This paper presents a new computational method for forward uncertainty quantification(UQ)analyses on large-scale structural systems in the presence of arbitrary and dependent random inputs.The method consists of a gen...This paper presents a new computational method for forward uncertainty quantification(UQ)analyses on large-scale structural systems in the presence of arbitrary and dependent random inputs.The method consists of a generalized polynomial chaos expansion(GPCE)for statistical moment and reliability analyses associated with the stochastic output and a static reanalysis method to generate the input-output data set.In the reanalysis,we employ substructuring for a structure to isolate its local regions that vary due to random inputs.This allows for avoiding repeated computations of invariant substructures while generating the input-output data set.Combining substructuring with static condensation further improves the computational efficiency of the reanalysis without losing accuracy.Consequently,the GPCE with the static reanalysis method can achieve significant computational saving,thus mitigating the curse of dimensionality to some degree for UQ under high-dimensional inputs.The numerical results obtained from a simple structure indicate that the proposed method for UQ produces accurate solutions more efficiently than the GPCE using full finite element analyses(FEAs).We also demonstrate the efficiency and scalability of the proposed method by executing UQ for a large-scale wing-box structure under ten-dimensional(all-dependent)random inputs.展开更多
In this paper,an adaptive polynomial chaos expansion method(PCE)based on the method of moments(MoM)is proposed to construct surrogate models for electromagnetic scattering and further sensitivity analysis.The MoM is a...In this paper,an adaptive polynomial chaos expansion method(PCE)based on the method of moments(MoM)is proposed to construct surrogate models for electromagnetic scattering and further sensitivity analysis.The MoM is applied to accurately solve the electric field integral equation(EFIE)of electromagnetic scattering from homogeneous dielectric targets.Within the bistatic radar cross section(RCS)as the research object,the adaptive PCE algorithm is devoted to selecting the appropriate order to construct the multivariate surrogate model.The corresponding sensitivity results are given by the further derivative operation,which is compared with those of the finite difference method(FDM).Several examples are provided to demonstrate the effectiveness of the proposed algorithm for sensitivity analysis of electromagnetic scattering from homogeneous dielectric targets.展开更多
The uncertainty quantification of flows around a cylinder is studied by the non-intrusive polynomial chaos method. Based on the validation with benchmark results, discussions are mainly focused on the statistic proper...The uncertainty quantification of flows around a cylinder is studied by the non-intrusive polynomial chaos method. Based on the validation with benchmark results, discussions are mainly focused on the statistic properties of the peak lift and drag coefficients and base pressure drop over the cylinder with the uncertainties of viscosity coefficient and inflow boundary velocity. As for the numerical results of flows around a cylinder, influence of the inflow boundary velocity uncertainty is larger than that of viscosity. The results indeed demonstrate that a five-order degree of polynomial chaos expansion is enough to represent the solution of flow in this study.展开更多
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 researc...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.展开更多
One of the open problems in the field of forward uncertainty quantification(UQ)is the ability to form accurate assessments of uncertainty having only incomplete information about the distribution of random inputs.Anot...One of the open problems in the field of forward uncertainty quantification(UQ)is the ability to form accurate assessments of uncertainty having only incomplete information about the distribution of random inputs.Another challenge is to efficiently make use of limited training data for UQ predictions of complex engineering problems,particularly with high dimensional random parameters.We address these challenges by combining data-driven polynomial chaos expansions with a recently developed preconditioned sparse approximation approach for UQ problems.The first task in this two-step process is to employ the procedure developed in[1]to construct an"arbitrary"polynomial chaos expansion basis using a finite number of statistical moments of the random inputs.The second step is a novel procedure to effect sparse approximation via l1 minimization in order to quantify the forward uncertainty.To enhance the performance of the preconditioned l1 minimization problem,we sample from the so-called induced distribution,instead of using Monte Carlo(MC)sampling from the original,unknown probability measure.We demonstrate on test problems that induced sampling is a competitive and often better choice compared with sampling from asymptotically optimal measures(such as the equilibrium measure)when we have incomplete information about the distribution.We demonstrate the capacity of the proposed induced sampling algorithm via sparse representation with limited data on test functions,and on a Kirchoff plating bending problem with random Young’s modulus.展开更多
Face stability is an essential issue in tunnel design and construction.Layered rock masses are typical and ubiquitous;uncertainties in rock properties always exist.In view of this,a comprehensive method,which combines...Face stability is an essential issue in tunnel design and construction.Layered rock masses are typical and ubiquitous;uncertainties in rock properties always exist.In view of this,a comprehensive method,which combines the Upper bound Limit analysis of Tunnel face stability,the Polynomial Chaos Kriging,the Monte-Carlo Simulation and Analysis of Covariance method(ULT-PCK-MA),is proposed to investigate the seismic stability of tunnel faces.A two-dimensional analytical model of ULT is developed to evaluate the virtual support force based on the upper bound limit analysis.An efficient probabilistic analysis method PCK-MA based on the adaptive Polynomial Chaos Kriging metamodel is then implemented to investigate the parameter uncertainty effects.Ten input parameters,including geological strength indices,uniaxial compressive strengths and constants for three rock formations,and the horizontal seismic coefficients,are treated as random variables.The effects of these parameter uncertainties on the failure probability and sensitivity indices are discussed.In addition,the effects of weak layer position,the middle layer thickness and quality,the tunnel diameter,the parameters correlation,and the seismic loadings are investigated,respectively.The results show that the layer distributions significantly influence the tunnel face probabilistic stability,particularly when the weak rock is present in the bottom layer.The efficiency of the proposed ULT-PCK-MA is validated,which is expected to facilitate the engineering design and construction.展开更多
