Bimodal Function Research Articles

Research of the array spacing effect on wake interaction of tidal stream turbines

Understanding the wake interactions among turbines is important for developing efficient tidal farm configurations. The present study compares the power and wake generated by a 2-row by 1-column array separated at three lateral spacings with those of a single tidal stream turbine. According to the results, power fluctuations increase with the tip speed ratio, with mean values corresponding to the inverse-u shape prediction of the blade element methods. For a single turbine, the cloud diagram of the lateral velocity deficit follows a half-elliptical contour with its center at the hub and initial covertices at the tip. The twin turbines largely exhibit these symmetric transversal distributions; however, they are doubled, with wake fusion occurring at the mid-width and closer with turbines approximation. The superposition of Gaussian-type velocity distributions matches the experimental bimodal functions with horizontally decreasing heights only after x > 6D. Vertically, the TIx in all conditions displays two strong areas, just above and below the turbine, owing to the interaction of the tip with tower vortices and the free and bed surface. In the two-unit arrangement, the blockage effect and the velocity of the mid-passage flow increase with the lateral separation of the turbines. This has a direct effect on wake combination, width, and recovery.

Two statistical regimes in the transition to filamentation.

We experimentally investigate fluctuations in the spectrum of ultrashort laser pulses propagating in air, close to the critical power for filamentation. Increasing the laser peak power broadens the spectrum while the beam approaches the filamentation regime. We identify two regimes for this transition: In the center of the spectrum, the output spectral intensity increases continuously. In contrast, on the edges of the spectrum the transition implies a bimodal probability distribution function for intermediate incident pulse energies, where a high-intensity mode appears and grows at the expense of the original low-intensity mode. We argue that this dual behavior prevents the definition of a univoquial threshold for filamentation, shedding a new light on the long-standing lack of explicit definition of the boundary of the filamentation regime.

Bimodal Distribution Function with Fuzzy Regression in Predicting Random Truckload Patterns

A bimodal distribution function with statistical parameters obtained using fuzzy regression is presented for predicting random truckload patterns that include all load ranges including overloads. Overload trucks often appear as a sizable portion of truck populations on highways. In applications when damage estimation of transportation facilities such as pavements and bridges is desired, theoretical models providing a reasonable representation of truckload populations including overloads will be useful. Load populations mostly exhibit inconsistent patterns, often with two or more distinct peaks. This is because of a combination of loaded and empty trucks as well as overloads in the population. As such, a mixed distribution model instead of a simple statistical distribution is used to portray a realistic representation of truckload populations. In this paper, using the bimodal model, theoretical distributions are developed to (1) represent the entire truckload population with weigh-in-motion (WIM) data, and (2) predict the population with limited data. If no truckload population is available, the bimodal distribution model can still be used, with certain assumptions, based on fuzzy regression using certain parameters pertaining to the traffic data such as the average daily truck traffic and an overall estimate of the percentage of overload data in the population. This paper represents new directions in modeling truckload patterns with capabilities to offer the continuity as well as the bimodal feature of the limited load data.

Heterogeneity Aware Emission Macroscopic Fundamental Diagram (e-MFD)

Transportation sector is one of the major producers of greenhouse gases which are responsible for climate change. Finding an appropriate emission estimation tool for large-scale networks is essential for developing efficient emission mitigation strategies. This paper presents an advanced version of the emission macroscopic fundamental diagram (e-MFD) which improves the stability and accuracy of the previous model. A bi-modal function is applied to separate free-flow and congested branches of the e-MFD. The accuracy of the proposed e-MFD is evaluated with both a synthetic grid network and a real-world city-level network. The study also assesses the model’s stability under directional traffic demands and road incidents. A comparison with the original e-MFD also verifies the superiority of the proposed model with higher accuracy. Standard deviation of density used in the proposed model to boost the performance. It is worth mentioning the standard deviation can be recorded with the existing hardware, such as loop detectors, and does not impose a considerable computational complexity. The proposed model can be employed for emission measurement in large-scale networks and hierarchical traffic control systems for more homogeneous congestion distribution and emission control.

