Closed-form Formulation Research Articles

From the anomalous diffusion impedance to the closed-form, infinite-series and integral formulations of the voltammetric response of thin-film insertion materials under restricted diffusion conditions. A modelling contribution based on the anomalous mass transfer function

Abstract The voltammetric response of one-step reversible electrochemical reactions (involving dissolved redox species) to a linear potential ramp can be modelled using the calculation method proposed in our previous work (J. Electroanal. Chem. 818 (2018) 84) provided that the mass transfer function for the species involved in the electrochemical reaction is known in closed form under one-dimensional mass transport conditions. The above approach applies in this short communication to the direct insertion reaction in thin-film planar materials under both anomalous and spatially restricted (confined) diffusion conditions. The Faradaic current is obtained first as an alternating infinite series well suited to formal analysis, and then as an integral formulation well suited to numerical computation. Finally, closed-form expressions of the voltammograms are derived at very low or very high values of the potential sweep rate, respectively. Two anomalous diffusion models taken from the electrochemical literature are investigated and compared, based on their voltammetric responses. The voltammetric peak coordinates obtained from the best model are thoroughly examined, and closed-form approximations are proposed for the voltammetric peak current and potential at very low or very high potential sweep rates. A zone diagram is plotted to illustrate the conditions for observation of those limiting situations. Finally, following a referee's request, an alternative modelling approach is proposed in Appendix, which does not require the use of scientific computation software. This approach involves both the Integral Equation method formalism and numerical inversion of Laplace transform.

Investigation of the equivalent material properties and failure stress of the re-entrant composite lattice structures using an analytical model

In the present study, a novel theoretical model is developed, based on classical laminate theory, to predict the equivalent mechanical properties of the re-entrant lattice structures, which composed of continuous fiber reinforced composite struts. Three main mechanism of stretching, flexing and hinging are considered and a general closed-form formulation is derived to estimate the auxetic honeycomb’s elastic and shear modulus as well as Poisson’s ratios. In spite of previous studies in which the response of honeycomb structures is modeled using beam theory, here, each strut of unit cell is expressed as a composite laminate with orthotropic mechanical properties and classical laminate theory is implemented to calculate the mechanical constants. In addition, using the available relations, the elastic buckling stress and failure load of the auxetic unit cell is evaluated. The proposed model is validated using experimental results, which are available from the previous studies as well as FE simulations. A parametric study is also conducted on the model to study the effects of cell angle and honeycomb’s density on the mechanical properties and failure stresses of the re-entrant structure. The results show that, the proposed analytical model can finely predict the equivalent and failure properties of the auxetic honeycomb structures. Furthermore, it is observed that, for the considered re-entrant composite structures with the cell angles between −25 to 25, the vertical strut fails due to the damage evolution and for the unit cells with other cell angles, the buckling failure is determinative.

Fast numerical algorithms with universal matrices for finding element matrices of quadrilateral and hexahedral elements

In this paper, a new closed-form formulation with universal matrices strongly recommends that Weighted Richardson Extrapolation (WRE) with robust and hourglass controlled one-point quadrature can absolutely replace the conventional Gauss Quadrature in terms of efficiency, accuracy, and speed to find element stiffness matrices of quadrilateral and hexahedral elements. Linearizing the geometric transformation and averaging the material property over an element, using sampling point at the origin (0,0) of a standard mapped 2-square (ξ,η) plane for two-dimensional analysis and origin (0,0,0) of a standard mapped 2-cube (ξ,η,ζ) system in the hexahedral element (mid-point rule), helps to obtain element stiffness matrix quickly and explicitly through constant universal matrices. This technique is used independently to each of the eight sub-cubes of the mapped 2-cube (ξ,η,ζ) of the element for three-dimensional problems and four sub-squares of the mapped 2-square of the element for two-dimensional problems. These matrices are assembled appropriately for a second and better approximation. A weighted addition of the two approximations produces a stiffness matrix as accurate as from conventional Gauss Quadrature. However, finding the stiffness matrix in this way, due to explicit integrations, demands only a third of the time needed for Gauss Quadrature.

