Method For Numerical Solution Research Articles

Patterns of air mixture movement in the operating area for the annular ejector of pneumatic transportation system

Purpose. To establish the regularities of two-phase flow of “gas-solid particles” in the operating area of an annular ejector where the following processes take place: air mixture ejecting, compressed air outflow from the ejector nozzle, air mixture flows mixing in the transport pipeline. In the work, the velocity distribution is also examined for dispersed phase and air phase of air mixture during its loading and accelerating in the transport pipeline of the pneumatic transport system. Methodology. The research is based on the fundamental approaches of mass point dynamics, aerodynamics, the theory of jet flows and iteration methods of numerical solution of equations. Findings. The mechanics of the air mixture flow under the ejection and aerodynamic force in the operating area of an annular ejector and at the beginning of transport pipeline is analyzed using the method of iterations. The impact of air mixture flow in the operating area of an annular ejector on energy performance of the pneumatic transport system is evaluated. Originality. The originality is that, for the first time, the regularities describing two-phase “gas–solid particles” flow at the loading area of the pneumatic transport system with an annular ejector have been obtained. This made it possible to characterize the velocity distribution of the dispersed and air phases of the air mixture during their loading and aerodynamic acceleration in the transport pipeline. Also, an innovative approach to the effectiveness of the use of pipeline pneumatic transport is the assessment of the energy indicators of the use of ejector-type pneumatic transport equipment depending on the rate of compressed air outflow from the ejector. Practical value. The implementation of the results in the modernization of existing and in the creation of new pneumatic transport systems with an annular ejector makes it possible to increase the efficiency of their use in the technological processes of moving dispersed materials at mining and metallurgical enterprises and in other areas of technology.

FEATURES OF DEFORMING OF COMPOSITE INDUCTORS DURING ELECTROMAGNETIC PROCESSING OF MATERIALS

Introduction. The development of many branches of mechanical engineering at the current stage requires the development and application of new technologies that are characterized by high energy efficiency and a small amount of waste. Such technologies include those that use electromagnetic field energy. The energy of the electromagnetic field can be used to change the shape of workpieces, to connect individual structural elements (for example, by welding), to control the physical properties of the material (for example, by induction heating), etc. Carrying out calculation studies at the stage of designing and proving the appropriate technological equipment allows determining the rational values of key parameters, which ensures the achievement of the goals of the technological operation and the operability of the equipment. Purpose. When the energy of the electromagnetic field is used to exert a force on the processed workpiece, the technological equipment is equally subject to force. The purpose of the article is to create effective methods of calculating the strength of technological equipment together with the workpiece, and conducting the corresponding calculation analysis is an urgent task in the scientific and practical sense. Results. The article proposes an effective method of analyzing the elastic-plastic deformation of composite structures under the influence of an electromagnetic field. The general mathematical formulation of the coupled problem of the deformation of conductive bodies in the presence of an electromagnetic field is considered. To construct a numerical solution method, the initial problem is reduced to finding the minimum of the total energy of the system. The finite element method is used as a numerical solution method. The proposed method is applied to the analysis of the deformation of the "inductor-workpiece" system of the technological operation of magnetic pulse processing of metals. Conclusions. Some results are presented that allow making certain recommendations regarding the design and application of technological operations of a similar class.

Vibration of smart sandwich plate with an auxetic core and dual-FG nanocomposite layers integrated with piezoceramic actuators

In the current research, a thorough investigation of the free vibrational characteristics of a novel smart sandwich plate has been performed utilizing two different analytical and numerical solution techniques. The proposed structure is a sandwich plate with an auxetic core and super stiff dual-FG nanocomposite layers integrated with piezoceramic actuators. The nanocomposite layers of the structure are made of FG metal-ceramic matrix to which carbon nanotubes (CNTs) are added through various FG distribution patterns. Besides, two thin piezoceramic actuators are attached to the nanocomposite layers, which causes the structure to be controllable by applying an electrical voltage. Using such super stiff dual-FG nanocomposite layers together with piezoceramic actuators and ultra-light auxetic core with negative Poisson’s ratio results in the light and smart sandwich plate with high structural stiffness, which its natural frequency responses will be analyzed in this study. For this purpose, the governing equations are derived according to the first-order shear deformation theory (FSDT) and then solved using two different strategies, including numerical and analytical solution methods. The responses obtained by these solution methods by the name of generalized differential quadrature (GDQ) and Navier are compared and verified with each other and other reputed references in the literature. Accordingly, detailed parametric studies in the framework of tabulated and graphical results are performed to reveal the impacts of all involved parameters on the natural frequency responses of the proposed smart sandwich plate.

