.92±0.02 MeV for potassium and copper with uncertainty propagation,[A.Gandhi,Aman Sharma,Rebecca Pachuau et al.,Chin.Phys.C 46,014002(2022)]

Experimentally measured neutron activation cross sections are presented for the^(65)Cu(n,α)^(62m)Cu,^(41)K(n,α)^(38)Cl,and^(65)Cu(n,2n)^(64)Cu reactions with detailed uncertainty propagation.The neutron cross sections were measured at an incident energy of 14.92±0.02 MeV,and the neutrons were based on the t(d,n)αfusion reaction.The^(27)Al(n,α)^(24)Na reaction was used as a reference reaction for the normalization of the neutron flux.The pre-calibrated lead-shielded HPGe detector was used to detect the residues'γ-ray spectra.The data from the measured cross sections are compared to the previously measured cross sections from the EXFOR database,theoretically calculated cross sections using the TALYS and EMPIRE codes,and evaluated nuclear data.

Dynamic system uncertainty propagation using polynomial chaos

The classic polynomial chaos method（PCM）, characterized as an intrusive methodology,has been applied to uncertainty propagation（UP） in many dynamic systems. However, the intrusive polynomial chaos method（IPCM） requires tedious modification of the governing equations, which might introduce errors and can be impractical. Alternative to IPCM, the non-intrusive polynomial chaos method（NIPCM） that avoids such modifications has been developed. In spite of the frequent application to dynamic problems, almost all the existing works about NIPCM for dynamic UP fail to elaborate the implementation process in a straightforward way, which is important to readers who are unfamiliar with the mathematics of the polynomial chaos theory. Meanwhile, very few works have compared NIPCM to IPCM in terms of their merits and applicability. Therefore, the mathematic procedure of dynamic UP via both methods considering parametric and initial condition uncertainties are comparatively discussed and studied in the present paper. Comparison of accuracy and efficiency in statistic moment estimation is made by applying the two methods to several dynamic UP problems. The relative merits of both approaches are discussed and summarized. The detailed description and insights gained with the two methods through this work are expected to be helpful to engineering designers in solving dynamic UP problems.

Nonlinear semi-analytical uncertainty propagation of trajectory under impulsive maneuvers

The usage of state transition tensors(STTs)was proved as an effective method for orbital uncertainty propagation.However,orbital maneuvers and their uncertainties are not considered in current STT-based methods.Uncertainty propagation of spacecraft trajectory with maneuvers plays an important role in spaceflight missions,e.g.,the rendezvous phasing mission.Under the effects of impulsive maneuvers,the nominal trajectory of a spacecraft will be divided into several segments.If the uncertainty is piecewise propagated using the STTs one after another,large approximation errors will be introduced.To overcome this challenge,a set of modified STTs is derived,which connects the segmented trajectories together and allows for directly propagating uncertainty from the initial time to the final time.These modified STTs are then applied to analytically propagate the statistical moments of navigation and impulsive maneuver uncertainties.The probability density function is obtained by combining STTs with the Gaussian mixture model.The proposed uncertainty propagator is shown to be efficient and affords good agreement with Monte Carlo simulations.It also has no dimensionality problem for high-dimensional uncertainty propagation.

A novel stochastic collocation method for uncertainty propagation in complex mechanical systems

This paper presents a novel stochastic collocation method based on the equivalent weak form of multivariate function integral to quantify and manage uncertainties in complex mechanical systems. The proposed method, which combines the advantages of the response surface method and the traditional stochastic collocation method, only sets integral points at the guide lines of the response surface. The statistics, in an engineering problem with many uncertain parameters, are then transformed into a linear combination of simple functions' statistics. Furthermore, the issue of determining a simple method to solve the weight-factor sets is discussed in detail. The weight-factor sets of two commonly used probabilistic distribution types are given in table form. Studies on the computational accuracy and efforts show that a good balance in computer capacity is achieved at present. It should be noted that it's a non-gradient and non-intrusive algorithm with strong portability. For the sake of validating the procedure, three numerical examples concerning a mathematical function with analytical expression, structural design of a straight wing, and flutter analysis of a composite wing are used to show the effectiveness of the guided stochastic collocation method.