Extended η-μ Fading Models

This article proposes an extension to the η-μ fading model by including a parameter that quantifies the clustering imbalance between in-phase and quadrature components. The inclusion of such a parameter renders the model enormously flexible, extending its reach of application in a wider range of scenarios. Despite the inclusion of an additional parameter, the equations obtained remain as mathematically tractable as those for the original η-μ model. Several new closed-form expressions for important statistics are then obtained, including: (i) probability density functions for both envelope and phase; (ii) cumulative distribution functions for both envelope and phase; (iii) arbitrary moments for the envelope; (iv) moment generating function for the envelope; (v) average bit error rate for coherent binary systems; (v) level crossing rate, average fading duration, and phase crossing rate; and (vi) several other joint distributions encompassing envelope, phase, and their respective time derivatives. It is noteworthy that many of the closed-form formulations presented here are not available even for the simple η-μ model.

M-APSK Constellation Optimization in the Presence of Phase Reference Error

Considering the analytical issue of ${M}$ -ary amplitude phase-shift keying design in the presence of a phase reference error (PRE), we present a systematic optimization formulation to first minimize the error floor of the optimum detector, which decides on the number of signal points on each ring for all possible numbers of rings. Then with a closed-form minimum symbol error probability formulation, we are able to determine the optimal number of rings and the corresponding ring radii for specific PRE variance and average signal-to-noise ratio. This approach involves purely numerical computation without time-consuming simulations, and works well for all $M\geq 16 $ .

Development and validation of a nonlinear dynamic model for tuned liquid multiple columns dampers

The tuned liquid column damper (TLCD), a passive damping device consisting of a large U-tube with oscillating liquid, has been shown to be effective at mitigating structural responses under natural hazards. Aside from their bandwidth-limited mitigation capabilities, a key limitation of TLCDs is in their large geometries that occupy large space often at prime locations. A solution is to implement multi-columned versions, termed tuned liquid multiple columns dampers (TLMCDs), which have the potential to be tuned to multiple frequencies and occupy less space by leveraging the multiple columns to allow fluid movement. However, mathematical models characterizing their dynamic behavior must be developed enabling proper tuning and sizing in the design process. In this paper, a new analytical model characterizing a TLMCD as a multiple degrees-of-freedom coupled nonlinear system is presented. The frequencies of free vibration and vibration modes of a TLMCD are identified in closed-form formulations. Results are validated using computational fluid dynamics simulations, and show that the analytical model can predict the damper’s liquid surface movements as well as its capability to reduce structural vibration when the structure is subjected to harmonic excitations. A parametric study is conducted to investigate the effect of head loss coefficients, column spacing, cross-section area ratios, and column numbers on mitigating structural response. It is found that, while TLMCDs are less effective than traditional TLCDs under an equal liquid mass, they can provide enhanced performance under geometric restrictions.

Generalized Closed-Form Sidebands’ Radiation Expressions for 4-D Antenna Arrays

In this communication, a closed-form formulation for calculating the radiated power at sidebands of four-dimensional (4-D) antenna arrays using a generalized element pattern of type (sin θ)u| cosθ| <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</sup> is derived. All the expressions exist when the exponents u, u lie in the region: u > -1 and u > -1/2. Compared with numerical integration, the proposed expression uses less computational resources and time. Examples show the efficiency and accuracy of proposed expression.

RETRACTED ARTICLE: A new nonlinear formulation-based prediction approach using artificial neural network (ANN) model for rubberized cement composite

In this paper, a nonlinear formulation approach using an artificial neural network (ANN) is proposed and applied for the first time to predict the behavior of rubberized cement composite for which there is neither a design formulation nor a closed-form prediction model. For this purpose, the experimental data on the rubberized composite was collected, which includes mix ingredients, curing condition, and age. The statistical analysis of the data was performed to well characterize the interdependencies. Then, the ANN model was built, and the parameters for the optimum ANN model were found. Afterward, based on the relationships used in the ANN architecture at different steps, such as inputs nodes, neurons of hidden layers, activation functions, and output calculations, the ANN model was formulated, and the optimum weights and biases were determined. Once the closed-form formulation was derived, of the model’s performance was assessed through its prediction correlation and error. It was found that the ANN formulation was very robust to predict the complex behavior of rubberized composites with R2 = 0.98. Finally, the normalized importance of the model’s inputs was determined and the parametric study of the ANN model was also performed to show the effect of each variable on the output. The results showed that the relationships between the inputs and the output were mostly nonlinear, indicating the fact that the system’s behavior is a complex problem, which was well captured by formulation-based ANN model.

