Parameter Continuation Method Research Articles

Buckling of Shell Panels Made of Fiberglass and Reinforced with an Orthogonal Grid of Stiffeners

The paper presents an approach to the stress-strain and buckling analysis in fiberglass cylindrical and conical panels reinforced from the concave side with an orthogonal grid of stiffeners. A mathematical model of the Timoshenko (Mindlin–Reissner) type is used. Transverse shears and geometric nonlinearity are taken into account. The stiffeners are introduced in two ways: using the method of refined discrete introduction and the method of structural anisotropy. We use a computational algorithm based on the Ritz method and the best parameter continuation method. We also provide buckling load values and make a comparison between two types of approaches to account for stiffeners, which shows good convergence.

Numerical approach towards gyrotactic microorganisms hybrid nanoliquid flow with the hall current and magnetic field over a spinning disk

The article explores the effect of Hall current, thermal radiation, and magnetic field on hybrid nanofluid flow over the surface of a spinning disk. The motive of the present effort is to upgrade the heat transmission rate for engineering and industrial purposes. The hybrid nanofluids as compared to the conventional fluids have higher thermal properties. Therefore, in the present article, a special class of nanoparticles known as carbon nanotubes (CNTs) and iron ferrite nanoparticles are used in the base fluid. The system of modeled equations is depleted into dimensionless differential equations through similarity transformation. The transform equations are further solved through the Parametric Continuation method (PCM). For the parametric study, the physical parameters impact on velocity, energy, mass transmission, and motile microorganism’s concentration profiles have been sketched. The obtained results are compared with the existing literature, which shows the best settlement. It concluded that the heat transmission rate reduces for Hall current and rises with radiative parameter. The results perceived that the addition of CNTs in carrier fluid is more efficacious than any other types of nanoparticles, due to its C–C bond. CNTs nanofluid can be more functionalized for the desired achievement, which can be utilized for a variety of applications by functionalization of non-covalent and covalent modification.

Harmonic balance-boundary element and continuation methods for steady-state wave scattering by interior and surface-breaking cracks with contact acoustic nonlinearity

A numerical method for nonlinear steady-state wave scattering due to a crack with contact acoustic nonlinearity (CAN) is developed in this article. The in-plane scattering problem is addressed here. The two systems studied are composed of an unbounded elastic solid that includes an interior crack and an elastic half plane that includes a surface-breaking crack. A time-harmonic plane wave is incident, and clapping and rubbing motions on the crack faces are induced as nonlinear phenomena. A harmonic balance-boundary element method (HB-BEM) is utilized to solve the nonlinear steady-state problem. The system of equations obtained by the HB-BEM consists of an elastostatic boundary integral equation (BIE) and frequency-domain elastodynamic BIEs of several different frequencies, which are coupled by nonlinear boundary conditions. To solve the system numerically, the nonlinear boundary conditions are regularized. In the regularized boundary conditions, traction is expressed by a smooth single-valued function with respect to crack-opening displacement. The solution of the steady-state system is tracked with a parameter continuation method to investigate bifurcations. The parameters that vary in the continuation method are the incident frequency and static field variables. To determine the stability of the steady-state solutions, a stability analysis method based on Hill’s infinite determinant is proposed.The stable solutions obtained by the proposed method agree with the conventional time-domain solutions after enough time has elapsed. A 1/2-order subharmonic resonance is observed with bifurcations for particular parameters. The interaction between the local resonant properties due to the crack and the CAN causes subharmonic resonance.

Numerical analysis of thermal conductive hybrid nanofluid flow over the surface of a wavy spinning disk

A three dimensional (3D) numerical solution of unsteady, Ag-MgO hybrid nanoliquid flow with heat and mass transmission caused by upward/downward moving of wavy spinning disk has been scrutinized. The magnetic field has been also considered. The hybrid nanoliquid has been synthesized in the presence of Ag-MgO nanoparticles. The purpose of the study is to improve the rate of thermal energy transmission for several industrial purposes. The wavy rotating surface increases the heat transmission rate up to 15%, comparatively to the flat surface. The subsequent arrangement of modeled equations is diminished into dimensionless differential equation. The obtained system of equations is further analytically expounded via Homotopy analysis method HAM and the numerical Parametric continuation method (PCM) method has been used for the comparison of the outcomes. The results are graphically presented and discussed. It has been presumed that the geometry of spinning disk positively affects the velocity and thermal energy transmission. The addition of hybrid nanoparticles (silver and magnesium-oxide) significantly improved thermal property of carrier fluid. It uses is more efficacious to overcome low energy transmission. Such as, it provides improvement in thermal performance of carrier fluid, which play important role in power generation, hyperthermia, micro fabrication, air conditioning and metallurgical field.

