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 weaken the influence of profile error on compressor aerodynamic performance, especially on pressure ratio and efficiency, a robust design method considering profile error is built to improve the robustness of aerod...To weaken the influence of profile error on compressor aerodynamic performance, especially on pressure ratio and efficiency, a robust design method considering profile error is built to improve the robustness of aerodynamic performance of the blade. The characteristics of profile error are random and small-scaled, which means that to evaluate the influence of profile error on blade aerodynamic performance is a time-intensive and high-precision work. For this reason, non-intrusive polynomial chaos(NIPC) and Kriging surrogate model are introduced in this robust design method to improve the efficiency of uncertainty quantification(UQ) and ensure the evaluate accuracy. The profile error satisfies the Gaussian distribution, and NIPC is carried out to do uncertainty quantification since it has advantages in prediction accuracy and efficiency to get statistical behavior of random profile error. In the integrand points of NIPC, several surrogate models are established based on Latin hypercube sampling(LHS)+ Kriging, which further reduces the costs of optimization design by replacing calling computational fluid dynamic(CFD) repeatedly. The results show that this robust design method can significantly improve the performance robustness in shorter time(40 times faster) without losing accuracy, which is meaningful in engineering application to reduce manufacturing cost in the premise of ensuring the aerodynamic performance. Mechanism analysis of the robustness improvement samples carried out in current work can help find out the key parameter dominating the robustness under the disturbance of profile error, which is meaningful to further improvement of compressor robustness.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
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 ...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.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
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
The Chebyshev polynomial approximation is applied to investigate the stochastic period-doubling bifurcation and chaos problems of a stochastic Duffing-van der Pol system with bounded random parameter of exponential pr...The Chebyshev polynomial approximation is applied to investigate the stochastic period-doubling bifurcation and chaos problems of a stochastic Duffing-van der Pol system with bounded random parameter of exponential probability density function subjected to a harmonic excitation. Firstly the stochastic system is reduced into its equivalent deterministic one, and then the responses of stochastic system can be obtained by numerical methods. Nonlinear dynamical behaviour related to stochastic period-doubling bifurcation and chaos in the stochastic system is explored. Numerical simulations show that similar to its counterpart in deterministic nonlinear system of stochastic period-doubling bifurcation and chaos may occur in the stochastic Duffing-van der Pol system even for weak intensity of random parameter. Simply increasing the intensity of the random parameter may result in the period-doubling bifurcation which is absent from the deterministic system.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
The author studies the effect of uncertain conductivity on the electroencephalography （EEG） forward problem. A three-layer spherical head model with different and random layer conductivities is considered. Polynomia...The author studies the effect of uncertain conductivity on the electroencephalography （EEG） forward problem. A three-layer spherical head model with different and random layer conductivities is considered. Polynomial Chaos （PC） is used to model the randomness. The author performs a sensitivity and correlation analysis of EEG sensors influenced by uncertain conductivity. The author addressed the sensitivity analysis at three stages： dipole location and moment averaged out, only the dipole moment averaged out, and both fixed. On average, the author observes the least influenced electrodes along the great longitudinal fissure. Also, sensors located closer to a dipole source, are of greater influence to a change in conductivity. The highly influenced sensors were on average located temporal. This was also the case in the correlation analysis. Sensors in the temporal parts of the brain are highly correlated. Whereas the sensors in the occipital and lower frontal region, though they are close together, are not so highly correlated as in the temporal regions. This study clearly shows that intrinsic sensor correlation exists, and therefore cannot be discarded, especially in the inverse problem. In the latter it makes it possible not to specify the conductivities. It also offers an easy but rigorous modeling of the stochastic propagation of uncertain conductivity to sensorial potentials （e.g., making it suited for research on optimal placing of these sensors）.展开更多
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...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.展开更多
