Explicit Closed-form Expressions Research Articles

Generalised Jeffery's equations for rapidly spinning particles. Part 1. Spheroids

The observed behaviour of passive objects in simple flows can be surprisingly intricate, and is complicated further by object activity. Inspired by the motility of bacterial swimmers, in this two-part study we examine the three-dimensional motion of rigid active particles in shear Stokes flow, focusing on bodies that induce rapid rotation as part of their activity. In Part 1 we develop a multiscale framework to investigate these emergent dynamics and apply it to simple spheroidal objects. In Part 2 (Dalwadi et al., J. Fluid Mech., vol. 979, 2024, A2) we apply our framework to understand the emergent dynamics of more complex shapes; helicoidal objects with chirality. Via a multiple scales asymptotic analysis for nonlinear systems, we systematically derive emergent equations of motion for long-term trajectories that explicitly account for the strong (leading-order) effects of fast spinning. Supported by numerical examples, we constructively link these effective dynamics to the well-known Jeffery's orbits for passive spheroids, deriving an explicit closed-form expression for the effective shape of the active particle, broadening the scope of Jeffery's seminal study to spinning spheroids.

Analysis of thermoelastic dissipation in microbeam resonators covered with multiple partial coatings

Thermoelastic dissipation (TED) is a mechanism for intrinsic energy dissipation that governs the upper limit on microresonators’ quality factor (Q-factor). The demands for high performance and diverse functionality have led to the development of MEMS resonators with increasingly complex structural geometries and materials. This work proposes an analytical framework to evaluate TED behavior in microbeam resonators coated with multiple partial coatings. By determining the one-way thermoelastic coupled equation in the thickness direction, the temperature function is obtained for each region, and TED model is derived by capturing the energy loss of each region within the framework of thermal energy method. The effectiveness of present framework is verified by comparing it with experimental data and finite element method (FEM) results. The developed TED model has an explicit closed-form expression that converges rapidly, and retaining only leading terms yields a simple TED model. Rules for applying the simple TED model to ensure accuracy are discussed. TED behaviors considering finite thermal contact conductance are comprehensively investigated. A methodology for a mixture of multiple partial coatings to suppress TED is proposed. This work is helpful in suppressing the TED and achieving high Q-factors for the microbeam resonators.

A coupled acoustic/elastic discontinuous Galerkin finite element method: Application to ultrasonic imaging of 3D-printed synthetic materials

We present the derivation of upwind numerical fluxes for the space discontinuous Galerkin (dG) finite element method applied to the numerical modeling of wave propagation in multidimensional coupled acoustic/elastic media. The space dG method is formulated using the first-order velocity-pressure and velocity-stress systems for acoustic and elastic media, respectively. After eigenanalysis of the first-order hyperbolic systems highlighting the eigenmodes of wave propagation, the upwind numerical fluxes on the acoustic/acoustic and acoustic/elastic interface are obtained in terms of exact solutions of relevant Riemann problems. Thanks to the proposed approach, explicit closed-form expressions of the upwind numerical fluxes are obtained on the acoustic/elastic interface for the general case of multidimensional anisotropic heterogeneous solid media coupled with acoustic fluids. The developed numerical fluxes are validated by analytical/numerical comparison considering the example of an acoustic domain with a circular elastic inclusion. Finally, the coupled solver is used to perform a multiparametric study on the microstructure's echogenicity in a 3D-printed synthetic material under ultrasonic imaging.

Simulation-based dynamic origin–destination matrix estimation on freeways: A Bayesian optimization approach

This study focuses on dynamic origin–destination demand estimation problem on freeway networks. Existing studies on this problem rely on high-coverage of traffic measurements and assumptions on travel times, exhibiting limitations in real-world applications. We formulate the problem as a bi-level programming model, where micro-simulations are incorporated to precisely model traffic flows/travel times on freeways. The bi-level programming model cannot provide explicit closed-form expressions for the objective function and its derivatives, and also intrinsically high-dimensional. Thus, it is highly challenging to find efficient solution algorithms. In this regard, a problem-specific and computationally efficient Bayesian optimization approach is designed. Herein, a novel surrogate model is proposed by embedding a physical surrogate model (it characterizes underlying physical mechanisms and provides global yet less precise approximations) into a functional surrogate model (it provides precise local approximations). The embedding provides problem-specific knowledge for the surrogate model. More importantly, it also restricts the feasible region, enabling the surrogate model to efficiently deal with high-dimensional problems. Gaussian process can be served as the functional surrogate model. Two linear physical surrogate models are proposed to capture interactions between travel demand and traffic measurements. To deal with constraints in the surrogate model, a projection-distance based acquisition function is designed. In searching for new points, the proposed acquisition function is capable of assigning unique weight of exploration to each feasible solution. The proposed approach is validated based on a freeway corridor example, which indicates its outperformance over existing dynamic origin–destination estimation methods in terms of computational efficiency and solution accuracy.