Метод расчета возмущенной траектории двухимпульсного перелета между гало-орбитой в окрестности точки L2 системы Солнце — Земля и окололунной орбитой

The paper introduces a new method for solving the problem of calculating the perturbed trajectory of a two-impulse flight between a near-lunar orbit and a halo orbit in the vicinity of the L2 point of the Sun — Earth system. Unlike traditional numerical methods, this method has better convergence. Accelerations from the gravitational forces of the Earth, the Moon and the Sun as point masses and acceleration from the second zonal harmonic of the geopotential are taken into account at all sections of the trajectory. The calculation of the flight path is reduced to solving a two-point boundary value problem for a system of ordinary differential equations. The developed method is based on the parameter continuation method and does not require the choice of an initial approximation for solving the boundary value problem. The last section of the paper provides examples and results of the analysis based on this method.

On the Parametric Continuation Method in $$\boldsymbol{R}^{\boldsymbol{n}}$$

We examine a way of using the parametric continuation method to compute mappings of the unit ball in $$n$$ -dimensional real space. The proposed approach yielded sufficient and necessary conditions for the global injectivity of mappings were obtained. It is established that these conditions actually coincide with the known features of the $$n$$ -dimensional complex space. The concretization of the method made here are the generalizations of some classes of functions analytic in the unit circle. In addition, the analogue of the Kaplan class was derived for mappings in $$n$$ -dimensional real space.

Determination of catastrophic sets of a tubular chemical reactor by two-parameter continuation method

AbstractThe work relates to development and presentation a two-parameter continuation method for determining catastrophic sets of stationary states of a tubular chemical reactor with mass recycle. The catastrophic set is a set of extreme points occurring in the bifurcation diagrams of the reactor. There are many large IT systems that use the parametric continuation method. The most popular is AUTO’97. However, its use is sometimes not convenient. The method developed in this work allows to eliminate the necessity to use huge IT systems from the calculations. Unlike these systems, it can be inserted into the program as a short subroutine. In addition, this method eliminates time-consuming iterations from the calculations.

Electrokinetic viscous rotating disk flow of Poisson-Nernst-Planck equation for ion transport

The main features of the present numerical model is to explore the behavior of an ionic transport electroviscous boundary layer flow over a rotating disk. For this purpose, the Nernst-Planck equation and Poisson's equation together with the traditional Navier Stokes equations are simulated for the conservation of ionic species. The Poisson-Nernst-Planck (PNP) equation neglects sterile effects and ion-ion interactions, which is widely recognized by the electro-chemists community, leading to advancement of various mathematical models. The modeled governing equations of the fluid flow are transformed to dimensionless ordinary differential equations under Von Karman's approach. The Parametric Continuation Method (PCM) is applied, in order to analyzed the numerical simulation of the problem. For validity of the method the results are compared with another numerical method (bvp4c) and some previous published work, seem to be in a very good agreement to each other. It is found that the potential gradient becomes smaller as Debye length parameter K increases and exposing more counter-ions to the flow which results an increase in the fluid velocity. The physical constraints impact on radial, axial and tangential velocity and on both positive and negative charge profile are sketched and briefly discussed.

Variable thickness flow over a rotating disk under the influence of variable magnetic field: An application to parametric continuation method

The present work explores the behavior of three-dimensional incompressible viscous fluid flow and heat transfer over the surface of a non-flat stretchable rotating disk. A variable thickness fluid is subjected under the influence of an external variable magnetic field and heat transfer. Navier–Stokes equation is coupled with Maxwell equations to examine the hydrothermal properties of fluid. The basic governing equations of motion are diminished to a system of nonlinear ordinary differential equations using appropriate similarity framework, which are further treated with numerical scheme known as parametric continuation method. The parametric continuation method has combined interesting characteristics of both shooting and implicit finite difference methods. For validity of the present numerical scheme, a comparison with the published work is performed and it is found that the results are in excellent agreement with each other. Numerical and graphical results for the velocity, temperature, and magnetic strength profiles as well as skin fractions and Nusselt number are presented and discussed in detail for various physical parameters. The heat transfer process is reduced with positive increment of no-flatness parameter [Formula: see text], while Prandtl number increases the heat transfer rate at the surface of the disk.

Refined model of stiffened shells

The paper addresses several existing versions of mathematical models describing stiffened shell structures (accounting for the interaction between stiffeners and skin “along a line”, and “along a strip”, and a method for “smearing” the stiffness of stiffeners over the entire structure). The authors suggest a new, most accurate version to account for stiffness properties of stiffeners, based on introducing various modular ratios along various coordinate axes. For stiffeners perpendicular to the direction under consideration, the authors introduce a modular ratio equal to the ratio between the width of stiffeners in this direction and the linear dimension of the shell in the direction under consideration. For analysis, the authors use an algorithm based on the Ritz method and the best parameter continuation method. Three versions of models, taking into account the discrete introduction of stiffeners, are compared. Analysis is performed for isotropic shallow doubly curved shells reinforced with an orthogonal grid of stiffeners.

