Unit Step Research Articles

Mixing enhancement with minimal thrust loss for the Mach 1.76 jet issuing from different beveled collar nozzles

The main objective of the present investigation is to enhance the jet mixing by introducing a nozzle collar exit inclination having bevel angles of 30°, 45°, and 60°. In addition to this, the present study also addresses the problem of investigating the jet which attains maximum mixing from beveled collar nozzle with the imposition of a minimal overall thrust loss penalty. The nozzles of the present study have been designed to deliver a uniform Mach 1.76 jet at the nozzle collar exit. The results thus obtained through numerical computations have been compared with that of the reference nozzle which has zero exit inclination. The investigations have been performed under both design and off-design conditions of the jet by varying the nozzle pressure ratio (NPR) from 4.5 to 6.5 with unit step size. It is seen that the rate of jet mixing is augmented with the increase in the bevel angle. Thus, the 60° beveled collar nozzle demonstrated the highest jet mixing in comparison to the reference nozzle, 30° and 45° beveled collar nozzle at each nozzle pressure ratio. However, 60° beveled collar nozzle has been substantially penalized with the highest thrust loss of about 6.5% at nozzle pressure ratio 4.5 and about 3.5% at nozzle pressure ratio 5.5 with respect to the reference nozzle and other nozzles of the present study. The 45° and 30° beveled collar nozzles reported the thrust penalty below 2.3% and 1%, respectively. The overall performance, which includes mixing enhancement along with the minimal thrust loss, has been found in the 45° beveled collar nozzle, which clearly justifies its preference over the other investigated nozzle configurations.

Some Estimates for Certain q-analogs of Gamma Integral Transform Operators

The aim of this work is to examine some q-analogs and differential properties of the gamma integral operator and its convolution products. The q-gamma integral operator is introduced in two versions in order to derive pertinent conclusions regarding the q-exponential functions. Also, new findings on the q-trigonometric, q-sine, and q-cosine functions are extracted. In addition, novel results for first and second-order q-differential operators are established and extended to Heaviside unit step functions. Lastly, three crucial convolution products and extensive convolution theorems for the q-analogs are also provided.

Computation of Deformable Interface Two‐Phase Flows: A Semi‐Lagrangian Finite Element Approach

ABSTRACTThis work aims at presenting a new computational approach to study two and three dimensional two‐phase flows and two dimensional coalescence phenomenon using direct numerical simulation. The flows are modeled by the incompressible Navier–Stokes equations, which are approximated by the finite element method. The Galerkin formulation is used to discretize the Navier–Stokes equations in the spatial domain and the semi‐Lagrangian method is used to discretize the material derivative. In order to satisfy the Ladyzhenskaya–Babuška–Brezzi condition, high‐order stable pair of elements are used, with pressure and velocity fields being calculated on different degrees of freedom in the unstructured mesh nodes. The interface is modeled by an unfitted adaptive moving mesh, where interface nodes are tracked in a Lagrangian fashion and moved with the velocity solution of the fluid motion equations. The surface tension is computed using the interface curvature and the gradient of a Heaviside function, and added in the momentum equations as a body force. In order to avoid undesired spurious modes at the interface due to high property ratios, a smooth transition between fluid properties is defined on the interface region. Several benchmark tests have been carried out to validate the proposed approach, and the obtained results have demonstrated agreement with analytical solutions and numerical results reported in the literature. A coalescence model is also proposed based on geometric criteria and results show interesting dynamics.

