Joint Probability Density Research Articles

Performance enhancement of snap-through vibration energy harvester with displacement amplifier

Vibration energy harvesters (VEH) based on bistable and snap-through mechanisms are being widely employed to harvest energy from different vibration sources. However, under weak ambient excitation, the performance of these systems is marginal. Multi-stability and enhancement techniques, such as displacement amplifier, are being used with bistable VEH to improve its performance. Similar to this, multi-stable snap-through VEH has also been developed; however, the usage of a displacement amplifier in the context of snap-through VEH has not yet been investigated. In this novel study, a displacement amplifier/ displacement amplification mechanism (DAM) is integrated into a snap-through VEH, and the enhanced performance under weak excitation is investigated. A generalized transformation is applied to develop a reduced-order model of the proposed system, and its dynamics and performance under harmonic and random excitation are investigated analytically and numerically. The frequency–amplitude relationship is derived under harmonic excitation, and a detailed investigation of the influence of system parameters on the frequency and force responses is performed. It is observed that by adjusting the mass and stiffness ratio between the VEH and DAM, it is possible to harvest more energy under harmonic excitation of low intensity. In the case of Gaussian white noise excitation, effective potential, and stochastic averaging methods are used to obtain marginal, joint, and stationary probability density functions and mean square power. The influence of noise intensity and other system parameters on the aforementioned measures is thoroughly investigated. It is observed that for specific parameter values, the proposed VEH can harvest 3.3 times more power under harmonic excitation and four times more power under random excitation compared to the snap-through VEH without DAM.

Estimation of multivariate design quantiles for drought characteristics using joint return period analysis, Vine copulas, and the systematic sampling method

Abstract The Wei River Basin has suffered from severe droughts. It is essential to build drought relief projects to cope with drought disasters. Traditionally, design quantiles have been estimated using univariate analysis, in which multiple characteristics of hydrological events are not considered. To design the more appropriate hydrological projects for the case study area, the Wei River Basin, it is essential to conduct research on multivariate analysis allowing multiple characteristics to be considered simultaneously. Here, the authors focus on hydrological drought (the basis for designing a hydrological project), and a framework to calculate the joint design quantiles of three drought characteristics is proposed. The most likely design quantiles relating to a specific return period, reflecting the highest occurrence probabilities among multiple combinations of variables, are derived by the maximum joint probability density function. Results show that compared to univariate analysis, design quantiles calculated via joint return period yield infrastructure with a smaller total storage capacity in the study area relating to a specific return period, i.e., reduces the economic input while maintaining the project safety. Proposed methods bring new sights to the design project. However, multi-method comparisons considering more uncertainties, inherent laws, investment, and other limited factors are still vital.

Configuration-dependent dynamics of non-spherical particles in a gas–solid fluidized bed

Configuration-dependent dynamics of non-spherical particles are investigated experimentally in a quasi-two-dimensional gas–solid fluidized bed. We show that the vertical velocity profiles are sensitive to the aspect ratio at low gas velocity, but not sensitive at high gas velocity. The velocity distribution, especially the rotational velocity distribution, deviates from the Maxwellian distribution. The joint probability density distribution of vertical and rotational velocity is asymmetric. We identify two high-probability particle configurations, two particles in a parallel or vertical arrangement, by calculating the radial distribution function, nematic correlation function and coordination number. We further explain the cause of this order appearance through entropy theory. By comparing the drag force of the particle configurations at various angles and Re, we demonstrate that it is necessary to put forward the configuration-dependent drag force for non-spherical particle fluidization.

Encounter-based approach to the escape problem.

We revise the encounter-based approach to imperfect diffusion-controlled reactions, which employs the statistics of encounters between a diffusing particle and the reactive region to implement surface reactions. We extend this approach to deal with a more general setting, in which the reactive region is surrounded by a reflecting boundary with an escape region. We derive a spectral expansion for the full propagator and investigate the behavior and probabilistic interpretations of the associated probability flux density. In particular, we obtain the joint probability density of the escape time and the number of encounters with the reactive region before escape, and the probability density of the first-crossing time of a prescribed number of encounters. We briefly discuss generalizations of the conventional Poissonian-type surface reaction mechanism described by Robin boundary condition and potential applications of this formalism in chemistry and biophysics.