Effective permeability of fractured porous media with fracture density near the percolation threshold

This study investigates the effective permeability of fractured porous media with fracture density near the percolation threshold. A new equivalent matrix-fracture network (EMFN) model was introduced, the code was used to efficiently deal with the huge number of modeling works for steady state flow in fractured porous media. The numerical model was verified by comparing with two previous seepage methods. We conducted a series of Monte Carlo simulations to explore the relationship between permeability discontinuity and three factors, namely fracture density, contrast between fracture and matrix permeability, and the scale of research domain. It was found that the effective permeability presents a bimodal distribution with specified fracture density in the transition regime. Each bimodal function can be subdivided into two well-fitting log-normal functions according to the percolation state to obtain analyzable average permeabilities. The results revealed significant differences between percolating and non-percolating configurations in the response to the change of each of the three factors mentioned above. More importantly, the finite-size effect plays an important role when fracture density is lower than or close to the percolation threshold; but this effect disappears gradually as the increase of fracture density. A stereoscopic approach is introduced to rationalize the scale effect.

Capture of Dark Matter Particles by a Galaxy in the Case of a Bimodal Distribution of Their Velocities

We have analyzed the rate of capture of dark matter (DM) particles by the galaxy in the case of the existence of two different types of DM or a bimodal velocity distribution function for DM. It is shown that, in addition to the scenario considered in our previous work which is based on the assumption of an unimodal distribution, more complex scenarios are possible in which the transition to the state of intense capture and/or exit from it can occur in two stages. A detailed description is given of the change in the curve describing the rate of capture of dark matter particles as a function of the rate of increase in the baryon mass of the galaxy for various values of the rate of decrease of the DM density.

Verification of the master curve concept (ASTM E1921) and inhomogeneity analysis of a German RPV weld for various loading rates

Fracture-mechanical investigations were carried out on the weld metal of a German reactor pressure vessel steel 22NiMoCr3-7. The crack tip loading rate and test temperature were varied in the brittle-ductile transition region, while the results were evaluated using the Master Curve methodology (ASTME1921), including inhomogeneity analyses in Appendix X5. The highly inhomogeneous datasets could be uniformly described with a bimodal distribution function with good accuracy compared to the standard procedure. However, this agreement decreases with increasing crack tip loading rate. In contrast, a modified Master Curve with a shape factor of p = 0.03/°C progressively matches the data better with increasing crack tip loading rate. The weld metal has a significantly higher toughness than the base material. SEM analysis suggests that the inhomogeneity is due to the stochastic distribution of sharp microcracks, while the observed inhomogeneity is additionally superimposed by dynamic effects (adiabatic heating and local crack arrest) at increased crack tip loading rates. It is argued that the microstructural inhomogeneity is hereby »neutralized« and could instead be interpreted as a possible artifact of the standard Master Curve under dynamic conditions.

Effects of body force on the statistical behaviour and modelling of scalar variance in turbulent premixed flames

The effects of body force/external pressure gradient on the statistical behaviours of the reaction progress variable variance and the terms of its transport equation have been investigated for different turbulence intensities using DNS data of statistically planar flames. Since the extent of flame wrinkling increases with the strengthening of body force promoting unstable stratification, the scalar variance has been found to decrease under strong body force promoting stability. This trend is particularly strong for low turbulence intensities where the probability density function of the reaction progress variable cannot be approximated by a bimodal distribution. Therefore, an algebraic relation for the reaction progress variable variance, derived based on a presumed bimodal probability density function of reaction progress variable, cannot be used for general flow conditions. The contributions of chemical reaction and scalar dissipation rates in the scalar variance transport equation remain leading order source and sink, respectively for all cases irrespective of the strength and direction of the body force. The counter-gradient type transport is found to weaken with increasing body force magnitude when the body force is directed from the heavier unburned gas to the lighter burned gas side of the flame brush, and vice versa. Although a scalar dissipation rate-based reaction rate closure can be utilised to model the reaction rate contribution to the scalar variance transport accurately, the dissipation rate contribution due to the gradient of the Favre-averaged reaction progress variable cannot be ignored and it plays a key role for large magnitudes of body force promoting stable stratification. An algebraic closure of the scalar dissipation rate, originally proposed for high Damköhler number combustion, has been modified for the thin reaction zones regime combustion by incorporating the effects of Froude number. This model has been shown to predict the scalar dissipation rate accurately for all cases considered here.

