Nonlinear Algebraic Equations Research Articles

Free vibrations of rotating pre-twisted blades including geometrically nonlinear pre-stressed analysis

The steady-state equilibrium deformations (SSEDs) caused by centrifugal force field in rotating blades are not necessarily perturbed disturbances. These deformations can be considered as large amplitude deformations, especially for high values of the rotating speed. Accordingly, in the current study, geometrically nonlinear terms are included in the static analysis under centrifugal forces (SACF) to accurately model the stiffening/softening effects in the vibrations of rotating pre-twisted blades. To this end, by developing a shell model based on first-order shear deformation theory (FSDT), nonlinear and linear integral boundary value problems (IBVPs) governing the SSEDs and vibrations of the blade are obtained, respectively. Multi-mode discretization of these IBVPs is carried out by the spectral Chebyshev technique. The discretization of the nonlinear IBVP results in nonlinear algebraic equations. By solving these equations, nonlinear pre-stressed analysis (NPA) is performed to achieve the SACF. Then, the free vibrations of the rotating pre-twisted blade about the determined equilibrium position is investigated. The numerical results show that the natural frequencies obtained in the presence of the nonlinear terms are extremely lower than those of the linear pre-stressed analysis.

An efficient hybrid numerical method for multi-term time fractional partial differential equations in fluid mechanics with convergence and error analysis

The fundamental purpose of this paper is to study the numerical solution of multi-term time fractional nonlinear Klein–Gordon equation, using regularized beta functions and fractional order Bernoulli wavelets. First, the exact formulas for the fractional integrals of the fractional order Bernoulli wavelets were obtained. Using properties of the regularized beta functions and their operational matrices the operational matrices of the fractional order Bernoulli wavelets were calculated. Through new operational matrices and appropriate collocation points, the time fractional nonlinear Klein–Gordon equation were transformed to a system of nonlinear algebraic equations. The convergence analysis and error bound of the proposed method were then performed. A sufficient number of numerical simulations were considered to show the effectiveness and validity of the presented numerical method and its theoretical analysis.

Dynamic snap-through of functionally graded shallow arches under rapid surface heating

In this paper, dynamic buckling response of a shallow arch constructed from functionally graded materials under thermal shock is investigated. It is assumed that one surface of the beam is subjected to sudden temperature elevation while the other surface is kept at reference temperature. To solve the one-dimensional Fourier heat transfer equation, Crank–Nicolson numerical method along with the central finite difference method are employed. In order to obtain the strain–deflection relations, Donnell theory of shallow arches with von Kármán simplifications are used. The governing equations are acquired based on well-known Hamilton’s principle that are converted to a set of non-linear algebraic equations using the Ritz method. Since the governing equations are non-linear equations, the Newton–Raphson method is applied and after that temporal evolution of displacements are obtained employing the Newmark time marching scheme. In the final stage, a theory to estimate the buckling load is required, which is the Bodiansky–Roth criterion in this research. To validate the formulation and solution method of this project, this work is compared with previous studies and good agreement is obtained. In addition, impacts of power law index, arch thickness, geometrical parameters, and temperature dependency on the dynamic buckling temperature are reported in this study.

Numerical investigation of geometrically nonlinear clamped uniform rods and rods with sections varying exponentially free vibration

The present paper is intended to investigate the problem of linear and non-linear longitudinal free vibration of uniform rods and rods whose cross-sections vary exponentially at large vibration amplitudes. The method adopted consists in discretizing the energy term on linear kij and non-linear rigidity tensor bijkl, as well as the mass tensor mij. Therefore, the formulation of this structure is based on Lagrange equations and the harmonic balance method so as to obtain the nonlinear algebraic equations. These latter are solved numerically and analytically through the explicit and linearized method. The response of Clamped-Clamped uniform and non-uniform rods on our structure are highlighted in the amplitude frequency and associated first three mode shapes. Moreover, this research leads to study the influence of the exponential slope on the maximum displacement, thus emphasizing the non-uniform bars usefulness. The obtained results are then compared with the available literature with a view to validating this theory. As a perspective, the method used in this paper would be pushed to study the FDM material, taking into account other parameters related to additive manufacturing, and later to be validated experimentally. Longitudinal vibrations are important in mechanical structures; therefore, the determination of their dynamic behaviour needs to be understood. In the present study, the effect of the displacement amplitude on the exponential slope of the structure was analysed, which led to the determination of the reduction range of the vibration amplitude under resonance. However, this should be taken into account in the design process. Besides, the usefulness of the non-linearity geometric effects was demonstrated to examine these structures by considering all the parameters involved. A linearized procedure is used to solve a nonlinear algebra equation. The use of this method leads to reduce calculation time contrary to iterative methods.

