Sparse Approximation of Data-Driven Polynomial Chaos Expansions: An Induced Sampling Approach

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.

Three-dimensional pseudo-dynamic reliability analysis of seismic shield tunnel faces combined with sparse polynomial chaos expansion

To address the seismic face stability challenges encountered in urban and subsea tunnel construction,an efficient probabilistic analysis framework for shield tunnel faces under seismic conditions is proposed.Based on the upper-bound theory of limit analysis,an improved three-dimensional discrete deterministic mechanism,accounting for the heterogeneous nature of soil media,is formulated to evaluate seismic face stability.The metamodel of failure probabilistic assessments for seismic tunnel faces is constructed by integrating the sparse polynomial chaos expansion method(SPCE)with the modified pseudo-dynamic approach(MPD).The improved deterministic model is validated by comparing with published literature and numerical simulations results,and the SPCE-MPD metamodel is examined with the traditional MCS method.Based on the SPCE-MPD metamodels,the seismic effects on face failure probability and reliability index are presented and the global sensitivity analysis(GSA)is involved to reflect the influence order of seismic action parameters.Finally,the proposed approach is tested to be effective by a engineering case of the Chengdu outer ring tunnel.The results show that higher uncertainty of seismic response on face stability should be noticed in areas with intense earthquakes and variation of seismic wave velocity has the most profound influence on tunnel face stability.

Generalized polynomial chaos expansion by reanalysis using static condensation based on substructuring

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.

Sensitivity Analysis of Electromagnetic Scattering from Dielectric Targets with Polynomial Chaos Expansion and Method of Moments

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.

A Non-Intrusive Stochastic Phase-Field for Fatigue Fracture in Brittle Materials with Uncertainty in Geometry and Material Properties

Understanding the probabilistic nature of brittle materials due to inherent dispersions in their mechanical properties is important to assess their reliability and safety for sensitive engineering applications.This is all the more important when elements composed of brittle materials are exposed to dynamic environments,resulting in catastrophic fatigue failures.The authors propose the application of a non-intrusive polynomial chaos expansion method for probabilistic studies on brittle materials undergoing fatigue fracture when geometrical parameters and material properties are random independent variables.Understanding the probabilistic nature of fatigue fracture in brittle materials is crucial for ensuring the reliability and safety of engineering structures subjected to cyclic loading.Crack growth is modelled using a phase-field approach within a finite element framework.For modelling fatigue,fracture resistance is progressively degraded by modifying the regularised free energy functional using a fatigue degradation function.Number of cycles to failure is treated as the dependent variable of interest and is estimated within acceptable limits due to the randomness in independent properties.Multiple 2D benchmark problems are solved to demonstrate the ability of this approach to predict the dependent variable responses with significantly fewer simulations than the Monte Carlo method.This proposed approach can accurately predict results typically obtained through 105 or more runs in Monte Carlo simulations with a reduction of up to three orders of magnitude in required runs.The independent random variables’sensitivity to the system response is determined using Sobol’indices.The proposed approach has low computational overhead and can be useful for computationally intensive problems requiring rapid decision-making in sensitive applications like aerospace,nuclear and biomedical engineering.The technique does not require reformulating existing finite element code and can perform the stochastic study by direct pre/post-processing.

Polynomial chaos surrogate and bayesian learning for coupled hydro-mechanical behavior of soil slope

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.

Over-limit risk assessment method of integrated energy system considering source-load correlation

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.

Data-driven sparse polynomial chaos expansion for models with dependent inputs

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.

Uncertainty Analysis of Seepage-Induced Consolidation in a Fractured Porous Medium

Numerical modeling of seepage-induced consolidation process usually encounters significant uncertainty in the properties of geotechnical materials.Assessing the effect of uncertain parameters on the performance variability of the seepage consolidation model is of critical importance to the simulation and tests of this process.To this end,the uncertainty and sensitivity analyses are performed on a seepage consolidation model in a fractured porous medium using the Bayesian sparse polynomial chaos expansion(SPCE)method.Five uncertain parameters including Young’s modulus,Poisson’s ratio,and the permeability of the porous matrix,the permeability within the fracture,and Biot’s constant are studied.Bayesian SPCE models for displacement,flow velocity magnitude,and fluid pressure at several reference points are constructed to represent the input-output relationship of the numerical model.Based on these SPCE models,the total and first-order Sobol’indices are computed to quantify the contribution of each uncertain input parameter to the uncertainty of model responses.The results show that at different locations of the porous domain,the uncertain parameters show different effects on the output quantities.At the beginning of the seepage consolidation process,the hydraulic parameters make major contributions to the uncertainty of the model responses.As the process progresses,the effect of hydraulic parameters decreases and is gradually surpassed by the mechanical parameters.This work demonstrates the feasibility to apply Bayesian SPCE approach to the uncertainty and sensitivity analyses of seepage-induced consolidation problems and provides guidelines to the numerical modelling and experimental testing of such problems.

Analysis of actuator delay and its effect on uncertainty quantification for real-time hybrid simulation

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.

A novel robust aerodynamic optimization technique coupled with adjoint solvers and polynomial chaos expansion

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.