A frequency domain analysis framework for assessing overturning risk of high-speed trains under crosswind

This study introduces a frequency domain analysis framework for assessing overturning risk of high-speed trains under the excitations of crosswind and track irregularities. Based on a comprehensive numerical computation, closed-form expressions are established for the power spectra and coherence of turbulence relative to a moving vehicle influenced by vehicle speed and direction. The wind turbulence at stationary positions fixed on the ground is characterized by Kaimal or von Karman spectra and Davenport coherence model or a more general coherence model. These closed-form expressions facilitate the quantification of vehicle dynamic response by the effective spectral analysis approach, and also provide essential turbulence information for wind tunnel studies of aerodynamic forces on stationary vehicle models. Closed-form formulations are also presented for computing vehicle dynamic response by focusing on two dominant modes relevant to vehicle overturning risk. The vehicle overturning risk analysis leads to a determination of characteristic wind speed curve for a given vehicle running speed with a target overturning probability. Examples are presented to illustrate analysis framework and to investigate the influence of various parameters including the vehicle model on the vehicle response characteristics.

Vibration of carbon nanotube-reinforced plates via refined nth-higher-order theory

This article presents the free vibration frequencies of composite plates reinforced with single-walled carbon nanotubes by using a refined simplified two-variable nth-higher-order theory. Four kinds of distribution of uniaxially aligned reinforcement material are presented. The most famous one is the uniform; in addition, three types of functionally graded distributions of carbon nanotubes in the through-thickness direction of the plates are investigated. The effective physical properties of composite media are given according to a refined rule of mixtures approach that contains the efficiency parameters. Exact closed-form formulation based on a refined simplified two-variable nth-higher-order plate theory that can be adapted to the vibration of such plates is investigated. Accuracy of presented approach is validated by comparing its results with those given by other investigators

Toward a Unified Approximate Analytical Representation for Spatially Running Spring-Loaded Inverted Pendulum Model

As a versatile template characterizing the center of mass movements in legged locomotion, the sagittal spring-loaded inverted pendulum (SLIP) model has been extensively explored in both biomechanics and robotics. Despite concise in mathematical formulation, the accurate analytical representation of the SLIP model is unaccessible due to its intrinsic nonlinearity. This article extends the traditional SLIP model from sagittal hopping into spatially running. A novel perturbation-based approach is proposed to obtain an analytical approximate solution for the 3-D-SLIP model, resulting in a straight-forward closed-form formulation wherein the numerical integration or iteration is avoided. The derived solution does not rely on the negligible gravity assumption as conventional simplification reported in the existing literature and offers satisfactory prediction performance in a wide range of model parameter combinations. The merits of the acquired approximate solution in both critical state of the apex return map and trajectory prediction with high accuracy have been demonstrated via performance evaluation, endowing this approximation the potential in motion planning and gait control for legged robots.

Optimisation of the transmission capacity of cognitive networks

Cognitive radio technology enables two wireless networks, namely a primary network and a secondary network, to operate in the same geographic region sharing the same spectrum. Primary users have priority to access the spectrum without taking into account the presence of secondary users, while secondary users have to limit their interference in order not to disturb the operation of primary users. This work analyses a cognitive network in which the secondary network can adapt its parameters, in order to exploit any available opportunity for secondary capacity improvement, under the restriction that these adjustments cannot cause any additional harm to primary users. Using Poisson point process modelling, the authors present a closed-form formulation to adjust the secondary modulation order, user density, and transmit power, in order to maximise the secondary capacity, without causing any additional disturbance to primary users, and keeping the quality of secondary transmissions above a minimum acceptable level. The formulation proposed allows us to gain meaningful insights regarding the interrelationship involving some network parameters and performance metrics.

