Closed-form Analytical Solution Research Articles

Closed-form analytical solution for the transfer matrix based on Pendry-MacKinnon discrete Maxwell's equations

Closed-form analytical solution for the transfer matrix based on Pendry-MacKinnon discrete Maxwell's equations

Plate buckling analysis in peridynamic framework

AbstractPeridynamics (PD) is a generalized continuum theory that is developed to account for long range internal force/moment interactions. PD equations of motion are of the integro‐differential type, and only several closed‐form analytical solutions are available for elementary structures. In this paper, governing equations for stability of Kirchhoff type (shear rigid) plates in the PD framework are derived. For several types of boundary conditions, closed form analytical solutions for the buckling loads are presented. The results are compared with the solutions according to the classical plate theory. A very good agreement between the non‐local and the classical theories is observed for the case of the small horizon sizes which shows the capability of the developed approach.

VSHARP: Variable Splitting Half-quadratic ADMM algorithm for reconstruction of inverse-problems

Medical Imaging (MI) tasks, such as accelerated parallel Magnetic Resonance Imaging (MRI), often involve reconstructing an image from noisy or incomplete measurements. This amounts to solving ill-posed inverse problems, where a satisfactory closed-form analytical solution is not available. Traditional methods such as Compressed Sensing (CS) in MRI reconstruction can be time-consuming or prone to obtaining low-fidelity images. Recently, a plethora of Deep Learning (DL) approaches have demonstrated superior performance in inverse-problem solving, surpassing conventional methods. In this study, we propose vSHARP (variable Splitting Half-quadratic ADMM algorithm for Reconstruction of inverse Problems), a novel DL-based method for solving ill-posed inverse problems arising in MI. vSHARP utilizes the Half-Quadratic Variable Splitting method and employs the Alternating Direction Method of Multipliers (ADMM) to unroll the optimization process. For data consistency, vSHARP unrolls a differentiable gradient descent process in the image domain, while a DL-based denoiser, such as a U-Net architecture, is applied to enhance image quality. vSHARP also employs a dilated-convolution DL-based model to predict the Lagrange multipliers for the ADMM initialization. We evaluate vSHARP on tasks of accelerated parallel MRI Reconstruction using two distinct datasets and on accelerated parallel dynamic MRI Reconstruction using another dataset. Our comparative analysis with state-of-the-art methods demonstrates the superior performance of vSHARP in these applications.

Mathematical theory and CFD solutions for internal convective cooling of gas turbine blades highlighting the effects of rotation

A quasi-one-dimensional mathematical theory is developed for determining the effects of rotation on the fluid flow and heat transfer in a cooling channel within a rotating gas turbine blade. The three-dimensional intricacies in the solutions of the same problem are captured through accurate computational fluid dynamic simulations. The derived analytical closed-form solutions predict the amount by which the relative total temperature of the coolant changes as a result of rotation. It is shown that such changes due to rotation are significant proportion of the change of coolant temperature due to heat transfer. The computed spatial evolutions of the primary velocity field, secondary velocity field and the force fields are presented as the coolant progresses from the inlet to the outlet so that the complexities and subtleties are exposed to a high degree of visualizability. Detailed considerations to various components of the centrifugal, Coriolis and pressure forces are given. The flow field is shown to depend strongly on whether the coolant moves in the spanwise outward or inward direction. The three-dimensionality of the flow field is reflected in complex circumferential variations of the blade temperature and the internal heat transfer coefficient. When the effects of the x–component of the Coriolis force interacts non-linearly with the effects of the z–component of the centrifugal force and a drastic x-gradient in fluid density due to a thin thermal boundary layer, then the primary velocity profile may exhibit great complexity, including flow separation. The present paper thus captures the strong, non-linear, two-way coupling between the fluid dynamics and heat transfer in the internal convective cooling occurring in a channel within a rotating gas turbine blade. Based on about 100 three-dimensional computational fluid dynamics simulations, new heat transfer correlations are developed for average Nusselt number for a rotating channel that will be of practical use.