Polynomial chaos expansions(PCEs)have been used in many real-world engineering applications to quantify how the uncertainty of an output is propagated from inputs by decomposing the output in terms of polynomials of t...Polynomial chaos expansions(PCEs)have been used in many real-world engineering applications to quantify how the uncertainty of an output is propagated from inputs by decomposing the output in terms of polynomials of the inputs.PCEs for models with independent inputs have been extensively explored in the literature.Recently,different approaches have been proposed for models with dependent inputs to expand the use of PCEs to more real-world applications.Typical approaches include building PCEs based on the Gram–Schmidt algorithm or transforming the dependent inputs into independent inputs.However,the two approaches have their limitations regarding computational efficiency and additional assumptions about the input distributions,respectively.In this paper,we propose a data-driven approach to build sparse PCEs for models with dependent inputs without any distributional assumptions.The proposed algorithm recursively constructs orthonormal polynomials using a set of monomials based on their correlations with the output.The proposed algorithm on building sparse PCEs not only reduces the number of minimally required observations but also improves the numerical stability and computational efficiency.Four numerical examples are implemented to validate the proposed algorithm.The source code is made publicly available for reproducibility.展开更多
As rainfall infiltrates into soil slopes,the hydraulic and mechanical behaviors of soils are interacted.In this study,an efficient probabilistic parameter estimation method for coupled hydro-mechanical behavior in soi...As rainfall infiltrates into soil slopes,the hydraulic and mechanical behaviors of soils are interacted.In this study,an efficient probabilistic parameter estimation method for coupled hydro-mechanical behavior in soil slope is proposed.This method integrates the Polynomial Chaos Expansion(PCE)method,the coupled hydro-mechanical modeling,and the Bayesian learning method.A coupled hydro-mechanical numerical model is established for the simulation of behaviors of unsaturated soil slope under rainfall infiltration,following by training a cheap-to-run PCE surrogate to replace it.Probabilistic estimation of soil parameters is conducted based on the Bayesian learning technique with the Markov Chain Monte Carlo(MCMC)simulation.A numerical example of an unsaturated slope under rainfall infiltration is presented to illustrate the proposed method.The effects of measurement durations and response types on parameter estimation are addressed.The result shows that with the increase of measurement duration,the uncertainties of soil parameters are significantly reduced.The uncertainties of hydraulic properties are reduced significantly using the pore water pressure data,while the uncertainties of soil strength parameters are reduced greatly using the measured displacement data.展开更多
The uncertainties can generate fluctuations with aerodynamic characteristics. Uncertainty Quantification(UQ) is applied to compute its impact on the aerodynamic characteristics.In addition, the contribution of each ...The uncertainties can generate fluctuations with aerodynamic characteristics. Uncertainty Quantification(UQ) is applied to compute its impact on the aerodynamic characteristics.In addition, the contribution of each uncertainty to aerodynamic characteristics should be computed by uncertainty sensitivity analysis. Non-Intrusive Polynomial Chaos(NIPC) has been successfully applied to uncertainty quantification and uncertainty sensitivity analysis. However, the non-intrusive polynomial chaos method becomes inefficient as the number of random variables adopted to describe uncertainties increases. This deficiency becomes significant in stochastic aerodynamic analysis considering the geometric uncertainty because the description of geometric uncertainty generally needs many parameters. To solve the deficiency, a Sparse Grid-based Polynomial Chaos(SGPC) expansion is used to do uncertainty quantification and sensitivity analysis for stochastic aerodynamic analysis considering geometric and operational uncertainties. It is proved that the method is more efficient than non-intrusive polynomial chaos and Monte Carlo Simulation(MSC) method for the stochastic aerodynamic analysis. By uncertainty quantification, it can be learnt that the flow characteristics of shock wave and boundary layer separation are sensitive to the geometric uncertainty in transonic region. The uncertainty sensitivity analysis reveals the individual and coupled effects among the uncertainty parameters.展开更多
A one-dimensional non-intrusive Polynomial Chaos (PC) method is applied in Uncertainty Quantification (UQ) studies for CFD-based ship performances simulations. The uncertainty properties of Expected Value (EV) a...A one-dimensional non-intrusive Polynomial Chaos (PC) method is applied in Uncertainty Quantification (UQ) studies for CFD-based ship performances simulations. The uncertainty properties of Expected Value (EV) and Standard Deviation (SD) are evaluated by solving the PC coefficients from a linear system of algebraic equations. The one-dimensional PC with the Legendre polynomials is applied to: (1) stochastic input domain and (2) Cumulative Distribution Function (CDF) image domain, allowing for more flexibility. The PC method is validated with the Monte-Carlo benchmark results in several high-fidelity, CFD-based, ship UQ problems, evaluating the geometrical, operational and environmental uncertainties for the Delft Catamaran 372. Convergence is studied versus PC order P for both EV and SD, showing that high order PC is not necessary for present applications. Comparison is carried out for PC with/without the least square minimization when solving the PC coefficients. The least square minimization, using larger number of CFD samples, is recommended for current test cases. The study shows the potentials of PC method in Robust Design Optimization (RDO) and Reliability-Based Design Optimization (RBDO) of ship hydrodynamic performances.展开更多
