Accurate Closed-form Solution Research Articles

Accurate Closed-Form Solutions for the Free Vibration and Supersonic Flutter of Laminated Circular Cylindrical Shells

According to the Donnell–Mushtari shell theory, this work presents a closed-form solution procedure for free vibration of open laminated circular cylindrical shells with arbitrary homogeneous boundary conditions (BCs). The governing differential equations of free vibration are derived from the Rayleigh quotient and solved by the iterative separation-of-variable (iSOV) method. In addition, considering axial aerodynamic pressure, simulated by the linear piston theory, the exact eigensolutions for the flutter of open laminated cylindrical shells with simply supported circumferential edges and closed laminated cylindrical shells are also achieved. The governing differential equations of cylindrical shell flutter are derived from the Hamilton variational principle and solved by the separation-of-variable (SOV) method. The influence of circumferential dimension on flutter speed is investigated for open cylindrical shells, which reveals that the number of circumferential waves in critical flutter mode increases with circumferential length, and there exists an infimum for flutter speed that is an invariant independent of circumferential length. The present results agree well with those obtained by the Galerkin method, the finite element method, and other analytical methods.

Buckling and free vibration of grid-stiffened composite conical panels using Extended Kantorovich Method

This paper develops very accurate closed-form solutions for buckling and free vibration of grid-stiffened composite conical panels using a semi-analytical method. On the base of first-order shear deformation theory (FSDT), the governing formulations are derived while an appropriate unit cell is used to derive the material properties of the lattice from their bulk properties. Multiple systems of ordinary differential equations (ODEs) are obtained by applying the Extended Kantorovich Method (EKM) to solve the governing system of partial differential equations (PDEs). The resulting ODEs are then solved using semi-analytical closed-form solutions to study buckling and free vibration of grid-stiffened panels. It is interesting to note that due to the general formulation of the conical panel, it is very easy to obtain other useful geometries including cylindrical panels, and even rectangular plates with general unit cell configurations. Regarding stability, fast convergence, and very low computing cost, the effectiveness of the proposed technique is investigated. The accuracy of the critical buckling loads and natural frequencies for various cases is studied which shows very good agreement in comparison with results obtained from the commercial finite element code ANSYS.

A simple and accurate closed-form analytical solution to the Boussinesq equation for horizontal flow

The classical problem used to model the response of an unconfined aquifer to a sudden change in boundary head is considered in this paper. This problem is usually modeled using the nonlinear groundwater Boussinesq equation. Due to the nonlinearity of this problem, no exact solutions exist. Solutions to the Boussinesq equation are therefore obtained using numerical techniques or approximate analytical methods. In this paper, we present a novel closed-form approximate analytical solution to this problem. The Boussinesq equation is converted to a first-order ordinary differential equation (ODE) by means of the Boltzmann transformation and by introducing a new variable related to the water flow. The first-order ODE is then solved analytically after introducing an intermediate approximation involving two fitting parameters. To avoid any numerical treatment, closed-form polynomial expressions of the fitting parameters are proposed. The final-form solution is simple to use and is obtained in terms of the incomplete gamma function, which is valid for both recharge and discharge. The derived solution is tested and compared to efficient numerical solutions, as well as to two types of analytical solutions: an accurate series expansion solution and an equivalent closed-form solution. The results show excellent agreement between the proposed solution and the numerical and series solutions. The proposed solution offers a key advantage in terms of both accuracy and simplicity; notably, it can be implemented using a simple spreadsheet.

Travelling wave solutions and modulation instability analysis of the nonlinear Manakov-system

In this paper, the accurate closed-form solutions of the Manakov-system are extracted via the extended ( G ′ G 2 ) -expansion method, the exp ⁡ ( − ϕ ( Ω ) ) -expansion method and generalized Kudryashov method. The solutions are obtained in the form of trigonometric, hyperbolic and rational functions. The dynamical behaviour of obtained solutions is discussed by plotting 3D graphs, 2D contour graphs and density graphs. The symbolic softwares such as Maple and Mathematica are used to plot the graphs of retrieved solutions. The modulation instability (MI) of Manakov-system is also discussed in this paper.

Complete Closed-Form and Accurate Solution to Pose Estimation From 3D Correspondences

Computing the pose from 3D data acquired in two different frames is of high importance for several robotic tasks like odometry, SLAM and place recognition. The pose is generally obtained by solving a least-squares problem given points-to-points, points-to-planes or points to lines correspondences. The non-linear least-squares problem can be solved by iterative optimization or, more efficiently, in closed-form by using solvers of polynomial systems. In this letter, a complete and accurate closed-form solution for a weighted least-squares problem is proposed. Adding weights for each correspondence allow to increase robustness to outliers. Contrary to existing methods, the proposed approach is complete since it is able to solve the problem in any non-degenerate case and it is accurate since it is guaranteed to find the global optimal estimate of the weighted least-squares problem. Simulations and experiments on real data demonstrate the superior accuracy and robustness of the proposed algorithm compared to previous approaches.