Modeling the presumed joint probability density function of conditioning variables in stratified turbulent flames

Experimental data from the Cambridge-Sandia burner are used to calculate the conditional average of temperature and the mass fractions of H2O, CO, and CO2 in the context of the Uniform Conditional State (UCS) model. The performance of various presumed joint PDF models are examined for this burner working with different swirl and stratification levels. All of the presented PDF models are based on a few statistical moments of the conditioning variables which are extracted from the data at each measuring point. Primarily, only mixture fraction and reaction progress variable are employed as the conditioning variables. The β function is employed for modeling the marginal PDF of mixture fraction in all cases. For the progress variable, the β function and Modified Laminar Flamelet (MLF)-PDF are tested for modeling the marginal PDF. It is shown that even though the MLF-PDF results in more accurate predictions overall for most of the cases, its superiority with respect to the β-PDF is relatively small in the stratified flames. The performance of the product, Plackett, and Gaussian copulas are also investigated for employing the cross-correlation of mixture fraction and progress variable in building their joint PDF. It is found that the Gaussian copula shows a superior performance over the other two. Next, normalized total enthalpy is employed as a third conditioning variable, and different PDF models are examined for the three-condition version of UCS. It is found that even though adding the third conditioning variable can improve the predictions by considering the effect of heat transfer on the chemistry manifold, such improvements depend on having a proper 3D PDF model for the conditioning variables. In the vicinity of the flame brush, the models employed for the 3D PDFs represent an inferior performance compared to the two-condition version with the use of a Gaussian copula due to complications in the marginal PDF of normalized total enthalpy in these regions. To benefit from considering the impact of heat transfer on chemistry while also avoiding complexities in modeling the 3D joint PDF near the reaction zone, a switch between the two- and three-conditional models which is triggered by the variance of progress variable is proposed. It is shown that the predictions obtained by the use of this switch are almost always superior to all other PDF models investigated in this study.

Statistical LOS/NLOS Classification for UWB Channels

Ultrawideband (UWB) technology has attracted a lot of attention for indoor and outdoor positioning systems due to its high accuracy and robustness in non-line-of-sight (NLOS) environments. However, UWB signals are affected by multipath propagation which causes errors in localization. To overcome this problem, researchers have proposed various techniques for NLOS identification and mitigation. One of the approaches is statistical LOS/NLOS classification, which uses statistical parameters of the received signal to distinguish between LOS and NLOS channels. In this paper, we formulated several techniques which can be used for effectively classifying Line of Sight (LOS) channel from a Non-Line of Sight (NLOS) channel. Various parameters obtained from Channel Impulse Response (CIR) like Skewness, Kurtosis, Root Mean Squared Delay Spread (RDS), Mean Excess Delay (MED), Energy, Energy Ratio and Mean of Covariance Matrix are used for channel classification. In addition to this, the Joint Probability Density Functions (PDFs) of various parameters are used to improve the accuracy of UWB LOS/NLOS channel classification. Two different criteria-Likelihood Ratio and Hypothesis Test are used for the identification of channel.

Encounter-based model of a run-and-tumble particle II: absorption at sticky boundaries