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) r...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.展开更多
Uncertainty is common in the life cycle of an aircraft, and Robust Aerodynamic Optimization(RAO) that considers uncertainty is important in aircraft design. To avoid the curse of dimensionality in surrogate-based opti...Uncertainty is common in the life cycle of an aircraft, and Robust Aerodynamic Optimization(RAO) that considers uncertainty is important in aircraft design. To avoid the curse of dimensionality in surrogate-based optimization, this study proposes an adjoint RAO technique called “R-Opt”. Polynomial Chaos Expansion(PCE) is coupled with the R-Opt technique to quantify uncertainty in the responses of the target(including its mean and standard deviation). Only one process of PCE model construction is required in each iteration, and the gradients of uncertainty can be inferred via chain rules. The proposed method is more efficient than prevalent methods,and avoids the problem of a disagreement over the best PCE basis from among a number of PCE models(especially in case of sparse PCE). It also supports the application of sparse PCE.Two benchmark tests and two airfoil cases were used to verify R-Opt, and the optimal solutions were deemed to be robust. It improved the mean aerodynamic performance and reduced the standard deviation of the target.展开更多
To obtain a universal model solving the uncertain acoustic field in shallow water, a non-intrusive model coupled polynomial chaos expansion (PCE) method with Helmholtz equa- tion is established, in which the polynom...To obtain a universal model solving the uncertain acoustic field in shallow water, a non-intrusive model coupled polynomial chaos expansion (PCE) method with Helmholtz equa- tion is established, in which the polynomial coefficients are solved by probabilistic collocation method (PCM). For the cases of Pekeris waveguide which have uncertainties in depth of water column, in both sound speed profile and depth of water column, and for the case of thermocline with lower limit depth uncertain, probability density functions (PDF) of transmission loss (TL) are calculated. The results show that the proposed model is universal for the acoustic propa- gation codes with high computational efficiency and accuracy, and can be applied to study the uncertainty of acoustic propagation in the shallow water en^-ironment with multiple parameters uncertain.展开更多
This paper is concerned with the application of generalized polynomial chaos (gPC) method to nonlinear random pantograph equations. An error estimation of gPC method is derived. The global error analysis is given fo...This paper is concerned with the application of generalized polynomial chaos (gPC) method to nonlinear random pantograph equations. An error estimation of gPC method is derived. The global error analysis is given for the error arising from finite-dimensional noise (FDN) assumption, projection error, aliasing error and discretization error. In the end, with several numerical experiments, the theoretical results are further illustrated.展开更多
In this paper,we consider the numerical stability of gravity-capillary waves generated by a localized pressure in water of finite depth based on the forced Korteweg-de Vries(FKdV)framework and the polynomial chaos.The...In this paper,we consider the numerical stability of gravity-capillary waves generated by a localized pressure in water of finite depth based on the forced Korteweg-de Vries(FKdV)framework and the polynomial chaos.The stability studies are focused on the symmetric solitary wave for the subcritical flow with the Bond number greater than one third.When its steady symmetric solitarywave-like solutions are randomly perturbed,the evolutions of some waves show stability in time regardless of the randomness while other waves produce unstable fluctuations.By representing the perturbation with a random variable,the governing FKdV equation is interpreted as a stochastic equation.The polynomial chaos expansion of the random solution has been used for the study of stability in two ways.First it allows us to identify the stable solution of the stochastic governing equation.Secondly it is used to construct upper and lower bounding surfaces for unstable solutions,which encompass the fluctuations of waves.展开更多
The polynomial chaos expansion(PCE)is an efficient numerical method for performing a reliability analysis.It relates the output of a nonlinear system with the uncertainty in its input parameters using a multidimension...The polynomial chaos expansion(PCE)is an efficient numerical method for performing a reliability analysis.It relates the output of a nonlinear system with the uncertainty in its input parameters using a multidimensional polynomial approximation(the so-called PCE).Numerically,such an approximation can be obtained by using a regression method with a suitable design of experiments.The cost of this approximation depends on the size of the design of experiments.If the design of experiments is large and the system is modeled with a computationally expensive FEA(Finite Element Analysis)model,the PCE approximation becomes unfeasible.The aim of this work is to propose an algorithm that generates efficiently a design of experiments of a size defined by the user,in order to make the PCE approximation computationally feasible.It is an optimization algorithm that seeks to find the best design of experiments in the D-optimal sense for the PCE.This algorithm is a coupling between genetic algorithms and the Fedorov exchange algorithm.The efficiency of our approach in terms of accuracy and computational time reduction is compared with other existing methods in the case of analytical functions and finite element based functions.展开更多
