Semi-analytical Method Research Articles

A numerical assembly technique for free vibration of plane frames using a shifted and deflated Newton method

A semi-analytic method, namely the Numerical Assembly Technique (NAT) is extended to compute natural frequencies of arbitrary planar frame structures. The frame structure is systematically defined by a set of nodes, beams, bearings, springs, and external loads and the corresponding boundary and interface conditions are formulated. We consider the Timoshenko-Ehrenfest beam theory and take axial, bending and shear deformations as well as linear and rotational inertia into account. Since we use the exact solutions of the governing differential equations in the NAT, the accuracy of the results is independent from the spatial discretization. However, the resulting eigenvalue problem is non-linear and requires a numerical algorithm to solve. We consider three different representations of exact homogeneous solutions and study the numerical stability of the resulting system matrix. To solve for the eigenvalues we propose a Newton method enhanced with a shifted deflation algorithm. The stability and the efficiency of this approach is illustrated in numerical examples.

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.

Semi-analytic modeling and experimental verification of arbitrary aero-engine complex spatial pipeline

In-depth research is imperative for complex spatial pipelines with fluid-structure interaction. A comprehensive and universal three-dimensional semi-analytical modeling method is first established for complex fluid-conveying spatial pipelines under the base excitation with arbitrary connections and configurations under different boundary conditions. The virtual spring technology is used to connect the straight and curved pipeline segments with spatial angles and simulate boundary conditions. For a thorough analysis, the actual stiffness of the bracket-clamp system is experimentally measured for the first time. The multi-level coupling dynamic model of part, component, and system levels under actual boundary conditions is established. An efficient method for solving the fluid-structure coupling pipeline system dynamics considering the base excitation is proposed based on energy method. The mid-plane displacement function of the Fourier series is introduced and solved using the principle of modal superposition. Hammer impact modal experiments, ANSYS software and vibration response experiments are performed on spatial pipelines supported by flexible bracket-clamps; the numerical and experimental results support the semi-analytical results. The degenerate model is verified with the simple plane pipelines in other references, and the errors are less than 5 %. The proposed method exhibits broad applicability and generalizability, making it suitable for various configurations and boundary conditions.

Resonance response and chaotic analysis for an irrational pendulum system

We propose a novel rotating pendulum system with irrational nonlinearity, bi-stable phenomenon and quasi-zero stiffness characteristic, which bears a transition from standard double well dynamics to discontinuous double well dynamics by adjusting a geometric parameter. Nonlinear dynamical behaviors of the proposed irrational pendulum system subjected to viscus damping and periodic excitation are studied. To analytically demonstrate the primary resonance response for this perturbed irrational pendulum system, we introduce an approximate irrational system which is significant parallels with the original pendulum system from the perspective of qualitative analysis and quantitative calculation. Averaging method is applied to investigate the dynamic response of the perturbed irrational pendulum system with quasi-zero stiffness at origin. The effect of the internal and external parameters on the response curves are discussed. Numerical fitting technique and semi-analytical Melnikov method are used to detect the chaotic boundaries of the perturbed irrational pendulum system with two types of homoclinic orbits. Different types of periodic solutions and chaos are clarified in the perturbed irrational pendulum system by using numerical simulations. It is found that a class of chaos with oscillatory and rotational motions bears significant similarities to paroxysmal chaos due to sudden rotational motion and two chaotic motions formed by period-doubling bifurcation merge into one chaos.

Investigating the Response Variability of Statically Determined Sandwich Beams considering two Random Fields of Elastic Modulus

In this paper, the displacement variation in sandwich beams is determined by employing a semi-analytical approach. The classical displacement is calculated by integration using Mohr’s equation, although the integration is complicated due to the inclusion of random fields in the inertial moment term. Using the trapezoidal rule to compute these integrals, the random fields are discretized into random variables at the nodal point of the beam segments. Thus, the expected displacement, standard deviation, and coefficient of variation can be computed. To validate the results, the random fields are simulated using a previously described spectral method. The results of numerical examples were compared with the semi-analytical method and the Monte Carlo simulation demonstrating the high accuracy of the proposed method. The results also illustrate the influence of the parameters of the random fields of elastic modulus on the variability of displacement.