This paper presents a framework for constructing surrogate models for sensitivity analysis of structural dynamics behavior.Physical models involving deformation,such as collisions,vibrations,and penetration,are devel-...This paper presents a framework for constructing surrogate models for sensitivity analysis of structural dynamics behavior.Physical models involving deformation,such as collisions,vibrations,and penetration,are devel-oped using the material point method.To reduce the computational cost of Monte Carlo simulations,response surface models are created as surrogate models for the material point system to approximate its dynamic behavior.An adaptive randomized greedy algorithm is employed to construct a sparse polynomial chaos expansion model with a fixed order,effectively balancing the accuracy and computational efficiency of the surrogate model.Based on the sparse polynomial chaos expansion,sensitivity analysis is conducted using the global finite difference and Sobol methods.Several examples of structural dynamics are provided to demonstrate the effectiveness of the proposed method in addressing structural dynamics problems.展开更多
This paper proposes a new non-intrusive hybrid interval method using derivative information for the dynamic response analysis of nonlinear systems with uncertain-but- bounded parameters and/or initial conditions. This...This paper proposes a new non-intrusive hybrid interval method using derivative information for the dynamic response analysis of nonlinear systems with uncertain-but- bounded parameters and/or initial conditions. This method provides tighter solution ranges compared to the existing polynomial approximation interval methods. Interval arith- metic using the Chebyshev basis and interval arithmetic using the general form modified affine basis for polynomials are developed to obtain tighter bounds for interval computation. To further reduce the overestimation caused by the ＂wrap- ping effect＂ of interval arithmetic, the derivative information of dynamic responses is used to achieve exact solutions when the dynamic responses are monotonic with respect to all the uncertain variables. Finally, two typical numerical examples with nonlinearity are applied to demonstrate the effective- ness of the proposed hybrid interval method, in particular, its ability to effectively control the overestimation for specific timepoints.展开更多
The discovery of chaos in the sixties of last century was a breakthrough in concept,revealing the truth that some disorder behavior,called chaos,could happen even in a deterministic nonlinear system under barely deter...The discovery of chaos in the sixties of last century was a breakthrough in concept,revealing the truth that some disorder behavior,called chaos,could happen even in a deterministic nonlinear system under barely deterministic disturbance.After a series of serious studies,people begin to acknowledge that chaos is a specific type of steady state motion other than the conventional periodic and quasi-periodic ones,featuring a sensitive dependence on initial conditions,resulting from the intrinsic randomness of a nonlinear system itself.In fact,chaos is a collective phenomenon consisting of massive individual chaotic responses,corresponding to different initial conditions in phase space.Any two adjacent individual chaotic responses repel each other,thus causing not only the sensitive dependence on initial conditions but also the existence of at least one positive top Lyapunov exponent(TLE) for chaos.Meanwhile,all the sample responses share one common invariant set on the Poincaré map,called chaotic attractor,which every sample response visits from time to time ergodically.So far,the existence of at least one positive TLE is a commonly acknowledged remarkable feature of chaos.We know that there are various forms of uncertainties in the real world.In theoretical studies,people often use stochastic models to describe these uncertainties,such as random variables or random processes.Systems with random variables as their parameters or with random processes as their excitations are often called stochastic systems.No doubt,chaotic phenomena also exist in stochastic systems,which we call stochastic chaos to distinguish it from deterministic chaos in the deterministic system.Stochastic chaos reflects not only the intrinsic randomness of the nonlinear system but also the external random effects of the random parameter or the random excitation.Hence,stochastic chaos is also a collective massive phenomenon,corresponding not only to different initial conditions but also to different samples of the random parameter or the random excitation.Thus,the unique common feature of deterministic chaos and stochastic chaos is that they all have at least one positive top Lyapunov exponent for their chaotic motion.For analysis of random phenomena,one used to look for the PDFs(Probability Density Functions) of the ensemble random responses.However,it is a pity that PDF information is not favorable to studying repellency of the neighboring chaotic responses nor to calculating the related TLE,so we would rather study stochastic chaos through its sample responses.Moreover,since any sample of stochastic chaos is a deterministic one,we need not supplement any additional definition on stochastic chaos,just mentioning that every sample of stochastic chaos should be deterministic chaos.We are mainly concerned with the following two basic kinds of nonlinear stochastic systems,i.e.one with random variables as its parameters and one with ergodical random processes as its excitations.To solve the stochastic chaos problems of these two kinds of systems,we first transform the original stochastic system into their equivalent deterministic ones.Namely,we can transform the former stochastic system into an equivalent deterministic system in the sense of mean square approximation with respect to the random parameter space by the orthogonal polynomial approximation,and transform the latter one simply through replacing its ergodical random excitations by their representative deterministic samples.Having transformed the original stochastic chaos problem into the deterministic chaos problem of equivalent systems,we can use all the available effective methods for further chaos analysis.In this paper,we aim to review the state of art of studying stochastic chaos with its control and synchronization by the above-mentioned strategy.展开更多