Numerical Study of Nonlinear Oscillations in a Clock Frequency MEMS-Generator

Under consideration is some mathematical model of a clock frequency generator, a device of the MEMS class (microelectromechanical systems). We numerically study the solution of the corresponding second-order ordinary differential equation with nonlinear right-hand side and show that there is a region of the model parameters in which the bounded solutions tend to a stable limit cycle in the phase plane and, therefore, the periodic oscillations are stable with respect to the external perturbations. To determine the boundary of the region, we use the parameter continuation method of the solution of the boundary value problem defining the limit cycle. The model leads to the numerical identification of the region of generator operability.

Minimum-Time Optimal Control for a Model of Biological Wastewater Treatment

We consider a mathematical model that describes biological treatment of wastewater. The model is a controlled nonlinear three-dimensional system of differential equations. A minimum-time optimal control problem is posed that achieves the required contaminant concentration in the shortest possible time. The problem is investigated by Pontryagin’s maximum principle. The optimal control in this problem is represented by a switching function and its properties are analyzed. The results of this analysis make it possible to reduce the solution of the problem to a successive application of finite-dimensional constrained minimization and the parameter continuation method. The application of these approaches is discussed in detail, and the relevant numerical results are reported.

The best parametrization for solving the boundary value problem for the system of differential-algebraic equations with delay

In this paper, we considered the numerical approach for solving a nonlinear boundary value problem for the system of differential-algebraic equations with delay argument. The shooting method is used to solve the boundary value problem. The Newton method is used to find the parameter of shooting. To overcome the difficulties associated with the choice of the initial approximation we apply E. Lahaye’s parameter continuation method. If the curve of the solution contains limit points, the method diverges. Then to find the parameter we used the method of continuation with respect to the best parameter - the length of the curve of the solution set. The solution is constructed by advancing the sequence of values of the parameter. With a discrete continuation, the initial-value problem is transformed by a finite-difference representation of the derivatives and entering the best argument and the corresponding equation of hypersphere. The resulting system is solved using the Newton method. To find the values of the functions at the delay point Lagrange polynomial with three points is used. An example of the behavior of an elastoviscoplastic rod is considered.

Mathematical simulation of stability of steady-state conditions for a road train

The article is devoted to the application of the parameter continuation method for solving the problem of maneuverability of a three-trailer road train. The mathematical model of a three-trailer road train with a wheelless intermediate link is worked out. The analysis of the equilibrium conditions of the system is performed. The structural instability of the model of the road train in the neighbourhood of the linear motion is presented. The presence of stable circular steady-state conditions is proved when moving along circles of a sufficiently small radius. It is established that for a given longitudinal velocity there is a certain range of changes in the rotation angle of the steering wheels of the tractor unit, within which stable steady-state conditions may be. The analysis of the stability of the steady-state conditions at θ = 0.38 rad. For the steady-state conditions, the configuration of the road train is drawn, where it can be seen that the without bearing link can only be oriented to the outer side of the turn. Also in the process of numerical integration, it is found that the resulting steady-state conditions are asymptotically stable.

Viscoelastic nonlinear dynamic behaviour of Timoshenko FG beams

Retaining rotary-inertia terms and translation/rotation, a nonlinear analysis is conducted on the dynamic behaviour of Timoshenko functionally graded (FG) beams. The shear/normal strain field components are formulated using the Timoshenko theory. The elastic/viscous stress components are related to the strain components, using the Kelvin-Voigt scheme, and the strain components to translation/rotation terms using the Timoshenko beam theory. The viscous effects are incorporated using a negative work expression. The through-thickness gradual changes in the properties are modelled using the Mori-Tanaka technique. Energy-loss/work/energy Hamilton’s principle for system’s energy balance is utilised and the coupled nonlinear sets of equations of motion are obtained. Reduced-order motion models are obtained using Galerkin’s discretisation method. Numerical simulations are conducted via a parameter-continuation method and the dynamical behaviour is examined.

Analytical Solution of Volterra–Fredholm Integral Equations Using Hybrid of the Method of Contractive Mapping and Parameter Continuation Method

In this paper, an analytical method is proposed to solve Volterra–Fredholm integral equations. The method is based on a hybrid of the method of contractive mapping and parameter continuation method. The existence and uniqueness of the solution will be discussed. Error analysis of the proposed method is also investigated. Finally, illustrative examples are given to demonstrate the validity and applicability of the proposed method.

