Joint Probability Density Function Research Articles

A flamelet-based Eulerian transported PDF method for the modeling and simulation of supersonic combustion

This paper presents a high-efficiency and high-fidelity approach to model supersonic combustion using the extended flamelet-generated manifold (FGM) and the Eulerian transported probability density function (PDF), also known as the Eulerian stochastic fields (ESF) method. The efficiency benefits from the FGM, where the compressibility effects induced by shock waves are considered using two extra control variables, i.e., the pressure (p) and the absolute internal energy of the oxidizer (Eox), in addition to the mixture fraction (Z) and progress variable (Yc) in traditional flamelet tables. The joint PDF for these control variables is modeled using transported PDF based on the ESF method. The ESF method enhances accuracy in the prediction of turbulence-chemistry interactions, avoiding complexity induced by the presumed PDF in the flamelet table and ad-hoc presumed and independent joint PDF assumptions, e.g., the typical presumed β-PDF for Z and δ-PDF for Yc. This FGM-ESF method is tested in large eddy simulations of two canonical hydrogen supersonic flames: a strut-stabilized hydrogen supersonic flame (DLR case) and a transverse hydrogen jet flame in a high-enthalpy incoming flow (Stanford case). For both cases, results show that including the compressibility effects in the FGM table is essential for properly describing the flame behaviors near shock waves. The sub-grid PDF of control variables significantly influences near-wall and shear-layer combustion, and the ESF method demonstrates superior performance in predicting the near-wall reaction zone and the shear-layer reaction zone compared to the perfectly micro-mixed sub-grid model (δ-PDF) for the Stanford case. This study marks the first application of the FGM-ESF approach with a Z-Yc-Eox-p FGM table to simulate supersonic flames, offering a novel perspective for future modeling efforts in this domain.Novelty and Significance StatementThe novelty of this research lies in the development and application of an efficient computational approach that for the first time couples the extended flamelet-generated manifold (FGM) method with the Eulerian stochastic fields (ESF) to simulate supersonic flames. This combined FGM-ESF method incorporates pressure and oxidizer energy as additional control variables in the FGM tabulation to account for compressibility effects. The ESF is used to describe the sub-grid scale joint probability density function without relying on complex presumed functions. Application to a canonical strut-stabilized hydrogen supersonic flame and transverse hydrogen jet flames in a high-enthalpy incoming flow demonstrate the superior predictive capabilities of the FGM-ESF method in capturing flame dynamics. The comparison between four-dimensional and two-dimensional FGM tables provides new insights into the importance of compressibility effects for these configurations.

Development of a new general class of bivariate distributions based on reversed hazard rate order

Motivated by real data sets to be analyzed in this paper, we develop a new general class of bivariate distributions that can model the effect of the so-called ‘load-sharing configuration’ in a system with two components based on the reversed hazard rate. Under such load-sharing configuration, after the failure of one component, the surviving component has to shoulder extra load, which eventually results in its failure at an earlier time than what is expected under the case of independence. In the developed class of bivariate distributions, it is assumed that the residual lifetime of the remaining component is shortened according to the reversed hazard rate order. We derive the joint survival function, joint probability density function and the marginal distributions. We discuss a bivariate ageing property of the developed class of distributions. Some specific families of bivariate distributions which can be usefully applied in practice are obtained. These families of bivariate distributions are applied to some real data sets to illustrate their usefulness.

Fatigue study of unbonded flexible risers for ocean thermal energy conversion based on the joint distribution of wind, wave, and current

This paper investigates flexible risers attached to the base of a semi-submersible platform. Using meteorological and oceanographic data, the wind-wave-current boundary distribution was fitted, and the c-vine copula method established the joint probability density function of wind, wave, and current. An improved grid-based method was used to account for these loads, selecting conditions that meet the confidence level by calculating occurrence probabilities. Time-domain analysis yielded the flexible riser stress time history, and the Miner linear damage accumulation method calculated damage at critical points. The overall lifespan pattern of the riser was analyzed with the joint probability distribution. Results indicate that large heave response amplitude and higher current speeds significantly reduce the riser's fatigue life. The improved grid-based method effectively reduces computational load, providing a feasible approach for designing the fatigue life of ocean thermal energy conversion risers under wind, wave, and current loads.