An enhanced sub-cycle statistical algorithm for inrush and fault currents classification in differential protection schemes

Differential protection is a widely-used approach for power transformer protection, the performance of which may be, at times, challenged due to the inrush current and current transformer (CT) saturation. This paper presents an enhanced statistical algorithm for discrimination of fault and inrush current signals under CT saturation scenarios. This paper proposes twofold discrimination indices centered on the concept of fourth-order statistic moment, the so-called kurtosis on the current signal. Mathematically proven, the algorithm performance only depends on the current signal phase angle and it does not require transformer parameters. The mathematical closed form formulation in this paper classifies fault condition and inrush currents in sub-cycle timeframes even in the CT saturation conditions. The performance of the proposed algorithm is verified under various fault and inrush current scenarios with and without CT saturation. The numerical results demonstrate that the proposed algorithm can successfully classify the disturbances and detects saturations quickly with promising accuracy.Index Terms: Inrush current; internal fault; current transformer saturation; differential protection

Eddy Current Loss Model Unifying the Effects of Reaction Field and Non-Homogeneous 3-D Magnetic Field

In this article, a new analytical method is proposed to estimate eddy currents inside linear conductive materials. The novel closed-form formulation takes into account the effects of both conductor reaction field that dominates at high frequencies as well as the spatially non-homogeneous nature of the magnetic field penetrating the material. Although the model is suitable for most kinds of permanent magnet machines, it is particularly useful for Surface-mounted Permanent Magnet (SPM) machines where both of these phenomena are prominent. The numerical implementation of the proposed model, which consists of a combination of the method of images and 4-D Fourier transform, is presented. The model is then validated against 2-D and 3-D finite element models (FEMs) for a simple magnetic circuit, showing good agreement. Finally, the eddy current magnet loss in an SPM synchronous machine is evaluated and results are discussed.

Analytical Formulation and Minimization of Voltage THD in Staircase Modulated Multilevel Inverters With Variable DC Ratios

Staircase Modulation (SCM) is a commonly employed switching strategy in Multilevel Inverters (MLI) with a high number of voltage levels (N). The main challenges in SCM are adjusting the Modulation Index (MI) to the desired value while minimizing the Total Harmonic Distortion (THD). This is achieved by solving an objective function for its optimum variables. These variables are the Switching Angles (SA) in the case of an MLI with Equal DC voltage Supply (EDCS), or both the SAs and the DC voltage supply Ratios (DCR), in the case of an MLI with Unequal DC voltage Supply (UDCS). Such an approach, which is referred to as Optimum Minimization of THD (OMTHD), relies on the accuracy of the THD expression used for the objective function. This paper reveals generic analytical closed-form formulations of both Phase-voltage THD (PTHD) for single-phase (1φ) MLIs and Line-voltage THD (LTHD) for three-phase (3φ) MLIs. The revealed THD expressions apply to SCM based MLIs of any topology, with either EDCS or UDCS voltage source configurations, which may be fixed or varying, and arbitrary value of N (odd and even). Given an arbitrary N value, the proposed unified formulations can be used to generate symbolic N-level PTHD or LTHD expressions, which are analytical functions of the SA and (optionally) DCR variables. The proposed THD expressions were used to form a generic OMTHD problem and solve for optimum SA and DCR variables, ensuring PTHD or LTHD minimization, subject to a set of optional constraints, such as the target MI, tolerable Modulation Error (ME), and the Maximum DCR (MDCR). Outcomes of the proposed OMTHD algorithm were successfully verified by Controller + Hardware-in-Loop (C-HIL) based experiments and compared against results of previous works, raveling significant improvement in THD accuracy and MLI's performance. A downloadable supplemental file containing Maple and MATLAB functions of the proposed THD expressions, as well as pre-calculated sets of optimum SAs and DCRs tables for 13 different values of N (4 ≤ N ≤ 16), is provided for readers' convenience.