A piecewise ellipsoidal reachable set estimation method for continuous bimodal piecewise affine systems

In this work, the issue of estimation of reachable sets in continuous bimodal piecewise affine systems is studied. A new method is proposed, in the framework of ellipsoidal bounding, using piecewise quadratic Lyapunov functions. Although bimodal piecewise affine systems can be seen as a special class of affine hybrid systems, reachability methods developed for affine hybrid systems might be inappropriately complex for bimodal dynamics. This work goes in the direction of exploiting the dynamical structure of the system to propose a simpler approach. More specifically, because of the piecewise nature of the Lyapunov function, we first derive conditions to ensure that a given quadratic function is positive on half spaces. Then, we exploit the property of bimodal piecewise quadratic functions being continuous on a given hyperplane. Finally, linear matrix characterizations of the estimate of the reachable set are derived.

Inverse estimation of hydraulic parameters of soils with rock fragments

The application of inverse modeling to determine the hydraulic properties of layered soils with rock fragments (RF > 2 mm) is associated with complex challenges such as selecting suitable hydraulic functions and defining a strategy to optimize many parameters. These two issues were addressed in this study by performing field drainage experiments in layered soil profiles (three layers) with different particle size distributions and different percentages of RF. Water contents (θ) and pressure heads (h) were monitored at different soil depths during drainage experiments (h ≥ −200 cm). Data of θ and h for a drier range (h < −5000 cm) were determined with a dew point potentiometer on disturbed soil samples. The unimodal and bimodal van Genuchten-Mualem functions were evaluated, and their parameters were optimized using the Hydrus-1D “Inverse solution”. The RETC curve fitting software was used to estimate the second set of parameters of the bimodal function. The parameters for the uni- and bimodal models (15 and 27 parameters, respectively) were repeatedly optimized by fixing some parameters while estimating others. The unimodal van Genuchten-Mualem function provided a satisfactory fit of θ and h measurements only when either drainage or dew point potentiometer measurements were used. On the other hand, the bimodal van Genuchten-Mualem function provided a satisfactory fit of θ and h measurements when both sets of data (drainage and dew point potentiometer measurements) were used simultaneously. Furthermore, the optimized parameters were physically consistent except for a few high saturated hydraulic conductivity values. Therefore, bimodal functions should be considered to represent the hydraulic properties of soils with RF. The problem of many calibrated parameters when using bimodal functions can be managed using sequential optimization, which allowed us to estimate 27 parameters for layered soils with RF successfully.

Phase-field model of grain boundary diffusion in nanocrystalline solids: Anisotropic fluctuations, anomalous diffusion, and precipitation

The anisotropic phase-filed model of grain boundary diffusion and precipitation of solute in nanocrystalline solids has been developed. In this model, the Cahn–Hilliard equation is generalized for the anisotropic phase-field diffusion of solute and anisotropic compositional fluctuations. It is found that dynamics of solute concentration profile demonstrates the anomalous diffusion behavior with scaling parameters depending on the mobility ratio and microstructure of a solid solution. It is noteworthy that the increase in source concentration can slow down the concentration front propagation due to uphill diffusion or formation of a new phase. Parameters of grain boundary diffusion control the precipitation dynamics. In particular, a decrease in transverse diffusion coefficient is responsible for longer incubation time, and lower rates of nucleation and nuclei growth in comparison with the case of isotropic solute transport near grain boundaries. Transport properties of boundary and bulk are responsible for the formation of the bimodal size distribution function of second phase particles and specific kinetics of average radius and number density.