Analytical, semi-analytical and numerical solutions for global stability analysis in buildings: Double-Beam Systems Timoshenko

In this article, two analytical solutions, one semi-analytical solution, and one numerical solution are proposed to calculate the global critical buckling load of buildings employing a continuum model parallel coupling two Timoshenko beams. The parallel coupling of two Timoshenko beams represents the four types of building behavior: global bending, global shear, local bending, and local shear, the latter of which has been widely ignored in the literature. The model is associated with one translational kinematic field and two rotational fields. The equilibrium equations, constitutive laws, and boundary conditions are derived using a variational approach by applying Hamilton’s principle to the relevant Lagrangian function. Specifically, to solve the classical problem of buildings with uniform properties along their height, the first analytical solution proposes exact closed-form equations for a vertical load applied at the top and a closed-form analytical solution based on the decomposition of two subsystems for a polynomial vertical load uniformly applied along the height of the beam. Meanwhile, the semi-analytical solution exactly solves the differential equation associated with a polynomial vertical load profile and is based on coefficient iteration, converging with only two or three iterations. To address the challenge of buildings with variable properties along their height, the combined application of the continuous method and the transfer matrix method allows us to propose an efficient numerical method leading to a 6 × 6 transfer matrix, constant regardless of the number of discretizations and derived analytically, which results in a computational advantage by drastically reducing computational costs. The numerical results are encouraging and show good agreement with the results of the finite element method, suggesting that the proposed method can be used for engineering purposes in both the preliminary and final phases of the structural analysis of tall buildings.

Quantum corrections to tunnelling amplitudes of neutral scalar fields

Though theoretical treatments of quantum tunnelling within single-particle quantum mechanics are well-established, at present, there is no quantum field-theoretic description (QFT) of tunnelling. Due to the single-particle nature of quantum mechanics, many-particle effects arising from quantum field theory are not accounted for. Such many-particle effects, including pair-production, have proved to be essential in resolving the Klein-paradox. This work seeks to determine how quantum corrections affect the tunnelling probability through an external field. We investigate a massive neutral scalar field, which interacts with an external field in accordance with relativistic quantum mechanics. To consider QFT corrections, we include another massive quantised neutral scalar field coupling to the original via a cubic interaction. This study formulates an all-order recursive expression for the loop-corrected scalar propagator, which contains only the class of vertex-corrected Feynman diagrams. This equation applies for general external potentials. Though there is no closed-form analytic solution, we also demonstrate how to approximate the QFT corrections if a perturbative coupling to the quantised field is assumed.

The fractional solitary wave profiles and dynamical insights with chaos analysis and sensitivity demonstration

This work investigates the dynamics of solitary waves in a thin-film ferroelectric material model, an essential component of optical systems. The study recognizes the limitations of the inverse scattering transform in solving the Cauchy problem for this conformable thin-film ferroelectric material model and applies a new Kudryashov approach and Bernoulli’s equation technique to ordinary differential equations. We apply these methods to the present model and obtain closed form analytical solutions. A variety of traveling wave and soliton solutions are constructed, including kink, bell, bright, and dark soliton solutions. It is acknowledged that the nonlinear evolution equation can be transformed into an ordinary differential equation by applying traveling wave transformations. Suitable physical parameters are chosen to create 2D, 3D, and contour plots that graphically depict the graphical dispersion of obtained fractional soliton solutions. The impact of the conformable fractional parameter is further demonstrated by the graphical representation of solitons propagation. Furthermore, the study develops the Hamiltonian function of the dynamical system to discuss the system’s total energy in terms of momentum and position. While a chaos analysis describes chaotic, quasi-periodic, and periodic behaviors, a sensitivity analysis demonstrates how responsive the model is to various initial conditions. Due to their dual-purpose nature as nonlinear ferroelectric and dielectric materials, thin ferroelectric films are widely used in modern electrical devices.

Solving inverse boundary value problem of Poisson equation by LS-SVM

Abstract In this paper, a new method based on least squares support vector machines (LS-SVM) is presented for solving the inverse boundary value problem of Poisson equation. The closed form analytical solution is obtained by optimizing the regression parameters. The core problem is to transform the parametric regression problem into a quadratic programming problem. To demonstrate the efficiency of the proposed algorithm, numerical experiments are conducted. The proposed method is found to be feasible for the inverse boundary value problem of Poisson equation.