Analytical, numerical and experimental study of the finite inflation of circular membranes

In the present work we derive an analytical expression for the pressure–deflection curve of circular membranes subjected to inflation. This problem has been studied mostly from a numerical point of view and there is still a lack of accurate closed-form solutions in nonlinear elasticity. The analytical formulation is developed with a semi-inverse method by setting a priori the kinematics of deformation of the membrane. A compressible Mooney–Rivlin material model is considered and a pressure–deflection relation is derived from the equilibrium. The kinematics is approximated and therefore the obtained solution is not exact. Consequently, the formulation is adjusted by introducing an additional polynomial function in the pressure–deflection equation. The polynomial is calibrated by fitting numerical solutions of the exact system of differential equilibrium equations. The calibration is done over a wide range of constitutive parameters that covers the response of all rubber materials for technological applications. As a result, a definitive and accurate expression of the applied pressure as a function of the deflection of the membrane is obtained. The formula is validated with finite element (FE) simulations and compared with other solutions available in the literature. The comparison shows that the present model is more accurate. In addition, unlike the other models, it can be applied to compressible materials. Experimental uniaxial and bulge tests are carried out on rubber materials and the model proposed is used to characterize the Mooney–Rivlin constitutive parameters. Since the pressure–deflection formula is accurate and easy-to-use, it is an innovative tool in engineering applications of inflated membranes.

Closed Form Option Pricing for Stocks with Fixed Dividends

This article discusses a simple, fast and accurate closed-form solution for European options and forwards on a stock with discrete, fixed-amount dividends, and robust w.r.t. a wide range of inputs. While many various precise expansion approaches exist, we are looking for a simple, easy to implement and fast method that works across a wide range of inputs. The option with dividends can be transformed into an average option that can be solved using well-known methods for average options, such as Curran. We find that this is the best compromise between accuracy, simplicity and speed for a wide range of practical applications.

Effect of Higher Transition Foundation on Dynamic Stability Boundaries of Simply Supported Beam

Accurate closed form solution to predict the dynamic stability behavior of simply supported (SS) beam on elastic foundation, subjected to an end axial periodic load, has been derived using simple single term trigonometric function which satisfies the geometric boundary conditions. The effect of higher transition foundation on dynamic stability regions is studied. The energy method is used in the present work to evaluate the dynamic stability boundaries. Euler - Bernoulli beam theory is used for the analysis. Numerical results are presented using proper non-dimensional parameters with varying foundation parameters below and above the second transition foundation value of elastic foundation, in both the digital and analogue forms for better comprehension of the dynamic stability behavior of the beams. The effect of the higher transition foundation value of the elastic foundation on the dynamic stability behavior is brought out in the present work.

Development and validation of a semi-analytical framework for droplet freezing with heterogeneous nucleation and non-linear interface kinetics

We present a novel, semi-analytical framework which accurately predicts every stage of droplet freezing. Supercooling degree of the freezing droplet is computed by coupling the 1-D transient heat conduction equation in spherical coordinates with a modified heterogeneous nucleation model. The coupling connects atomic-scale thermodynamic properties with bulk thermophysical properties and yields interesting insights into the freezing process. The study also develops a novel dendritic growth model for crystal propagation that takes into account the effects of non-linear interface kinetics and surface curvature. This non-equilibrium phase-change formulation during the recalescence stage is coupled with quasi-steady phase change approximation using perturbation series for equilibrium freezing stage to yield an accurate closed form solution of freezing dynamics. The temperature curves, nucleation rates, dendritic growth velocities and mass fraction evolution from the model are validated or verified with the available experimental or numerical data in the literature. The analytical framework developed is a reduced-order freezing model that is fast to compute and yields accurate results. The framework presented can be useful as a subgrid model for high-fidelity crystal growth and spray freezing simulations.

Accurate closed-form solution of the SIR epidemic model

An accurate closed-form solution is obtained to the SIR Epidemic Model through the use of Asymptotic Approximants (Barlow et al., 2017). The solution is created by analytically continuing the divergent power series solution such that it matches the long-time asymptotic behavior of the epidemic model. The utility of the analytical form is demonstrated through its application to the COVID-19 pandemic.