Optimal midcourse guidance law for the exo-atmospheric interceptor with solid-propellant booster

This paper focuses on the midcourse guidance law for the exo-atmospheric interceptor with solid-propellant booster to intercept a target in the Keplerian orbit, in which the thrust direction command is provided under the typical framework of predictor-corrector algorithm. Firstly, an analytical solution is derived to predict the burnout velocity and position vectors, in which the thrust vector is assumed to be along the velocity vector. Secondly, the predicted values are used to formulate a set of nonlinear algebraic equations to determine the desired burnout velocity direction to impact the target. Actually, those equations are transcendental because Lambert's problem with unspecified time-of-flight is involved. Then, Newton-Raphson method is used to solve the problem. In the calculation process, the Jacobian matrix corresponding to the equations is derived in an analytical manner, which significantly improves the computational efficiency. Thirdly, an optimal correction guidance law is constructed to steer the interceptor to achieve the desired burnout velocity direction. Finally, convergence evaluation, optimality verification, and Monte Carlo simulations are carried out to test the performance of the proposed guidance law. The results show that it not only performs well in providing the optimal guidance command, but also has high computational efficiency and guidance accuracy. Moreover, it has great robustness even in large dispersions and uncertainties.

A Novel Collision-Free Homotopy Path Planning for Planar Robotic Arms.

Achieving the smart motion of any autonomous or semi-autonomous robot requires an efficient algorithm to determine a feasible collision-free path. In this paper, a novel collision-free path homotopy-based path-planning algorithm applied to planar robotic arms is presented. The algorithm utilizes homotopy continuation methods (HCMs) to solve the non-linear algebraic equations system (NAES) that models the robot’s workspace. The method was validated with three case studies with robotic arms in different configurations. For the first case, a robot arm with three links must enter a narrow corridor with two obstacles. For the second case, a six-link robot arm with a gripper is required to take an object inside a narrow corridor with two obstacles. For the third case, a twenty-link arm must take an object inside a maze-like environment. These case studies validated, by simulation, the versatility and capacity of the proposed path-planning algorithm. The results show that the CPU time is dozens of milliseconds with a memory consumption less than 4.5 kB for the first two cases. For the third case, the CPU time is around 2.7 s and the memory consumption around 18 kB. Finally, the method’s performance was further validated using the industrial robot arm CRS CataLyst-5 by Thermo Electron.

High order asymptotic preserving discontinuous Galerkin methods for gray radiative transfer equations

In this paper, we will develop a class of high order asymptotic preserving (AP) discontinuous Galerkin (DG) methods for nonlinear time-dependent gray radiative transfer equations (GRTEs). Inspired by the works in [6,55], we propose to penalize the nonlinear GRTEs under the micro-macro decomposition framework by adding a weighted linear diffusive term. A hyperbolic, namely Δt=O(h) in the transport regime where Δt and h are the time step and mesh size respectively, and unconditional stability instead of parabolic time step restriction Δt=O(h2) in the diffusive regime are obtained, which are also free from the photon mean free path. We further employ a Picard iteration with a predictor-corrector procedure, to decouple the resulting global nonlinear system to a linear system with local nonlinear algebraic equations within each outer iterative loop. For the resulting scheme, only an implicit system for the macroscopic variable needs to be solved, while the microscopic variable can be updated explicitly. Besides, the nonlinear implicit system is decoupled to a linear positive definite system followed by nonlinear algebraic equations due to the Picard iteration. Namely, high dimension, implicit treatment and nonlinearity are all decoupled. Our scheme is shown to be AP and asymptotically accurate (AA). Numerical tests for one and two spatial dimensional problems are performed to demonstrate that our scheme is high order accurate, effective and efficient.

Performance analysis of relaxation Runge–Kutta methods

Recently, global and local relaxation Runge–Kutta methods have been developed for guaranteeing the conservation, dissipation, or other solution properties for general convex functionals whose dynamics are crucial for an ordinary differential equation solution. These novel time integration procedures have an application in a wide range of problems that require dynamics-consistent and stable numerical methods. The application of a relaxation scheme involves solving scalar nonlinear algebraic equations to find the relaxation parameter. Even though root-finding may seem to be a problem technically straightforward and computationally insignificant, we address the problem at scale as we solve full-scale industrial problems on a CPU-powered supercomputer and show its cost to be considerable. In particular, we apply the relaxation schemes in the context of the compressible Navier–Stokes equations and use them to enforce the correct entropy evolution. We use seven different algorithms to solve for the global and local relaxation parameters and analyze their strong scalability. As a result of this analysis, within the global relaxation scheme, we recommend using Brent’s method for problems with a low polynomial degree and of small sizes for the global relaxation scheme, while secant proves to be the best choice for higher polynomial degree solutions and large problem sizes. For the local relaxation scheme, we recommend secant. Further, we compare the schemes’ performance using their most efficient implementations, where we look at their effect on the timestep size, overhead, and weak scalability. We show the global relaxation scheme to be always more expensive than the local approach—typically 1.1–1.5 times the cost. At the same time, we highlight scenarios where the global relaxation scheme might underperform due to its increased communication requirements. Finally, we present an analysis that sets expectations on the computational overhead anticipated based on the system properties.