Measurement of(n,α)and(n,2n)reaction cross sections at a neutron energy of 14.92±0.02 MeV for potassium and copper with uncertainty propagation

Experimentally measured neutron activation cross sections are presented for the ^(65)Cu(n,0)^(62m)Cu,^(41) K(n,a)^(38C)l,and ^(65)Cu(n.2n)^(64)Cu reactions with detailed uncertainty propagation.The neutron cross secions were measured at an incident energy of 14.92±0.02 MeV,and the neutrons were based on the(d,n)a fusion reaction.The ^(27) Al(n,a)^(24)Na reaction was used as a reference reaction for the normalization of the neutron flux.The pre-calib-rated lead-shielded HPGe detector was used to detect the residues'γ-ray spetra.The data from the measured cross sections are compared to the previously measured cross sections from the EXFOR database,theoretically calculated cross sections using the TALYS and EMPIRE codes,and evaluated nuclear data.

Two-stage nested optimization-based uncertainty propagation method for model calibration

Model calibration is the procedure that adjusts the unknown parameters in order to fit the model to experimental data and improve predictive capability.However,it is difficult to implement the procedure because of the aleatory uncertainty.In this paper,a new method of model calibration based on uncertainty propagation is investigated.The calibration process is described as an optimization problem.A two-stage nested uncertainty propagation method is proposed to resolve this problem.Monte Carlo Simulation method is applied for the inner loop to propagate the aleatory uncertainty.Optimization method is applied for the outer loop to propagate the epistemic uncertainty.The optimization objective function is the consistency between the result of the inner loop and the experimental data.Thus,different consistency measurement methods for unary output and multivariate outputs are proposed as the optimization objective function.Finally,the thermal challenge problem is given to validate the reasonableness and effectiveness of the proposed method.

A non-probabilistic reliability topology optimization method based on aggregation function and matrix multiplication considering buckling response constraints

A non-probabilistic reliability topology optimization method is proposed based on the aggregation function and matrix multiplication.The expression of the geometric stiffness matrix is derived,the finite element linear buckling analysis is conducted,and the sensitivity solution of the linear buckling factor is achieved.For a specific problem in linear buckling topology optimization,a Heaviside projection function based on the exponential smooth growth is developed to eliminate the gray cells.The aggregation function method is used to consider the high-order eigenvalues,so as to obtain continuous sensitivity information and refined structural design.With cyclic matrix programming,a fast topology optimization method that can be used to efficiently obtain the unit assembly and sensitivity solution is conducted.To maximize the buckling load,under the constraint of the given buckling load,two types of topological optimization columns are constructed.The variable density method is used to achieve the topology optimization solution along with the moving asymptote optimization algorithm.The vertex method and the matching point method are used to carry out an uncertainty propagation analysis,and the non-probability reliability topology optimization method considering buckling responses is developed based on the transformation of non-probability reliability indices based on the characteristic distance.Finally,the differences in the structural topology optimization under different reliability degrees are illustrated by examples.

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.

Uncertainty evaluation and correlation analysis of single-particle energies in phenomenological nuclear mean field:an investigation into propagating uncertainties for independent model parameters

Based on the Monte Carlo approach and conventional error analysis theory,taking the heaviest doubly magic nucleus 208Pb as an example,we first evaluate the propagated uncertainties of universal potential parameters for three typical types of single-particle energy in the phenomenological Woods–Saxon mean field.Accepting the Woods–Saxon modeling with uncorrelated model parameters,we found that the standard deviations of singleparticle energy obtained through the Monte Carlo simulation and the error propagation rules are in good agreement.It seems that the energy uncertainty of the single-particle levels regularly evoluate with certain quantum numbers to a large extent for the given parameter uncertainties.Further,the correlation properties of the single-particle levels within the domain of input parameter uncertainties are statistically analyzed,for example,with the aid of Pearson’s correlation coefficients.It was found that a positive,negative,or unrelated relationship may appear between two selected single-particle levels,which will be extremely helpful for evaluating the theoretical uncertainty related to the single-particle levels(e.g.,K isomer)in nuclear structural calculations.