Asymmetric viscoelastic nonlinear vibrations of imperfect AFG beams

The nonlinear asymmetric viscoelastic vibration of imperfect axially functionally graded (AFG) beams is investigated within this study; this is for the first time. As the mechanical properties vary in the longitudinal direction, asymmetry in the vibration response appears. Viscosity (or internal friction between beam elements) dissipates the vibration energy; this is incorporated via the Kelvin-Voigt scheme. A stretching type nonlinearity due to clamped-clamped boundary conditions gives rise to geometric nonlinearities. A geometric imperfection, due to improper manufacturing, is incorporated. The Kelvin-Voigt type of viscosity relates the strain field to the stress one. The energy loss is dynamically balanced by the kinetic and potential energies using Hamilton’s principle. Two nonlinear coupled equations for transverse/axial motions are obtained, and then are solved using a parameter-continuation method. It is shown that how the gradient index, viscosity, and imperfection affect the asymmetric vibrations.

Effect of variable magnetic field on the flow between two squeezing plates

In this paper, a viscous fluid is considered between two horizontal and infinite squeezing plates. The unsteady equations of mass and momentum conservation is coupled with the variable magnetic field and energy equations. The governing system of equations along with entropy generation is solved by the Parametric Continuation Method (PCM). A parametric investigation is plotted through graphs for the velocity field and magnetic field components. Also, the entropy generation due to heat transfer, magnetic field and fluid friction is studied through graphs and tables. It is derived that an increase in magnetic Reynolds number, squeezing number or Hartman number increases the fluid temperature. The magnetic field components are also increasing with increase in magnetic flux. It is also derived that an increase in the rate of momentum diffusion increases the entropy generation and Bejan number due to maximum disorderness of molecules near the plates. Applications of the study include automotive magneto-rheological shock absorbers, novel aircraft landing gear systems, heating-up or cooling processes, biological sensor systems and biological prosthetic etc.

A METHOD FOR CALCULATING MECHANICAL CHARACTERISTICS OF INDUCTION MOTORS WITH SQUIRREL-CAGE ROTOR

Purpose. Development of a method for calculating mechanical characteristics of induction motors, taking into consideration saturation of the magnetic path and displacement of the current in the rotor bars. Methodology . The algorithm is based on calculating the steady-state mode of induction motor operation for a set slip, described by a system of non-linear algebraic equations of electrical equilibrium, whereas the mechanical characteristic is evaluated as a set of steady-state modes using parameter continuation method. The idea of the steady-state mode calculation consists in determining vectors of currents and flux linkages of the motor circuits, using which makes it possible to evaluate the electromagnetic torque, active and reactive powers, etc. Results . The study resulted in the development of a method and algorithm for calculating static characteristics of induction motors, which allows looking into the effect of different laws of voltage regulation on the mechanical characteristics, depending on the frequency change. Originality . An algorithm for calculating mechanical characteristics of the squirrel-cage induction motor was developed based on the mathematical model of the induction motor in which electromagnetic parameters are calculated using real saturation curves for the main magnetic flux and leakage fluxes, and displacement of the current in the rotor bars is evaluated by presenting the rotor winding as a multi-layer structure. Applying the transformation of the electrical equilibrium equations into the orthogonal axes enabled a significant reduction of calculation volume without impairing the accuracy of the results. Practical value . The developed algorithm allows studying the effect of different laws of scalar regulation of the voltage on the mechanical characteristics of the induction motor in order to obtain the necessary torque-speed curves for their optimization. It can be used for programming frequency converters.

Transforming Calculated Schemes in Geometrically Non-linear Mechanics Problems of the Sandwich Plates with Contour Reinforcing Beams

A numerical study of the geometrically nonlinear problem of deforming the cylindrical shape of a sandwich plate with a transversally soft core, supported in the end sections of absolutely rigid bodies, according to transforming calculation schemes as it is loaded, has been carried out. It is shown that such a transformation is necessary after the delamination of the adhesive layer connecting the end section of the core with the contour reinforcing beam, designed to ensure the transfer of external load on the carrier layers. The previously obtained equations were used, which are based on introducing into consideration, as unknown contact forces, the interaction of the outer layers with a core, as well as the external layers and the core with reinforcing bodies at all points of their interface surfaces. The problem under study required the sequential formulation and solving of two problems: on the stress-strain state of the structure until the beginning of the destruction of the adhesive layer between the core and the reinforcing beam and about determining the increments of the parameters of the stress-strain state after delamination. To simulate the delamination process, a method based on the parameter continuation method is proposed. This method provides for zeroing the tangential stresses formed in the adhesive layer at the first loading stage, by choosing the incremental tangent stresses specified above as the loading parameter. Numerical methods for solving the formulated problems were developed. They are based on a preliminary reduction of the initial problems to systems of integro-algebraic equations and the construction of their solutions by the finite sum method. To resolve the geometric nonlinearity, a two-layer iterative process is used.