In this paper we develop an encounter-based model of a run-and-tumble particle (RTP) confined to a finite interval with partially absorbing, sticky boundaries at both ends. We assume that the particle switches between two constant velocity states ±v at a rate α. Whenever the particle hits a boundary, it becomes stuck by pushing on the boundary until either a tumble event reverses the swimming direction or it is permanently absorbed. We formulate the absorption process by identifying the first passage time (FPT) for absorption with the event that the time A(t) spent attached to either wall up to time t (the occupation time) crosses some random threshold . Taking to be an exponential distribution, , we show that the joint probability density for particle position X(t) and velocity state satisfies a well-defined boundary value problem (BVP) with κ 0 representing a constant absorption rate. The solution of this BVP determines the so-called occupation time propagator, which is the joint probability density for the triplet . The propagator is then used to incorporate more general models of absorption based on non-exponential (non-Markovian) distributions . We illustrate the theory by calculating the mean FPT and splitting probabilities for absorption. We also show how our previous results for partially absorbing, non-sticky boundaries can be recovered in an appropriate limit. Absorption now depends on the number of collisions of the RTP with the boundary. Finally, we extend the theory by taking absorption to depend on the individual occupation times at the two ends.

Time-resolved low-pressure air-assisted spray performance and unsteadiness evaluation

The main advantages of air-assisted spray are its high-quality atomization at low injection pressures and insensitivity to the viscosity of atomized liquid. In this study, the droplet size and velocity of a low-pressure intermittent air-assisted spray were studied by using phase Doppler anemometry, and the effects of liquid fuel injection duration on time-resolved spray microscopic characteristics and spray unsteadiness were analyzed. Droplet size-velocity joint probability density functions were employed to characterize the droplet diameter-velocity distribution as well as the probability range. A comparison of the droplet Weber number with an empirical critical value indicates that atomized droplets hardly undergo secondary shear breakup. Based on the ideal spray theory of Edwards and Marx, an improved algorithm is proposed with the concept of iterative rejection of inter-particle arrival times to quantify the unsteadiness of air-assisted sprays by eliminating the dependence of the calculation results on droplet sampling data. The results show that intermittent air-assisted spray is an inherently unsteady process that can be influenced by fuel injection duration and spatial location, while independent of the droplet size. In addition, the spray unsteadiness exhibits noteworthy variations at different spray stages segmented by droplet velocity vs time. The relation between the potential internal gas–liquid two-phase status determined by fuel injection duration and the spray performance is elaborated.

Renewable Scenario Generation Based on the Hybrid Genetic Algorithm with Variable Chromosome Length

Determining the operation scenarios of renewable energies is important for power system dispatching. This paper proposes a renewable scenario generation method based on the hybrid genetic algorithm with variable chromosome length (HGAVCL). The discrete wavelet transform (DWT) is used to divide the original data into linear and fluctuant parts according to the length of time scales. The HGAVCL is designed to optimally divide the linear part into different time sections. Additionally, each time section is described by the autoregressive integrated moving average (ARIMA) model. With the consideration of temporal correlation, the Copula joint probability density function is established to model the fluctuant part. Based on the attained ARIMA model and joint probability density function, a number of data are generated by the Monte Carlo method, and the time autocorrelation, average offset rate, and climbing similarity indexes are established to assess the data quality of generated scenarios. A case study is conducted to verify the effectiveness of the proposed approach. The calculated time autocorrelation, average offset rate, and climbing similarity are 0.0515, 0.0396, and 0.9035, respectively, which shows the superior performance of the proposed approach.

The impact of ammonia addition on soot formation in ethylene flames

The present work investigates computationally the impact of blending ammonia with ethylene on soot formation in four laminar diffusion flames and the corresponding turbulent partially premixed flames. The conditions conform to experimental investigations (Bennett et al., Combust. Flame, 2020, and Boyette et al., Combust. Flame, 2021) with both sets of flames featuring identical fuel blends. A mass and number density preserving sectional method was applied to all cases with the soot surface growth and oxidation models updated to include reactions with oxides of nitrogen. The gas phase chemistry was extended to include a comprehensively validated ammonia sub-mechanism. For the laminar flames, the inception of soot particles is based on detailed chemistry with pyrene treated as representative of the PAH pool. The accuracy of fitting a global acetylene-based inception step is evaluated for the different fuel blends and subsequently applied to the turbulent flames using a fully coupled transported joint probability density function method featuring an 84-dimensional joint-scalar space. Results obtained for laminar flames show that detailed chemistry coupled with the sectional soot model provides excellent agreement with the measured suppression of the soot volume fraction with increased use of ammonia. Computed particle size distributions (PSDs) show an increase in smaller soot particles under such conditions, consistent with experimental observations. The experimentally observed reduced impact in turbulent flames is also reproduced computationally. The suppression of soot is principally caused by changes in the radical pool leading to reduced soot surface growth and, to a lesser extent, soot inception. The contribution of oxides of nitrogen to soot oxidation is modest. Computed PSDs in laminar and turbulent flames highlight the importance of differences in flame structures and flowfield timescales.