Accurate Closed-Form Solution for Moving Underwater Vehicle Localization Using Two-Way Travel Time

Position information is essential for an underwater vehicle that can affect the performance of many other applications. Vehicle motion during the two-way travel time (TWTT) using an acoustic positioning system can affect the localization accuracy of the estimators based on the traditional static model. A new time measurement model for moving vehicle localization is presented, which compensates for the vehicle motion. The Cramér–Rao lower bounds (CRLBs) of the proposed model are derived for two different cases, where the depth of the underwater vehicle is unknown or known. Then, closed-form solutions for the two cases using the proposed model are derived and the solutions are shown analytically to reach the CRLBs. Simulations collaborate the theoretical performance of the proposed estimators and the moving model significantly improves the localization accuracy in comparison with the static model.

Highly accurate closed-form solutions for the free in-plane vibration of rectangular plates with arbitrary homogeneous boundary conditions

In this paper, highly accurate closed-form solutions for the free in-plane vibration of rectangular isotropic plates with arbitrary homogeneous boundary conditions are achieved by an iterative separation-of-variable method, and all solutions are presented in an elegant explicit form. For plates with at least a pair of opposite edges simply supported, the present solutions are the exact solutions. For other homogeneous boundary conditions, the accurate closed-form solutions agree well with the results of the fine finite element method. This work shows that the iterative separation-of-variables method is a powerful method to find the closed-form eigensolutions of partial differential equations. The present solutions can be taken as the benchmarks for the validation of numerical methods, a guide in parametric design of structure and the basis of constructing new numerical methods.

High-Speed Distance Relaying of the Entire Length of Transmission Lines Without Signaling

Short-circuit faults close to either end of a transmission line, are normally cleared instantaneously by the distance relay at that end and after hundreds of milliseconds, i . e ., in Zone 2 operating time, by the relay at the opposite end of the line. This sequential tripping can be accelerated on condition that a reliable communication link is available for signaling between the two line ends. This paper proposes a novel non-communication method providing high-speed distance relaying over the entire length of the protected transmission line. The inputs to the method are the protected line parameters and local voltage and current signals measured by the relay, similar to those to conventional distance relays. The proposed method accomplishes Accelerated Sequential Tripping (AST) within a couple of cycles after the opening of the remote-end circuit breaker (ORCB) of the line. To achieve this, an accurate closed-form solution is derived for the fault distance in terms of post-ORCB voltage and current phasors. For the detection of the ORCB instant, a set of proper indices are proposed. This is to verify the fault distance calculated by the relay, before issuing a trip command. The proposed method is successfully validated by conducting more than 20000 hardware-in-the-loop (HIL) tests, and also using real-life data.

Some modifications on RCAM for getting accurate closed-form approximate solutions of Duffing- and Lienard-type equations

In this work, authors propose some modifications Adomian decomposition method to get some accurate closed form approximate or exact solutions of Duffing- and Li´enard-type nonlinear ordinary differential equations.Results obtained by the revised scheme have been exploited subsequently to derive constraints among parameters to get the solutions to be bounded. The present scheme appears to be efficient and may be regarded as the confluence of apparently different methods for getting exact solutions for a variety of nonlinear ordinary differential equations appearing as mathematical models in several physical processes.

Highly accurate closed-form solutions for free vibration and eigenbuckling of rectangular nanoplates

This work presents the highly accurate closed-form solutions for free vibration and eigenbuckling of isotropic rectangular nanoplates with arbitrary homogeneous boundary conditions based on Eringen’s nonlocal theory and classical thin plate theory.The iterative separation-of-variable (iSOV) method based on the Rayleigh quotient, which is the most accurate closed-form solution method among all separation-of-variable methods, is used to obtain the highly accurate solutions, including the exact well-known Navier and Levy types of solutions. The highly accurate closed-form solutions for free vibration of rectangular nanoplates and for eigenbuckling of different scale rectangular plates with arbitrary homogeneous boundary conditions are achieved for the first time, and all solutions are presented in excellent explicit forms.The present solutions coincide well with analytical and numerical solutions in literature, verifying the accuracy of the present method. The influences of nonlocal parameters, boundary conditions and lengths of nanoplates on frequencies and critical buckling loads are studied, and nonlocal effects are explained in physical sense. The present solutions can be taken as the benchmarks for the validation of numerical methods, a guide in parametric design of structure, and the basis of constructing new numerical methods.