By using a series of canonical transformations (Birkhoff's series), an approximate integral of a conservative compound pendulum is evaluated. Level lines of this approximate integral are compared with the numerica...By using a series of canonical transformations (Birkhoff's series), an approximate integral of a conservative compound pendulum is evaluated. Level lines of this approximate integral are compared with the numerical simulation results. It is seen clearly that with a raised energy level, the nearly integrable system becomes non-integrable, i.e. the regular motion pattern changes to the chaotic one. Experiments with such a pendulum device display the behavior mentioned above.展开更多
In this paper, we analyze chaotic dynamics of nonlinear systems and study chaos synchronization of Lorenz system. We extend our study by discussing other methods available in literature. We propose a theorem followed ...In this paper, we analyze chaotic dynamics of nonlinear systems and study chaos synchronization of Lorenz system. We extend our study by discussing other methods available in literature. We propose a theorem followed by a lemma in general and another one for a particular case of Lorenz system. Numerical simulations are given to verify the proposed theorems.展开更多
Full lifecycle high fidelity digital twin is a complex model set contains multiple functions with high dimensions and multiple variables.Quantifying uncertainty for such complex models often encounters time-consuming ...Full lifecycle high fidelity digital twin is a complex model set contains multiple functions with high dimensions and multiple variables.Quantifying uncertainty for such complex models often encounters time-consuming challenges,as the number of calculated terms increases exponentially with the dimensionality of the input.This paper based on the multi-stage model and high time consumption problem of digital twins,proposed a sparse polynomial chaos expansions method to generate the digital twin dynamic-polymorphic uncertainty surrogate model,striving to strike a balance between the accuracy and time consumption of models used for digital twin uncertainty quantification.Firstly,an analysis and clarification were conducted on the dynamic-polymorphic uncertainty of the full lifetime running digital twins.Secondly,a sparse polynomial chaos expansions model response was developed based on partial least squares technology with the effectively quantified and selected basis polynomials which sorted by significant influence.In the end,the accuracy of the proxy model is evaluated by leave-one-out cross-validation.The effectiveness of this method was verified through examples,and the results showed that it achieved a balance between maintaining model accuracy and complexity.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
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
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support of the National Natural Science Foundation of China (NSFC) under the Grant No. 51790512the Overseas Expertise Introduction Project for Discipline Innovation (111 Project) under Grant No. B17037Industry-University-Research Cooperation Project of Aero Engine Corporation of China (AECC) under Grant No. HFZL2018CXY011-1 and MIIT
文摘To weaken the influence of profile error on compressor aerodynamic performance, especially on pressure ratio and efficiency, a robust design method considering profile error is built to improve the robustness of aerodynamic performance of the blade. The characteristics of profile error are random and small-scaled, which means that to evaluate the influence of profile error on blade aerodynamic performance is a time-intensive and high-precision work. For this reason, non-intrusive polynomial chaos(NIPC) and Kriging surrogate model are introduced in this robust design method to improve the efficiency of uncertainty quantification(UQ) and ensure the evaluate accuracy. The profile error satisfies the Gaussian distribution, and NIPC is carried out to do uncertainty quantification since it has advantages in prediction accuracy and efficiency to get statistical behavior of random profile error. In the integrand points of NIPC, several surrogate models are established based on Latin hypercube sampling(LHS)+ Kriging, which further reduces the costs of optimization design by replacing calling computational fluid dynamic(CFD) repeatedly. The results show that this robust design method can significantly improve the performance robustness in shorter time(40 times faster) without losing accuracy, which is meaningful in engineering application to reduce manufacturing cost in the premise of ensuring the aerodynamic performance. Mechanism analysis of the robustness improvement samples carried out in current work can help find out the key parameter dominating the robustness under the disturbance of profile error, which is meaningful to further improvement of compressor robustness.
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Project([2018]3010)supported by the Guizhou Provincial Science and Technology Major Project,China。
文摘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.
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Project supported by the National Natural Science Foundation of China （Grants Nos 10472091 and 10332030）.
