High-dimensional uncertainty quantification of projectile motion in the barrel of a truck-mounted howitzer based on probability density evolution method

This paper proposed an efficient research method for high-dimensional uncertainty quantification of projectile motion in the barrel of a truck-mounted howitzer.Firstly,the dynamic model of projectile motion is established considering the flexible deformation of the barrel and the interaction between the projectile and the barrel.Subsequently,the accuracy of the dynamic model is verified based on the external ballistic projectile attitude test platform.Furthermore,the probability density evolution method(PDEM)is developed to high-dimensional uncertainty quantification of projectile motion.The engineering example highlights the results of the proposed method are consistent with the results obtained by the Monte Carlo Simulation(MCS).Finally,the influence of parameter uncertainty on the projectile disturbance at muzzle under different working conditions is analyzed.The results show that the disturbance of the pitch angular,pitch angular velocity and pitch angular of velocity decreases with the increase of launching angle,and the random parameter ranges of both the projectile and coupling model have similar influence on the disturbance of projectile angular motion at muzzle.

Irradiation-induced void evolution in iron：A phase-field approach with atomistic derived parameters

A series of material parameters are derived from atomistic simulations and implemented into a phase field（PF） model to simulate void evolution in body-centered cubic（bcc） iron subjected to different irradiation doses at different temperatures.The simulation results show good agreement with experimental observations — the porosity as a function of temperature varies in a bell-shaped manner and the void density monotonically decreases with increasing temperatures; both porosity and void density increase with increasing irradiation dose at the same temperature. Analysis reveals that the evolution of void number and size is determined by the interplay among the production, diffusion and recombination of vacancy and interstitial.

Influence of process parameters on the microstructural evolution of a rear axle tube during cross wedge rolling

In the shaping process of cross wedge rolling（CWR）, metal undergoes a complex microstructural evolution, which affects the quality and mechanical properties of the product. Through secondary development of the DEFORM-3D software, we developed a rigid plastic finite element model for a CWR-processed rear axle tube, coupled with thermomechanical and microstructural aspects of workpieces. Using the developed model, we investigated the microstructural evolution of the CWR process. Also, the influence of numerous parameters, including the initial temperature of workpieces, the roll speed, the forming angle, and the spreading angle, on the grain size and the grain-size uniformity of the rolled workpieces was analyzed. The numerical simulation was verified through rolling and metallographic experiments. Good agreement was obtained between the calculated and experimental results, which demonstrated the reliability of the model constructed in this work.

Discrete differential evolution algorithm for integer linear bilevel programming problems

A discrete differential evolution algorithm combined with the branch and bound method is developed to solve the integer linear bilevel programming problems, in which both upper level and lower level variables are forced to be integer. An integer coding for upper level variables is adopted, and then a discrete differential evolution algorithm with an improved feasibility-based comparison is developed to directly explore the integer solution at the upper level. For a given upper level integer variable, the lower level integer programming problem is solved by the existing branch and bound algorithm to obtain the optimal integer solution at the lower level. In the same framework of the algorithm, two other constraint handling methods, i.e. the penalty function method and the feasibility-based comparison method are also tested. The experimental results demonstrate that the discrete differential evolution algorithm with different constraint handling methods is effective in finding the global optimal integer solutions, but the improved constraint handling method performs better than two compared constraint handling methods.

Probabilistic Response and Short-Term Extreme Load Estimation of Offshore Monopile Wind Turbine Towers by Probability Density Evolution Method

A new analysis framework based on probability density evolution method(PDEM)and its Chebyshev collocation solution are introduced to predict the dynamic response and short-term extreme load of offshore wind turbine(OWT)towers subjected to random sea state.With regard to the stochastic responses,random function method is employed to generate samples of sea elevation,the probability density evolution equation(PDEE)is solved to calculate time-variant probability density functions of structural responses.For the probabilistic load estimation,a FAST model of NREL 5MW offshore turbine is established to obtain samples of bending moment at the tower base.The equivalent extreme event theory is used to construct a virtual stochastic process(VSP)to assess the short-term extreme load.The results indicate that the proposed approach can predict time-variant probability density functions of the structural responses,and shows good agreement with Monte Carlo simulations.Additionally,the predicted short-term extreme load can capture the fluctuation at the tail of the extreme value distribution,thus is more rational than results from the typical distribution models.Overall,the proposed method shows good adaptation,precision and efficiency for the dynamic response analysis and load estimation of OWT towers.

An asymptotic solving method for the periodic solution of a class of disturbed nonlinear evolution equation

A class of disturbed evolution equation is considered using a simple and valid technique. We first introduce the periodic traveling-wave solution of a corresponding typical evolution equation. Then the approximate solution for an original disturbed evolution equation is obtained using the asymptotic method. We point out that the series of approximate solution is convergent and the accuracy of the asymptotic solution is studied using the fixed point theorem for the functional analysis.