Effect of the uncertainty of autocorrelation estimation on trend analysis

Autocorrelation is ubiquitous in real-life time series. For trend analysis, positive autocorrelation mainly increases the statistical uncertainty of the detected trend. The theoretical framework has been established to quantify how the autocorrelation can impact trend analysis if its characteristic is known precisely. However, if the autocorrelation is also uncertain like that estimated in real-life data, then the consequences are yet unclear. The situation is further complicated by the concurrence of both short- and long-range correlation (SLRC), e.g., in air temperature. In this study, we use an appropriate minimal model with one short-range autoregressive parameter and one long-range fractional parameter to account for such SLRC, and then accordingly propose a framework to integrate the effect of the uncertainty of autocorrelation estimation into trend analysis, on the basis of the detrended fluctuation analysis and Akaike weights. This framework can also obtain a joint probability density function of the estimated autoregressive and fractional parameters for measuring their uncertainty, identifying their combined optimal estimates, uncovering their strong interdependence, and determining their two-dimensional confidence region. The effectiveness and applicability of the proposed framework are verified by a numerical experiment and a case study of daily air temperature anomalies at the Potsdam station. This study can provide a solid theoretical basis to underpin our understanding of the autocorrelation effect on trend analysis in theory and applications. Published by the American Physical Society 2024

High-efficient non-iterative reliability-based design optimization based on the design space virtually conditionalized reliability evaluation method

Dynamic-reliability-based design optimization (DRBDO) is a promising methodology to address the significant challenge posed by the new generation of structural design theories centered around reliability considerations. Solving DRBDO problems typically requires iterations ranging from a dozen to several hundreds, with each iteration dedicated to updating the values of design variables. Furthermore, DRBDO necessitates hundreds of or even more representative structural analyses at each iteration to compute the reliability measure, which serves as a foundation for determining the search direction in the subsequent iteration. This results in the double-loop problem confronted by DRBDO, leading to substantial computational costs for structural re-computing, particularly in the cases involving complex nonlinear stochastic dynamical systems. In the present paper, a non-iterative DRBDO paradigms is proposed by combining a novel virtually conditional reliability evaluation and the newly proposed decoupled multi-probability density evolution method (M-PDEM). By leveraging the decoupled M-PDEM, a series of one-dimensional partial differential equations (PDEs) named Li-Chen equations are solved to calculate the joint PDF of multiple responses. This enables efficient computation of the joint probability density function (PDF) of design variables and extreme response as well as the conditional PDF of the extreme response given the values of design variables based on finite representative structural analyses. Then, the reliability of different designs can be regarded as the integral of the conditional PDF, which yields the reliability feasible domain. For problems that the objective function is monotonic to each design variables, by combining with a direct search technique, this method transforms the optimization process into true iteration-free calculations, and thereby eliminates the significant computational burden associated with structural re-computing at different intermediate designs in optimization iterations. Finally, the accuracy and effectiveness of this novel method are validated through numerical examples.

Bivariate Generalized Exponential Distribution with Positive Probability at (X = 0, Y = 0) and Estimation of Parameters for a Particular Case