Higher order Haar wavelet method for numerical solution of integral equations

Higher order Haar wavelet method for numerical solution of integral equations

Trivariate Spline Collocation Methods for Numerical Solution to 3D Monge-Ampère Equation

Trivariate Spline Collocation Methods for Numerical Solution to 3D Monge-Ampère Equation

Models of plate tectonics with the Lattice Boltzmann Method

Modern geodynamics is based on the study of a large set of models, with the variation of many parameters, whose analysis in the future will require Machine Learning to be analyzed. We introduce here for the first time how a formulation of the Lattice Boltzmann Method capable of modeling plate tectonics, with the introduction of plastic non-linear rheology, is able to reproduce the breaking of the upper boundary layer of the convecting mantle in plates. Numerical simulation of the earth’s mantle and lithospheric plates is a challenging task for traditional methods of numerical solution to partial differential equations (PDE’s) due to the need to model sharp and large viscosity contrasts, temperature dependent viscosity and highly nonlinear rheologies. Nonlinear rheologies such as plastic or dislocation creep are important in giving mantle convection a past history. We present a thermal Lattice Boltzmann Method (LBM) as an alternative to PDE-based solutions for simulating time-dependent mantle dynamics, and demonstrate that the LBM is capable of modeling an extremely nonlinear plastic rheology. This nonlinear rheology leads to the emergence plate tectonic like behavior and history from a two layer viscosity model. These results demonstrate that the LBM offers a means to study the effect of highly nonlinear rheologies on earth and exoplanet dynamics and evolution.

FBG-Based Deformation Reconstruction Method of Multipoint Elastic Support Thin Plate

For the multipoint elastic support thin plate deformation reconstruction problem, combined with the relationship between strain and deformation. A recursive numerical solution method of curvature incorporating displacement information is proposed. Estimation of boundary strains by elastic boundary conditions and interpolation algorithms. The detailed measurement principle of curvature and the curvature deformation mathematical model is derived and solved by the Runge–Kutta numerical solution method. The effect of cumulative errors can be greatly reduced. Finite element method (FEM)-based simulations for thin plate, the fiber Bragg grating (FBG) sensors placement strategy is determined, and have been conducted to preliminary validate the effectiveness of the proposed approach. To further verify the validity and repeatability of the proposed approach, the deflection obtained by the reconstruction algorithm and the actual deflection measured by laser displacement sensors were compared. Experimental results indicate that after the average strain coefficient correction strain, the proposed approach can accurately reconstruct the deformation of the thin plate, the average of relative error with a value of less than 9% under different loadings.

Mathematical modelling and simulation of leak detection system in crude oil pipeline

A first-order differential leak detection model that accurately detects leaks in a crude oil pipeline is presented. This model incorporates a leak factor KL in the axial direction, which is simulated by applying the finite element method of numerical solution using COMSOL multi-physics software. Additionally, the model includes the transport equation for turbulent kinetic energy and the rate of kinetic energy model. Eigenvalues for velocities and pressures were determined and plotted against time for various pipe segments. The system is stable when the Eigenvalue is zero, but a leak is declared when the Eigenvalue for pressure or velocity is less than one. The study shows that pressure measurements are more sensitive parameters for detecting leaks than velocity measurements, and the sinusoidal waveform characterizes leak behaviours for velocity.

THE NUMERICAL METHODS FOR SOLVING NONLINEAR INTEGRAL EQUATIONS

Volterra integral equations are a special type of integrative equations; they are divided into two categories referred to as the first and second type. This thesis will deal with the second type which has wide range of the applications in physics and engineering problems. The aim of this paper is to compare between analytical solution and numerical solution to solve Integral Equations. Numerical examples are presented to illustrate the applications of this methods and to compare the computed results with other numerical methods for analytical solutions. Finally by comparison of numerical results, Simplicity and efficiency of this method be shown.

On projection method for numerical solution of hypersingular integral equations of the first kind combined with quadrature methods

The Fredholm integral equations of the first kind with two regular and hypersingular kernels are considered on [−1, 1]. The hypersingular kernel is considered smooth enough with no additional conditions. A projection method based on second kind Chebyshev polynomials approximation, combined with quadrature integration method, is developed to obtain high accurate approximations. The proposed method reduces the underlying integral equation to a system of algebraic equations. Several illustrative examples are provided to show the efficiency of the method.

A Novel Collocation Method for Numerical Solution of Hypersingular Integral Equation with Singular Right-Hand Function

In this study, the Fredholm hypersingular integral equation of the first kind with a singular right-hand function on the interval − 1 , 1 is solved. The discontinuous solution on the domain − 1 , 1 is approximated by a piecewise polynomial, and a collocation method is introduced to evaluate the unknown coefficients. This method, which can be applied to both linear and nonlinear integral equations, is very simple and straightforward. The presented illustrations relate that the results are very accurate compared to the other methods in the literature.