The redshift number density evolution of Mg Ⅱ absorption systems

We make use of the recent large sample of 17 042 Mg Ⅱ absorption systems from Quider et al. to analyze the evolution of the redshift number density. Regardless of the strength of the absorption line, we find that the evolution of the redshift number density can be clearly distinguished into three different phases. In the intermediate redshift epoch （0.6 ≤ z ≤ 1.6）, the evolution of the redshift number density is consis- tent with the non-evolution curve, however, the non-evolution curve over-predicts the values of the redshift number density in the early （z ≤ 0.6） and late （z ≥ 1.6） epochs. Based on the invariant cross-section of the absorber, the lack of evolution in the red- shift number density compared to the non-evolution curve implies the galaxy number density does not evolve during the middle epoch. The fiat evolution of the redshift number density tends to correspond to a shallow evolution in the galaxy merger rate during the late epoch, and the steep decrease of the redshift number density might be ascribed to the small mass of halos during the early epoch.

A novel refined dynamic model of high-speed maglev train-bridge coupled system for random vibration and running safety assessment

Running safety assessment and tracking irregularity parametric sensitivity analysis of high-speed maglev train-bridge system are of great concern,especially need perfect refinement models in which all properties can be well characterized based on various stochastic excitations.A three-dimensional refined spatial random vibration analysis model of high-speed maglev train-bridge coupled system is established in this paper,in which multi-source uncertainty excitation can be considered simultaneously,and the probability density evolution method(PDEM)is adopted to reveal the system-specific uncertainty dynamic characteristic.The motion equation of the maglev vehicle model is composed of multi-rigid bodies with a total 210-degrees of freedom for each vehicle,and a refined electromagnetic force-air gap model is used to account for the interaction and coupling effect between the moving train and track beam bridges,which are directly established by using finite element method.The model is proven to be applicable by comparing with Monte Carlo simulation.By applying the proposed stochastic framework to the high maglev line,the random dynamic responses of maglev vehicles running on the bridges are studied for running safety and stability assessment.Moreover,the effects of track irregularity wavelength range under different amplitude and running speeds on the coupled system are investigated.The results show that the augmentation of train speed will move backward the sensitive wavelength interval,and track irregularity amplitude influences the response remarkably in the sensitive interval.

Extreme value distribution and reliability of nonlinear stochastic structures

A new approach to evaluate the extreme value distribution （EVD） of the response and reliability of general multi-DOF nonlinear stochastic structures is proposed. The approach is based on the recently developed probability density evolution method, which enables the instantaneous probability density functions of the stochastic responses to be captured. In the proposed method, a virtual stochastic process is first constructed to satisfy the condition that the extreme value of the response equals the value of the constructed process at a certain instant of time. The probability density evolution method is then applied to evaluate the instantaneous probability density function of the response, yielding the EVD. The reliability is therefore available through a simple integration over the safe domain. A numerical algorithm is developed using the Number Theoretical Method to select the discretized representative points. Further, a hyper-ball is imposed to sieve the points from the preceding point set in the hypercube. In the numerical examples, the EVD of random variables is evaluated and compared with the analytical solution. A frame structure is analyzed to capture the EVD of the response and the dynamic reliability. The investigations indicate that the proposed approach provides reasonable accuracy and efficiency.

Probabilistic model and analysis of coupled train-ballasted track-subgrade system with uncertain structural parameters

Random dynamic responses caused by the uncertainty of structural parameters of the coupled train-ballasted track-subgrade system under train loading can pose safety concerns to the train operation.This paper introduced a computational model for analyzing probabilistic dynamic responses of three-dimensional(3D)coupled train-ballasted track-subgrade system(TBTSS),where the coupling effects of uncertain rail irregularities,stiffness and damping properties of ballast and subgrade layers were simultaneously considered.The number theoretical method(NTM)was employed to design discrete points for the multi-dimensional stochastic parameters.The time-histories of stochastic dynamic vibrations of the TBSS with systematically uncertain structural parameters were calculated accurately and efficiently by employing the probability density evolution method(PDEM).The model-predicted results were consistent with those by the Monte Carlo simulation method.A sensitivity study was performed to assess the relative importance of those uncertain structural parameters,based on which a case study was presented to explore the stochastic probability evolution mechanism of such train-ballasted track-subgrade system.