Kundu and Gupta, 2009, J Multivar Anal, 100, 581–593, provided a bivariate generalized exponential distribution (BVGED [Formula: see text]), which is a bivariate continuous lifetime distribution. In practice, however, sometimes both the components may fail instantaneously. Thus, we get a mixture bivariate distribution with positive probability at instantaneous failure of each component. The main aim of this article is to define a BVGED with positive probability at instantaneous failure of each component (BVGEDP [Formula: see text]), where PX=0P(X = 0) and P(Y = 0) is a function of parameter [Formula: see text]. It is observed that the joint probability density function and the cumulative distribution function can be expressed in compact forms. Several properties of the distribution have been discussed. For a particular case (BVGEDP ([Formula: see text] 1, 1, 1)), moment type estimators are obtained which are in closed form and maximum likelihood estimators of parameters ([Formula: see text]) are obtained using an iterative procedure. Also, their finite sample and asymptotic properties are discussed using simulation. A final section presents an application to a real data set on which the proposed BVGEDP ([Formula: see text], 1, 1, 1) distribution is fitted. AMS Subject Classification: 62E99

A New Bivariate Survival Model: The Marshall-Olkin Bivariate Exponentiated Lomax Distribution with Modeling Bivariate Football Scoring Data

This paper focuses on applying the Marshall-Olkin approach to generate a new bivariate distribution. The distribution is called the bivariate exponentiated Lomax distribution, and its marginal distribution is the exponentiated Lomax distribution. Numerous attributes are examined, including the joint reliability and hazard functions, the bivariate probability density function, and its marginals. The joint probability density function and joint cumulative distribution function can be stated analytically. Different contour plots of the joint probability density function and joint reliability and hazard rate functions of the bivariate exponentiated Lomax distribution are given. The unknown parameters and reliability and hazard rate functions of the bivariate exponentiated Lomax distribution are estimated using the maximum likelihood method. Also, the Bayesian technique is applied to derive the Bayes estimators and reliability and hazard rate functions of the bivariate exponentiated Lomax distribution. In addition, maximum likelihood and Bayesian two-sample prediction are considered to predict a future observation from a future sample of the bivariate exponentiated Lomax distribution. A simulation study is presented to investigate the theoretical findings derived in this paper and to evaluate the performance of the maximum likelihood and Bayes estimates and predictors. Furthermore, the real data set used in this paper comprises the scoring times from 42 American Football League matches that took place over three consecutive independent weekends in 1986. The results of utilizing the real data set approve the practicality and flexibility of the bivariate exponentiated Lomax distribution in real-world situations, and the bivariate exponentiated Lomax distribution is suitable for modeling this bivariate data set.

Low-carbon economic optimization strategy for industrial loads in parks considering source-load-price multivariate uncertainty

A low-carbon economic optimization method is proposed for industrial loads in parks considering multivariate uncertainty. Model of dynamic load node carbon intensity is proposed considering the uncertainty of clean energy and dynamic conventional units carbon intensity based on system currents. The robustness is improved based on multi-interval uncertainty set. The gap is filled by hybrid copula model with Generalized Autoregressive Conditional Heteroskedasticity (GARCH) prices in the uncertainty model. The GARCH is used to built the marginal distribution function of prices, and a hybrid copula model is proposed to obtain the joint probability density function based on the weights calculated by Euclidean distance. The LNCI probability distributions of the regulation potential of loads are proposed with the virtual energy storage and the virtual power model to improve the descriptive ability of uncertainty based on the expansion and contraction functions. Finally, the typical scenarios are extracted based on Monte Carlo and Wasserstein measures. The Column sum Constraint Generation algorithm and strong duality theory are used to get the robust-stochastic optimization. The method proposed in the paper has a positive effect on enterprises to rationalize their production schedules, reduce electricity and carbon emissions costs and increase the revenue from participating in grid regulation.