Application of a Closed-Form Analytical Solution to Model Overland Flow and Sediment Transport Using Rainfall Simulator Data

Application of a Closed-Form Analytical Solution to Model Overland Flow and Sediment Transport Using Rainfall Simulator Data

Closed form analytical solution and scan pattern shaping theory of three-element Risley prism for LiDAR systems

Three-element Risley prism are emerging technologies in LiDAR. It can convert the FoV into a rectangular shape with high coverage rates in specific conditions, making it suitable for applications in autonomous driving scenarios. The addition of a third prism has been proven effective in eliminating blind spots and singularity problems. One challenge facing the use of three-element scanners in autonomous driving is the accurate positioning control of light beams during high-speed motion, especially in specific scan patterns such as near rectangles. This work introduces a closed-form analytical solution and parameters for the scan patterns of the three-element scanner through a non-paraxial tracing method. The closed-form solution includes two groups of coefficients representing all possible configurations of the three-element scanner. The parameters consist of ten degrees of freedom: angular velocity ratio, M1, M2, deflection angle ratio, k1, k1, initial phases, θ0q(q=1,2,3), distance between the two prisms dair1, dair2, and distance from the screen, P. To our knowledge, this is the first study to discuss these degrees of freedom in the context of the closed-form solution. We obtain its scan patterns by MATLAB 2019a and explore a configuration of three-element Risley prism in near-rectangular scan patterns, achieving a wide FoV and high point cloud density while avoiding blind spots and singularity issues. To validate the simulation results, a three-element Risley prism scanner system based on brushless motor is developed. Our study suggests that the three-element rotating prism scanner holds potential superiority in forward-looking LiDAR solutions.

Asymptotic solution of electromagnetic heating of skin tissue with lateral heat conduction

We study the temperature evolution in the three-dimensional skin tissue exposed to an electromagnetic beam of millimeter wavelength. The skin absorption coefficient of the beam frequency determines how deep the electromagnetic energy penetrates into the skin tissue, which gives a sub-millimeter penetration depth for a 94 GHz wave. In contrast, in the lateral directions perpendicular to the depth, the beam size is usually much larger than the penetration depth. Based on this separation of length scales, we establish an asymptotic formulation in which each term has separable dependences on the depth coordinate and on the lateral coordinates. We solve it analytically to obtain a two-term asymptotic solution of the temperature distribution in the three-dimensional skin tissue. This closed-form analytical solution provides a practical and accurate way of predicting the temperature. When the beam size is moderately larger than the penetration depth (a ratio of 20), the effect of lateral heat conduction is well captured in the asymptotic solution with maximum error less than 0.0017 in the normalized temperature of magnitude well above 1.

Dynamic Analysis of a Uniform Microbeam Resting on a Nonlinear Foundation Considering Its Curvature Subjected to a Mechanical Impact and Electromagnetic Actuation.

This study proposes an investigation into the nonlinear vibration of a simply supported, flexible, uniform microbeam associated with its curvature considering the mechanical impact, the electromagnetic actuation, the nonlinear Winkler-Pasternak foundation, and the longitudinal magnetic field. The governing differential equations and the boundary conditions are modeled within the framework of a Euler-Bernoulli beam considering an element of the length of the beam at rest and using the second-order approximation of the deflected beam and the Galerkin-Bubnov procedure. In this work, we present a novel characterization of the microbeam and a novel method to solve the nonlinear vibration of the microactuator. The resulting equation of this complex problem is studied using the Optimal Homotopy Asymptotic Method, employing some auxiliary functions derived from the terms that appear in the equation of motion. An explicit closed-form analytical solution is proposed, proving that our procedure is a powerful tool for solving a nonlinear problem without the presence of small or large parameters. The presence of some convergence-control parameters assures the rapid convergence of the solutions. These parameters are evaluated using some rigorous mathematical procedures. The present approach is very accurate and easy to implement, even for complicated nonlinear problems. The local stability near the primary resonance is studied.

Hybrid technique for multi-dimensional fractional diffusion problems involving Caputo–Fabrizio derivative

In this study, a precise and analytical method, namely Shehu transform decomposition method (STDM), is applied to examine multi-dimensional fractional diffusion equations, which will describe density dynamics in a material undergoing diffusion. The fractional derivative is taken into account by the Caputo–Fabrizio (CF) derivative. The proposed approach combines the Shehu transformation (ST) with the Adomian decomposition method (ADM), employing Adomain polynomials to handle nonlinear terms. Fixed point theorem has been used to prove the uniqueness and existence of the solution of governing differential equation. To validate the accuracy of proposed method, we present the analytical solutions of three applications of multi-dimensional fractional diffusion equations. Furthermore, we compute the graphical and numerical results using MATLAB to demonstrate the close-form analytical solution in the comparison of the exact solution. The obtained findings are promising and suitable for the solution of multi-dimensional diffusion problems with time-fractional derivatives. The main advantage is that our developed scheme does not require assumptions or restrictions on variables that ruin the actual problem. This scheme plays a significant role in finding the solution and overcoming the restriction of variables that may cause difficulty in modeling the problem.

