Simulation Probability Density Function Research Articles

A generalized method for the stationary probabilistic response of nonlinear dynamical system

Statistical properties of random response essentially depend on the system and excitation features. The mathematical expressions for the response statistical properties should include the information on the system and excitation. In order to adequately illustrate the dependence of random response on the system and excitation features, only the stationary response is concerned. The exponential-polynomial-closure (EPC) method is further generalized and explicitly includes this information in the response probability density function (PDF) approximation. The unknown coefficients in the EPC approximate solution, which are constants in the previous stationary expression, are generalized to be functions of the system and excitation parameters. With the consideration of the independence among these parameters, each unknown coefficient is finally expressed as the product of several functions of each parameter. Based on the simulated response PDF results, the function of each parameter was determined by the least-squares method. Such a generalization deeply depicts the essentials of the response statistical properties and uniquely determines the response PDF expression. Three typical nonlinear systems are taken as examples to verify the efficiency of the proposed method. Numerical results show that approximate results obtained by the proposed method agree well with the simulated or exact ones. In addition, the evolution of the response PDF to the system or excitation parameters is obtained, which can be utilized for stochastic sensitivity analysis.

The generalized EPC method for the non-stationary probabilistic response of nonlinear dynamical system

It is well known that the non-stationary response probability density function (PDF) plays an important role in the reliability and failure analysis. With current approximate or numerical methods, the non-stationary approximate PDF is generally obtained in terms of the ones at the discrete time instants. Repeated computation must be conducted for the ones at other time instants since the response PDF at only one time instant can be obtained after performing the solution procedure. Thus, the computational efficiency suffers a major setback. In this paper, the exponential polynomial closure (EPC) method is further improved and enhanced to obtain the completely non-stationary PDF solution, which is distributed continuously in the time domain. It takes the temporal base function into the PDF approximation. The unknown coefficients in the EPC solution are generalized to be explicit functions of the time parameter. With the least squares method, the explicit time functions can be determined based on the few simulated response PDF values. This implementation makes the continues PDF distribution in the time domain available, which greatly improves the computational efficiency without the repeating computation. Two typical nonlinear systems under stationary and non-stationary random excitations are taken as examples to illustrate the efficiency of the proposed method. Numerical results show that the results obtained by the proposed method agree well with the simulated results. In addition, the relationship between the explicit time function and the modulate function is discussed.

Intelligent overall planning model of underground space based on digital twin

Against the background of the rapid development of social economy and advanced industrial manufacturing, emerging technologies are sweeping the world. The overall planning of urban space is also gradually developing in the direction of "intelligence". The real estate market has been booming in recent years. As a result, the available land resources on the surface are declining. It first gives a general introduction to the digital twin technology and operating rules, and then introduces the analytic hierarchy process and the complexity evolution process model for specific analysis. It first determines the relative weight vector, and then verifies the consistency of the judgment matrix to obtain the simulated probability density function. For the calculation of underground space safety accident risk assessment, the Bayesian network is used for evaluation. This facilitates disaster warnings of possible and ongoing accidents and reduces the adverse consequences of accidents. Then, the digital twin technology is applied to the analysis of underground space visualization, underground construction and underground disaster prevention monitoring. During the construction stage, the bottom of the upper structure column has changed a lot. The difference between the difference between the foundation difference between the adjacent color spectrum column of the upper structure is relatively mild. When digging a large area of ​​ground soil, the foundation dropped to 5.6 mm. When the soil on the side of the pile was removed, the ground building was basically settle down, 0.8 mm. When the structure of the pile side soil is completed, it indicates that the technology of the pile foundation base of the surface building is basically. Finish. Subsequent sidewalk soil has almost no effect on the upper structure. At this stage, the final deposition difference is 2.427 mm, which meets the requirements of the neighboring column base in the current regulations. Finally, I studied the prevention of disaster prevention in the underground space, simulated the accident scene, and obtained the most suitable imaging equipment for the monitoring of disaster. After the prevention of disaster testing and simulation accidents for underground space, the most suitable imaging equipment for preventing disaster testing was obtained. Its formal digital twin technology.