Efficient Probabilistic Load Flow Calculation Considering Vine Copula⁃Based Dependence Structure of Renewable Energy Generation

Correlations among random variables make significant impacts on probabilistic load flow(PLF)calculation results.In the existing studies,correlation coefficients or Gaussian copula are usually used to model the correlations,while vine copula,which describes the complex dependence structure(DS)of random variables,is seldom discussed since it brings in much heavier computational burdens.To overcome this problem,this paper proposes an efficient PLF method considering input random variables with complex DS.Specifically,the Rosenblatt transformation(RT)is used to transform vine copula⁃based correlated variables into independent ones;and then the sparse polynomial chaos expansion(SPCE)evaluates output random variables of PLF calculation.The effectiveness of the proposed method is verified using the IEEE 123⁃bus system.

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.

A polynomial chaos expansion method for the uncertain acoustic field in shallow water

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.

Response prediction using the PC-NARX model for SDOF systems with degradation and parametric uncertainties

Building collapses during recent earthquakes have brought up the need for research on factors pertaining to collapse and the safety of structures.This requires response replication of structures that account for uncertainties from ground motions and structural properties.Structural collapse often implies that the structural system is no longer capable of maintaining its gravity load-carrying capacity,which often points to factors involving strength and stiffness degradation.In this study,the polynomial chaos nonlinear autoregressive with exogenous input form(PC-NARX)model is explored for dynamic response replication of a nonlinear single-degree-of-freedom(SDOF)structure.The generalized hysteretic Bouc-Wen model is applied to emulate stiffness and strength degradation for an SDOF structure close to collapse.A stochastic ground motion model is used to represent the uncertainties in seismic excitation.The PC-NARX model is employed and further evaluated for response replication of an SDOF system with inherent uncertain structural properties.A generic algorithm(GA)is used to select the terms for structural dynamics,and polynomial chaos expansion(PCE)is used to incorporate uncertain parameters into NARX model coefficients.It is demonstrated that the PC-NARX model provides good accuracy to account for both ground motion and structural uncertainties into response replication of SDOF structures with significant strength and stiffness degradation.The PC-NARX model thus presents a promising technique for collapse safety analysis of structures.

An Efficient Sampling Method for Regression-Based Polynomial Chaos Expansion

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.

A Polynomial Chaos Expansion Trust Region Method for Robust Optimization

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.

Methods and advances in the study of aeroelasticity with uncertainties

Uncertainties denote the operators which describe data error, numerical error and model error in the mathematical methods. The study of aeroelasticity with uncertainty embedded in the subsystems, such as the uncertainty in the modeling of structures and aerodynamics, has been a hot topic in the last decades. In this paper, advances of the analysis and design in aeroelasticity with uncertainty are summarized in detail. According to the non-probabilistic or probabilistic uncer- tainty, the developments of theories, methods and experiments with application to both robust and probabilistic aeroelasticity analysis are presented, respectively. In addition, the advances in aeroelastic design considering either probabilistic or non-probabilistic uncertainties are introduced along with aeroelastic analysis. This review focuses on the robust aeroelasticity study based on the structured singular value method, namely the ~t method. It covers the numerical calculation algo- rithm of the structured singular value, uncertainty model construction, robust aeroelastic stability analysis algorithms, uncertainty level verification, and robust flutter boundary prediction in the flight test, etc. The key results and conclusions are explored. Finally, several promising problems on aeroelasticity with uncertainty are proposed for future investigation.

A comparison of sensitivity indices for tolerance design of a transmission mechanism

Sensitivity analysis is used to quantify the contribution of the uncertainty of input variables to the uncertainty of systematic output responses.For tolerance design in manufacturing and assembly,sensitivity analysis is applied to help designers allocate tolerances optimally.However,different sensitivity indices derived from different sensitivity analysis methods will always lead to conflicting results.It is necessary to find a sensitivity index suitable for tolerance allocation to transmission mechanisms so that the sensitivity results can truly reflect the effects of tolerances on kinematic and dynamic performances.In this paper,a variety of sensitivity indices are investigated and compared based on hybrid simulation.Firstly,the hybrid simulation model of the crank-slider mechanism is established.Secondly,samples of the kinematic and dynamic responses of the mechanism with joint clearances and link length errors are obtained,and the surrogate model established using polynomial chaos expansion(PCE).Then,different sensitivity indices are calculated based on the PCE model and are further used to evaluate the effect of joint clearances and link length errors on the output response.Combined with the tolerance-cost function,the corresponding tolerance allocation schemes are obtained based on different sensitivity analysis results.Finally,the kinematic and dynamic responses of the mechanism adopting different tolerance allocation schemes are simulated,and the sensitivity index corresponding to the optimal response is determined as the most appropriate index.

Conditional Simulation of Flow in Heterogeneous Porous Media with the Probabilistic Collocation Method

A stochastic approach to conditional simulation of flow in randomly heterogeneous media is proposed with the combination of the Karhunen-Loeve expansion and the probabilistic collocation method(PCM).The conditional log hydraulic conductivity field is represented with the Karhunen-Loeve expansion,in terms of some deterministic functions and a set of independent Gaussian random variables.The propagation of uncertainty in the flow simulations is carried out through the PCM,which relies on the efficient polynomial chaos expansion used to represent the flow responses such as the hydraulic head.With the PCM,existing flow simulators can be employed for uncertainty quantification of flow in heterogeneous porous media when direct measurements of hydraulic conductivity are taken into consideration.With illustration of several numerical examples of groundwater flow,this study reveals that the proposed approach is able to accurately quantify uncertainty of the flow responses conditioning on hydraulic conductivity data,while the computational efforts are significantly reduced in comparison to the Monte Carlo simulations.