Nonlocal strain gradient-based nonlinear vibration analysis of nonlinear FG three-phase HJCNT/MWCNT/epoxy composite microbeam resonators using a modified micromechanical model

This paper aims to study nonlinear dynamic behavior of functionally graded (FG) three-phase composite microbeam resonators made of an epoxy matrix and two reinforcements namely multi-walled carbon nanotubes (MWCNTs) and hetero-junction carbon nanotubes (HJCNTs). The effective mechanical properties of the composite microbeam are obtained using the modified Halpin-Tsai micromechanical model. The microbeam surrounding medium is simulated using a two-parameter elastic foundation. The von-Karman’s geometric nonlinearity relations are incorporated and the equations of motion are derived based on the nonlocal strain gradient Euler–Bernoulli beam model. A new closed-form analytical solution is obtained using the homotopy perturbation method. The effects of vibration amplitude, nanofiber volume fraction, nanofiber distribution pattern, small-scale parameters and the foundation parameters on the nonlinear frequency and deflection of the FG three-phase composite microbeams are studied in detail. The findings of the paper are valuable for researchers in the field of microbeam resonators.

A Constitutive Framework for Modeling of the Visco-Hyperelastic Materials: Application to the Mechanical Behavior of Cylinders Made of the Rate-Dependent Elastomers

In this paper, the formulation of the mechanical behavior of visco-hyperelastic materials is developed in a thermo-dynamically compatible framework. In this viewpoint, the stress power is assumed to be equal to the sum of the reversible (elastic) energy rate stored and the irreversible energy rate dissipated. Based on this assumption and using the second law of thermodynamics, the constitutive modeling framework is obtained for the rate-dependent materials. Inspired by the constitutive law of a Newtonian fluid and also, models of density strain energy of an elastic material, a framework for the dissipative energy rate related to the irreversible part is proposed for viscoelastic materials. In this framework, the reversible energy part is considered a function of the right Cauchy–Green deformation tensor, and the irreversible energy part is considered a function of the same tensor’s rate. To evaluate the proposed model for nonlinear viscoelastic behavior modeling, the results of tests performed on non-compressible materials at different rates are used, and the proposed model provided acceptable results. Finally, the proposed model is implemented to find a closed-form analytical solution for mechanical behavior modeling of the thick-walled cylindrical shells made of visco-hyperelastic materials, also, stress and stability analyses are assessed for these types of cylinders. To achieve this goal, the effect of pre-stretch on the stability of cylinders has been substantially investigated. Furthermore, based on the proposed model, the effect of the parameters such as the wall thickness and behavior of the material used in the cylinder are examined on the stability of open-ended and closed-ended cylinders.

Propagation attenuation of elastic waves in multi-row infinitely periodic pile barriers: A closed-form analytical solution

Periodically arranged media exhibit attenuation zones for elastic waves, and the related periodic structures play a crucial role in the design of seismic isolation and vibration mitigation. From the perspective of wave theory, this paper presents a closed-form analytical solution for the propagation attenuation of elastic waves through infinitely periodic, multi-row pile barriers. The innovative approach begins by deriving the extended Graf’s addition theorem to transform wave potentials across arbitrary coordinate systems, overcoming the constraints of traditional formulas that limit pile centers to linear configurations. Subsequently, it completes the wave function expansion by utilizing the phase differences in the frequency domain of wave fields around different periodic elements, thus skillfully integrating the effects of incident and scattered wave fields. For the first time, this study delivers exact expressions for the scattering of various types of plane waves (SH, SV, and P) by multi-row pile barriers with an infinite periodic array, addressing the complexities of large pile numbers in previous theoretical studies. The proposed solution increases computational efficiency by tens of times and markedly reduces the majority of resource usage, improving the analysis of complex vibration isolation systems involving numerous piles. Building on the accuracy verification and convergence analysis, several numerical examples are conducted to explore the effects of pile row, pile spacing, and pile arrangement on the propagation attenuation. The findings elucidate the mechanisms driving vibration reduction design using periodic wave barrier, furthering its application in engineering structures.