MHD Casson non-Newtonian fluid flow in a channel with expanding/contracting porous walls in the presence of thermal radiation

In this research, we explore the impact of changing viscosity in the context of the asymmetric laminar flow of Casson fluid with thermal radiation. The flow occurs within a channel that can either expand or contract, featuring porous walls. The flow equations are converted into ordinary differential equations (ODEs) through appropriate transformations to simplify the analysis. The velocity field and temperature profile expressions are obtained using Akbari-Ganji's method (AGM) and finite-element method (FEM). The graphs show the impact of different parameters on the flow characteristics, especially the viscosity-dependent parameter, Casson fluid parameter, and expansion ratio. The fluid velocity near the lower wall increases with the Casson fluid parameter but decreases after the mid-point. The fluid velocity is more affected by the temperature-dependent viscosity parameter for Newtonian fluid than non-Newtonian fluid. The results showed that as the Prandtl number increases, the temperature decreases, while the temperature increases with an increase in the Radiation parameter. Also, An increase in the wall expansion ratio leads to higher wall temperatures, whereas a decrease in the power-law index results in reduced temperatures. The results we've acquired were cross-referenced with prior studies, revealing that the employed methods exhibit high precision. The novelty of the present work is obtaining new results through the use of Akbari-Ganji's semi-analytical method.

Semianalytical Method for Controlling the Deformation of Retaining Structures Subjected to Asymmetrical Loads

Semianalytical Method for Controlling the Deformation of Retaining Structures Subjected to Asymmetrical Loads

The new analytical and numerical analysis of 2D stretching plates in the presence of a magnetic field and dependent viscosity

This study explores heat transfer in a system involving Jeffery fluid of MHD flow and a porous stretching sheet. The mathematical representation of this system is initially described using a partial differential equation (PDE), which is then converted into an ordinary differential equation (ODE) through numerical techniques such as Lie similarity and transformation methods, along with the shooting approach. The results indicate that altering the variables of Jeffery fluid, heat source, porosity on a stretching sheet, and the physical characteristics of the magnetic field within the system leads to an upward trend. Implementing this enhanced heat transfer system can yield benefits across various domains, including industrial machinery, mass data storage units, electronic device cooling, etc., thereby enhancing heating and cooling processes. Furthermore, the study also utilized Akbari-Ganji’s Method, a new semi-analytical method designed to solve nonlinear differential equations of heat and mass transfer. The results obtained from this method were compared with those from the finite element method for accuracy, efficiency, and simplicity. This research provides valuable insights into heat transfer dynamics in complex systems and offers potential applications in various industrial settings. It also contributes to developing more efficient and effective heat transfer techniques.

Mathematical modeling of droplet collisions in sprays under microgravity conditions

Two-phase flows under microgravity conditions play a crucial role in Space technology and science. Studying the behavior of dispersed droplets suspended in a gas flow and their influence on the characteristics of the carrier flow is the main task of computational modeling of turbulent two-phase flows. To understand the features and predict the behavior of such flows, along with experimental studies, it is necessary to develop mathematical models, which adequately describe all the basic physical and chemical processes in a gas-droplet environment at different scales.In this paper we obtain solutions for the problem of two droplets collision, using solutions for cumulative jets, and propose droplet collision outcome criterion. This research will focus on the process of two interacting droplets coalescence into one and coalescence with subsequent separation into two or more droplets. A semi-analytical method has been developed for application of the solution of the cumulative jet problem to the problem of droplet collision. A differential equation has been obtained that describes the following possible outcomes of droplet collision: coalescence and coalescence with subsequent separation. A criterion for determining the outcome of on-head droplet collision has been obtained. In accordance with the developed criterion, the number and size of formed droplets are obtained. The effect of collision on moving droplet evaporation in a heated atmosphere was regarded.