Probabilistic memory-enhanced recognition-primed decision model and its application to pilot decision-making in midair encounters

The recognition-primed decision model is renowned for its adaptability and superior performance, making it extensively utilized across various domains, including artificial agents, social systems, medical diagnostic systems, and industrial operations. This paper presents a novel approach, namely the probabilistic memory model, which integrates the probabilistic characteristics inherent in human memory into the recognition-primed decision model and enhances the recognition-primed decision model’s ability to process continuous information effectively. The memory structure in the proposed model is represented by a succession of joint probability density functions, each delineating a prototype in human memory. Leveraging these prototypes, four memory functions, i.e., memory encoding, prototype abstraction, memory indexing, and memory retrieval, are designed. As a case study, the proposed model is applied to analyze pilot decision-making in midair encounter scenarios. The findings substantiate the superior suitability of the proposed model in handling continuous information. Furthermore, empirical evidence supports the notion that the proposed model excels in retaining information from experiences, thereby exhibiting improved accuracy compared to existing models. Additionally, this paper studies the sensitivity of the model to input noises and the prototype abstraction number. Finally, the proposed model is evaluated against realistic midair encounter scenarios. The evaluation results demonstrate that the proposed model can effectively predict pilot behavior during midair encounters, outperforming existing models in realistic midair situations.

Higher-order stochastic averaging for investigating a vehicle suspension system with nonlinear damping and stiffness

The paper deals with a quarter-car model with nonlinear damping and stiffness under white noise base excitation using the higher-order stochastic averaging method for analyzing approximate responses. Recently, a novel higher-order averaging procedure has been developed to find analytically the first-, second-, and third-order stationary joint probability density functions (PDF) of amplitude and full phase by solving the corresponding Fokker-Planck-Kolmogorov (FPK) equation, and it will be extended to the nonlinear quarter-car model. Accordingly, the mean-square responses such as the displacement, and velocity of the sprung mass can be obtained analytically. The influences of excitation intensity on the dynamic responses, as well as the variations of linear and nonlinear damping, are analyzed. A very satisfactory agreement is found between the accuracy of the solutions corresponding to higher-order stochastic averaging and that of Monte Carlo simulation.

Design considerations for a continuous mussel farm in New England Offshore waters. Part I: Development of environmental conditions for engineering design

Sustainable aquaculture in nearshore waters faces challenges such as stakeholder conflicts, environmental pollution, and spatial constraints. Offshore aquaculture offers a promising solution but requires robust engineering design to withstand extreme weather conditions. This study develops the environmental conditions essential for engineering the design of a continuous mussel dropper system in New England offshore waters. A potential farm location was identified using criteria including water depth, federal boundaries, seafloor suitability, farm size, and proximity to ports, based on bathymetric and sedimentary maps. Historical data from five wave monitoring stations and two current velocity stations were analyzed to model extreme environmental conditions, as waves and currents pose primary threats. The extreme wave and current conditions are modeled using the Weibull distribution, on the annual maximum hourly significant wave height data and the largest 0.3 % of current speeds. A newly proposed method combining Spalding’s wall function with a fourth-order polynomial is used to enhance the current profile analysis. Additionally, a joint probability density function was developed for wave height and current velocity at a specific depth, providing insights into wave height, period, and wavelength for various return periods such as 10, 25, 50, and 100 years. The results suggest a 10-yr wave of 8 m significant wave height and a current speed of 1.68 m/s, while a 50-yr values are 9.4 m and 1.96 m/s respectively. These findings offer critical data on extreme wave and current conditions in New England’s offshore waters, providing practical guidance for the engineering design of offshore mussel farms. This research advances offshore mussel farming and benefits the development of all types of offshore aquaculture systems.

Optical Möbius strips in isotropic random non-paraxial light

Abstract The statistics of Möbius strips with various topologies, formed by the axes of polarization ellipses as they are traced along a closed circular contour of small size passing through the center of a solitary circular polarization singularity line (C-line), have been investigated both analytically and numerically in a random isotropic electromagnetic field. Found are the analytical expressions for the joint probability density function of the differential characteristics of the random isotropic electromagnetic field, which allow for the determination of the topological properties of diagrams of polarization ellipses and the normal vectors to them, as well as the optical strips that arise in the space around C-lines.