APPLICATION OF THE WEIERSTRASS–MANDELBROT FUNCTION TO THE SIMULATION OF ATMOSPHERIC SCALAR TURBULENCE: A STUDY FOR CARBON DIOXIDE

The statistical simulation of atmospheric carbon dioxide based on time series analysis plays an important role in the development of refined carbon emission assessment and traceability. A new statistical method based on the random Weierstrass–Mandelbrot function is proposed to simulate the high-Reynolds-number and high-Péclet-number scalar turbulence in the atmospheric flow. In this paper, we focus on the carbon dioxide, which is the most important scalar for the global climate changes. The new statistical method is used to simulate the time series of carbon dioxide concentrations in the high-frequency inertial subrange. Results show that the simulated power spectral densities are almost the same as the observed ones in the inertial subrange. Compared with the Gaussian distribution, the simulated probability density functions are more similar to the observed ones with the skewness characteristic. More importantly, the fractal characteristics of the time series that cannot be simulated by traditional methods in the meteorological engineering can also be well simulated by the new method.

Efficient phase-shift error compensation algorithm based on probability density function

In non-digital projection systems, the inaccuracy of phase shifts is one of the most important nonlinear errors. In this paper, an efficient and universal compensation algorithm is proposed, which does not need to find the exact phase shift amounts, but to compensate the periodical errors by a universal two-coefficient error model. With this model, only two error coefficients should to be searched for any multi-step phase shift algorithm. To find the error coefficients, a correlation process and a tool known as the probability density function (PDF) are used. First, a set of pre-simulated PDF curves are generated with the different error coefficients. Second, a practical PDF curve from captured fringes is calculated. Finally, the correlation process is carried out to locate the most similar simulated PDF curve, whose error coefficients are the practical ones. In this paper, we demonstrate the compensation for three, four and five-step phase-shifting algorithms separately. Experimental results show that this method has high universality, and can meet the requirements of rapid and high accuracy measurement.

Molecular dynamics simulation of defect production in Fe due to irradiation

Molecular dynamics simulations were performed to investigate displacement cascades in Fe as a function of primary knock-on atom (PKA) energy. The statistics of defect production due to displacement cascades was focused in this study, where 1000 cases were simulated for individual PKA energies. It was found in our simulations that the simulated probability density function of the number of athermal point defects produced by the displacement cascade exhibits approximately a normal distribution. For PKA energies below 100 eV, the average number of surviving point defects is in proportion to PKA energy with a constant standard deviation. On the other hand, for higher energies, the average and the standard deviation individually show a power dependence on PKA energy. The PKA energy dependences of the average number of surviving point defects are consistent with those of the NRT and Bacon models for lower and higher PKA energies, respectively. Statistical characteristics of displacement cascades will help to describe the changes in the microstructure and mechanical properties of materials under irradiation.

Modeling of turbulent flames with the large eddy simulation–probability density function (LES–PDF) approach, stochastic fields, and artificial neural networks

This work proposes a chemical mechanism tabulation method using artificial neural networks (ANNs) for turbulent combustion simulations. The method is employed here in the context of the Large-Eddy Simulation (LES)–Probability Density Function (PDF) approach and the method of stochastic fields for numerical solution, but can also be employed in other methods featuring real-time integration of chemical kinetics. The focus of the paper is on exploring an ANN architecture aiming at improved generalization, which uses a single multilayer perceptron (MLP) for each species over the entire training dataset. This method is shown to outperform previous approaches which take advantage of specialization by clustering the composition space using the Self-Organizing Map (SOM). The ANN training data are generated using the canonical combustion problem of igniting/extinguishing one-dimensional laminar flamelets with a detailed methane combustion mechanism, before being augmented with randomly generated data to produce a hybrid random/flamelet dataset with improved composition space coverage. The ANNs generated in this study are applied to the LES of a turbulent non-premixed CH4/air flame, Sydney flame L. The transported PDF approach is used for reaction source term closure, while numerical solution is obtained using the method of stochastic fields. Very good agreement is observed between direct integration (DI) and the ANNs, meaning that the ANNs can successfully replace the integration of chemical kinetics. The time taken for the reaction source computation is reduced 18-fold, which means that LES–PDF simulations with comprehensive mechanisms can be performed on modest computing resources.