Robust optimization is an approach for the design of a mechanical structure which takes into account the uncertainties of the design variables.It requires at each iteration the evaluation of some robust measures of th...Robust optimization is an approach for the design of a mechanical structure which takes into account the uncertainties of the design variables.It requires at each iteration the evaluation of some robust measures of the objective function and the constraints.In a previous work,the authors have proposed a method which efficiently generates a design of experiments with respect to the design variable uncertainties to compute the robust measures using the polynomial chaos expansion.This paper extends the proposed method to the case of the robust optimization.The generated design of experiments is used to build a surrogate model for the robust measures over a certain trust region.This leads to a trust region optimization method which only requires one evaluation of the design of experiments per iteration(single loop method).Unlike other single loop methods which are only based on a first order approximation of robust measure of the constraints and which does not handle a robust measure for the objective function,the proposed method can handle any approximation order and any choice for the robust measures.Some numerical experiments based on finite element functions are performed to show the efficiency of the method.展开更多
In an integrated energy system,source-load multiple uncertainties and correlations lead to an over-limit risk in operating state,including voltage,temperature,and pressure over-limit.Therefore,efficient probabilistic ...In an integrated energy system,source-load multiple uncertainties and correlations lead to an over-limit risk in operating state,including voltage,temperature,and pressure over-limit.Therefore,efficient probabilistic energy flow calculation methods and risk assessment theories applicable to integrated energy systems are crucial.This study proposed a probabilistic energy flow calculation method based on polynomial chaos expansion for an electric-heat-gas integrated energy system.The method accurately and efficiently calculated the over-limit probability of the system state variables,considering the coupling conditions of electricity,heat,and gas,as well as uncertainties and correlations in renewable energy unit outputs and multiple types of loads.To further evaluate and quantify the impact of uncertainty factors on the over-limit risk,a global sensitivity analysis method for the integrated energy system based on the analysis of covariance theory is proposed.This method considered the source-load correlation and aimed to identify the key uncertainty factors that influence stable operation.Simulation results demonstrated that the proposed method achieved accuracy to that of the Monte Carlo method while significantly reducing calculation time.It effectively quantified the over-limit risk under the presence of multiple source-load uncertainties.展开更多
An efficient and accurate uncertainty propagation methodology for mechanics problems with random fields is developed in this paper. This methodology is based on the stochastic response surface method (SRSM) which ha...An efficient and accurate uncertainty propagation methodology for mechanics problems with random fields is developed in this paper. This methodology is based on the stochastic response surface method (SRSM) which has been previously proposed for problems dealing with random variables only. This paper extends SRSM to problems involving random fields or random processes fields. The favorable property of SRSM lies in that the deterministic computational model can be treated as a black box, as in the case of commercial finite element codes. Numerical examples are used to highlight the features of this technique and to demonstrate the accuracy and efficiency of the proposed method. A comparison with Monte Carlo simulation shows that the proposed method can achieve numerical results close to those from Monte Carlo simulation while dramatically reducing the number of deterministic finite element runs.展开更多
Uncertainties in structure properties can result in different responses in hybrid simulations. Quantification of the effect of these tmcertainties would enable researchers to estimate the variances of structural respo...Uncertainties in structure properties can result in different responses in hybrid simulations. Quantification of the effect of these tmcertainties would enable researchers to estimate the variances of structural responses observed from experiments. This poses challenges for real-time hybrid simulation (RTHS) due to the existence of actuator delay. Polynomial chaos expansion (PCE) projects the model outputs on a basis of orthogonal stochastic polynomials to account for influences of model uncertainties. In this paper, PCE is utilized to evaluate effect of actuator delay on the maximum displacement from real-time hybrid simulation of a single degree of freedom (SDOF) structure when accounting for uncertainties in structural properties. The PCE is first applied for RTHS without delay to determine the order of PCE, the number of sample points as well as the method for coefficients calculation. The PCE is then applied to RTHS with actuator delay. The mean, variance and Sobol indices are compared and discussed to evaluate the effects of actuator delay on uncertainty quantification for RTHS. Results show that the mean and the variance of the maximum displacement increase linearly and exponentially with respect to actuator delay, respectively. Sensitivity analysis through Sobol indices also indicates the influence of the single random variable decreases while the coupling effect increases with the increase of actuator delay.展开更多
基金Project supported by the National Research Foundation of Korea(Nos.NRF-2020R1C1C1011970 and NRF-2018R1A5A7023490)。
文摘This paper presents a new computational method for forward uncertainty quantification(UQ)analyses on large-scale structural systems in the presence of arbitrary and dependent random inputs.The method consists of a generalized polynomial chaos expansion(GPCE)for statistical moment and reliability analyses associated with the stochastic output and a static reanalysis method to generate the input-output data set.In the reanalysis,we employ substructuring for a structure to isolate its local regions that vary due to random inputs.This allows for avoiding repeated computations of invariant substructures while generating the input-output data set.Combining substructuring with static condensation further improves the computational efficiency of the reanalysis without losing accuracy.Consequently,the GPCE with the static reanalysis method can achieve significant computational saving,thus mitigating the curse of dimensionality to some degree for UQ under high-dimensional inputs.The numerical results obtained from a simple structure indicate that the proposed method for UQ produces accurate solutions more efficiently than the GPCE using full finite element analyses(FEAs).We also demonstrate the efficiency and scalability of the proposed method by executing UQ for a large-scale wing-box structure under ten-dimensional(all-dependent)random inputs.