Inertial amplification as a performance enhancement method for snap-through vibration energy harvester

In recent years, there has been a lot of interest in exploiting the bistable behavior of snap-through systems to harvest energy from vibration sources. The efficient operation of any bistable VEH depends on its ability to exhibit large-amplitude interwell motion. Under weak ambient excitation, bistable VEH performs marginally because of the confinement of motion to a single well. Frequency up-conversion, multi-stability, and adaptive techniques are some of the performance enhancement strategies suggested for bistable-VEH. Considering the VEH's space constraints, the above designs are hard to implement in practical cases. This study introduces an inertial amplification mechanism (IAM) as a simple passive strategy to enhance the performance of a snap-through VEH, a concept not explored in previous studies. The addition of IAM increases the effective mass without increasing the physical mass and thereby enhances energy harvesting, especially from weak ambient excitation sources. The dynamics and performance of the enhanced snap-through VEH are investigated analytically and numerically under harmonic and random excitations. The harmonic balance method (HBM) derives the frequency-amplitude relationship, which shows a hardening behavior and an increase in bandwidth. The effective potential method provides a closed-form expression for the joint probability density function (Joint PDF), which is governed by the Fokker-Planck equation. The joint PDF shows a transition from bimodal to unimodal with an increase in the value of the geometrical parameter. The stochastic averaging method is employed to obtain the stationary probability density function, which defines the long-term dynamics of the VEH. The effects of noise intensity, mass ratio, and inertial amplifier angle on the dynamics are investigated. Finally, the performance of the proposed VEH is compared with a conventional snap-through VEH, an equivalent linear VEH, and a multistable VEH under harmonic and random excitation conditions. The findings suggest that the snap-through VEH with the IAM has advantages over the linear and multistable nonlinear VEH in terms of extracting energy from low-intensity harmonic and random excitation sources. This simple augmentation strategy preserves the original system's bistability, eliminating the need for the complex design of a multistable VEH.

Stochastic stability of random stacking of blocks

The paper explores the stability of a tower obtained by stacking identical rectangular blocks on top of each other with an inherent randomness due to slight positional offsets between successive blocks. With probabilistic modeling techniques, the diffusive behavior of the stacking process is studied and the collapse is seen as a first passage time problem. In the considered model, alignment errors are idealized as independent Gaussian random variables with zero mean and given standard deviation. We derive expressions for the joint probability density functions of block positions, and analyze their correlation. The study extends to the stochastic stability of a stack of given height, by exploring the statistical characteristics of the center of gravity of the part of tower above each block. Eventually, the probabilistic analysis of collapse is developed to quantify the statistics of the number of blocks that can be heaped up before the tower topples. Although this problem may initially appear playful, it offers an illustrated introduction to first passage problems on a non homogenous process. From a practical standpoint, this analysis offers a simple understanding of the influence of alignment errors on the overall stability of a stack, which may find several fields of application.

A joint time–space extrapolation approach within the Wiener path integral technique for efficient stochastic response determination of nonlinear systems

A joint time–space extrapolation approach within the Wiener path integral (WPI) technique is developed for determining, efficiently and accurately, the non-stationary stochastic response of diverse nonlinear dynamical systems. The approach can be construed as an extension of a recently developed space-domain extrapolation scheme to account also for the temporal dimension. Specifically, based on a variational principle, the WPI technique yields a boundary value problem (BVP) to be solved for determining a most probable path corresponding to specific final boundary conditions. Further, the most probable path is used for evaluating, approximately, a point of the system response joint probability density function (PDF) corresponding to a specific time instant. Remarkably, the BVP exhibits two unique features that are exploited in this paper for developing an efficient joint time–space extrapolation approach. First, the BVPs corresponding to two neighboring grid points in the spatial domain of the response PDF not only share the same equations, but also the boundary conditions differ only slightly. Second, information inherent in the time-history of an already determined most probable path can be used for evaluating points of the response PDF corresponding to arbitrary time instants, without the need for solving additional BVPs. In a nutshell, relying on the aforementioned unique and advantageous features of the WPI-based BVP, the complete non-stationary response joint PDF is determined, first, by calculating numerically a relatively small number of PDF points, and second, by extrapolating in the joint time–space domain at practically zero additional computational cost. Compared to a standard brute-force implementation of the WPI technique, the developed extrapolation approach reduces the associated computational cost by several orders of magnitude. Two numerical examples relating to an oscillator with asymmetric nonlinearities and fractional derivative elements, and to a nonlinear structure under combined stochastic and deterministic periodic loading are considered for demonstrating the reliability of the extrapolation approach. Juxtapositions with pertinent Monte Carlo simulation data are included as well.