Coping with uncertainties through an automated workflow for 3D reservoir modelling of carbonate reservoirs

Reliable 3D modelling of underground hydrocarbon reservoirs is a challenging task due to the complexity of the underground geological formations and to the availability of different types of data that are typically affected by uncertainties. In the case of geologically complex depositional environments, such as fractured hydrocarbon reservoirs, the uncertainties involved in the modelling process demand accurate analysis and quantification in order to provide a reliable confidence range of volumetric estimations. In the present work, we used a 3D model of a fractured carbonate reservoir and populated it with different lithological and petrophysical properties. The available dataset also included a discrete fracture network(DFN) property that was used to model the fracture distribution. Uncertainties affecting lithological facies, their geometry and absolute positions(related to the fault system), fracture distribution and petrophysical properties were accounted for. We included all different types of uncertainties in an automated approach using tools available in today’s modelling software packages and combining all the uncertain input parameters in a series of statistically representative geological realizations. In particular, we defined a specific workflow for the definition of the absolute permeability according to an equivalent, single porosity approach, taking into account the contribution of both the matrix and the fracture system. The results of the analyses were transferred into a 3D numerical fluid-dynamic simulator to evaluate the propagation of the uncertainties associated to the input data down to the final results, and to assess the dynamic response of the reservoir following a selected development plan. The "integrated approach" presented in this paper can be useful for all technicians involved in the construction and validation of 3D numerical models of hydrocarbon-bearing reservoirs and can potentially become part of the educational training for young geoscientists and engineers, since an integrated and well-constructed workflow is the backbone of any reservoir study.

Interval Arithmetic and Its Application to Electrical Circuits

Interval arithmetic is an elegant tool for practical work with inequalities, approximate numbers, error bounds, and more generally with certain convex and bounded sets. In this section we give a number of simple examples showing where intervals and ranges of functions over intervals arise naturally. Interval mathematics is a generalization in which interval numbers replace real numbers, interval arithmetic replaces real arithmetic, and interval analysis replaces real analysis. Interval is limited by two bounds： lower bound and upper bound. The present paper introduces some of the basic notions and techniques from interval analysis needed in the sequel for presenting various uses of interval analysis in electric circuit theory and its applications. In this article we address the representation of uncertain and imprecise information, the interval arithmetic and its application to electrical circuits.

Fitting dynamic models to epidemic outbreaks with quantified uncertainty:A primer for parameter uncertainty,identifiability,and forecasts

Mathematical models provide a quantitative framework with which scientists can assess hypotheses on the potential underlying mechanisms that explain patterns in observed data at different spatial and temporal scales,generate estimates of key kinetic parameters,assess the impact of interventions,optimize the impact of control strategies,and generate forecasts.We review and illustrate a simple data assimilation framework for calibrating mathematical models based on ordinary differential equation models using time series data describing the temporal progression of case counts relating,for instance,to population growth or infectious disease transmission dynamics.In contrast to Bayesian estimation approaches that always raise the question of how to set priors for the parameters,this frequentist approach relies on modeling the error structure in the data.We discuss issues related to parameter identifiability,uncertainty quantification and propagation as well as model performance and forecasts along examples based on phenomenological and mechanistic models parameterized using simulated and real datasets.

Input database related uncertainty of Biome-BGCMuSo agroenvironmental model outputs

Gridded model assessments require at least one climatic and one soil database for carrying out the simulations.There are several parallel soil and climate database development projects that provide sufficient,albeit considerably different,observation based input data for crop model based impact studies.The input database related uncertainty of the Biome-BGCMuSo agro-environmental model outputs was investigated using three and four different gridded climatic and soil databases,respectively covering an area of nearly 100.000 km2 with 1104 grid cells.Spatial,temporal,climate and soil database selection related variances were calculated and compared for four model outputs obtained from 30-year-long simulations.The choice of the input database introduced model output variability that was comparable to the variability the year-to-year change of the weather or the spatial heterogeneity of the soil causes.Input database selection could be a decisive factor in carbon sequestration related studies as the soil carbon stock change estimates may either suggest that the simulated ecosystem is a carbon sink or to the contrary a carbon source on the long run.Careful evaluation of the input database quality seems to be an inevitable and highly relevant step towards more realistic plant production and carbon balance simulations.