基金supported by the Young Scientists Fund of the National Natural Science Foundation of China(No.62102444)a Major Research Project in Higher Education Institutions in Henan Province(No.23A560015).
文摘In this paper,an adaptive polynomial chaos expansion method(PCE)based on the method of moments(MoM)is proposed to construct surrogate models for electromagnetic scattering and further sensitivity analysis.The MoM is applied to accurately solve the electric field integral equation(EFIE)of electromagnetic scattering from homogeneous dielectric targets.Within the bistatic radar cross section(RCS)as the research object,the adaptive PCE algorithm is devoted to selecting the appropriate order to construct the multivariate surrogate model.The corresponding sensitivity results are given by the further derivative operation,which is compared with those of the finite difference method(FDM).Several examples are provided to demonstrate the effectiveness of the proposed algorithm for sensitivity analysis of electromagnetic scattering from homogeneous dielectric targets.
基金Supported by the National Natural Science Foundation of China under Grant No 11371069the Young Foundation of Institute of Applied Physics and Computational Mathematics under Grant No ZYSZ1518-13the Science Foundation of China Academy of Engineering Physics under Grant No 2013A0101004
文摘The uncertainty quantification of flows around a cylinder is studied by the non-intrusive polynomial chaos method. Based on the validation with benchmark results, discussions are mainly focused on the statistic properties of the peak lift and drag coefficients and base pressure drop over the cylinder with the uncertainties of viscosity coefficient and inflow boundary velocity. As for the numerical results of flows around a cylinder, influence of the inflow boundary velocity uncertainty is larger than that of viscosity. The results indeed demonstrate that a five-order degree of polynomial chaos expansion is enough to represent the solution of flow in this study.
基金Supported by National Natural Science Foundation of China(Grant No.51875256)Open Platform Fund of Hunan Institute of Technology of China(Grant No.KFA20009)Hong Kong,Macao and Taiwan Science and Technology Cooperation Project in Jiangsu Province of China(Grant No.BZ2020050)。
文摘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.
基金supported by the NSF of China(No.11671265)partially supported by NSF DMS-1848508+4 种基金partially supported by the NSF of China(under grant numbers 11688101,11571351,and 11731006)science challenge project(No.TZ2018001)the youth innovation promotion association(CAS)supported by the National Science Foundation under Grant No.DMS-1439786the Simons Foundation Grant No.50736。
文摘One of the open problems in the field of forward uncertainty quantification(UQ)is the ability to form accurate assessments of uncertainty having only incomplete information about the distribution of random inputs.Another challenge is to efficiently make use of limited training data for UQ predictions of complex engineering problems,particularly with high dimensional random parameters.We address these challenges by combining data-driven polynomial chaos expansions with a recently developed preconditioned sparse approximation approach for UQ problems.The first task in this two-step process is to employ the procedure developed in[1]to construct an"arbitrary"polynomial chaos expansion basis using a finite number of statistical moments of the random inputs.The second step is a novel procedure to effect sparse approximation via l1 minimization in order to quantify the forward uncertainty.To enhance the performance of the preconditioned l1 minimization problem,we sample from the so-called induced distribution,instead of using Monte Carlo(MC)sampling from the original,unknown probability measure.We demonstrate on test problems that induced sampling is a competitive and often better choice compared with sampling from asymptotically optimal measures(such as the equilibrium measure)when we have incomplete information about the distribution.We demonstrate the capacity of the proposed induced sampling algorithm via sparse representation with limited data on test functions,and on a Kirchoff plating bending problem with random Young’s modulus.
基金supported by Science and Technology Project of Yunnan Provincial Transportation Department(Grant No.25 of 2018)the National Natural Science Foundation of China(Grant No.52279107)The authors are grateful for the support by the China Scholarship Council(CSC No.202206260203 and No.201906690049).