Accurate Global Atmospheric State Analysis Using Objective Error Statistics Including Observation Error Dependence on Water Substance Field

AbstractAtmospheric state analysis is a difficult scientific problem but essential for atmospheric sciences. Data assimilation can generate accurate analyses by integrating information on the atmospheric state using probability density functions (PDFs), where the Gaussian approximation is typically used and PDFs are described by error covariance matrices (ECMs). However, ECMs have been estimated empirically, and dependency of the ECMs on meteorological conditions (flow) is only partially represented. These limit atmospheric state analysis accuracy. This is especially problematic for water substance‐sensitive microwave radiances (WS‐MWRs) because of their strong flow dependence. We objectively estimated ECMs of all data including flow‐dependence of the ECMs of WS‐MWRs. Since the ECM of each data is a component of one ECM representing one joint PDF as a whole, it is theoretically better to objectively estimate ECMs of all data, not just a particular data. For WS‐MWRs, we categorized flow into four using water substance amount and estimating an ECM for each category. Numerical experiments using the new ECMs on an operational global numerical weather prediction system show the followings. The new error standard deviations are generally smaller than those of empirical. Standard deviations and interchannel correlations of observation errors of WS‐MWRs increase with water substance amount. The effects of WS‐MWRs on analysis were approximately doubled. The analysis fields differ systematically such as increase of low‐level clouds over cold oceans. The forecast accuracy improved with 95% statistical significance up to 9%. Both the flow dependence of correlation and variance of WS‐MWRs contributed to the improvement of forecast accuracy.

Practical method for evaluating wind influence on autonomous ship operations (2nd report)

Recently, a considerable number of research and development projects have focused on automatic vessels. A highly realistic simulator is needed to validate control algorithms for autonomous vessels. For instance, when considering the automatic berthing/unberthing of a vessel, the effect of wind in such low-speed operations cannot be ignored because of the low rudder performance during slow harbor maneuvers. Therefore, a simulator used to validate an automatic berthing/unberthing control algorithm should be able to reproduce the time histories of wind speed and wind direction realistically. Therefore, in our first report on this topic, to obtain the wind speed distribution, we proposed a simple algorithm to generate the time series and distribution of wind speed only from the mean wind speed. However, for wind direction, the spectral distribution could not be determined based on our literature surveys, and hence, a simple method for estimating the coefficients of the stochastic differential equation (SDE) could not be proposed. In this study, we propose a new methodology for generating the time history of wind direction based on the results of Kuwajima et al.’s work. They proposed a regression equation of the standard deviation of wind direction variation for the mean wind speed. In this study, we assumed that the wind direction distribution can be represented by a linear filter as in our previous paper, and its coefficients are derived from Kuwajima’s proposed equation. Then, as in the previous report, the time series of wind speed and wind direction can be calculated easily by analytically solving the one-dimensional SDE. The joint probability density functions of wind speed and wind direction obtained by computing them independently agree well with the measurement results.