Large Eddy Simulation of Supersonic Combustion Using the Eulerian Stochastic Fields Method

The development of supersonic combustion engines (scramjet-type) presents several challenges. Numerical simulations of scramjet engine combustion have become an attractive alternative to experimental investigations. However, supersonic combustion simulations still have several challenges: such as adequate modelling of shock/boundary layer and turbulence/chemistry interactions. The present work presents two large-eddy simulation probability density function (LES-PDF) novel formulations developed for high-speed applications. One is a conservative joint-scalar approach, where the joint-probability for reactive scalars and energy is solved, while the other is a joint velocity-scalar. The LES-PDF transport equations are solved using the Eulerian stochastic fields technique implemented in a density-based compressible solver. The performance of the models is verified through the simulations of two and three-dimensional supersonic reacting mixing layers and compared against DNS data from the literature. The results show that the joint-scalar formulation is accurate and robust, while the joint velocity-scalar closures require further development.

Simple Degree-of-Freedom Modeling of the Random Fluctuation Arising in Human–Bicycle Balance

In this study, we propose a new simple degree-of-freedom fluctuation model that accurately reproduces the probability density functions (PDFs) of human–bicycle balance motions as simply as possible. First, we measure the time series of the roll angular displacement and velocity of human–bicycle balance motions and construct their PDFs. Next, using these PDFs as training data, we identify the model parameters by means of particle swarm optimization; in particular, we minimize the Kolmogorov–Smirnov distance between the human PDFs from the participants and the PDFs simulated by our model. The resulting PDF fitnesses were over 98.7 % for all participants, indicating that our simulated PDFs were in close agreement with human PDFs. Furthermore, the Kolmogorov–Smirnov statistical hypothesis testing was applied to the resulting human–bicycle fluctuation model, showing that the measured time responses were much better supported by our model than the Gaussian distribution.

Performance analysis of spatial modulation over generalized [formula omitted] fading distribution

In this paper, moment generating function (MGF) based mathematical expression is derived to compute bit error rate (BER) for spatial modulation (SM) system over generalized α−κ−μ fading channel distribution. The derivation exploits the MGF of the square of α−κ−μ fading distribution and Gauss Quadrature formulas to find a numerical solution of the intractable mathematical integrals. The Monte Carlo simulations are performed to analyze the BER curves of the SM system for M-quadrature amplitude modulation (QAM) signal over different values of α−κ−μ fading distribution. Besides, the effect of antenna configuration on the performance of the investigated system is also highlighted. Furthermore, the theoretical and simulated probability density function (PDF) curves for different values of α−κ−μ distribution are also drawn.

Modeling Subgrid Variability of Snow Depth Using the Fokker‐Planck Equation Approach

AbstractA physically based subgrid variability model for snow process using the Fokker‐Planck equation (FPE) approach was proposed. This FPE can express the evolution of the probability density function (PDF) of snow depth within a finite area, possibly a grid cell of distributed models or a small basin, whose shape can be irregular. The main advantage of this approach is that it does not rely on a given PDF but dynamically computes the PDF through an advection‐diffusion‐type equation, the FPE, which was derived from point‐scale process‐based governing equations. Snow depth was treated as a random variable, while the snow redistribution and snowmelt rate were treated as the sources of stochasticity. The main challenge in solving this FPE is evaluating the time‐space covariances appearing in the diffusion coefficient. In this study, approximations to evaluate the covariance terms, accounting for snowmelt and snow redistribution, were proposed. The simulated results of the FPE model were validated by the measured time series of snow depth at one site and the spatial distributions of snow depth measured by ground penetrating radar and airborne light detection and ranging (Lidar). It was shown that the point‐observed snow depth fell within the simulated range during most of the 2‐year study period. The simulated PDFs of snow depth within the study area were similar to the observed PDFs of snow depth by ground penetrating radar and Lidar. In summary, these results demonstrate the efficacy of the proposed FPE model representing the subgrid variability of snow depth.