文摘Face stability is an essential issue in tunnel design and construction.Layered rock masses are typical and ubiquitous;uncertainties in rock properties always exist.In view of this,a comprehensive method,which combines the Upper bound Limit analysis of Tunnel face stability,the Polynomial Chaos Kriging,the Monte-Carlo Simulation and Analysis of Covariance method(ULT-PCK-MA),is proposed to investigate the seismic stability of tunnel faces.A two-dimensional analytical model of ULT is developed to evaluate the virtual support force based on the upper bound limit analysis.An efficient probabilistic analysis method PCK-MA based on the adaptive Polynomial Chaos Kriging metamodel is then implemented to investigate the parameter uncertainty effects.Ten input parameters,including geological strength indices,uniaxial compressive strengths and constants for three rock formations,and the horizontal seismic coefficients,are treated as random variables.The effects of these parameter uncertainties on the failure probability and sensitivity indices are discussed.In addition,the effects of weak layer position,the middle layer thickness and quality,the tunnel diameter,the parameters correlation,and the seismic loadings are investigated,respectively.The results show that the layer distributions significantly influence the tunnel face probabilistic stability,particularly when the weak rock is present in the bottom layer.The efficiency of the proposed ULT-PCK-MA is validated,which is expected to facilitate the engineering design and construction.
基金This work was supported in part by the U.S.National Science Foundation(NSF grants CMMI-1824681,DMS-1952781,and BCS-2121616).
文摘Polynomial chaos expansions(PCEs)have been used in many real-world engineering applications to quantify how the uncertainty of an output is propagated from inputs by decomposing the output in terms of polynomials of the inputs.PCEs for models with independent inputs have been extensively explored in the literature.Recently,different approaches have been proposed for models with dependent inputs to expand the use of PCEs to more real-world applications.Typical approaches include building PCEs based on the Gram–Schmidt algorithm or transforming the dependent inputs into independent inputs.However,the two approaches have their limitations regarding computational efficiency and additional assumptions about the input distributions,respectively.In this paper,we propose a data-driven approach to build sparse PCEs for models with dependent inputs without any distributional assumptions.The proposed algorithm recursively constructs orthonormal polynomials using a set of monomials based on their correlations with the output.The proposed algorithm on building sparse PCEs not only reduces the number of minimally required observations but also improves the numerical stability and computational efficiency.Four numerical examples are implemented to validate the proposed algorithm.The source code is made publicly available for reproducibility.
基金supported by the National Key Research and Development Program of China(2021YFB2600700)the National Natural Science Foundation of China(Project Nos.52025094,52088102,51979158)+1 种基金support from Shanghai Municipal Education Commission(Project No.2021-01-07-00-02-E00089)Key Projects for Intergovernmental Cooperation in International Science,Technology and Innovation(Grant No.2018YFE0125100).
文摘As rainfall infiltrates into soil slopes,the hydraulic and mechanical behaviors of soils are interacted.In this study,an efficient probabilistic parameter estimation method for coupled hydro-mechanical behavior in soil slope is proposed.This method integrates the Polynomial Chaos Expansion(PCE)method,the coupled hydro-mechanical modeling,and the Bayesian learning method.A coupled hydro-mechanical numerical model is established for the simulation of behaviors of unsaturated soil slope under rainfall infiltration,following by training a cheap-to-run PCE surrogate to replace it.Probabilistic estimation of soil parameters is conducted based on the Bayesian learning technique with the Markov Chain Monte Carlo(MCMC)simulation.A numerical example of an unsaturated slope under rainfall infiltration is presented to illustrate the proposed method.The effects of measurement durations and response types on parameter estimation are addressed.The result shows that with the increase of measurement duration,the uncertainties of soil parameters are significantly reduced.The uncertainties of hydraulic properties are reduced significantly using the pore water pressure data,while the uncertainties of soil strength parameters are reduced greatly using the measured displacement data.
基金supported by the National Natural Science Foundation of China(No.11572252)the ‘‘111" Project of China(No.B17037)the National Science Fund for Excellent Young Scholars(No.11622220)
文摘The uncertainties can generate fluctuations with aerodynamic characteristics. Uncertainty Quantification(UQ) is applied to compute its impact on the aerodynamic characteristics.In addition, the contribution of each uncertainty to aerodynamic characteristics should be computed by uncertainty sensitivity analysis. Non-Intrusive Polynomial Chaos(NIPC) has been successfully applied to uncertainty quantification and uncertainty sensitivity analysis. However, the non-intrusive polynomial chaos method becomes inefficient as the number of random variables adopted to describe uncertainties increases. This deficiency becomes significant in stochastic aerodynamic analysis considering the geometric uncertainty because the description of geometric uncertainty generally needs many parameters. To solve the deficiency, a Sparse Grid-based Polynomial Chaos(SGPC) expansion is used to do uncertainty quantification and sensitivity analysis for stochastic aerodynamic analysis considering geometric and operational uncertainties. It is proved that the method is more efficient than non-intrusive polynomial chaos and Monte Carlo Simulation(MSC) method for the stochastic aerodynamic analysis. By uncertainty quantification, it can be learnt that the flow characteristics of shock wave and boundary layer separation are sensitive to the geometric uncertainty in transonic region. The uncertainty sensitivity analysis reveals the individual and coupled effects among the uncertainty parameters.