Nonlinear step responses and how to find them

AbstractThe unit step response is a standard tool for experimental identification; its shape is equivalent to a solution of an appropriate governing equation. In this study, we investigate two variants of nonlinear first-order governing equation, $${y'=k(1-y^\beta )/\beta }$$ y ′ = k ( 1 - y β ) / β , and mainly its rescaled version $${y'=k\left( 1-y^\beta \right) }$$ y ′ = k 1 - y β . We explore a continuum of response shapes, ranging from constant zero to constant one, including standard shapes such as $$1-\exp (-t)$$ 1 - exp ( - t ) and $${\tanh (t)}$$ tanh ( t ) , but also uncommon shapes such as a part of the logarithmic integral. Two methods for estimating the parameter $$\beta $$ β are proposed: one based on analyzing the curvature of a single response and the other based on analyzing initial slopes for multiple responses, while varying the amplitude of the step. Our investigation is conducted in the context of gas permeation through a homogeneous porous medium, where nonlinear response shapes have been observed. This context allows for a meaningful interpretation of the results, consistent with expectations for specific thermodynamic processes. However, neither method is constrained by this context, and either might be considered for any first-order system that fails to have amplitude-scale invariance or for which the shapes seem to exhibit an effect of saturation.

Preventing mass loss in the standard level set method: New insights from variational analyses

For decades, the computational multiphase flow community has grappled with mass loss in the level set method. Numerous solutions have been proposed, from fixing the reinitialization step to combining the level set method with other conservative schemes. However, our work reveals a more fundamental culprit: the smooth Heaviside and delta functions inherent to the standard formulation. Even if reinitialization is done exactly, i.e., the zero contour interface remains stationary, the use of smooth functions lead to violation of mass conservation. We propose a novel approach using variational analysis to incorporate a mass conservation constraint. This introduces a Lagrange multiplier that enforces overall mass balance. Notably, as the delta function sharpens, i.e., approaches the Dirac delta limit, the Lagrange multiplier approaches zero. However, the exact Lagrange multiplier method disrupts the signed distance property of the level set function. This motivates us to develop an approximate version of the Lagrange multiplier that preserves both overall mass and signed distance property of the level set function. Our framework even recovers existing mass-conserving level set methods, revealing some inconsistencies in prior analyses. We extend this approach to three-phase flows for fluid-structure interaction (FSI) simulations. We present variational equations in both immersed and non-immersed forms, demonstrating the convergence of the former formulation to the latter when the body delta function sharpens. Rigorous test problems confirm that the FSI dynamics produced by our simple, easy-to-implement immersed formulation with the approximate Lagrange multiplier method are accurate and match state-of-the-art solvers.

Determination of the plastic J integral of ductile material using the XFEM with only Heaviside function and variable-node elements

In this paper, the extended finite element method (XFEM) with only Heaviside function is proposed for the elastic–plastic fracture mechanics (EPFM) modeling. The proposed method removes tip enrichment functions depending on the polar coordinates at the crack front, and a step function only determined by the level set function is utilized to track the crack front. To alleviate the volumetric locking phenomena caused by the plastic incompressibility, the B-bar method is incorporated into the three dimensional (3D) XFEM program. Therefore, the fully integration scheme can be chosen to ensure the accuracy when addressing large plastic deformation in EPFM analysis. Additionally, the material tangent stiffness matrix of Ramberg-Osgood constitutive is given, and the local refinement technique using variable-node elements is adopted to reduce the number of elements and nodes for efficient analysis. A Newton-Raphson iterative algorithm is developed to solve the nonlinear algebraic equations caused by material nonlinearity. Several numerical examples including the determination of crack opening displacement, and the fully plastic J integral in the ductile materials are presented to test the performance of the proposed method. Comparisons with the results from the existing methodologies show that the new enrichment scheme can save computational cost and obtain sufficient accuracy even in the case of 3D curved crack.