Investigation on the threshold control of safety blasting vibration velocity for the extraction of complicated orebody under railway

The threshold control of safety blasting vibration velocity is a significant process for the underground mining of complicated ore deposit under construction,road,and water.According to the equivalent principle of displacement and velocity of mass point,differential evolution is put forward based on 3DEC dynamic analysis,making the calculation more efficient and accurate.The 3DEC model of the complicated orebody under railway is established according to the topographic maps and geological data of the eastern Pyrite Mine.The stimulus-response distribution of internal stress and displacement fields are demonstrated by analyzing the on-site monitoring vibration displacement and velocity data of the mass point.The reliability of parameter selection,such as blasting simulation waveforms,rock damping,is identified.The safety vibration velocity of railway is set to 4.5 cm/s in line with the requirement of safety blasting rules.Thus,the maximum amount of single-stage explosive in this region is 44.978 kg.The simulation result is in good agreement with the on-site monitoring datum.No displacement and settlement of the 701 railway special line was achieved by choosing the critical amount of the single-stage explosive.

Serviceability evaluation of water supply networks under seismic loads utilizing their operational physical mechanism

The serviceability of water supply networks(WSNs)under seismic loads has significant importance for estimating the probable losses and the impact of diminished functionality on affected communities.The innovation presented in this paper is suggesting a new strategy to evaluate the seismic serviceability of WSNs,utilizing their operational physical mechanism.On one hand,this method can obtain the seismic serviceability of each node as well as entire WSNs.On the other hand,this method can dynamically reflect the propagation of randomness from ground motions to WSNs.First,a finite element model is established to capture the seismic response of buried pipe networks,and a leakage model is suggested to obtain the leakage area of WSNs.Second,the transient flow analysis of WSNs with or without leakage is derived to obtain dynamic water flow and pressure.Third,the seismic serviceability of WSNs is analyzed based on the probability density evolution method(PDEM).Finally,the seismic serviceability of a real WSN in Mianzhu city is assessed to illustrate the method.The case study shows that randomness from the ground motions can obviously affect the leakage state and the probability density of the nodal head during earthquakes.

Orthogonal expansion of ground motion and PDEM-based seismic response analysis of nonlinear structures

This paper introduces an orthogonal expansion method for general stochastic processes. In the method, a normalized orthogonal function of time variable t is first introduced to carry out the decomposition of a stochastic process and then a correlated matrix decomposition technique, which transforms a correlated random vector into a vector of standard uncorrelated random variables, is used to complete a double orthogonal decomposition of the stochastic processes. Considering the relationship between the Hartley transform and Fourier transform of a real-valued function, it is suggested that the first orthogonal expansion in the above process is carried out using the Hartley basis function instead of the trigonometric basis function in practical applications. The seismic ground motion is investigated using the above method. In order to capture the main probabilistic characteristics of the seismic ground motion, it is proposed to directly carry out the orthogonal expansion of the seismic displacements. The case study shows that the proposed method is feasible to represent the seismic ground motion with only a few random variables. In the second part of the paper, the probability density evolution method （PDEM） is employed to study the stochastic response of nonlinear structures subjected to earthquake excitations. In the PDEM, a completely uncoupled one-dimensional partial differential equation, the generalized density evolution equation, plays a central role in governing the stochastic seismic responses of the nonlinear structure. The solution to this equation will yield the instantaneous probability density function of the responses. Computational algorithms to solve the probability density evolution equation are described. An example, which deals with a nonlinear frame structure subjected to stochastic ground motions, is illustrated to validate the above approach.

Joint probability density function of the stochastic responses of nonlinear structures

The joint probability density fimction （PDF） of different structural responses is a very important topic in the stochastic response analysis of nonlinear structures. In this paper, the probability density evolution method, which is successfully developed to capture the instantaneous PDF of an arbitrary single response of interest, is extended to evaluate the joint PDF of any two responses. A two-dimensional partial differential equation in terms of the joint PDF is established. The strategy of selecting representative points via the number theoretical method and sieved by a hyper-ellipsoid is outlined. A two-dimensional difference scheme is developed. The free vibration of an SDOF system is examined to verify the proposed method, and a flame structure exhibiting hysteresis subjected to stochastic ground motion is investigated. It is pointed out that the correlation of different responses results from the fact that randomness of different responses comes from the same set of basic random parameters involved. In other words, the essence of the probabilistic correlation is a physical correlation.

Dynamic Reliability Evaluation of Double-Layer Cylindrical Latticed Shell under Multi-Support Excitations

To overcome the excessive computational cost and/or bad accuracy of traditional approaches,the probabilistic density evolution method(PDEM) is introduced.The dynamic reliability of a double-layer cylindrical latticed shell is evaluated by applying PDEM and Monte Carlo Method(MCM) respectively,and four apparent wave velocities(100 m/s,500 m/s,800 m/s and 1 200 m/s) and five thresholds(0.1 m,0.2 m,0.3 m,0.4 m and 0.5 m) are taken into consideration.Only the difference between threshold and maximal deformation...