Electric-field-dependent bimodal distribution functions for electrons in argon, xenon, and krypton owing to the Ramsauer-Townsend minima in the electron-atom momentum-transfer cross sections

This paper considers solutions of the linear Fokker-Planck equation for the steady velocity distribution functions of electrons dilutely dispersed in an excess of either argon, xenon, or krypton. Owing to the large Ramsauer-Townsend minima in the electron-atom momentum-transfer cross sections for these atoms, the steady electron distribution, often referred to as the Davydov distribution, exhibits a bimodal distribution which varies dramatically with the strength of the external electric field. This effect provides a unique verification of the location and shape of the Ramsauer-Townsend minima in the cross sections. It is anticipated that current experimental techniques that probe the details of nonequilibrium electron distributions will verify the results reported in this paper. In addition, these nonequilibrium, non-Maxwellian distributions cannot be rationalized in terms of either the Gibbs-Boltzmann entropy nor the Tsallis nonextensive entropy.

NEW DIRECTIONS IN PREDICTING AND MODELING TRUCK LOAD POPULATIONS INCLUDING OVERLOADS

Overload trucks often appear as a sizeable portion of truck populations on highways. Within most transportation jurisdictions in the United States, overloads refer to truck weights in excess of 356 kN (80 kips). The load populations exhibit an inconsistent pattern-often with two or more distinct peaks. This paper represents new directions in modeling truck load patterns with capabilities to offer the continuity as well as the bimodal feature of the load data. The method presented is data-driven and uses a bimodal distribution function. The extension of the model to predict the truck load population when data are incomplete or not available is also presented. The parameters required in the model are estimated via the fuzzy regression analysis in the proposed method. The fuzzy regression analysis allows the utilization of subjective and limited information. The Weigh-in-Motion (WIM) data obtained from several stations were used to quantify the parameters of this bimodal distribution function by employing the fuzzy regression analysis. The procedure on obtaining an appropriate distribution function for truck load population is explained along with illustrative examples.

Soot particle size distribution reconstruction in a turbulent sooting flame with the split-based extended quadrature method of moments

The Method of Moments (MOM) has largely been applied to investigate sooting laminar and turbulent flames. However, the classical MOM is not able to characterize a continuous particle size distribution (PSD). Without access to information on the PSD, it is difficult to accurately take into account particle oxidation, which is crucial for shrinking and eliminating soot particles. Recently, the Split-based Extended Quadrature Method of Moments (S-EQMOM) has been proposed as a numerically robust alternative to overcome this issue [Salenbauch et al., “A numerically robust method of moments with number density function reconstruction and its application to soot formation, growth, and oxidation,” J. Aerosol Sci. 128, 34–49 (2019)]. The main advantage is that a continuous particle number density function can be reconstructed by superimposing kernel density functions (KDFs). Moreover, the S-EQMOM primary nodes are determined individually for each KDF, improving the moment realizability. In this work, the S-EQMOM is combined with a large eddy simulation/presumed-probability density function flamelet/progress variable approach for predicting soot formation in the Delft Adelaide Flame III. The target flame features low/high sooting propensity/intermittency and comprehensive flow/scalar/soot data are available for model validation. Simulation results are compared with the experimental data for both the gas phase and the particulate phase. Good quantitative agreement has been obtained especially in terms of the soot volume fraction. The reconstructed PSD reveals predominantly unimodal/bimodal distributions in the first/downstream portion of this flame with particle diameters smaller than 100 nm. By investigating the instantaneous and statistical sooting behavior at the flame tip, it has been found that the experimentally observed soot intermittency is linked to mixture fraction fluctuations around its stoichiometric value that exhibits a bimodal probability density function.

