Random vibration of pretensioned rectangular membrane structures under heavy rainfall excitation

An efficient method for solving high-dimension stationary FPK equation of strongly nonlinear systems under additive and/or multiplicative white noise

The Fokker–Planck–Kolmogorov (FPK) equation, as a well investigated partial differential equation, is of great significance to stochastic dynamics due to its theoretical rigor and exactness. However, practical difficulties with the FPK method are encountered when analysis of multi-degree-of-freedom (MDOF) systems with arbitrary nonlinearity is required. In the present paper, a cell renormalized method (CRM) which is based on a numerical determination of response statistical moments of the FPK equation is developed. Specifically, by invoking the concept of equivalence of probability flux, a cell renormalization procedure and a reconstruction scheme of derivative moments are introduced to divide the continuous state space into a discretized region of cells so that numerical derivative moments is derived. Subsequently, the Cell Renormalized FPK (CR-FPK) equation can be solved by a finite difference algorithm. Two numerical examples are included, and the effectiveness of the proposed method is assessed.

In this paper, the method proposed recently by the author for the solution of probability density function (PDF) of nonlinear stochastic systems is presented in detail and extended for more general problems of stochastic differential equations (SDE), therefore the Fokker-Planck-Kolmogorov (FPK) equation is expressed in general form with no limitation on the degree of nonlinearity of the SDE, the type of δ-correlated excitations, the existence of multiplicative excitations, and the dimension of SDE or FPK equation. Examples are given and numerical results are provided for comparing with known exact solution to show the effectiveness of the method.

The damage (in real terms after adjusting for inflation) caused by extreme weather events globally has increased dramatically over the past few decades. This is a result of an increase in the amplitude and frequency of weather extremes, as well as of human factors causing a widespread increase in levels of exposure and vulnerability. There are a number of reasons to consider that, in many regions of the globe, weather extremes (e.g. heat waves, droughts, forest fires, intense rainfall, floods and landslides) are becoming both yet more extreme and more frequent. Projections for the future based on climate and impact models point to a further strengthening of this trend. There has already been an increase in rainfall intensity in conditions of a warmer climate, and a continuation of this trend is expected, with adverse consequences for flood risk. However, the development of flood-prone areas and increase in damage potential are often the dominant factors underpinning growing flood damage and flood risk. In warmer climates, an increased risk of river and flash flooding caused by heavy rainfall, as well as an increasing risk of coastal flooding associated with sea level rise can be expected over large areas. By the same token, a reduction in the risk of snowmelt flooding events is projected in the warmer climate. Projections also indicate an increased risk of drought in many areas. The projections for climate change in Poland point to several risks associated with an increase in the frequency, intensity and severity of weather extremes (heat waves, intensive rainfall, flooding and landslides, coastal surges, drought during the growing season and winter, strong winds and pathogens associated with warming). Heat waves will become more frequent, more intense and more troublesome for the ageing population of Poland.

In this paper, the probabilistic solutions of the multi-degree-of-freedom (MDOF) or large-scale stochastic dynamic systems with polynomial type of nonlinearity and excited by Gaussian white noise excitations are obtained and investigated with the subspace method proposed recently by the authors. The space of the state variables of large-scale nonlinear stochastic dynamic (NSD) system excited by white noises is separated into two subspaces. Both sides of the Fokker–Planck–Kolmogorov (FPK) equation corresponding to the NSD system is then integrated over one of the subspaces. The FPK equation for the joint probability density function of the state variables in another subspace is formulated. Therefore, the FPK equation in low dimensions is obtained from the original FPK equation in high dimensions and it makes the problem of obtaining the probabilistic solutions of large-scale NSD systems solvable with the exponential polynomial closure (EPC) method. A simple flexural beam on nonlinear elastic springs is analyzed with the subspace method to show the effectiveness of the subspace-EPC method in this case.

Impacts of radiance data assimilation on the Beijing 7.21 heavy rainfall

In this work, a novel shape control approach of the probability density function (PDF) for nonlinear stochastic systems is presented. First, we provide the formula for the PDF shape controller without devising the control law of the controller. Then, based on the exact analytical solution of the Fokker-Planck-Kolmogorov (FPK) equation, the product function of the polynomial and the exponential polynomial is regarded as the stationary PDF of the state response. To validate the performance of the proposed control approach, we compared it with the exponential polynomial method and the multi-Gaussian closure method by implementing comparative simulation experiments. The results show that the novel PDF shape control approach is effective and feasible. Using an equal number of parameters, our method can achieve a similar or better control effect as the exponential polynomial method. By comparison with the multi-Gaussian closure method, our method has clear advantages in PDF shape control performance. For all cases, the integral of squared error and the errors of first four moments of our proposed method were very small, indicating superior performance and promising good overall control effects of our method. The approach presented in this study provides an alternative for PDF shape control in nonlinear stochastic systems.