Nuclear structure investigations of 84Sr and 86Sr using γ-ray spectroscopic methods

To investigate the low-spin structure of 84Sr and 86Sr, two fusion-evaporation experiments using the (p,2n) reaction have been performed. The data have been evaluated in terms of γ-ray spectroscopy, i.e. the level scheme was built up, branching intensities were determined, nuclear state spin and parity assignments were made after γγ angular correlation analyses. The previously known level scheme information was extended. In addition, lifetimes were determined by means of the Doppler Shift Attenuation Method (DSAM). In 86Sr, a new method for lifetime determinations, the Spectral Difference Method (SDM), was applied for the first time. In cases where all necessary input variables were known, reduced transition strengths were calculated with Monte Carlo simulated probability density functions. It is stressed that linear error propagation for the calculated errors of reduced transition strengths is not universally applicable. As both nuclei lie in the vicinity of the N=50 neutron shell closure and on a possible proton subshell closure of the 2p3/2 orbital at Z=38, Nuclear Shell Model calculations for 84Sr, 86Sr, and 88Sr have been performed in the truncated (2p3/2, 1f5/2, 2p1/2, 1g9/2) model space for protons and neutrons with the empirical JUN45 residual interaction. The comparison between experimental results and theoretical predictions show a large degree of consistency.

Observations and simulations of wave runup during a laboratory dune erosion experiment

XBeach, a process-based numerical model designed to simulate morphologic change during extreme storm events, was calibrated and compared to observations from a large-scale laboratory dune erosion experiment. Wave conditions along the tan, wave runup at the beach, and 3-dimensional position of the dune were measured throughout the experiment. Seiching modes in the tank were simulated and matched the observations once the model was calibrated. Simulated waves were sensitive to γ, the ratio of wave height to water depth, and cf., the coefficient of friction. Simulated runup explained 50–87% of the observed variance in observations. However, the magnitude of simulated wave runup was underestimated throughout the experiment. Errors in simulated runup were composed of a high bias and gain error in mean water level, low bias in the infragravity swash, and low bias and gain errors in incident swash. Observed probability density functions of swash were statistically consistent between times when swash was confined to the foreshore and times when swash interacted with the dune. However, simulated probability density functions of swash were statistically different during the collision regime. Despite the systematic underestimation of wave runup, modeled dune erosion compared well with observations after the sediment transport parameters were calibrated. Modeled dune erosion was sensitive to the critical slope parameters over the wet and dry regions of the beach, the depth of the interface between the wet and dry regions of the beach, and the threshold depth for sediment transport and return flow.

Analysis of diagnostic climate model cloud parametrizations using large‐eddy simulations

Current climate models often predict fractional cloud cover on the basis of a diagnostic probability density function (PDF) describing the subgrid‐scale variability of the total water specific humidity, qt, favouring schemes with limited complexity. Standard shapes are uniform or triangular PDFs, the widths of which are assumed to scale with the grid‐box mean qt or the grid‐box mean saturation specific humidity, qs. In this study, the qt variability is analysed from large‐eddy simulations for two stratocumulus, two shallow cumulus, and one deep convective cases. We find that, in most cases, triangles are a better approximation to the simulated PDFs than uniform distributions. In 2 of the 24 slices examined, the actual distributions were so strongly skewed that the simple symmetric shapes could not capture the PDF at all. The distribution width for either shape scales acceptably well with both the mean values of qt and qs, the former being a slightly better choice. The qt variance is underestimated by the fitted PDFs, but overestimated by the existing parametrizations. While the cloud fraction is in general relatively well diagnosed from fitted or parametrized uniform or triangular PDFs, it fails to capture cases with small partial cloudiness, and in 10–30% of the cases misdiagnoses clouds in clear skies or vice versa. The results suggest choosing a parametrization with a triangular shape, where the distribution width would scale with the grid‐box mean qt using a scaling factor of 0.076. However, this is subject to the caveat that the reference simulations examined here were partly for rather small domains and driven by idealised boundary conditions.