Uncertainty analysis for integrated water system simulations using GLUE with different acceptability thresholds

Integrated water quantity and quality simulations have become a popular tool in investigations on global water crisis.For integrated and complex models,conventional uncertainty estimations focus on the uncertainties of individual modules,e.g.,module parameters and structures,and do not consider the uncertainties propagated from interconnected modules.Therefore,this study investigated all the uncertainties of integrated water system simulations using the GLUE(i.e.,generalized likelihood uncertainty estimation)method,including uncertainties associated with individual modules,propagated uncertainties associated with interconnected modules,and their combinations.The changes in both acceptability thresholds of GLUE and the uncertainty estimation results were also investigated for different fixed percentages of total number of iterations(100000).Water quantity and quality variables(i.e.,runoff and ammonium nitrogen)were selected for the case study.The results showed that module uncertainty did not affect the runoff simulation performance,but remarkably weakened the water quality responses as the fixed percentage increased during calibration and validation periods.The propagated uncertainty from hydrological modules could not be ignored for water quality simulations,particularly during validation.The combination of module and propagated uncertainties further weakened the water quality simulation performance.The uncertainty intervals became wider owing to an increase in the fixed percentages and introduction of more uncertainty sources.Moreover,the acceptability threshold had a negative nonlinear relationship with the fixed percentage.The fixed percentages(20.0%-30.0%)were proposed as the acceptability thresholds owing to the satisfactory simulation performance and noticeably reduced uncertainty intervals they produced.This study provided methodological foundations for estimating multiple uncertainty sources of integrated water system models.

Adjoint-based robust optimization design of laminar flow wing under flight condition uncertainties

It is an inherent uncertainty problem that the application of laminar flow technology to the wing of large passenger aircraft is affected by flight conditions.In order to seek a more robust natural laminar flow control effect,it is necessary to develop an effective optimization design method.Meanwhile,attention must be given to the impact of crossflow(CF)instability brought on by the sweep angle.This paper constructs a robust optimization design framework based on discrete adjoint methods and non-intrusive polynomial chaos.Transition prediction is implemented by coupled Reynolds-Averaged Navier-Stokes(RANS)and simplified e^(N)method,which can consider both Tollmien-Schlichting(TS)wave and crossflow vortex instability.We have performed gradient enhancement processing on the general Polynomial Chaos Expansion(PCE),which is advantageous to reduce the computational cost of single uncertainty propagation.This processing takes advantage of the gradient information obtained by solving the coupled adjoint equations considering transition.The statistical moment gradient solution used for the robust optimization design also uses the derivatives of coupled adjoint equations.The framework is applied to the robust design of a 25°swept wing with infinite span in transonic flow.The uncertainty quantification and sensitivity analysis on the baseline wing shows that the uncertainty quantification method in this paper has high accuracy,and qualitatively reveals the factors that dominate in different flow field regions.By the robust optimization design,the mean and standard deviation of the drag coefficient can be reduced by 29%and 45%,respectively,and compared with the deterministic optimization design results,there is less possibility of forming shock waves under flight condition uncertainties.Robust optimization results illustrate the trade-off between the transition delay and the wave drag reduction.

Reliability measure approach considering mixture uncertainties under insufficient input data

Reliability analysis and reliability-based optimization design require accurate measurement of failure probability under input uncertainties.A unified probabilistic reliability measure approach is proposed to calculate the probability of failure and sensitivity indices considering a mixture of uncertainties under insufficient input data.The input uncertainty variables are classified into statistical variables,sparse variables,and interval variables.The conservativeness level of the failure probability is calculated through uncertainty propagation analysis of distribution parameters of sparse variables and auxiliary parameters of interval variables.The design sensitivity of the conservativeness level of the failure probability at design points is derived using a semi-analysis and sampling-based method.The proposed unified reliability measure method is extended to consider p-box variables,multi-domain variables,and evidence theory variables.Numerical and engineering examples demonstrate the effectiveness of the proposed method,which can obtain an accurate confidence level of reliability index and sensitivity indices with lower function evaluation number.