文摘The Chebyshev polynomial approximation is applied to investigate the stochastic period-doubling bifurcation and chaos problems of a stochastic Duffing-van der Pol system with bounded random parameter of exponential probability density function subjected to a harmonic excitation. Firstly the stochastic system is reduced into its equivalent deterministic one, and then the responses of stochastic system can be obtained by numerical methods. Nonlinear dynamical behaviour related to stochastic period-doubling bifurcation and chaos in the stochastic system is explored. Numerical simulations show that similar to its counterpart in deterministic nonlinear system of stochastic period-doubling bifurcation and chaos may occur in the stochastic Duffing-van der Pol system even for weak intensity of random parameter. Simply increasing the intensity of the random parameter may result in the period-doubling bifurcation which is absent from the deterministic system.
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.
文摘The author studies the effect of uncertain conductivity on the electroencephalography （EEG） forward problem. A three-layer spherical head model with different and random layer conductivities is considered. Polynomial Chaos （PC） is used to model the randomness. The author performs a sensitivity and correlation analysis of EEG sensors influenced by uncertain conductivity. The author addressed the sensitivity analysis at three stages： dipole location and moment averaged out, only the dipole moment averaged out, and both fixed. On average, the author observes the least influenced electrodes along the great longitudinal fissure. Also, sensors located closer to a dipole source, are of greater influence to a change in conductivity. The highly influenced sensors were on average located temporal. This was also the case in the correlation analysis. Sensors in the temporal parts of the brain are highly correlated. Whereas the sensors in the occipital and lower frontal region, though they are close together, are not so highly correlated as in the temporal regions. This study clearly shows that intrinsic sensor correlation exists, and therefore cannot be discarded, especially in the inverse problem. In the latter it makes it possible not to specify the conductivities. It also offers an easy but rigorous modeling of the stochastic propagation of uncertain conductivity to sensorial potentials （e.g., making it suited for research on optimal placing of these sensors）.
文摘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.
基金support from the National Natural Science Foundation of China(Grant Nos.52174123&52274222).
文摘This paper presents a framework for constructing surrogate models for sensitivity analysis of structural dynamics behavior.Physical models involving deformation,such as collisions,vibrations,and penetration,are devel-oped using the material point method.To reduce the computational cost of Monte Carlo simulations,response surface models are created as surrogate models for the material point system to approximate its dynamic behavior.An adaptive randomized greedy algorithm is employed to construct a sparse polynomial chaos expansion model with a fixed order,effectively balancing the accuracy and computational efficiency of the surrogate model.Based on the sparse polynomial chaos expansion,sensitivity analysis is conducted using the global finite difference and Sobol methods.Several examples of structural dynamics are provided to demonstrate the effectiveness of the proposed method in addressing structural dynamics problems.
文摘This paper proposes a new non-intrusive hybrid interval method using derivative information for the dynamic response analysis of nonlinear systems with uncertain-but- bounded parameters and/or initial conditions. This method provides tighter solution ranges compared to the existing polynomial approximation interval methods. Interval arith- metic using the Chebyshev basis and interval arithmetic using the general form modified affine basis for polynomials are developed to obtain tighter bounds for interval computation. To further reduce the overestimation caused by the ＂wrap- ping effect＂ of interval arithmetic, the derivative information of dynamic responses is used to achieve exact solutions when the dynamic responses are monotonic with respect to all the uncertain variables. Finally, two typical numerical examples with nonlinearity are applied to demonstrate the effective- ness of the proposed hybrid interval method, in particular, its ability to effectively control the overestimation for specific timepoints.