Analysis of extinction in a non-premixed turbulent flame using large eddy simulation and the chemical explosion mode analysis

The paper presents computational results of a turbulent non-premixed flame with large extinction, using the large eddy simulation probability density function methodology. The simulation captures local flame extinction at different flame locations, and major species predictions show good agreement with experimental data. A chemical explosive mode analysis is used to determine the flame structure including the explosive modes and the Damköhler number distribution. Due to the nature of the turbulence combustion model, statistical information on the sub-grid flame structure is available: such as sub-grid explosive modes. The analysis suggests that, in the present flame, sub-grid structures are only relevant close to the inlet nozzle, and that downstream extinction is governed by large-scale interactions.

Change of scaling and appearance of scale-free size distribution in aggregation kinetics by additive rules

The idealized general model of aggregate growth is considered on the basis of the simple additive rules that correspond to a one-step aggregation process. The two idealized cases were analytically investigated and simulated by the Monte Carlo method in the Desktop Grid distributed computing environment to analyze “pile-up” and “wall” cluster distributions in different aggregation scenarios. Several aspects of aggregation kinetics (change of scaling, change of size distribution type, and appearance of scale-free size distribution) driven by the “zero cluster size” boundary condition were determined by the analysis of evolving cumulative distribution functions. The “pile-up” case with a minimum active surface (singularity) could imitate piling up aggregations of dislocations, and the case with a maximum active surface could imitate arrangements of dislocations in walls. The change of scaling law (for pile-ups and walls) and availability of scale-free distributions (for walls) were analytically shown and confirmed by scaling, fitting, moment, and bootstrapping analyses of simulated probability density and cumulative distribution functions. The initial “singular” symmetric distribution of pile-ups evolves by the “infinite” diffusive scaling law and later it is replaced by the other “semi-infinite” diffusive scaling law with asymmetric distribution of pile-ups. In contrast, the initial “singular” symmetric distributions of walls initially evolve by the diffusive scaling law and later it is replaced by the other ballistic (linear) scaling law with scale-free exponential distributions without distinctive peaks. The conclusion was made as to possible applications of such an approach for scaling, fitting, moment, and bootstrapping analyses of distributions in simulated and experimental data.

Stochastic control of Indian megadroughts and megafloods

A multi-millennial run of the CSIRO Mark2 coupled climatic model has been used to investigate megadroughts and megafloods during the Indian summer monsoon (June–September). These extreme events were defined as having rainfall anomalies at least two standard deviations from normal. More than ten megafloods and more than twenty megadroughts, so-defined, were found to occur in a 5,000-year period of the simulation. The simulation replicated most of the major features of the observed summer monsoon, but a comparison of observed and simulated probability density functions suggests that the limited observed rainfall time series to date does not adequately sample the possible range of Indian monsoonal rainfall. An investigation of causal mechanisms of Indian rainfall variability reproduced the observed negative correlation with ENSO events, but it was found that neither extreme ENSO events or extremes of a range of other climatic phenomena coincided with the simulated, extreme megadroughts and megafloods. This disconnect between these events is succinctly illustrated with examples related to ENSO events in particular. Autoregressive and FFT analysis of observed and simulated Indian summer monsoon rainfall time series revealed them to consist of white noise. Since these time series therefore consist of random outcomes, it is apparent that these Indian megadroughts and megafloods are the consequence of stochastic influences. Thus, it is concluded that the interannual variability of Indian summer monsoonal rainfall cannot be predicted in general, nor can megadroughts and megafloods in particular.