A homotopy-based computational scheme for two-dimensional fractional cable equation

In this paper, we examine the time-dependent two-dimensional cable equation of fractional order in terms of the Caputo fractional derivative. This cable equation plays a vital role in diverse areas of electrophysiology and modeling neuronal dynamics. This paper conveys a precise semi-analytical method called the q-homotopy analysis transform method to solve the fractional cable equation. The proposed method is based on the conjunction of the q-homotopy analysis method and Laplace transform. We explained the uniqueness of the solution produced by the suggested method with the help of Banach’s fixed-point theory. The results obtained through the considered method are in the form of a series solution, and they converge rapidly. The obtained outcomes were in good agreement with the exact solution and are discussed through the 3D plots and graphs that express the physical representation of the considered equation. It shows that the proposed technique used here is reliable, well-organized and effective in analyzing the considered non-homogeneous fractional differential equations arising in various branches of science and engineering.

A semi-analytical approach for estimating inland water inherent optical properties and chlorophyll a using airborne hyperspectral imagery

The inversion of inherent optical properties (IOPs) and chlorophyll a (Chla) is one of the key objectives in water color remote sensing, and hyperspectral remote sensing with rich spectral information makes precise inversion possible. In this study, we developed a semi-analytical estimation method for inland water IOPs based on the quasi-analytical algorithm (QAA). Considering the complex optical characteristics of inland waters, empirical parameter regional optimization was conducted. Furthermore, a dual-band joint inversion strategy and Gaussian function fitting method were utilized to optimize the solution processes for the backscattering coefficient of particles (bbp) and the absorption coefficient of phytoplankton pigments (aph), respectively. This approach overcomes the limitations of single-band bbp inversion in inland waters. In addition, it directly decomposes the absorption coefficient to obtain aph using a Gaussian function, which can reduce the intermediate steps and errors caused by indirect inversion. For Chla, we constructed a binary inversion model using aph(677) and remote sensing reflectance (Rrs), where the coefficient of determination (R2) exceeded 0.8. We also constructed an airborne hyperspectral image correction process, including vicarious calibration, atmospheric correction, and bidirectional reflectance distribution function (BRDF) correction, obtaining high-precision Rrs images. The ground models were successfully applied to the airborne hyperspectral images, mapping the spatial distribution of IOPs and Chla concentration in the study area. The experiments demonstrated that the proposed semi-analytical method using airborne hyperspectral imagery exhibits a good performance in terms of modeling accuracy and mapping analysis, and successfully applied to long-term monitoring using satellite hyperspectral images, highlighting the significant potential of hyperspectral remote sensing for high-precision monitoring of regional water bodies.

Agile earth observation satellite constellations scheduling for large area target imaging using heuristic search

Demand for imaging of large area targets by space-based assets such as constellations of Earth Observation Satellites (EOS) is on the rise for different applications. The area targets coverage cannot be achieved by a single satellite in a single observation opportunity, as the region for imaging spreads wider than the coverage swaths of imaging sensors. Hence, the imaging satellites are scheduled in a cooperative way to achieve maximum coverage of the area in a given planning time horizon. The scheduling algorithms also aim to maximize or minimize some specific objectives of interest while simultaneously complying with all resource constraints for an efficient use of precious space resources. As the current generation satellites are highly agile, there are many ways of choosing an imaging location in the target area for each single observation opportunity of a satellite. Even though the possible imaging region in continuous space is discretized into a finite number of strips for each observation, the huge combinatorial search space formed with all satellites’ imaging opportunities makes the problem intractable. Thus, the area target imaging scheduling is a complex combinatorial NP-hard optimization problem. Many exact and heuristic methods were evolved to provide a solution to this scheduling problem. Even though the exact methods solve the problem to optimality, they are limited to smaller problem instances because of the high computational complexities involved in those methods. More focus is seen in the literature on heuristic approaches as they guarantee scheduling performance with feasible solutions within an acceptable computational complexity. This paper presents our efficient heuristic approach proposed for the scheduling problem for imaging a large area target using a constellation of multiple satellites. Our approach is an extension of a widely used greedy approach in combination with insights from dynamic programming. In this paper, first we addressed the scheduling problem by describing it in detail and formulating it into a non-linear integer programming problem, considering the multi-objectives of achieving maximum coverage and minimum imaging resolution. The solution to the problem comprises two phases, namely the area decomposition phase and the scheduling phase. In the area decomposition phase, the area target is divided into strips dynamically for each observation opportunity of a satellite. Then exact observation period of each strip is computed using a simplified semi-analytical computational method. In the scheduling phase, we applied our heuristic search strategy, namely Forward and Backward Heuristic Search (FBHS) to obtain a near optimal solution within an acceptable computational time. Through the extensive simulations conducted under various scenarios, the effectiveness of the proposed method is verified in comparison with the baseline greedy approach.