基金Project supported by the National Natural Science Foundation of China(Grant No.50979060)
文摘A one-dimensional non-intrusive Polynomial Chaos (PC) method is applied in Uncertainty Quantification (UQ) studies for CFD-based ship performances simulations. The uncertainty properties of Expected Value (EV) and Standard Deviation (SD) are evaluated by solving the PC coefficients from a linear system of algebraic equations. The one-dimensional PC with the Legendre polynomials is applied to: (1) stochastic input domain and (2) Cumulative Distribution Function (CDF) image domain, allowing for more flexibility. The PC method is validated with the Monte-Carlo benchmark results in several high-fidelity, CFD-based, ship UQ problems, evaluating the geometrical, operational and environmental uncertainties for the Delft Catamaran 372. Convergence is studied versus PC order P for both EV and SD, showing that high order PC is not necessary for present applications. Comparison is carried out for PC with/without the least square minimization when solving the PC coefficients. The least square minimization, using larger number of CFD samples, is recommended for current test cases. The study shows the potentials of PC method in Robust Design Optimization (RDO) and Reliability-Based Design Optimization (RBDO) of ship hydrodynamic performances.
基金supported by the National Natural Science Foundation of China (No. 51105034)the Doctoral Thesis Build Project of Beijing 2012 (China)
文摘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.
基金National Natural Science Foundation of China(No.11721202)。
文摘Uncertainty is common in the life cycle of an aircraft, and Robust Aerodynamic Optimization(RAO) that considers uncertainty is important in aircraft design. To avoid the curse of dimensionality in surrogate-based optimization, this study proposes an adjoint RAO technique called “R-Opt”. Polynomial Chaos Expansion(PCE) is coupled with the R-Opt technique to quantify uncertainty in the responses of the target(including its mean and standard deviation). Only one process of PCE model construction is required in each iteration, and the gradients of uncertainty can be inferred via chain rules. The proposed method is more efficient than prevalent methods,and avoids the problem of a disagreement over the best PCE basis from among a number of PCE models(especially in case of sparse PCE). It also supports the application of sparse PCE.Two benchmark tests and two airfoil cases were used to verify R-Opt, and the optimal solutions were deemed to be robust. It improved the mean aerodynamic performance and reduced the standard deviation of the target.
文摘To obtain a universal model solving the uncertain acoustic field in shallow water, a non-intrusive model coupled polynomial chaos expansion (PCE) method with Helmholtz equa- tion is established, in which the polynomial coefficients are solved by probabilistic collocation method (PCM). For the cases of Pekeris waveguide which have uncertainties in depth of water column, in both sound speed profile and depth of water column, and for the case of thermocline with lower limit depth uncertain, probability density functions (PDF) of transmission loss (TL) are calculated. The results show that the proposed model is universal for the acoustic propa- gation codes with high computational efficiency and accuracy, and can be applied to study the uncertainty of acoustic propagation in the shallow water en^-ironment with multiple parameters uncertain.
基金Supported by the National Natural Science Foundation of China(No.11501427,11571128)
文摘This paper is concerned with the application of generalized polynomial chaos (gPC) method to nonlinear random pantograph equations. An error estimation of gPC method is derived. The global error analysis is given for the error arising from finite-dimensional noise (FDN) assumption, projection error, aliasing error and discretization error. In the end, with several numerical experiments, the theoretical results are further illustrated.
基金The authors are grateful to the anonymous referees for their valuable comments and suggestions.This research of Kim was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF)funded by the Ministry of Education,Science and Technology(20110005272).
文摘In this paper,we consider the numerical stability of gravity-capillary waves generated by a localized pressure in water of finite depth based on the forced Korteweg-de Vries(FKdV)framework and the polynomial chaos.The stability studies are focused on the symmetric solitary wave for the subcritical flow with the Bond number greater than one third.When its steady symmetric solitarywave-like solutions are randomly perturbed,the evolutions of some waves show stability in time regardless of the randomness while other waves produce unstable fluctuations.By representing the perturbation with a random variable,the governing FKdV equation is interpreted as a stochastic equation.The polynomial chaos expansion of the random solution has been used for the study of stability in two ways.First it allows us to identify the stable solution of the stochastic governing equation.Secondly it is used to construct upper and lower bounding surfaces for unstable solutions,which encompass the fluctuations of waves.
基金funding from the Walloon region of Belgium,convention number 5856,subvention FIRST-ENTREPRISE.