Generating probabilistic estimates of hydrological response for Irish catchments using a weather generator and probabilistic climate change scenarios

AbstractIn accounting for uncertainties in future simulations of hydrological response of a catchment, two approaches have come to the fore: deterministic scenario‐based approaches and stochastic probabilistic approaches. As scenario‐based approaches result in a wide range of outcomes, the role of probabilistic‐based estimates of climate change impacts for policy formulation has been increasingly advocated by researchers and policy makers. This study evaluates the impact of climate change on seasonal river flows by propagating daily climate time series, derived from probabilistic‐based climate scenarios using a weather generator (WGEN), through a set of conceptual hydrological models. Probabilistic scenarios are generated using two different techniques. The first technique used probabilistic climate scenarios developed from statistically downscaled scenarios for Ireland, hereafter called SDprob. The second technique used output from 17 global climate models (GCMs), all of which participated in CMIP3, to generate change factors (hereafter called CF). Outputs from both the SDprob and the CF approach were then used in combination with WGEN to generate daily climate scenarios for use in the hydrological models. The range of simulated flow derived with the CF method is in general larger than those estimated with the SDprob method in winter and vice versa because of the strong seasonality in the precipitation signal for the 17 GCMs. Despite this, the simulated probability density function of seasonal mean streamflow estimated with both methods is similar. This indicates the usefulness of the SDprob or probabilistic approach derived from regional scenarios compared with the CF method that relies on sampling a diversity of response from the GCMs. Irrespective of technique used, the probability density functions of seasonal mean flow produced for four selected basins is wide indicating considerable modelling uncertainties. Such a finding has important implications for developing adaptation strategies at the catchment level in Ireland. Copyright © 2011 John Wiley & Sons, Ltd.

Validation of the ENSEMBLES global climate ­models over southwestern Europe using probability density functions, from a downscaling perspective

In this study we analyzed the performance of 12 state-of-the-art global climate models (GCMs) from 2 different model generations used in the ENSEMBLES project (a European Commission- funded climate-change research project) over southwestern Europe. For this purpose, we assessed the similarity of the simulated and quasi-observed (reanalysis) probability density functions for circulation, temperature, and humidity variables at various pressure levels, which we chose from a statistical- downscaling point of view. Our main goals were to assess which GCM variables can be reliably used as predictors for downscaling, and which GCMs perform especially well over the region under study. Results showed that specific humidity is as reliably reproduced as circulation and temperature vari- ables, and that overall performance is best for the Hadley Centre's HADGEM2 model. Secondary goals were to estimate the skillful scale of the models, and to measure the added value of bias correction, a post-processing step commonly used in practice. We found that all models lack performance at the scale of individual grid boxes, indicating that they are not robustly skillful at their smallest scale. We also found that model performance generally improves after removing monthly bias. However, model errors at higher-order moments, which cannot be removed by simply correcting the bias, were common in some models.

Fast Simulation Method for Turbo Codes over Additive White Class A Noise Channel

This study shows a fast simulation method for turbo codes over an additive white class A noise (AWAN) channel. The reduction of the estimation time is achieved by applying importance sampling (IS) which is one of the variance reduction simulation methods. In order to adapt the AWAN channel, we propose a design method of a simulation probability density function (PDF) utilized in IS. The proposed simulation PDF is related to the Bhattacharyya bound to evaluate wider area of the signal space than the conventional method. Since the mean translation method, which is a conventional design method of the simulation PDF used in IS, is optimized for an additive white Gaussian noise channel, the evaluation time of the error performance of turbo codes over the AWAN channel can not be reduced. To evaluate BER of 10 ―8 , the simulation time of the proposed method can be reduced to 1 / 1 04 under the condition of the same accuracy of the traditional Monte Carlo simulation method.