Approximating the time-weighted return: The case of flows at unknown time

We consider the problem of approximating the true time-weighted return when a cash flow occurs at an unknown time during the estimation period, which is usually the case of a traditional portfolio evaluated on a daily basis. We aim to provide the best approximation, in terms of mean square error (MSE), under the following main assumptions: the distribution of the log-returns belongs to a subclass of elliptical distributions; a single flow occurs at a uniformly distributed random time; the amount of the flow and the returns of the period are independent. We derive a closed-form formulation for high evaluation frequencies when the returns satisfy the popular assumption of a Geometric Brownian Motion. Besides, with the further assumption of small flows, the Original Dietz return can be obtained as an approximation of our optimal estimator. This implies that under the above-mentioned conditions the Original Dietz return has a MSE close to the minimum. Although further improvements of the MSE seem to be possible only by increasing the estimation frequency, which in turn is usually infeasible, our model provides a rigorous way to handle large flows, which are especially frequent in applications such as performance attribution.

Exploiting System Performance in AF non-orthogonal multiple access network under impacts of imperfect SIC and imperfect hardware

The conventional non-orthogonal multiple access (NOMA) networks with perfect successive interference cancellation (SIC) is studied in many system models in the literature. With regards to SIC operation, this paper considers existence of hardware noise and interference of imperfect SIC operation and then performance degradation under affected factors can be addressed. To examine the system performance of the proposed system, the closed-form formulation of each users are presented in term of outage probability. In high signal to noise ratio (SNR) regime, the corresponding asymptotic outage probability is further considered to provide meaningful insights in our proposed system. In addition, the performance of the proposed scheme is further evaluated on the throughput performance. The numerical examples are deployed by comparing with analytical results to validate the effectiveness of the proposed scheme. Finally, this paper compare system performance in some scenarios in term of the outage/throughput behavior by varying simulation parameters.

Size and temperature effect of Young’s modulus of boron nitride nanosheet

Boron nitride nanosheets (BNNSs), a new type of wide bandgap nanomaterial, has attracted great attention due to their excellent properties and potential applications. Thus, it is necessary to have a comprehensive understanding of the mechanical properties of BNNSs in various working conditions. This paper presents an analytical model based on molecular mechanics to study the size effect and temperature effect on the Young’s modulus of BNNSs. A closed-form formulation is derived for Young’s modulus as a function of the length of B–N bonds and the out-plane displacement. It is shown that the chirality and the size of the BNNSs affect the length of BN bonds in molecular dynamic (MD) simulations. It is also found that the length of BN bonds and the out-plane displacement in a monolayer BNNS is remarkably temperature dependent. Therefore, the sizes and the temperatures can affect the Young’s modulus of BNNSs. The expressions developed in this paper are employed to investigate the Young’s modulus for zigzag and armchair BNNS with various sizes and temperatures. The present model, associating with a beam model, provides a simple method to calculate elastic properties which takes into account all bonded energies and force coefficient changes with atoms distance. The results of size effect and temperature effect are in good agreement with data of simulation performed in finite element method (FEM) simulation and MD simulation. The present study provides a molecular mechanics model to predict the Young’s modulus of a monolayer BNNS, and the present model may be applied to other two-dimensional (2D) materials in further study.

Higher Order Statistics for Composite Fading Models

This letter presents a novel, general second-order statistics framework for composite fading models. The aim is to obtain expressions for the level crossing rate and average fade duration statistics in any fading scenario. Two formulations are presented: 1) one that leads to exact expressions and 2) another that leads to approximate ones. By using the former, even for very simple fading scenarios, no closed-form formula can be found. By using the latter, as proposed here, closed-form formulations may be found. Interestingly, it is shown that, in practical terms, the results in 1) and 2) yield results that are almost indistinguishable from each other when applied to multipath-shadowing environments. As an application example, the $\alpha -\mu $ scenario for both multipath and shadowing has been exercised.

Direct-Sequence Ultrasonic Wideband Technology for Intra-Body Communications

This letter proposes a direct-sequence ultrasonic wideband (DS-UsWB) for intra-body communications (IBCs). The DS technology superimposed channel coding is proposed to enable simultaneous multi-node communications in body area networks (BANs). We simulate the channel impulse response model in a human arm by $k$ -Wave, and analyze the chip duration and pulse duration requirements to avoid an inter-symbol interference (ISI) and an inter-pulse interference (IPI) for the DS-UsWB. We then derive the closed-form formulations for data rate and bit error rate (BER) for the DS-UsWB system. Simulation results verify our theoretical analysis, and show that the proposed DS-UsWB system performance is better than the existing ultrasonic wideband (UsWB) system.