An experimental study on dynamic processes of ephemeral gully erosion in loess landscapes

Low-level convergence induced by land surface heterogeneities can have substantial influence on atmospheric convection and rainfall. Analyses of heavy convective rainfall in the Baltimore–Washington metropolitan area are performed using the Weather Research and Forecasting (WRF) Model, coupled with the Princeton Urban Canopy Model (PUCM) that resolves urban subfacet heterogeneity. Analyses center on storms that produced heavy rainfall and record urban flooding in Baltimore on 1 June 2012. The control simulation using PUCM shows a better performance in reproducing the surface energy balance and rainfall than the simulation using a traditional slab model for the urban area. Sensitivity experiments are carried out to identify the role of the land surface heterogeneities, arising from land–water and urban–nonurban contrasts in the Baltimore–Washington metropolitan area, on heavy rainfall from organized thunderstorm systems. The intersection of low-level convergence zones from thunderstorm downdrafts and from the bay breeze from the Chesapeake Bay enhances the upward motion of preexisting convective storms. The larger sensible heating from the urban area modifies the low-level temperature and wind fields, which in turn modifies the bay breeze. The enhanced moisture supply in the deepened bay-breeze inflow layer due to urban heating promotes intense convection and heavy rainfall in conjunction with the enhanced upward motion at intersecting convergence zones. This study suggests that better representations of surface heat and moisture fluxes in urban areas along major water bodies are required to better capture the timing and location of severe thunderstorms and heavy rainfall.

Probabilistic solutions of nonlinear oscillators excited by combined colored and white noise excitations

An intensity-modulated Gaussian process {x(t)} is defined as the product of a non-negative stationary random process {σx(t)} and a stationary Gaussian process {z(t)} that is statistically independent of {σx(t)}. When such an intensity-modulated process is the excitation to a linear time-invariant system, the response process {y(t)}, conditioned on the excitation-modulating function σx(t), is a generally nonstationary Gaussian process. This property is used to develop an approximation to the probability density function (PDF) of the system response by assuming that the time-varying conditional response variance σ2y(t), conditioned on the excitation-modulating function σx(t), is governed by the gamma PDF. The resulting response process PDF is characterized by two parameters: the mean and variance of the conditional response variance σ2y(t). An alternative approximation to the response PDF, characterized by these same two parameters, is developed by using a Taylor’s series representation in the variable σ2y of the response PDF conditioned on σ2y. This latter PDF approximation is shown to be identical to the Gram–Charlier series of type A through terms involving the Hermite polynomial of degree 6. The two response PDF approximations are shown to give essentially identical results for values of the coefficient of variation of σ2y of one-half or less. Simple, easily evaluated expressions for the mean and variance of σ2y are derived in terms of measurable quantities by utilizing the assumption that fluctuations in σx(t) occur slowly in comparison with those of z(t).

Reliability assessment of stochastic dynamical systems using physics informed neural network based PDEM

The stability of the anchorage slope on the Baiyang Yangtze River Highway Bridge in Yichang, China, was investigated under different rainfall conditions using model test, numerical simulation, and factor analysis. The results of the study are as follows: (1) with the increase of rainfall intensity, the change of earth pressure can be divided into three stages. However, when the rainfall intensity was larger than a certain value, the change of earth pressure of cut slope became two stages; with the increase of rainfall intensity, pore water pressure increased with the increase of rainfall time, while at a certain stage after the rainfall, the pore water pressure in the cut slope did not decrease immediately, but increased for a period of time. (2) When the rainfall stopped, the stability coefficient of the anchorage slope continued to decrease, then slowly increased, and finally tended to be gentle. Meanwhile, when the rainstorm reached a certain intensity, the main factor that restricted the rainfall infiltration rate became the geotechnical permeability coefficient of the cut slope, which was no longer the rainfall intensity. (3) Factor analysis shows that the rainfall intensity and rainfall duration were the most important factors for anchorage slope stability, while earth pressure, pore water pressure and slope displacement were much less significant.

In this paper, the probability density function (PDF) and the mean up-crossing rate of nonlinear ship rolling in random beam seas are investigated. The excitation of stationary random sea waves is approximated as a second-order linear filtered white noise. The Fokker–Planck–Kolmogorov (FPK) equation governing the probability density function of ship rolling is a four-dimensional linear partial differential equation with varying coefficients, and obtaining its exact solution is much more sophisticated. The exponential-polynomial closure (EPC) method is applied to solve the corresponding FPK equation of the system. In numerical examples, linear-plus-cubic damping model and linear-plus-quadratic damping model with three different sea states are further examined. Comparison with the equivalent linearisation (EQL) method and Monte Carlo simulated results show that the proposed procedure is effective to obtain a satisfactory probability density function solution, especially in the tail region.

The rapid development of cities puts forward higher requirements for the resilience of the road network. As one of the unfavorable weathers, rainfall has a great impact on the operation of road traffic. Using weather and vehicle speed data of a city in China, this paper compares and analyzes changes in traffic operation indicators such as the standard speed and the percentage reduction of the standard speed under different rainfall intensities. Results indicate that under the same rainfall conditions and time period, the traffic operation of the elevated road is least affected by rainfall, and the traffic operation of underpass tunnels and flat roads is more affected by rainfall. With the increase of rainfall intensity, the reduction percentage of the standard speed of the underpass tunnel shows an increasing trend; however, it is less affected in light rain and moderate rain, and more affected in heavy rain and rainstorm. With the increase of rainfall intensity, the standard speed of the elevated road has no obvious change trend, and it is less affected by light rain and moderate rain, but more by rainstorm. In each time period, the standard speed reduction percentage of flat roads shows an obvious increasing trend with the increase of rainfall intensity.