Analysis of the human liver model through semi-analytical and numerical techniques with non-singular kernel

This work consists of the study of the time-fractional human liver model with the Caputo–Fabrizio fractional derivative. The existence and uniqueness of the proposed model are shown using fixed point theory. Also, the stability of the considered model is shown using the Ulam Hyres theorem and the Lyapunov function. The solution of the proposed model is obtained using a semi-analytical and numerical scheme. The series solution obtained from the semi-analytical method gives the proper result at any time, similarly, the numerical scheme gives the solution for a long time. The obtained numerical results are compared with real clinical data and earlier published work and found to be very close to real data than earlier published work. Results in the graphs and tables show that the proposed fractional-order model is superior to the traditional model.

A semi-analytical wavelet finite element method for wave propagation in rectangular rods

Prior knowledge of the dispersion curves and mode shapes of guided waves provides valuable information for wave mode selection and excitation in the field of non-destructive evaluation (NDE) and structural health monitoring (SHM). They are typically computed by the matrix methods, the finite element (FE) and semi-analytical finite element (SAFE) methods. However, the former is prone to numerical instability, and the latter two are limited by the refinement level of the FE mesh. In this paper, a semi-analytical wavelet finite element (SAWFE) method is presented to characterize wave propagation in rectangular rods. The piecewise polynomial interpolation functions of the SAFE method are replaced by two-dimensional scaling functions of the B-spline wavelet on the interval (BSWI). To demonstrate the accuracy of the proposed SAWFE technique, the propagation of guided waves in an aluminium plate is studied first. Then, the propagation of guided waves in rectangular rods of arbitrary aspect ratio is investigated. The results of this work clearly show that the SAWFE method presented here has higher accuracy and efficiency than the SAFE method.

Internet of things-based robust semi-analytical over ubiquitous data for indoor positioning geomagnetic

In the era of the Internet of Things (IoT), there is an increasingly urgent demand for high-precision and ubiquitous indoor positioning. However, robust technical solutions have yet to emerge, with one challenge being the effective use of geomagnetic information to address indoor personnel orientation. As known, outdoor environments with open spaces can utilize geomagnetic resolution-based methods for orientation, but indoor environments often suffer from significant magnetic distortions, rendering such methods impractical. To address this issue, this paper proposes a semi-analytical orientation method. The method initially integrates the analytical orientation results with the geometric relationship of corridor structures extracted based on spatial context information, obtaining corrected orientation results. This is achieved by measuring magnetic distortions and determining the fusion coefficient. Furthermore, the paper thoroughly analyzes the impact of different fusion coefficients on the orientation results. Test results indicate that compared to existing methods, the proposed approach exhibits better robustness, effectively improving orientation accuracy, and is widely applicable to path-based or corridor-based positioning and orientation scenarios.

Robust approximation rules for critical electric field of dielectric gas mixtures

A semi-analytic method for quickly approximating the density-reduced critical electric field for arbitrary mixtures of gases is proposed and validated. Determination of this critical electric field is crucial for designing and testing alternatives to SF6 for insulating high voltage electrical equipment. We outline the theoretical basis of the approximation formula from electron fluid conservation equations, and demonstrate how for binary mixtures the critical electric field can be computed from the transport data of electrons in the pure gases. We demonstrate validity of the method in mixtures of N2 and O2, and SF6 and O2. We conclude with an application of the method to approximate the critical electric field for mixtures of SF6 and HFO1234ze(E), which is a high interest mixture being actively studied for high voltage insulation applications.