Dimension-reduction of FPK equation via equivalent drift coefficient

The Fokker–Planck–Kolmogorov（FPK） equation plays an essential role in nonlinear stochastic dynamics. However, neither analytical nor numerical solution is available as yet to FPK equations for high-dimensional systems. In the present paper, the dimension reduction of FPK equation for systems excited by additive white noise is studied. In the proposed method, probability density evolution method（PDEM）, in which a decoupled generalized density evolution equation is solved, is employed to reproduce the equivalent flux of probability for the marginalized FPK equation. A further step of constructing an equivalent coefficient finally completes the dimension-reduction of FPK equation. Examples are illustrated to verify the proposed method.

The relationship between the properties of PAHs and AGN activities in type-I AGNs

In order to explore the relationship between properties of Polycyclic Aromatic Hydrocarbons （PAHs） and AGN activities in Type-I AGNs, we compiled a sample of 47 Type-I AGNs with measured PAH 11.3 μm and 7.7 μm emission lines. The PAH emission and optical properties of these AGNs are taken from the literature. It is found that the equivalent width （EW） of 11.3 μm emission from PAHs shows a weak correlation with the ratio of the EWs of the FelI complex between λ4434 and A4684 to H/3 （RFe）. The PAH 11.3/7.7 ratio is correlated with various elements defined by the first eigenvector space, which are RFe, [OIII]A5007 luminosity and H/3 asymmetry. These correlations infer that AGNs with high RFe, weak [OIII] emission and a strong H/3 blue-wing are likely to have low PAH ionization, and hence a larger amount of neutral PAHs.

Multicolor photometry of the galaxy cluster A98: substructures and star formation properties

An optical photometric observation with the Beijing-Arizona-Taiwan- Connecticut （BATC） multicolor system is carried out for A98 （z =0.104）, a galaxy cluster with two large enhancements in X-ray surface brightness. Spectral energy distributions （SEDs） covering 15 intermediate bands are obtained for all sources detected down to V - 20 mag in a field of 58′× 58′. After star-galaxy separation with colorcolor diagrams, a photometric redshift technique is applied to the galaxy sample for further membership determination. The color-magnitude relation is taken as a further restriction of the early-type cluster galaxies. As a result, a list of 198 faint member galaxies is achieved. Based on the newly generated sample of member galaxies, the dynamical substructures, A98N, A98S, and A98W, are investigated in detail. A separate galaxy group, A98X, is also found to the south of the main concentration of A98, which is gravitationally unbound to A98. For 74 spectroscopically confirmed member galaxies, the environmental effect on the star formation history is investigated. The bright galaxies in the core region are found to have shorter time scales of star formation, longer mean stellar ages, and higher interstellar medium metallicities, which can be interpreted in the context of the hierarchical cosmological scenario.

Modeling the dynamical friction timescale of a sinking satellite

When a satellite galaxy falls into a massive dark matter halo, it suffers from the dynamical friction force which drags it into the halo＇s center, where it finally merges with the central galaxy. The time interval between entry and merger is called the dynamical friction timescale （Tdf）. Many studies have been dedicated to deriving Tdf using analytical models or N-body simulations. These studies have obtained qualitative agreements on how Zdf depends on the orbital parameters, and the mass ratio between the satellite and the host＇s halo. However, there are still disagreements on deriving an accurate form for Tdf. We present a semi-analytical model to predict Tdf and we focus on interpreting the discrepancies among different studies. We find that the treatment of mass loss from the satellite by tidal stripping dominates the behavior of Tdf. We also identify other model parameters which affect the predicted Tdf.

A Statistical Model for Predicting the Average Abundance Patterns of Heavier Elements in Metal-Poor Stars

We have collected nearly all the available observed data of the elements from Ba to Dy in halo and disk stars in the metallicity range -4.0 <[Fe/H]< 0.5. Based on the observed data of Ba and Eu, we evaluated the least-squares regressions of [Ba/Fe] on [Fe/H], and [Eu/H] on [Ba/H]. Assuming that the heavy elements (heavier than Ba) are produced by a combination of the main components of s- and r-processes in metal-poor stars, and choosing Ba and Eu as respective representative elements of the main s- and the main r-processes, a statistical model for predicting the Galactic chemical evolution of the heavy elements is presented. With this model, we calculate the mean abundance trends of the heavy elements La, Ce, Pr, Nd, Sm, and Dy with the metallicity. We compare our results with the observed data at various metallicities, showing that the predicted trends are in good agreement with the observed trends, at least for the metallicity range [Fe/H]≥ -2.5. Finally, we discuss our results and deduce some important information about the Galactic chemical evolution.