Nonlinear Vlasov and Fokker-Planck Dynamics in Confined Systems with Drag: A Numerical Study

Abstract We explore numerically the behavior of a one-dimensional many-body system consisting of particles that interact through short range repulsive forces, and are also under the effects of drag forces, and of an external confining potential. The statistical dynamics of systems of this kind exhibits interesting links with the thermostatistical formalism based on the Sq non-additive entropies. In the regime of overdamped motion, these systems admit an effective description in terms of a non-linear Fokker-Planck equation. When the overdamped condition is relaxed, and inertial effects are explicitly taken into account, the system can be described by a Vlasov-like effective mean field dynamics. The Vlasov-like description of this type of systems has been recently investigated in the literature from an analytical point of view. In the present contribution we explore the behaviour of these system numerically, through direct molecular dynamics simulations. We consider examples of systems with four different short range repulsive forces, with a dependence on distance given by exponential, Heaviside, Lorentzian, and Bessel functions. The results of our numerical simulations are fully consistent with the predictions derived from the Vlasov-like mean field description. In particular, we verify that, in the asymptotic limit of large times, the system evolves towards a state exhibiting a spatial distribution of particles that coincides with the stationary solution of an appropriate nonlinear Fokker-Planck equation. This limit spatial distribution has the form of a q-Gaussian.

Analysis of a Class of Minimization Problems Lacking Lower Semicontinuity

The minimization of nonlower semicontinuous functions is a difficult topic that has been minimally studied. Among such functions is a Heaviside composite function that is the composition of a Heaviside function with a possibly nonsmooth multivariate function. Unifying a statistical estimation problem with hierarchical selection of variables and a sample average approximation of composite chance constrained stochastic programs, a Heaviside composite optimization problem is one whose objective and constraints are defined by sums of possibly nonlinear multiples of such composite functions. Via a pulled-out formulation, a pseudostationarity concept for a feasible point was introduced in an earlier work as a necessary condition for a local minimizer of a Heaviside composite optimization problem. The present paper extends this previous study in several directions: (a) showing that pseudostationarity is implied by (and thus, weaker than) a sharper subdifferential-based stationarity condition that we term epistationarity; (b) introducing a set-theoretic sufficient condition, which we term a local convexity-like property, under which an epistationary point of a possibly nonlower semicontinuous optimization problem is a local minimizer; (c) providing several classes of Heaviside composite functions satisfying this local convexity-like property; (d) extending the epigraphical formulation of a nonnegative multiple of a Heaviside composite function to a lifted formulation for arbitrarily signed multiples of the Heaviside composite function, based on which we show that an epistationary solution of the given Heaviside composite program with broad classes of B-differentiable component functions can in principle be approximately computed by surrogation methods. Funding: The work of Y. Cui was based on research supported by the National Science Foundation [Grants CCF-2153352, DMS-2309729, and CCF-2416172] and the National Institutes of Health [Grant 1R01CA287413-01]. The work of J.-S. Pang was based on research supported by the Air Force Office of Scientific Research [Grant FA9550-22-1-0045].

Studying the transient process of an intermittent control system from its response property

As we all know, the output of a system is affected by its input and response properties. When the input switches, there must exist a transient process in the output and this transient process is different for different systems due to their different response properties and different dynamic process. However, the response property and dynamic process have rarely been studied in the obtained achievements about the transient process of an intermittent control system. The obtained achievements cannot agree with the real physical process and cannot be applied to study the transient process in engineering.By introducing the unit step function and taking the intermittent input signal as a piecewise signal, we have studied the transient process. Our research shows that the transient process is related to the response characteristics, historical dynamic information, and control parameters, which agrees well with the real system and can be applied to analyze and optimize the transient process in engineering. Some examples in our paper verify our theoretical achievements.