A pitchfork-like relationship between reduced Barents-Kara sea ice and Ural atmospheric circulation

Atmospheric circulation monthly anomalies over the Ural region are key indicators of Eurasian climate anomalies. Here, whether there exists a one-to-two correspondence relationship that generally agrees with the supercritical pitchfork bifurcation model, referred to as a pitchfork-like relationship, between reduced sea ice concentration (SIC) in the Barents-Kara Seas in specific months and the lagging Ural circulation anomalies is explored. Based on the monthly observational SIC data and two reanalyses during 1979/1980 − 2020/2021, two typical examples are found by estimating the joint probability density function. Results show that when the gradually reduced SIC in September (January) passes a critical threshold, the preferred Ural circulation patterns in October (February) exhibit a regime transition from the flat zonal westerlies to wavy westerlies with a Ural trough and wavy westerlies with a Ural ridge. Because both the barotropic and baroclinic conversion of energy from the climatological-mean flow to Ural circulation anomalies exhibit a regime transition from one regime to two regimes. It might be associated with the increased both positive and negative shear vorticity of background westerly wind over the Ural region before the regime transition, contributed by the thermodynamic effect of the SIC reduction. After the regime transition, positive and negative anomaly events of Ural atmospheric circulation occur with equal probability under the same SIC. Our results suggest an increased incidence of both positive and negative anomalies of Ural atmospheric circulation and also the Siberian High, under the recent SIC reduction, which implies a low predictability of Eurasian climate anomalies in October and February.

Solving discrete first-order matrix linear control problems with general parametric uncertainties: A probability-density-based approach

This paper deals with the probabilistic analysis of discrete first-order linear control models with uncertainties. For the sake of generality in our stochastic analysis, we assume that all model parameters (the initial and final states, the matrix containing the free dynamics part, and the control’s coefficient) are random variables with an arbitrary joint probability density function. We then combine some results from classical Control Theory with Probability Theory to obtain, under very general hypotheses, the first probability density function of the control and the solution, which are parametric stochastic processes. To illustrate our theoretical findings, we also show two numerical examples and a classical discrete macroeconomic model whose parameters are treated as random variables.

Modified Weighted Uniform Distribution And Its Bivariate Extension

In this paper a new version of weighted uniform distribution is constructed and studied. The statistical properties of the new distribution including the behavior of hazard and reversed hazard functions, moments, the central moments, moment generating function, mean, variance, coefficient of skewness, coefficient of kurtosis, median, mode, quantile , stochastic ordering and order statistics are also obtained, a simulation study and real data applications are performed. Moreover, a bivariate extension of the new distribution named the bivariate modified uniform (BMWU) distribution is introduced. The proposed bivariate distribution is of type Farlie–Gumbel–Morgenstern (FGM) copula. The BMWU distribution has modified weighted uniform marginal distributions. The joint cumulative distribution function, the joint survival function, the joint probability density function, the joint hazard rate function and the statistical properties of the BMWU distribution are also derived.