A bimodal extension of the ARYA&amp;PARIS approach for predicting hydraulic properties of structured soils

• Bimodal Arya&Paris hydraulic properties were developed based on soil structure. • The bimodal PTF provides better prediction of un/saturated hydraulic conductivity. • The bimodal PTF predicts well the spatial variability of the hydraulic properties. The main purpose of this paper is to develop a bimodal pedotransfer function to obtain soil water retention (WRC) and hydraulic conductivity (HCC) curves. The proposed pedo-transfer function (PTF) extends the Arya and Paris (AP) approach, which is based on particle size distribution (PSD), by incorporating aggregate-size distribution (ASD) into the PTF to obtain the bimodal WRC. A bimodal porosity approach was developed to quantify the fraction of each of the porous systems (matrix and macropores) in overall soil porosity. Saturated hydraulic conductivity, K 0 , was obtained from WRC using the Kozeny-Carman equation, whose parameters were inferred from the behaviour of the bimodal WRC close to saturation. Finally, the Mualem model was applied to obtain the HCC. In order to calibrate the PTF, measured soil physical and hydraulic properties data were used, coming from field infiltration experiments from an irrigation sector of 140 ha area in the “Sinistra Ofanto” irrigation system in Apulia, southern Italy. The infiltration data were fitted by using both bimodal and unimodal hydraulic properties by an inverse solution of the Richards equation. The bimodal “measured” hydraulic properties were then used to calibrate the scaling parameter (α AP ) of the proposed bimodal AP ( bimAP ) PTF. Similarly, for the sake of comparison with the bimodal results, the unimodal hydraulic properties were used to calibrate the α AP of the classical unimodal AP ( unimAP ) PTF. Compared to the unimAP PTF, the proposed bimAP significantly improves the predictions of the mean WRC parameters and K 0 , as well as the prediction of the shape of the whole HCC. Moreover, compared to the unimodal approach, it also allows to reproduce statistics of the hydraulic parameters (for example, variance) similar to those observed in the calibration dataset. Multiple linear regression (MLR) was also applied to analyse the sensitivity of the bimodal α AP parameter to textural and structural features, confirming significant predictive effects of soil structure.

Wind-tunnel experiments of Aeolian sand transport reveal a bimodal probability distribution function for the particle lift-off velocities

The probability distribution function (PDF) of the lift-off velocities of Aeolian (wind-blown) sand particles – i.e., the velocities with which the particles rebound or are ejected from the bed owing to a granular-bed collision – is an important parameter in models of sediment transport and dust emission. However, in order to reliably describe this distribution function, measurements of the individual particle trajectories near the sand bed during sediment transport are required. Such measurements are challenging due to the high particle concentration near the bed, which makes it difficult to track the dynamics of the particles involved in the lift-off process. Here we seek to overcome this difficulty by applying an experimental method for grain trajectory detection, which allows us to obtain a large number of near-bed lift-off particle trajectories during sand transport in a boundary layer wind tunnel.The PDFs of horizontal and vertical lift-off velocities all were found to display a bimodal distribution at a height smaller than 1 mm from the bed. The first and second maxima in the distribution curve of horizontal and vertical lift-off velocities occur at around 0.25 and 1.0 ms−1, and at 0 and 1.0 ms−1, respectively. Moreover, the first maximum is mainly associated with splashed particles, whereas the second peak is mainly caused by rebound particles. However, the lift-off velocities far from the bed follow a unimodal distribution, in agreement with previous experiments. Our results are providing, thus, unprecedented insights about near-bed transport processes over wind-blown soils, and have implication for modelling granular-bed interactions in Aeolian sand transport and geomorphodynamic models.