Performance analysis of a fluid flow system modulated by a single server queue prone to catastrophic failures and repairs

<p style='text-indent:20px;'>This article deals with a new class of fluid flow queue regulated by an <inline-formula><tex-math id="M1">\begin{document}$ M/M/1 $\end{document}</tex-math></inline-formula> queueing system in the presence of active catastrophic failures and subsequent repairs of the server. The distribution of the stationary buffer-content is determined in terms of modified Bessel function of first kind. Besides, an explicit closed-form analytical expression for the probability density function of the fluid level in the stationary regime is derived in an elegant manner. Some important and essential performance measures are obtained. Finally, numerical results are provided to illustrate the behaviour of the buffer content distribution and the performance measures for the proposed fluid flow queueing model.</p>

Clarifying some common misconceptions about the expansion of the universe

Modern cosmology is based on a 6-parameter model, called Lambda-CDM, which provides an accurate description of the expansion of the Universe on the basis of measured parameters. However, there are attempts to give ‘phenomenological’ descriptions of the dynamics of the Universe that are inconsistent with this model and sometimes lead to misconceptions. They stem from a Newtonian approach that interprets the expansion as due to an ‘initial kick’ that placed all matter and energy in the Universe on ‘inertial trajectories’. These descriptions do not conform to the relativistic approach. Here we show that the Einstein field equations trace the expansion of cosmic space directly to the increase in the energy content of the observable universe caused by the additional space entering it as a consequence of the forward motion of its horizon. We derive here the connection between variations in the scale factor and the corresponding variations in the energy of the observable universe, in the case of the most general cosmological model. We then obtain explicit closed-form expressions of this relationship both in the case of the Universe of any curvature and containing matter and radiation, and in the case of the de Sitter universe, characterized by the cosmological constant only. Some interesting features are pointed out, both in terms of the energy growth rate of the observable universe in the various cosmological models, and in terms of the dynamics of the scale factor. Finally our demonstration is corroborated by an analogy between the expansion of cosmic space and the gravitational stretching of space at the center of a sphere of uniform density whose radius increases with time. This analogy, which allowed us to obtain the novel closed-form expression of the stretching of space at the centre of such a sphere, confirmed that the paradigm that ‘gravity expands space’ is valid in both systems.

An elliptical inhomogeneity under nonuniform heat flux

We use complex variable techniques to study the decoupled two-dimensional steady-state heat conduction and thermoelastic problems associated with an elliptical elastic inhomogeneity perfectly bonded to an infinite matrix subjected to a nonuniform heat flux at infinity. Specifically, the nonuniform remote heat flux takes the form of a linear distribution. It is found that the internal temperature and thermal stresses inside the elliptical inhomogeneity are quadratic functions of the two in-plane coordinates. Explicit closed-form expressions of the analytic functions characterizing the temperature and thermoelastic field in the matrix are derived.

Accuracy of a DtN Finite Element Approach for the Elastic Half-Space

In this work, the accuracy of a DtN finite element approach is numerically assessed. This procedure allows an efficient and accurate numerical solution of boundary-value problems of axisymmetric elasticity in semi-infinite domains. For this kind of problems, no explicit closed-form expression for the associated DtN map exists, so a suitable semi-analytical approximation of it, in series form, is used to impose exact boundary conditions on a semi-spherical artificial boundary. The procedure is tested by solving the classical Boussinesq problem, whose exact solution is known analytically. The computed numerical solution is compared to the analytical solution, achieving an excellent agreement between both solutions, both for displacements and stresses. The convergence of the analytical solution to the numerical solution is numerically demonstrated, in terms of artificial boundary location, series truncation order and finite element mesh size.

An integrated multitiered supply chain network model of competing agricultural firms and processing firms: The case of fresh produce and quality

In this paper, we develop an integrated multitiered competitive agricultural supply chain network model in which agricultural firms and processing firms compete to sell their differentiated products. The focus here is on fresh produce and minimally processed such agricultural products, with quality also captured. The competition among agricultural firms and processing firms is studied through game theory, where the governing Cournot–Nash equilibrium conditions correspond to a variational inequality problem. The algorithm, at each iteration, yields explicit closed form expressions for the agricultural product path flows, the agricultural product shipments from agricultural firms to the processing firms, and the Lagrange multipliers. A numerical study consisting of several supply chain disruption scenarios demonstrates the applicability of our modeling framework.