Green Road is open: Economic Pathway with a carbon price escalator

The paper develops the concept of “Economic Pathways” (EPs), which characterizes theory-based scenarios for an economy that strives for decarbonization by the middle of the century. The theoretical framework derives closed-form analytical solutions for consumption, innovation, emissions, and population. The EPs differ in the stringency of assumed policies and associated income and emission development. Unlike the well-known “Shared Socioeconomic Pathways”, they allow the inclusion of important causalities between the economy and the environment and thereby considerably narrow down the scope of likely future developments. The quantitative part serves to illustrate the long-term consequences of climate policy. I show that deep decarbonization only moderately delays economic development, but requires increasing escalation of the carbon price. The paper argues that the adoption of more stringent climate policies becomes more likely as the phase-out of fossil fuels increases. The “Green Road” is not only feasible but also attractive and realistic.

A fresh perspective on solution to fragmentation problems

The population balances offer a continuum framework to model fragmentation of discrete size drops, bubbles, particles, crystals, polymer chains, etc. We have examined the available approaches to solve the resulting population balances and the older discrete kinetic equations. The analytical solutions have enhanced our understanding of breakup processes. We offer in this work new analytical solutions using a fresh and counter-intuitive approach. The approximate numerical solutions approach analytical solutions as the fineness of discretization increases. We looked for scaling patterns in coarse numerical solutions and harnessed the observed scaling to obtain closed-form analytical solutions in the limit of infinitely fine discretization. The approach provides analytical solutions for a family of breakup functions, including evolving populations of infinite number of particles. The success of the new approach is contingent on the discretization step having internal consistency concerning particle number and mass. The approach holds promise for empirical breakage functions.

An analytical study on the influence of magnetic field-dependent viscosity on viscous and ohmic dissipative second-grade fluid boundary layers

This article aims to investigate the magnetohydrodynamic (MHD) boundary layer flow of a second-grade non-Newtonian fluid over a horizontally stretching sheet, considering magnetic field-dependent (MFD) viscosity, as well as viscous and Ohmic dissipations. Analytical solutions for previously unexplored momentum and heat transfer equations involving MFD viscosity are derived. Two boundary conditions, namely Prescribed Surface Temperature (PST) and Prescribed Heat Flux (PHF), are taken into account. Governing dimensional partial differential equations (PDEs) are transformed into non-dimensional ordinary differential equations (ODEs) using similarity transformations. Closed-form analytical solutions for flow are derived, considering stretching velocity, MFD viscosity, second-grade fluid properties, and suction impacts. Heat transfer equations are transformed into Gauss-hypergeometric form, yielding solutions in terms of confluent hypergeometric functions. Analytical expressions for skin friction, local Nusselt number, and non-dimensional wall temperature are derived. A unique solution is obtained for the flow equation. It is found that both second-grade and magnetic viscosity parameters expand the momentum boundary layer while the magnetic parameter reduces thickness. Thermal boundary layer thickness increases with a higher second-grade parameter, magnetic viscosity, and Eckert number. Moreover, analytical solutions are validated against published results for a special case.

Closed-form asymptotic solution for the transport of chlorine concentration in composite pipes

Recently, some researchers have revisited the analysis of chlorine transportation in cylindrical pipes by deploying a coupling between the Laplace transform method and the complex analysis’ residue approach for inverting complex integrals. This method yielded interesting results after the incorporation of root-finding numerical schemes. Thus, away from incorporating numerical tools, the present study makes consideration of the same formulation of chlorine transport in a single-layered pipe and further extends it to the case of a bi-layered pipe using the hybrid of the Laplace transform method and the asymptotic approximations method. The need for asymptotic approximations for the modified Bessel functions, which arise in the reduced ordinary differential equations, necessitates the quest for closed-form analytical solutions, which are largely considered benchmark solutions for numerical investigations. Moreover, the obtained closed-form asymptotic solutions have been examined graphically; where it was observed that both the radial diffusion coefficient η and the spatial radial variable are contributory in the transport of chorine concentration in the media.