基金Project supported by National Natural Science Foundation of China (10872165)Northwestern Polytechnical University (CX200712)
文摘The discovery of chaos in the sixties of last century was a breakthrough in concept,revealing the truth that some disorder behavior,called chaos,could happen even in a deterministic nonlinear system under barely deterministic disturbance.After a series of serious studies,people begin to acknowledge that chaos is a specific type of steady state motion other than the conventional periodic and quasi-periodic ones,featuring a sensitive dependence on initial conditions,resulting from the intrinsic randomness of a nonlinear system itself.In fact,chaos is a collective phenomenon consisting of massive individual chaotic responses,corresponding to different initial conditions in phase space.Any two adjacent individual chaotic responses repel each other,thus causing not only the sensitive dependence on initial conditions but also the existence of at least one positive top Lyapunov exponent(TLE) for chaos.Meanwhile,all the sample responses share one common invariant set on the Poincaré map,called chaotic attractor,which every sample response visits from time to time ergodically.So far,the existence of at least one positive TLE is a commonly acknowledged remarkable feature of chaos.We know that there are various forms of uncertainties in the real world.In theoretical studies,people often use stochastic models to describe these uncertainties,such as random variables or random processes.Systems with random variables as their parameters or with random processes as their excitations are often called stochastic systems.No doubt,chaotic phenomena also exist in stochastic systems,which we call stochastic chaos to distinguish it from deterministic chaos in the deterministic system.Stochastic chaos reflects not only the intrinsic randomness of the nonlinear system but also the external random effects of the random parameter or the random excitation.Hence,stochastic chaos is also a collective massive phenomenon,corresponding not only to different initial conditions but also to different samples of the random parameter or the random excitation.Thus,the unique common feature of deterministic chaos and stochastic chaos is that they all have at least one positive top Lyapunov exponent for their chaotic motion.For analysis of random phenomena,one used to look for the PDFs(Probability Density Functions) of the ensemble random responses.However,it is a pity that PDF information is not favorable to studying repellency of the neighboring chaotic responses nor to calculating the related TLE,so we would rather study stochastic chaos through its sample responses.Moreover,since any sample of stochastic chaos is a deterministic one,we need not supplement any additional definition on stochastic chaos,just mentioning that every sample of stochastic chaos should be deterministic chaos.We are mainly concerned with the following two basic kinds of nonlinear stochastic systems,i.e.one with random variables as its parameters and one with ergodical random processes as its excitations.To solve the stochastic chaos problems of these two kinds of systems,we first transform the original stochastic system into their equivalent deterministic ones.Namely,we can transform the former stochastic system into an equivalent deterministic system in the sense of mean square approximation with respect to the random parameter space by the orthogonal polynomial approximation,and transform the latter one simply through replacing its ergodical random excitations by their representative deterministic samples.Having transformed the original stochastic chaos problem into the deterministic chaos problem of equivalent systems,we can use all the available effective methods for further chaos analysis.In this paper,we aim to review the state of art of studying stochastic chaos with its control and synchronization by the above-mentioned strategy.
文摘By using a series of canonical transformations (Birkhoff's series), an approximate integral of a conservative compound pendulum is evaluated. Level lines of this approximate integral are compared with the numerical simulation results. It is seen clearly that with a raised energy level, the nearly integrable system becomes non-integrable, i.e. the regular motion pattern changes to the chaotic one. Experiments with such a pendulum device display the behavior mentioned above.
文摘In this paper, we analyze chaotic dynamics of nonlinear systems and study chaos synchronization of Lorenz system. We extend our study by discussing other methods available in literature. We propose a theorem followed by a lemma in general and another one for a particular case of Lorenz system. Numerical simulations are given to verify the proposed theorems.
基金co-supported by the National Natural Science Foundation of China(Nos.51875014,U2233212 and 51875015)the Natural Science Foundation of Beijing Municipality,China(No.L221008)+1 种基金the Science,Technology Innovation 2025 Major Project of Ningbo of China(No.2022Z005)the Tianmushan Laboratory Project,China(No.TK-2023-B-001).
文摘Full lifecycle high fidelity digital twin is a complex model set contains multiple functions with high dimensions and multiple variables.Quantifying uncertainty for such complex models often encounters time-consuming challenges,as the number of calculated terms increases exponentially with the dimensionality of the input.This paper based on the multi-stage model and high time consumption problem of digital twins,proposed a sparse polynomial chaos expansions method to generate the digital twin dynamic-polymorphic uncertainty surrogate model,striving to strike a balance between the accuracy and time consumption of models used for digital twin uncertainty quantification.Firstly,an analysis and clarification were conducted on the dynamic-polymorphic uncertainty of the full lifetime running digital twins.Secondly,a sparse polynomial chaos expansions model response was developed based on partial least squares technology with the effectively quantified and selected basis polynomials which sorted by significant influence.In the end,the accuracy of the proxy model is evaluated by leave-one-out cross-validation.The effectiveness of this method was verified through examples,and the results showed that it achieved a balance between maintaining model accuracy and complexity.
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.