On the Capacity of Gaussian MIMO Channels With Memory

The operational capacity of Gaussian MIMO channels with memory was obtained by Brandenburg and Wyner (1974) under certain mild assumptions on the channel impulse response and its noise covariance matrix, which essentuially require channel memory to be not too strong. This channel was also considered by Tsybakov (2006) and its information capacity was obtained in some cases. It was further conjectured, based on numerical evidence, that these capacities are the same in all cases. This conjecture is proved here. An explicit closed-form expression for the optimal input power spectral density matrix is also given. The obtained result is further extended to the case of joint constraints, including per-antenna and interference power constraints as well as energy harvesting constraints. These results imply the information-theoretic optimality of OFDM-type transmission systems for such channels with memory.

Curvature invariants for accelerating Kerr–Newman black holes in (anti-)de Sitter spacetime

The curvature scalar invariants of the Riemann tensor are important in general relativity because they allow a manifestly coordinate invariant characterisation of certain geometrical properties of spacetimes such as, among others, curvature singularities, gravitomagnetism. We calculate explicit analytic expressions for the set of Zakhary–McIntosh curvature invariants for accelerating Kerr–Newman black holes in (anti-)de Sitter spacetime as well as for the Kerr–Newman–(anti-)de Sitter black hole. These black hole metrics belong to the most general type D solution of the Einstein–Maxwell equations with a cosmological constant. Explicit analytic expressions for the Euler–Poincare density invariant, which is relevant for the computation of the Euler–Poincare characteristic χ(M), and the Kretschmann scalar are also provided for both cases. We perform a detailed plotting of the curvature invariants that reveal a rich structure of the spacetime geometry surrounding the singularity of a rotating, electrically charged and accelerating black hole. These graphs also help us in an exact mathematical way to explore the interior of these black holes. Our explicit closed form expressions show that the above gravitational backgrounds possess a non-trivial Hirzebruch signature density. Possible physical applications of this property for the electromagnetic duality anomaly in curved spacetimes that can spoil helicity conservation are briefly discussed.

Parametric Resonance for Enhancing the Rate of Metastable Transition

This work is devoted to quantifying how periodic perturbation can change the rate of metastable transition in stochastic mechanical systems with weak noises. A closed-form explicit expression for approximating the rate change is provided, and the corresponding transition mechanism can also be approximated. Unlike the majority of existing relevant works, these results apply to kinetic Langevin equations with high-dimensional potentials and nonlinear perturbations. They are obtained based on a higher-order Hamiltonian formalism and perturbation analysis for the Freidlin-Wentzell action functional. This tool allowed us to show that parametric excitation at a resonant frequency can significantly enhance the rate of metastable transitions. Numerical experiments for both low-dimensional toy models and a molecular cluster are also provided. For the latter, we show that vibrating a material appropriately can help heal its defect, and our theory provides the appropriate vibration.

Performance analysis of the low‐complexity adaptive channel estimation algorithms over non‐stationary multipath Rayleigh fading channels under carrier frequency offsets

The use of selective partial updating method for reducing computational complexity of adaptive algorithms has been proposed recently. However, its performance analysis has not been studied as extensively and exactly as LMS or RLS algorithms for time-varying channel estimation. This paper provides performance analysis of the low-complexity family of affine projection algorithms based on selective partial update method on the estimation of multipath Rayleigh fading channels in the presence of carrier frequency offsets (CFO) and random channel variations. The analysis is based on the calculation of the error correlation matrix of the estimation, the mean-square weight error (MSWE) and the mean-square estimation error (MSE) parameters. The analysis does not use strong assumptions like small or large step-size, and explicit closed-form expressions for the MSE of estimation are obtained only from common hypotheses in wireless communication context. In this paper, the optimum step-size parameters minimizing the MSE of estimation are analytically derived without any simplifying assumptions. For the sake of comparison with other analytical approaches, the performance of the introduced algorithms is also investigated using the energy conservation relation. Likewise, for exact performance analysis, all the moment terms that appear in closed form expressions for the MSE of estimation are evaluated. Simulations are conduced to corroborate the presented studies and show that the theoretical results agree well with the simulation results over non-stationary multipath Rayleigh fading channels.

On elastic clamping boundary conditions in plate models describing detaching bilayers

Elastic clamping boundary conditions are derived, which describe root rotations and root displacements in plate models used to analyze detaching bimaterial layers subjected to arbitrary end loadings. The elasticity technique uses the reciprocity theorem and J-integral calculations and is applicable to isotropic and orthotropic bimaterial layers of different thicknesses, also with extreme elastic constants and mismatch. Explicit closed form expressions for all clamping coefficients are derived for bimaterial isotropic layers with mid-thickness cracks and zero second Dundurs’ parameter; the derivation relies on existing solutions for energy release rate and local fields. Closed form expressions are derived for the most relevant coefficients in homogeneous symmetric orthotropic layers and thin isotropic layers on isotropic half planes. The technique is applicable to general cases using analytical or numerical results for the energy release rate.