The impact of suction/injection with radiation effect on natural convection flow over a vertical channel in the existence of point/line heat source configuration

The present study manifests the significance of a heat source concentrated at a point or along a line on the natural convection flow featuring suction/injection and radiation effect within a vertical channel. The consistent heat source is modeled with the Heaviside step function and converted to a point/line heat source. The governing equations modeling the flow are resolved by the Laplace transform technique. Shear stress, Nusselt coefficient, and mass flow rate, along with respective values, are examined through the analysis presented in tables whilst the physical components like suction/injection ( K ), Prandtl number (Pr), heat source (S), and radiation (N) are explored graphically on the velocity and the temperature field. Results illustrate that the temperature and the fluid velocity intensify with the escalation of point/heat source attributes. However, as the parameters Pr, K and N increase, the velocity distribution decelerates. In addition, the heat transfer rate gradually diminishes with an elevation in radiation, and reducing the line heat source to that of point heat source parameters corresponds to a decrease in the rate of heat transfer. Also, in the absence of radiation, the current finding aligns seamlessly with the established research in the literature.

On the convergence of Fourier representations and Schwartz distributions

While Fourier theory remains a cornerstone for analyzing and interpreting the spectral content of diverse signals, its limitations often surface in practical applications. Notably, popular signals like sinusoids, Dirac deltas, signums, and unit steps lack convergent Fourier representations within the conventional framework. This discrepancy necessitates distribution theory to build and interpret suitable representations for such signals. However, existing literature in signal processing and communication engineering often glosses over these intricacies, leaving researchers grappling with obscure concepts regarding the very existence of Fourier representations for non-conforming signals. This work bridges this critical gap by offering a comprehensive exploration of the conditions guaranteeing the existence of Fourier representations. We introduce a novel linear space – the Gauss–Schwartz (GS) function space – and its corresponding class of tempered superexponential (TSE) distributions. We demonstrate that the Fourier transform (FT) acts as an isomorphism on the GS space of test functions and, by duality, on TSE distributions. Crucially, the GS space proves to be minimal in the sense that its dual, encompassing TSE distributions, represents the largest possible linear space over which the FT can be defined through duality. This theoretical advancement signifies a twofold contribution. Firstly, it clarifies the existence and interpretation of Fourier representations for prevalent signals that evade conventional analysis. Secondly, the introduction of the GS-TSE framework expands the reach of the FT to encompass a broader spectrum of functions, enabling novel applications in diverse fields. Ultimately, this work paves the way for a more robust and accessible understanding of Fourier analysis, empowering researchers and practitioners to leverage its full potential in exploring and processing a wider range of signals.

The influence of student interaction on GPA in social networks

The kinetic theory is utilized to study the impact of student interaction on the grade point average (GPA) in social networks. The interactions between students can not only be through offline means but also social networks. To analyze the evolution of the student's GPA on social networks, we set up a joint GPA-connectivity density function and two Boltzmann models, which consider the general connectivity distribution of individuals and describe the structure of social networks statistically. In the model, we use the standard Heaviside function to divide the GPA into two parts, corresponding to the students’ different learning attitudes. In addition, the number of contacts for each student is assumed to determine the probability of their learning attitudes being transmitted and successfully influencing another student. We investigate the evolution of the overall students’ average GPA through the kinetic models and find the conditions for the reversal of GPA. Finally, numerical simulations reveal how social networks affect the students’ GPA.

Exact analysis of hydromagnetic non‐Gray optically thick heat absorbing fluid with nonlinear parabolic ramped conditions

AbstractThe article presents a solution to a complex problem involving a transient MHD‐free convective chemically reactive flow. The flow involves a viscous incompressible electrically conducting non‐Gray optically thick fluid moving past a semi‐infinite vertical plate that has suddenly started but is temporarily accelerated. The plate has nonlinear parabolic ramped conditions, and the flow is exposed to thermal radiation, heat absorbing sink, and diffusion‐thermo effect. For the current study, the model fluid being used is moist air. The governing equations are obtained using the Laplace transform technique with the help of the Heaviside step function. This study is the first to consider parabolic ramped motion, temperature and concentration simultaneously. Effects of the pertinent parameters on the Sherwood number and Nusselt number are visualized using 3D surface plotting. Findings of the problem manifest that concentration, temperature, and velocity profiles in case of ramped conditions are less than in isothermal conditions. The rate of heat transfer of mercury is less than oxygen, air, water, and ethanol at a room temperature of 22°C–25°C. The ramped plate has the tendency to augment the heat transfer rate. The present study is of great interest in numerous fields of industry and machine‐building applications.