Modified Weighted Rayleigh Distribution and Its Bivariate Extension

In this paper, a new version of weighted Rayleigh distribution is constructed and studied. The statistical properties of the new distribution including the behavior of hazard and reversed hazard functions, moments, the central moments, moment generating function, mean, variance, coefficient of skewness, coefficient of kurtosis, median, mode, quantiles, stochastic ordering, exact information matrix and order statistics are also obtained, a simulation study and real data applications are performed. Furthermore, a bivariate extension of the new distribution called the bivariate modified Rayleigh (BMWR) distribution is introduced. The proposed bivariate distribution is of type Farlie–Gumbel–Morgenstern (FGM) copula. The BMWR distribution has modified weighted Rayleigh marginal distributions. The joint cumulative distribution function, the joint survival function, the joint probability density function, the joint hazard rate function and the statistical properties of the BMWR distribution are also derived.

On the impact of differential diffusion between soot and gas phase species in turbulent flames

The molecular diffusivities of larger PAHs and soot particles approach zero leading to differential diffusion with gas-phase species. The present work systematically quantifies the impact on soot moments, soot related statistical correlations and Particle Size Distributions (PSDs) using a fully coupled transported joint probability density function (JPDF) method featuring a 78-dimensional joint-scalar space, including enthalpy, gas phase species with the PSD discretised using 62 size classes via a mass and number density preserving sectional method. Differential diffusion of soot (DDS) is treated via a gradual decline of diffusivity among soot sections maintaining realisability and the expected exponential decay of variance. The solution of the flow field features a time-dependent second moment closure and an elliptic solver. The turbulent non-premixed Sandia C2H4 flame from the International Sooting Flame (ISF) data base was selected as a target along with the KAUST (C2H4/N2) variant of the same flame. Results show that reduced soot diffusion leads to a significant increase in the soot volume fraction RMS and that the correlation coefficient between soot volume fraction and temperature is further reduced in a particle-size-dependent manner. Similar observations are made for correlations between the soot volume fraction and the mass fractions of gas-phase species such as CO, OH, H and C2H2. The results suggest that computational methods that presume explicit (e.g. flame structure related) correlations between such scalars and with soot face leading order modeling challenges. It is also shown that the correlation between CO and soot increases due to oxidation of soot and that DDS leads to a modest downstream shift of PSDs towards larger particles.

Data-driven reliability assessment with scarce samples considering multidimensional dependence

This study proposes a data-driven method for assessing reliability, based on the scarce input dataset with multidimensional correlation. Since considering the distribution parameters estimated from the scarce dataset as those of the population may lead to epistemic uncertainty, the bootstrap resampling algorithm is adopted to infer the distribution parameters as interval parameters. To account for the variable dependence, vine copula theory is utilized to construct the joint probability density function (PDF) of input variables, and maximum likelihood estimation (MLE) and Akaike information criterion (AIC) analysis are employed to select optimal copulas based on the samples for the vine structure. Subsequently, the failure probability bounds of a response function are calculated based on the constructed joint PDF with interval distribution parameters by the active learning Kriging (AK) method combining the sparse grid integration (SGI) method. Finally, several examples are provided to demonstrate the feasibility and efficiency of the proposed method.

Nonlinear stochastic dynamics of an array of coupled micromechanical oscillators

AbstractThe stochastic response of a multi‐degree‐of‐freedom nonlinear dynamical system is determined based on the recently developed Wiener path integral (WPI) technique. The system can be construed as a representative model of electrostatically coupled arrays of micromechanical oscillators, and relates to an experiment performed by Buks and Roukes. Compared to alternative modeling and solution treatments in the literature, the paper exhibits the following novelties. First, typically adopted linear, or higher‐order polynomial, approximations of the nonlinear electrostatic forces are circumvented. Second, for the first time, stochastic modeling is employed by considering a random excitation component representing the effect of diverse noise sources on the system dynamics. Third, the resulting high‐dimensional, nonlinear system of coupled stochastic differential equations governing the dynamics of the micromechanical array is solved based on the WPI technique for determining the response joint probability density function. Comparisons with pertinent Monte Carlo simulation data demonstrate a quite high degree of accuracy and computational efficiency exhibited by the WPI technique. Further, it is shown that the proposed model can capture, at least in a qualitative manner, the salient aspects of the frequency domain response of the associated experimental setup.