Correlation analysis and joint probability density function model of wind pressures: Focusing on multivariate wind loads field on low-rise building under typhoon climate

The characteristics of the multivariate wind loads field on the roof are crucial to the wind-resistant design of low-rise buildings, which contain the correlation characteristics in space and probability characteristics in the time domain. This paper proposes a framework for constructing a Joint Probability Density Function (Joint PDF) model for a multivariate wind loads field. It provides a detailed correlation analysis for the first time. This paper employs wind pressure data collected from the roof of a low-rise building during Typhoon Muifa. It was found that the correlation becomes more robust with increasing roof pitch and the wind pressures are strongly correlated with a correlation coefficient exceeding 0.50 when the roof pitch is above 15°. The mixture distribution model is applied to the probability density function fitting procedure of wind pressure time series under typhoon climate, and the fitting effect is significantly better than other classical probability density functions. The optimal copula function is determined according to the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) for estimating the Joint PDF. The results reveal that Gumbel-copula and Student-copula have the highest proportion in optimal copula functions, accounting for over 90% of the total copula functions. Then, a bivariate Joint PDF for wind pressures are established with the optimal copula function. Additionally, the comparison between measured bivariate Joint PDF and that constructed using copula functions verifies the accuracy of the proposed framework for constructing Joint PDFs. The Joint PDF of wind pressures can enhance the understanding of the stochastic characteristics of local wind load fields on roofs, and the correlation characteristics in space provide crucial references for improving the accuracy of wind load random field simulation and saving the cost of wind resistance design.

Survival probability surfaces of hysteretic fractional order structures exposed to non-stationary code-compliant stochastic seismic excitation

A novel first-passage probability stochastic incremental dynamics analysis (SIDA) methodology tailored for hysteretic fractional order structural systems under a fully non-stationary seismic excitation vector consistently designated with contemporary aseismic codes provisions (e.g., Eurocode 8) is developed. Specifically, the vector of the imposed seismic excitations is characterised by evolutionary power spectra that stochastically align with aseismic codes elastic response acceleration spectra, defined for specified modal damping ratios and scaled ground accelerations. Leveraging the concepts of stochastic averaging and statistical linearization, the approximative non-stationary response displacement joint probability density function (PDF) is derived, retaining the particularly coveted attribute of computational efficacy. Subsequently, the coupling with the survival probability model allows for the efficient determination of the response first-passage time probability density surfaces and the survival probability surfaces across various limit-state rules and scalable intensity measures. The first-passage time probability serves as a robust engineering demand parameter, effectively monitoring structural behaviour by considering both intensity and timing information, while inherently aligned with pertinent limit-state requirements. Notably, the associated low computational cost and the ability to handle a wide range of complex nonlinear/hysteretic structural behaviours, coupled with its compliance with modern aseismic codes, underscore its potential for applications in the fields of structural and earthquake engineering. A nonlinear system endowed with fractional derivative elements is used to exemplify the method’s reliability. The accuracy of the proposed method is validated in a Monte Carlo-based context, conducting nonlinear response time–history analyses with an extensive ensemble of accelerograms compatible with Eurocode 8 response acceleration spectra.

Implementation of the P1 Radiation Model in a Velocity-Turbulent Frequency-Composition Joint PDF Method

ABSTRACT This research employs a more detailed radiation treatment to improve the accuracy of the velocity-turbulent frequency-composition joint probability density function (PDF) methods in numerical solution of the turbulent non-premixed flames. The radiation heat transfer in the traditional velocity-frequency-composition joint PDF methods has been disregarded or treated simply using an emission-only radiation model, even though radiation is important in the combustion systems. In the present PDF method, radiation is modeled using the P1-approximation and the radiative properties are calculated by the weighted-sum-of-gray-gases (WSGG) model. The results of present method are first validated by solving a laboratory turbulent piloted jet flame and comparing them with the measured data. Then, two constructed large-scale flames are modeled and simulated. Results reveal that reabsorption of emitted radiation increases from 2.98% in a laboratory-scale flame to 56.8% in the scaled-up flame and has a strong effect on the temperature distribution in the scaled-up flame. It is shown that reabsorption of emitted radiation increases the peak temperature on the flame axis by 119 degrees. Furthermore, radiation becomes more significant with the increase of the flame scales, so that the radiation fraction increases from 4.15% in the small-scale flame to 16.6% in the large-scale flame.