Study on the dynamic characteristics of the suspender with additional dampers and a frequency-based multiple parameter identification method for the system

The measurement of tensile forces inside suspenders plays an important role in safety inspection and monitoring during construction and operation of large structural engineering projects, including arch bridges and suspension bridges. The frequency-based method is one of the most widely used methods for tension force measurement of such structure components, but it is affected by many factors, such as the additional damping from a damper and various boundary conditions. In this study, the equation of damped motion for a suspender-damper system is derived, a numerical solution method based on the finite difference scheme is given, and a frequency-based multiple parameter identification method is proposed based on the derivative of the system eigenvalue. The study shows that the position and damping coefficient of the damper exert an important effect on the frequency and the plural mode of the suspender-damper system. The influence mechanism is discussed in the study. The numerical examples indicate that the proposed parameter identification algorithm can identify multiple system parameters of the suspender with a high precision and the maximum error is no more than 1%. The influence of the errors in the frequencies is also discussed in this study.

An Efficient Spectral Trust-Region Deflation Method for Multiple Solutions

It is quite common that a nonlinear partial differential equation (PDE) admits multiple distinct solutions and each solution may carry a unique physical meaning. One typical approach for finding multiple solutions is to use the Newton method with different initial guesses that ideally fall into the basins of attraction confining the solutions. In this paper, we propose a fast and accurate numerical method for multiple solutions comprised of three ingredients: (i) a well-designed spectral-Galerkin discretization of the underlying PDE leading to a nonlinear algebraic system (NLAS) with multiple solutions; (ii) an effective deflation technique to eliminate a known (founded) solution from the other unknown solutions leading to deflated NLAS; and (iii) a viable nonlinear least-squares and trust-region (LSTR) method for solving the NLAS and the deflated NLAS to find the multiple solutions sequentially one by one. We demonstrate through ample examples of differential equations and comparison with relevant existing approaches that the spectral LSTR-Deflation method has the merits: (i) it is quite flexible in choosing initial values, even starting from the same initial guess for finding all multiple solutions; (ii) it guarantees high-order accuracy; and (iii) it is quite fast to locate multiple distinct solutions and explore new solutions which are not reported in literature.

Numerical Analysis of Target Scattering Characteristics Based on Singularity Expansion Method

Based on the physical meaning of wave equation and Singularity Expansion Method, a numerical solution method is designed to extract the pole distributions of infinite cylinder, sphere and ellipsoid. By analysing the wave equation and boundary conditions of the target scattering characteristics, the extractions of poles of target scattering field are transformed into the solution of Hankel function and its derivative roots. Then, the T-matrix method is improved to make the numerical solution process of roots more accurate and faster. The numerical results are compared with strict analytical solutions and experimental results to fully verify the correctness of the method mentioned in this paper this study provides a basis for understanding the scattering mechanism of surface waves, and also provides a method and idea for solving problems such as acoustic scattering and electromagnetic scattering in engineering practice.

Characterization of high-power mechanical quality factor of piezoelectric ceramic discs under self-heating condition

We have developed a method to characterize high-power mechanical quality factor Qm of a thickness-poled piezoelectric disc under self-heating condition. In order to obtain Qm from temperature and vibration measurements, a heat transfer model for a thin piezoelectric disc vibrating in its fundamental radial mode was built, and a numerical solution method based on the finite difference approach was developed to investigate the temperature rise of the piezoelectric disc as a function of position and time. Numerical simulations were performed to verify the validity of the model and to quantify the influence of different material parameters on the temperature rise. A theoretical formula for characterizing Qm of a piezoelectric disc was deduced according to the definition of the mechanical quality factor, in which the relationship among the self-heating parameters (hg), the sample vibration level (v0) and Qm was established. Furthermore, using a self-designed experimental equipment, we have characterized the high-power mechanical quality factor Qm of a PZT8 disc sample under different vibration levels using a tone-burst excitation, which can increase the operating electric field strength limits. The disc results show that Qm degrades drastically with increasing vibration amplitude. Especially, the Qm values measured under self-heating conditions have lower magnitudes than those measured using transient/burst method, indicating self-heating effects do significantly contribute to Qm characterization.

Symmetry analysis, laws of conservation, and numerical and approximate analysis of Burger’s fractional order differential equation

The computation of fractional order differential and integration equations is highly utilized in numerous fields, such as mathematics in biology, ecology, physics, chemistry, and so on. This article aims at presenting methods for analytical and numerical solutions, Lie’s symmetry analysis, computing conservation laws for Burger’s fractional order differential equation (FODE), and reducing time-fractional cylindrical-Burgers equation order. Hence, we have employed a generalized (G′/G) expansion method for analytical solutions and a non-standard finite difference scheme for numerically solving Burger’s FODE. Additionally, the fractional derivative generalization of Noether's theorem has been utilized to compute the equation’s conservation laws. Numerical results have been reported here to approve theoretical results acquired in non-standard finite difference schemes. The proposed generalized (G′/G) expansion method is a simple, accurate, and practical approach to problem solving. Additionally, this method can be employed to execute non-linear wave equations. Programs including MATLAB and Maple have been utilized to simplify the process of solving complex equations.