Robust Localization Based on Constrained Total Least Squares in Wireless Sensor Networks

Source localization based on signal strength measurements has become very popular due to its wide applications. This paper focuses on differential received signal strength-based localization with model uncertainties in case of unknown transmit power. The error caused by the measurement noise and the reference power offset and the first-order approximate error generated in the process of DRSS receiving power linearization are merged into the constrained total least square regression matrix equation after linearization. The location optimization problem is formed by minimizing the influence of the errors. Based on the explicit closed form expressions of the first step vector and the second derivative matrix of the optimization problem, an iterative technique based on Newton’s method is proposed. The simulation results show that the root mean square error of the new algorithm is closer to the Cramér–Rao lower bound.

Exact Traveling Waves of a Generalized Scale-Invariant Analogue of the Korteweg–de Vries Equation

In this paper, we study a generalized scale-invariant analogue of the well-known Korteweg–de Vries (KdV) equation. This generalized equation can be thought of as a bridge between the KdV equation and the SIdV equation that was discovered recently, and shares the same one-soliton solution as the KdV equation. By employing the auxiliary equation method, we are able to obtain a wide variety of traveling wave solutions, both bounded and singular, which are kink and bell types, periodic waves, exponential waves, and peaked (peakon) waves. As far as we know, these solutions are new and their explicit closed-form expressions have not been reported elsewhere in the literature.

The marginal distribution function of threshold-type processes with central symmetric innovations

This paper addresses the problem of finding exact and explicit (closed-form) expressions for the stationary marginal distribution of threshold-type time series processes, their associated moments, autocovariance and autocorrelation coefficients. The innovation process of the models under consideration follows three central symmetric distribution functions: Gaussian, Laplace, and Cauchy. Theoretical results for both two- and three-regime threshold-type models are derived. Various examples give rise to a deeper understanding of certain features of the stationary process structure. Exact results for the stationary density, central moments, and autocorrelations of threshold-type processes are compared with approximate density and moment results obtained through an existing numerical methods.

Fifth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (5th-CASAM-N): II. Paradigm Application to a Bernoulli Model Comprising Uncertain Parameters

This work presents the application of the recently developed “Fifth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (5th-CASAM-N)” to a simplified Bernoulli model. The 5th-CASAM-N builds upon and incorporates all of the lower-order (i.e., the first-, second-, third-, and fourth-order) adjoint sensitivities analysis methodologies. The Bernoulli model comprises a nonlinear model response, uncertain model parameters, uncertain model domain boundaries and uncertain model boundary conditions, admitting closed-form explicit expressions for the response sensitivities of all orders. Illustrating the specific mechanisms and advantages of applying the 5th-CASAM-N for the computation of the response sensitivities with respect to the uncertain parameters and boundaries reveals that the 5th-CASAM-N provides a fundamental step towards overcoming the curse of dimensionality in sensitivity and uncertainty analysis.

Recurrence relations arising from confluent hypergeometric functions

The aim of this paper is to present some recurrence relations arising from confluent hypergeometric functions. In addition, an explicit closed-form expression for a sequence associated to the hypergeometric series in terms of Bell partition polynomials is proposed. Several examples are given to illustrate our results.

Isotherm penetration depth under a moving Gaussian surface heat source on a thick substrate

This paper presents novel explicit exDEFAULTpressions for the penetration depth of an isotherm induced by a moving heat Gaussian source on a semi-infinite solid. This work is novel in its methodology of dimensional analysis, asymptotics, and blending of more than one parameter, and provides closed form expressions of high accuracy compared to the exact solution. The solutions obtained are based on the governing equations, not on empirical fittings, and are valid for any material and heat source that are captured by the original governing differential equations. Previous scaling laws for penetration depth of temperature due to moving heat sources were based on point heat sources, and predicted a finite penetration under all conditions, which is against observations that penetration becomes zero for heat sources that are too weak, too distributed, or too fast. This discrepancy is especially relevant for modern processes such as laser heat treatment, laser cladding, and additive manufacturing of metals using lasers; the analysis and expressions presented here account for all those effects. The dimensionless maximum isotherm penetration depth depends on two dimensionless groups: the heat source distribution parameter and the Rykalin number associated with the velocity of the heat source. Previous work on blending involved a single parameter, and a novel extension of the blending technique is presented here to tackle the increased complexity. Maximum isotherm penetration depth is determined for the first time with explicit closed-form expressions over a wide range of heat distributions and Rykalin numbers with an error below 9.7% against the exact solution. The proposed equations are validated against experiments reported in the literature. The expressions developed can be evaluated using a handheld calculator or a spreadsheet and are suitable for practitioners in industry to optimize process parameters and for researchers to validate numerical models.