Application of pseudo-parameter iteration method to nonlinear deflection analysis of an infinite beam with variable beam cross-sections on a nonlinear elastic foundation

In this study, the nonlinear deflection of an infinite beam with variable beam cross-sections on a nonlinear elastic foundation was analyzed using the pseudo-parameter iteration method (PIM), which is a novel iterative semi-analytic method for solving ordinary/partial differential equations. To do this, we set six types of infinite beams with concave and convex shapes under static loading conditions. To calculate the nonlinear deflection of the infinite beam with variable cross-sections, the Bernoulli-Euler beam equation (fourth-order ordinary differential equation) considering changing beam flexural rigidity was introduced, and the PIM was adopted to this equation. Through the numerical experiment, it was confirmed that the nonlinear deflections calculated via the PIM are quite close to the exact solution within a few iterations. In addition, the graph of error quickly reaches the steady state error for all cases as the number of iterations increases.

A Weakly Nonlinear System for Waves and Sheared Currents over Variable Bathymetry

The wave–current–seabed interaction problem is studied by using a coupled-mode system developed for modeling wave scattering by non-homogeneous, sheared currents in variable bathymetry regions. The model is based on a modal series expansion of wave velocity based on vertical eigenfunctions, dependent on local depth and flow parameters, including propagating and evanescent modes. The latter representation is able to accurately satisfy the wave flow continuity condition and the no-entrance boundary condition on the sloping parts of the seabed. A new derivation of a simplified nonlinear system is introduced using decomposition to a mean flow and a perturbative wave field. To force the system to consider incoming waves at the inlet, boundary knowledge of periodic, travelling nonlinear water waves over a flat bottom is required. For this purpose, specific solutions are derived using the semi-analytical method based on the stream function formulation, for cases of water waves propagating above linearly and exponentially sheared currents. Results obtained by the application of the CMS concerning the propagation of waves and currents—in particular, examples characterized by depth inhomogeneities—are presented and discussed, illustrating the applicability and performance of the method.

Multiple SSO Space Debris Flyby Trajectory Design Based on Cislunar Orbit

This paper investigates the trajectory design problem in the scenario of a multiple Sun-synchronous Orbit (SSO) space debris flyby mission from a DRO space station. At first, the characteristics of non-planar transfer from DRO to SSO in the Earth–Moon system are analyzed. The methods of large-scale ergodicity and pruning are utilized to investigate single-impulse and two-impulse DRO–Earth transfers. Using a powered lunar flyby, the two-impulse DRO–Earth transfer is able to fly by SSO debris while satisfying the requirements of the mission. After the local optimization, the optimal result of two-impulse DRO–Earth transfer and flyby is obtained. A multi-objective evolutionary algorithm is used to design the Pareto-optimal trajectories of multiple flybys. The semi-analytical optimization method is developed to provide the estimations of the transfer parameters in order to reduce the computations caused by the evolutionary algorithm. Simulations show that transferring from the 3:2 resonant DRO to a near-coplanar flyby of a SSO target debris using a powered lunar gravity assist needs a 0.47 km/s velocity increment. The mission’s total velocity increment is 1.39 km/s, and the total transfer time is 2.23 years.

Magnetohydrodynamic study of Micropolar fluid flow in the porous walled channel with variable viscosity and thermal conductivity: HAM Solution

This article examines the production of entropy and the heat transfer rate in the passage of micropolar fluid through a channel. The micropolar fluid takes place through the porous walled channel under the presence of an external magnetic field acting in the perpendicular direction of the flow. In this work, the thermal conductivity and viscosity of the fluid is considered variable and are the function of the temperature. Here, we consider two-dimensional creeping flow with two different forms of temperature boundary conditions, namely, Newtonian heating (NH) boundary constraints and specified surface temperature (SST) boundary conditions. The flow of the micropolar fluid in the proposed problem is governed by the coupled non-linear PDEs. These PDEs are first transformed into the ODEs by the use of some suitable similarity transformations and then obtained ODEs are further solved by using the semi-analytic Homotopy Analysis Method (HAM). This article shows the effect of various flow parameters on the fluid’s velocity, heat transfer rate, temperature of the fluid, and entropy production for both types of thermal boundary conditions. Through this article, it is concluded that the entropy production is minimal for Newtonian heating (NH) boundary conditions as compared to specified surface temperature (SST) boundary conditions. This result suggests that the NH boundary condition is more suitable for electrical appliances as compared to the SST boundary condition. This type of model is applicable in nuclear reactors, engineering appliances, heat reservoirs, etc.