Numerical solution of critical force of n-step telescopic boom with superlift device

The installation of a superlift device on the telescopic boom of a crane is the main method to improve the stress condition of the telescopic boom and to effectively improve the lifting performance of the crane. However, there is a lack of an effective and accurate numerical method to solve the critical force of a crane telescopic boom with a superlift device. Therefore, the telescopic boom with a superlift device is reasonably simplified as a stepped column and double cable model. Based on the longitudinal and transverse bending theory, the differential equations of the deflection of the n-stepped column with double cables model are established, the buckling characteristic equation and its recurrence formula of the n-stepped column with double cables model are derived, and the combined superlift equations are established. The Levenberg–Marquardt optimization algorithm is used to solve the critical force and length coefficient. A large amount of data were compared and analyzed between the numerical solution results and the ANSYS finite element simulation results. The error result analysis proved the correctness of the deduced buckling characteristic equation and its derived formula and the accuracy of the numerical solution. The numerical solution method in this paper can be used to calculate the critical force and length coefficient under any number of boom sections, boom section length, sectional moment of inertia, cable length, and included angle between the two cables of the telescopic boom with a superlift device. The numerical solution method can provide a technical support for the structural design of a telescopic boom with a superlift device in practical projects.

Mesh-free least-squares reverse-time migration

AbstractThe reflectivity of the subsurface can be precisely determined using least-squares reverse-time migration (LSRTM). As LSRTM necessitates solving the wave equation, the numerical solution method of the wavefield directly determines the quality of the migration image. The conventional LSRTM method usually uses the finite difference method based on a regular grid to calculate the wavefield. Due to the stepwise approximation of an irregular surface with a regular grid, scattering noise may occur in the propagation of the wavefield, which affects the quality of the image. In addition, the conventional LSRTM cannot effectively handle the models with rugged topography. The finite difference method generated by radial basis functions (FD-RBF) is a mesh-free method and can construct interpolation functions to solve the wave equations numerically according to arbitrarily distributed spatial coordinate points. Therefore, we use the FD-RBF method to develop a mesh-free LSRTM approach to eliminate the influence of the inherent limitation of a regular grid on the imaging. Numerical examples show that the mesh-free LSRTM method can better represent the curved or steep interface within the model and is also suitable for models with rugged topography. The LSRTM method can provide higher-quality images and effectively reduce the memory required for calculations.

Series Solution of the Time-Dependent Schrödinger–Newton Equations in the Presence of Dark Energy via the Adomian Decomposition Method

The Schrödinger–Newton model is a nonlinear system obtained by coupling the linear Schrödinger equation of canonical quantum mechanics with the Poisson equation of Newtonian mechanics. In this paper, we investigate the effects of dark energy on the time-dependent Schrödinger–Newton equations by including a new source term with energy density proportional to the cosmological constant Λ, in addition to the particle-mass source term. The resulting Schrödinger–Newton–Λ (S-N-Λ) system cannot be solved exactly, in closed form, and one must resort to either numerical or semianalytical (i.e., series) solution methods. We apply the Adomian Decomposition Method, a very powerful method for solving a large class of nonlinear ordinary and partial differential equations, to obtain accurate series solutions of the S-N-Λ system, for the first time. The dark energy dominated regime is also investigated in detail. We then compare our results to existing numerical solutions and analytical estimates and show that they are consistent with previous findings. Finally, we outline the advantages of using the Adomian Decomposition Method, which allows accurate solutions of the S-N-Λ system to be obtained quickly, even with minimal computational resources. The extensive use of the Adomian Decomposition Method in the field of quantum mechanics and quantum field theory may open new mathematical, and physical, perspectives on obtaining semi-analytical solutions for some complex problems of quantum theory.

An adaptive splitting method for the Cox-Ingersoll-Ross process

We propose a new splitting method for strong numerical solution of the Cox-Ingersoll-Ross model. For this method, applied over both deterministic and adaptive random meshes, we prove a uniform moment bound and strong error results of order 1/4 in L1 and L2 for the parameter regime κθ>σ2. We then extend the new method to cover all parameter values by introducing a soft zero region (where the deterministic flow determines the approximation) giving a hybrid type method to deal with the reflecting boundary. From numerical simulations we observe a rate of order 1 when κθ>σ2 rather than 1/4. Asymptotically, for large noise, we observe that the rates of convergence decrease similarly to those of other schemes but that the proposed method making use of adaptive timestepping displays smaller error constants.