Efficient global sensitivity analysis for high-dimensional outputs combining data-driven probability models and dimensionality reduction

• GSA for applications involving complex numerical models and high-dimensional outputs is examined • Two different, recently developed, GSA formulations are combined • Principal component analysis (PCA) is firstly adopted as a dimensionality reduction technique. • Relevant latent output GSA statistics are then calculated by extending probability model-based GSA. • Extension pertains to estimation of necessary covariance statistics This paper examines the efficient variance-based global sensitivity analysis (GSA), quantified by estimating first-/higher-order and total-effect Sobol’ indices, for applications involving complex numerical models and high-dimensional outputs. Two different, recently developed, techniques are combined to address the associated challenges. Principal component analysis (PCA) is first considered as a dimensionality reduction technique. The GSA for the original output vector is then formulated by calculating variance and covariance statistics for the low-dimensional latent output space identified by PCA. These statistics are efficiently approximated by extending recent work on data-driven, probability model-based GSA (PM-GSA). The extension, constituting the main novel contribution of this work, pertains to the estimation of covariance statistics beyond the variance statistics examined in the original PM-GSA formulation. Specifically, a Gaussian mixture model (GMM) is developed to approximate the joint probability density function between some subset of the input vector, and each latent output, or each pair of latent outputs. The GMM is then utilized to estimate the aforementioned statistics. Results across two natural hazards engineering examples show that the dimensionality reduction and transformation of output space established through PCA do not impact the overall accuracy of the PM-GSA, and that the proposed implementation accommodates highly-efficient GSA estimates.

Adaptive risk-sensitive filter for Markovian jump linear systems

In this paper, we present divergence-based minimax methods to derive the adaptive risk-sensitive filter for Markovian jump linear systems with parameter uncertainties. The risk-sensitive parameter is treated as a random variable rather than chosen as a prior, which reflects the hypothesis merging property. To tackle the proposed joint probability density function due to the introduction of the random variable, a method that projects orthogonally the original probability measure into a new independent conditional probability space in the Kullback–Leibler (KL) sense is then presented. The exact close-form expression can be obtained via optimization that is of a form suitable in a recursive procedure. The proposed filter can improve robustness by increasing the mixed covariance, whose risk-sensitive like parameter can be updated adaptively using the measurements to fit different situations well. Simulation examples illustrate that the proposed methods can be regarded as competitive alternatives to existing robust approaches against modeling uncertainties.

Uncertainty Quantification in Constitutive Models of Highway Bridge Components: Seismic Bars and Elastomeric Bearings

Bridges are essential structures in the logistic chain of countries, making it critical to design them to be as resilient as possible. One way to achieve this is through performance-based seismic design (PBSD), which involves using nonlinear Finite Element (FE) models to predict the response and potential damage of different structural components under earthquake excitations. Nonlinear FE models need accurate constitutive models of material and components. Among them, seismic bars and laminated elastomeric bearings play an important role in a bridge's response to earthquakes; therefore, properly validated and calibrated models should be proposed. Only default parameter values from the early development of the constitutive models widely used by researchers and practitioners for these components tend to be used, and low identifiability of its governing parameters and the high cost of generating reliable experimental data have prevented a thorough probabilistic characterization of their model parameters. To address this issue, this study implements a Bayesian probabilistic framework using Sequential Monte Carlo (SMC) for updating the parameters of constitutive models of seismic bars and elastomeric bearings and proposes joint probability density functions (PDF) for the most influential parameters. The framework is based on actual data from comprehensive experimental campaigns. The PDFs are obtained from independent tests conducted on different seismic bars and elastomeric bearings, to then consolidate all the information in a single PDF for each modeling parameter by means of the conflation methodology, where the mean, coefficient of variation, and correlation between calibrated parameters are obtained for each bridge component. Finally, findings show that the incorporation of model parameter uncertainty through a probabilistic framework will allow for a more accurate prediction of the response of bridges under strong earthquakes.