Kamal Transform of Unit Step Functions

Kamal Transform of Unit Step Functions

Convolution Theorem for (p,q)-Gamma Integral Transforms and Their Application to Some Special Functions

This article introduces (p,q)-analogs of the gamma integral operator and discusses their expansion to power functions, (p,q)-exponential functions, and (p,q)-trigonometric functions. Additionally, it validates other findings concerning (p,q)-analogs of the gamma integrals to unit step functions as well as first- and second-order (p,q)-differential operators. In addition, it presents a pair of (p,q)-convolution products for the specified (p,q)-analogs and establishes two (p,q)-convolution theorems.

The coupled dynamics of information-behavior-epidemic propagation considering the heterogeneity of adoption thresholds and network structures in multiplex networks

During epidemics, individual responses to prevention and control information vary significantly, affecting the execution of preventive measures. This paper introduces a novel model to analyze the co-evolution of the information-behavior-epidemic dynamic, accounting for the heterogeneity in individual behavioral adoption thresholds and network structures. The model consists of three layers: the upper layer, utilizing scale-free social networks to represent the diffusion of information; the middle and lower layers, employing community networks, to illustrate individuals’ decision-making and epidemic transmission processes, respectively. We applied the Heaviside step function to depict the effects of information and local environmental factors on individuals’ decisions. Through the microscopic Markov chain approach (MMCA), we determined the epidemic threshold. Our methodology also included extensive Monte Carlo (MC) simulations to validate our theoretical predictions. Our results reveal several key insights: (i) reducing community mobility and the threshold for adopting preventive behaviors can significantly restrain the scale of the epidemic and delay outbreaks; (ii) the level of individual activity in the physical contact network has a significant influence on epidemic transmission; (iii) integrating individuals’ behavior adoption thresholds induces a two-stage phase change, deviating from the continuous phase transition typically observed in epidemic transmission.

Novel cell-based smoothed extended finite element method for simulating the interactions of ultrasonic waves with randomly distributed cracks in solid structures

The conventional extended finite element method (XFEM) is a powerful tool for simulating crack-related problems in materials; however, several limitations exist, such as numerical instabilities and convergence issues. These problems arise because enriched elements have higher stiffness than standard elements, and this difference can cause computational difficulties. To overcome these limitations, we developed the cell-based smoothed extended finite element method (CS-XFEM), an advanced computational technique designed to simulate the intricate interactions between ultrasonic waves and randomly distributed cracks within solid materials. This innovative approach integrates a cell-based smoothing technique into the XFEM, effectively softening the stiffness of the enriched elements around crack tips. Therefore, the CS-XFEM eliminates numerical instability, providing a more stable and reliable computational framework. In this study, numerical experiments were conducted in which plasticity properties were assigned to both the crack bodies and tips to reflect crack yielding. Further, frictional contact in the crack body elements was formulated using the Heaviside function, and deformation around the crack tips was approximated using a singular function. Through comprehensive numerical investigations, we demonstrated that the conventional XFEM fails to converge and, instead, diverges when ultrasonic waves interact with randomly distributed cracks. By contrast, our proposed CS-XFEM method demonstrates strong convergence capabilities, rendering it well-suited for exploring the interactions between ultrasonic waves and randomly distributed cracks under varying crack quantities, lengths, and friction coefficients. Overall, the proposed CS-XFEM is an efficient, accurate, and robust method for investigating the acoustic nonlinearity induced by randomly distributed cracks with frictional contact in solid structures.