In-plane non-linear elastic stability of parabolic arches with different rise-to-span ratios

Investigations of in-plane nonlinear elastic buckling and postbuckling of parabolic shallow arches have been reported in some literature, where a basic assumption that curve arc differential being equaled to horizontal line differential was used to facilitate the derivation. However, this assumption can be accepted only when the rise-to-span ratio of an arch is quite small, and when the rise-to-span ratio of an arch is greater than 1/12.5, the analytical predictions for buckling and postbuckling responses of the arch based on the assumption have significant deviations from the corresponding finite element results. It is known that parabolic arches having a rise-to-span ratio from 1/8–1/4 are often used in the engineering practice, especially in the arch bridges. This paper aims to improve accuracy of nonlinear analyses for parabolic arches with different rise-to-span ratios. A more precise strain expression that includes an approximate curve arc differential item in the Cartesian coordinate system is derived in this paper. Based on the strain expressions and principle of virtual work, nonlinear differential equilibrium equations are derived and more accurate analytical solutions for the buckling and postbuckling behavior are derived. Parabolic arches with the rise-to-span ratio ranging from 1/12.5–1/3 are taken as examples to investigate the effectiveness of the proposed method for prediction of the buckling and postbuckling behavior by comparisons with the predictions of a finite element method. The comparisons show that the solutions are in good agreements with the finite element results and that the new strain expression can lead to more accurate closed-form analytical solutions for the buckling and postbuckling behavior of parabolic arches with different rise-to-span ratios. It is found that the rise-to-span ratio has a significant influence on the in-plane nonlinear buckling behavior of parabolic arches.

Queueing Analysis of Unicast IPTV With Adaptive Modulation and Coding in Wireless Cellular Networks

Unicast IPTV services that can support live TV, video-on-demand (VoD), video conferencing, and online gaming applications over broadband wireless cellular networks have been becoming popular in recent years. However, video streaming services significantly impact the performance of wireless cellular networks because they are bandwidth hogs. To maintain the system performance, effective admission control and resource allocation mechanisms based on an accurate mathematical analysis are required. On the other hand, the quality of a wireless link usually changes with time due to the user mobility or time-varying channel characteristics. To counteract such time-varying channels and improve the spectral efficiency, adaptive modulation and coding (AMC) scheme can be adopted in offering unicast IPTV services for mobile users. In this paper, closed-form solutions for the bandwidth usage, blocking rate, and dropping rate of unicast IPTV services over wireless cellular networks were derived based on the novel queueing model that considers both user mobility and AMC. Simulations were also conducted to validate the accuracy of analytical results. Numerical results demonstrate that the presented analytical results are accurate. Based on the accurate closed-form solutions, network providers can implement precise admission control and resource allocation for their networks to enhance the system performance.

Analysis of propagation delay and repeater insertion in single-walled carbon nanotube bundle interconnects

A new closed-form expression of 50% propagation delay for distributed RLC interconnects is proposed using the multivariable curve fitting method, with a maximum error of 4% with respect to SPICE results. Then accurate closed-form solutions for the optimum repeater number and size to minimize the propagation delay are further derived. The performance of single-walled carbon nanotube (SWCNT) bundle interconnects is evaluated using the proposed models in the intermediate and global levels at the 22- and 32-nm technology nodes, and compared against traditional Cu interconnects. It is shown that the performance of SWCNT bundle interconnects in propagation delay can outperform Cu interconnects, and the improvement will be enhanced with technology scaling and wire length increasing. On the other hand, the propagation delay of SWCNT bundle interconnects is super-linearly dependent on the wire length similar to Cu interconnects, indicating that the method of repeater insertion to reduce the propagation delay can also apply to SWCNT bundle interconnects. The results shown that repeater insertion can really reduce the propagation delay of SWCNT bundle interconnects effectively, and the optimum repeater number is much smaller than that of Cu interconnects.

A short remark on the integrability of a nonlinear reaction–diffusion equation arising in mathematical biology: Compatibility analysis

An analytical approach based on the compatibility concept is employed to solve a nonlinear reaction–diffusion model arising in mathematical biology. The solution process makes it extremely easy to obtain a relatively accurate closed-form solution of the model. The pencil-and-paper solution procedure can be extended to other class of nonlinear problems of similar kind.

Design analysis of sandwiched structures experiencing differential thermal expansion and differential free-edge stretching

Compared to the theory of elasticity solutions the strength of material solutions offer closed form solutions that is favoured by practising engineers for performing design analysis. However, the existing strength of material solutions for a sandwich structure experiencing differential thermal strains have principally ignored the free edge conditions; and for the very limited publications that have enforced the free edge conditions, the solutions have been inaccurate. Understandably, design analysis using such solutions is unreliable. This manuscript describe a solution technique that enforces the nil shear stress condition at the free edge using a high power exponential function resulting in a simple yet accurate closed form solutions for the interfacial shear stress. The interfacial peeling stress is made up of two components: the mean and the amplitude of variations of the transverse normal stress in the bonding layer; the latter is the dominant component whose magnitude is linearly proportional to the gradient of the interfacial shear stress. Following validation by finite element analysis, design analysis for debonding, fracturing, and out-of-plane deformation are performed using the concise closed form solutions.