文摘The polynomial chaos expansion(PCE)is an efficient numerical method for performing a reliability analysis.It relates the output of a nonlinear system with the uncertainty in its input parameters using a multidimensional polynomial approximation(the so-called PCE).Numerically,such an approximation can be obtained by using a regression method with a suitable design of experiments.The cost of this approximation depends on the size of the design of experiments.If the design of experiments is large and the system is modeled with a computationally expensive FEA(Finite Element Analysis)model,the PCE approximation becomes unfeasible.The aim of this work is to propose an algorithm that generates efficiently a design of experiments of a size defined by the user,in order to make the PCE approximation computationally feasible.It is an optimization algorithm that seeks to find the best design of experiments in the D-optimal sense for the PCE.This algorithm is a coupling between genetic algorithms and the Fedorov exchange algorithm.The efficiency of our approach in terms of accuracy and computational time reduction is compared with other existing methods in the case of analytical functions and finite element based functions.
基金funding from the Walloon region of Belgium,convention number 5856,subvention FIRST-ENTREPRISE.
文摘Robust optimization is an approach for the design of a mechanical structure which takes into account the uncertainties of the design variables.It requires at each iteration the evaluation of some robust measures of the objective function and the constraints.In a previous work,the authors have proposed a method which efficiently generates a design of experiments with respect to the design variable uncertainties to compute the robust measures using the polynomial chaos expansion.This paper extends the proposed method to the case of the robust optimization.The generated design of experiments is used to build a surrogate model for the robust measures over a certain trust region.This leads to a trust region optimization method which only requires one evaluation of the design of experiments per iteration(single loop method).Unlike other single loop methods which are only based on a first order approximation of robust measure of the constraints and which does not handle a robust measure for the objective function,the proposed method can handle any approximation order and any choice for the robust measures.Some numerical experiments based on finite element functions are performed to show the efficiency of the method.
基金supported in part by the National Natural Science Foundation of China(No.51977005)。
文摘In an integrated energy system,source-load multiple uncertainties and correlations lead to an over-limit risk in operating state,including voltage,temperature,and pressure over-limit.Therefore,efficient probabilistic energy flow calculation methods and risk assessment theories applicable to integrated energy systems are crucial.This study proposed a probabilistic energy flow calculation method based on polynomial chaos expansion for an electric-heat-gas integrated energy system.The method accurately and efficiently calculated the over-limit probability of the system state variables,considering the coupling conditions of electricity,heat,and gas,as well as uncertainties and correlations in renewable energy unit outputs and multiple types of loads.To further evaluate and quantify the impact of uncertainty factors on the over-limit risk,a global sensitivity analysis method for the integrated energy system based on the analysis of covariance theory is proposed.This method considered the source-load correlation and aimed to identify the key uncertainty factors that influence stable operation.Simulation results demonstrated that the proposed method achieved accuracy to that of the Monte Carlo method while significantly reducing calculation time.It effectively quantified the over-limit risk under the presence of multiple source-load uncertainties.
基金The project supported by the National Natural Science Foundation of China(10602036)
文摘An efficient and accurate uncertainty propagation methodology for mechanics problems with random fields is developed in this paper. This methodology is based on the stochastic response surface method (SRSM) which has been previously proposed for problems dealing with random variables only. This paper extends SRSM to problems involving random fields or random processes fields. The favorable property of SRSM lies in that the deterministic computational model can be treated as a black box, as in the case of commercial finite element codes. Numerical examples are used to highlight the features of this technique and to demonstrate the accuracy and efficiency of the proposed method. A comparison with Monte Carlo simulation shows that the proposed method can achieve numerical results close to those from Monte Carlo simulation while dramatically reducing the number of deterministic finite element runs.
基金National Science Foundation of China under grant No.51378107Fundamental Research Funds for the Central Universities and Doctoral Research Fund by Southeast University under Grant No.YBJJ-1442
文摘Uncertainties in structure properties can result in different responses in hybrid simulations. Quantification of the effect of these tmcertainties would enable researchers to estimate the variances of structural responses observed from experiments. This poses challenges for real-time hybrid simulation (RTHS) due to the existence of actuator delay. Polynomial chaos expansion (PCE) projects the model outputs on a basis of orthogonal stochastic polynomials to account for influences of model uncertainties. In this paper, PCE is utilized to evaluate effect of actuator delay on the maximum displacement from real-time hybrid simulation of a single degree of freedom (SDOF) structure when accounting for uncertainties in structural properties. The PCE is first applied for RTHS without delay to determine the order of PCE, the number of sample points as well as the method for coefficients calculation. The PCE is then applied to RTHS with actuator delay. The mean, variance and Sobol indices are compared and discussed to evaluate the effects of actuator delay on uncertainty quantification for RTHS. Results show that the mean and the variance of the maximum displacement increase linearly and exponentially with respect to actuator delay, respectively. Sensitivity analysis through Sobol indices also indicates the influence of the single random variable decreases while the coupling effect increases with the increase of actuator delay.