NEW DIRECTIONS IN PREDICTING AND MODELING TRUCK LOAD POPULATIONS INCLUDING OVERLOADS

Overload trucks often appear as a sizeable portion of truck populations on highways. Within most transportation jurisdictions in the United States, overloads refer to truck weights in excess of 356 kN (80 kips). The load populations exhibit an inconsistent pattern-often with two or more distinct peaks. This paper represents new directions in modeling truck load patterns with capabilities to offer the continuity as well as the bimodal feature of the load data. The method presented is data-driven and uses a bimodal distribution function. The extension of the model to predict the truck load population when data are incomplete or not available is also presented. The parameters required in the model are estimated via the fuzzy regression analysis in the proposed method. The fuzzy regression analysis allows the utilization of subjective and limited information. The Weigh-in-Motion (WIM) data obtained from several stations were used to quantify the parameters of this bimodal distribution function by employing the fuzzy regression analysis. The procedure on obtaining an appropriate distribution function for truck load population is explained along with illustrative examples.

Auctions: A New Method for Selling Objects with Bimodal Density Functions

In this paper we define a new auction, called the Draw auction. It is based on the implementation of a draw when a minimum price of sale is not reached. We find that a Bayesian Nash equilibrium is reached in the Draw auction when each player bids his true personal valuation of the object. Furthermore, we show that the expected profit for the seller in the Draw auction is greater than in second-price auctions, with or without minimum price of sale. We make this affirmation for objects whose valuation can be modeled as a bimodal density function in which the first mode is much greater than the second one. Regarding the Myerson auction, we show that the expected profit for the seller in the Draw auction is nearly as good as the expected profit in the optimal auction, with the difference that our method is much more simple to implement than Myerson’s one. All these results are shown by computational tests, for whose development we have defined an algorithm to calculate Myerson auction.

A novel bi-modal extended Huber loss function based refined mask RCNN approach for automatic multi instance detection and localization of breast cancer.

Breast cancer is an extremely aggressive cancer in women. Its abnormalities can be observed in the form of masses, calcification and lumps. In order to reduce the mortality rate of women its detection is needed at an early stage. The present paper proposes a novel bi-modal extended Huber loss function based refined mask regional convolutional neural network for automatic multi-instance detection and localization of breast cancer. To refine and increase the efficacy of the proposed method three changes are casted. First, a pre-processing step is performed for mammogram and ultrasound breast images. Second, the features of the region proposal network are separately mapped for accurate region of interest. Third, to reduce overfitting and fast convergence, an extended Huber loss function is used at the place of SmoothL1(x) in boundary loss. To extend the functionality of Huber loss, the delta parameter is automated by the aid of median absolute deviation with grid search algorithm. It provides the best optimum value of delta instead of user-based value. The proposed method is compared with pre-existing methods in terms of accuracy, true positive rate, true negative rate, precision, F-score, balanced classification rate, Youden's index, Jaccard Index and dice coefficient on CBIS-DDSM and ultrasound database. The experimental result shows that the proposed method is a better suited approach for multi-instance detection, localization and classification of breast cancer. It can be used as a diagnostic medium that helps in clinical purposes and leads to a precise diagnosis of breast cancer abnormalities.

Separate retrieval of microphysical characteristics in aerosol fractions from laser sensing data

A procedure is proposed for deriving the aerosol microphysical characteristics of the spherical particle, the complex refractivity index m = mreal + i*mimag and the bimodal size distribution function U(r), using laser sensing data at the wavelengths within 355-1064 nm. A possibility of a separate fraction-wise assessment of m + U(r) is studied for weakly absorbing particles for mimag < 0.015, when mfine ≠ mcoarse holds. The reconstruction of the mode-averaged mmean is used in the preliminary stage and provides information on the mmean + U(r, mmean) parameters and the mode difference of mreal in the current observation session. A subsequent transformation of the scattering efficiency factors (integral equation kernels) divides the refractive index with respect to the fractions and gives a correct estimation of mcoarse at different sought-for values of mrealfine and mrealcoarse. A change of the kernels and the region of allowed values of the sought-for U(r) ensures its admissible estimation for both fractions. The algorithms are tested for one mfine (1.50+i*0.01) and nine mcoarse (mreal = 1.40, 1.50, 1.60; mimag = 0.0001, 0.001, 0.01). In order to include the influence of the contribution from the modal particles into the total concentration, 462 empirical models of U(r) are used.