Research on the characteristics of electro-hydraulic position servo system of RBF neural network under fuzzy rules

A radial basis function neural network PID controller under fuzzy rules (FUZZY-RBF-PID) was designed for the electro-hydraulic position servo system under the influence of uncertain factors such as load mutation, and load stiffness change. Firstly, the mathematical model of the system is established, and the frequency domain and time domain analysis of the system are carried out. Secondly, based on the analysis results, a radial basis function (RBF) neural network PID controller is designed, and fuzzy rules are innovatively used to adjust the learning rate of PID parameters in the RBF neural network learning algorithm in real time. Thirdly, the simulation results show that under the action of the FUZZY-RBF-PID controller, the unit step response of the system has high steady-state accuracy, fast response speed, and under the condition of large load stiffness, the system can recover to the steady-state value faster after being disturbed. At the same time, when the input signal is the sinusoidal signal of 10 HZ, the system under the action of the FUZZY-RBF-PID controller has no obvious phase lag phenomenon, and the tracking error is minimal. The proposed method can effectively improve the comprehensive performance of the electro-hydraulic position servo system under the influence of uncertain factors.

Particle swarm optimization solution for roll-off control in radiofrequency ablation of liver tumors: Optimal search for PID controller tuning.

The study investigates the efficacy of a bioinspired Particle Swarm Optimization (PSO) approach for PID controller tuning in Radiofrequency Ablation (RFA) for liver tumors. Ex-vivo experiments were conducted, yielding a 9th order continuous-time transfer function. PSO was applied to optimize PID parameters, achieving outstanding simulation results: 0.605% overshoot, 0.314 seconds rise time, and 2.87 seconds settling time for a unit step input. Statistical analysis of 19 simulations revealed PID gains: Kp (mean: 5.86, variance: 4.22, standard deviation: 2.05), Ki (mean: 9.89, variance: 0.048, standard deviation: 0.22), Kd (mean: 0.57, variance: 0.021, standard deviation: 0.14) and ANOVA analysis for the 19 experiments yielded a p-value ≪ 0.05. The bioinspired PSO-based PID controller demonstrated remarkable potential in mitigating roll-off effects during RFA, reducing the risk of incomplete tumor ablation. These findings have significant implications for improving clinical outcomes in hepatocellular carcinoma management, including reduced recurrence rates and minimized collateral damage. The PSO-based PID tuning strategy offers a practical solution to enhance RFA effectiveness, contributing to the advancement of radiofrequency ablation techniques.

A comparative study on PI-PD controller design using stability region centroid methods for unstable, integrating and resonant systems with time delay

In this paper, controller tuning methods based on stability region centroid methods reported in the literature are used to design PI-PD controllers for unstable, integrating and resonant systems with time delay. By analyzing the stability boundary locus (SBL) for the PD controller, which is utilized in the inner loop of this structure, the controller parameters are obtained using three methods which are the Weighted Geometric Center (WGC), Centroid of Convex Stability Region (CCSR), and Stability Triangle Approach (STA). These techniques were applied analytically, step by step. For the closed loop transfer function obtained in the inner loop of the controlled system, these three methods were utilized to design the PI controller in the outer loop, individually. Unit step responses of the controlled system, using PI-PD gains determined by each method, have been obtained. Furthermore, a disturbance of a certain time and amplitude was added to the systems to test the disturbance rejection behavior and the robustness performance of the methods. Perturbed responses were obtained through changing the model parameters at a certain rate. Time domain performance metrics were analyzed to compare the responses. The simulations were evaluated using settling time, rise time, and percentage overshoot as the assessment criteria. As a result of this study, the effectiveness of three methods, namely WGC, CCSR, and STA, in controller design for unstable, integrating, and resonant time-delay systems has been demonstrated. In addition, a comparison of time domain performance metrics is presented for the nominal and perturbed systems. Based on these comparisons, it is concluded that the methods outperform each other only in some time response performance measures. The presented results showed the advantages of these methods over each other in terms of some performance criteria. The contribution of this study to the literature is the comparative analysis of these three analytical methods.