Vector model for solving the inverse kinematics problem in the system of external adaptive control of robotic manipulators

A new approach that allows to solve the problems of forward and inverse kinematics for robotic manipulators is proposed by the authors. The approach is to represent the kinematic model of a robotic arm with six rotational joints in the form of a vector model. The first 4 vectors make their movements and turn in the same plane, because of the geometric features of the robot. Thus, on the basis of a vector equation, a system of nonlinear algebraic equations was formed, which were solved by the numerical Newton method, at each iteration of which a system of linear algebraic equations was solved by the Seidel method. The algorithm was tested on a trajectory set in the form of an Archimedean spiral approximated by straight line segments. Trajectory was checked by a real robot KUKA KR6 R900 and it was shown that deviation between calculated and real position of flange point was less than 0.005 mm. It is shown that in comparison with the common Denavit–Hartenberg method the method proposed by the authors has an order of magnitude fewer mathematical operations (790 versus 52).

Numerical solution of nonlinear integral equation arising in optometry

In this paper, a mathematical model of the eye corneal shape is introduced which is a nonlinear boundary value problem. This boundary value problem is converted to a nonlinear integral equation and by means of the collocation method, the problem is reduced into a system of the nonlinear algebraic equations. The approximate solutions are obtained by solving this nonlinear system and numerical results are provided to show that introduced method is efficient. Also, comparisons with other works are made to confirm the reliability of the method.

AN EFFECTIVE AND SIMPLE SCHEME FOR SOLVING NONLINEAR FREDHOLM INTEGRAL EQUATIONS

In this paper, a simple scheme is constructed for finding approximate solution of the nonlinear Fredholm integral equation of the second kind. To this end, the Lagrange interpolation polynomials together with the Gauss-Legendre quadrature rule are used to transform the source problem to a system of nonlinear algebraic equations. Afterwards, the resulting system can be solved by the Newton method. The basic idea is to choose the Lagrange interpolation points to be the same as the points for the Gauss-Legendre integration. This facilitates the evaluation of the integral part of the equation. We prove that the approximate solution converges uniformly to the exact solution. Also, stability of the approximate solution is investigated. The advantages of the method are simplicity, fastness and accuracy which enhance its applicability in practical situations. Finally, we provide some test examples.

Proposed Section Methods to Find Root of the Single Variable Non-Linear Algebraic and Transcendental Equations

The main purpose of numerical analysis is to develop a numerical algorithm to find approximate values. The main aim of this paper is to present Proposed Section Method-I and Proposed Section Method-II (Proposed Section Methods). These method makes some slight difference in well-known bisection method to solve single variable nonlinear equation. In well-known Bisection Method, the interval in which the root of the equation lies is bisected using Arithmetic Mean (AM). These Proposed Section Method splits the interval using Geometric Mean (GM) and Harmonic Mean (HM). These method gives the roots of the equation which are closed to the root given by well-known Bisection Method. In order to analyze the results some examples related with algebraic and transcendental functions are considered. It was concluded that if the root of the single variable non-linear equation exists by all these methods then the solutions are very close to each other.

On the Wavelet Collocation Method for Solving Fractional Fredholm Integro-Differential Equations

An efficient algorithm is proposed to find an approximate solution via the wavelet collocation method for the fractional Fredholm integro-differential equations (FFIDEs). To do this, we reduce the desired equation to an equivalent linear or nonlinear weakly singular Volterra–Fredholm integral equation. In order to solve this integral equation, after a brief introduction of Müntz–Legendre wavelets, and representing the fractional integral operator as a matrix, we apply the wavelet collocation method to obtain a system of nonlinear or linear algebraic equations. An a posteriori error estimate for the method is investigated. The numerical results confirm our theoretical analysis, and comparing the method with existing ones demonstrates its ability and accuracy.

Code of the Multidimensional Fractional Pseudo-Newton Method using Recursive Programming

The following paper presents one way to define and classify the fractional pseudo-Newton method through a group of fractional matrix operators, as well as a code written in recursive programming to implement this method, which through minor modifications, can be implemented in any fractional fixed-point method that allows solving nonlinear algebraic equation systems.

Code of the Multidimensional Fractional Quasi-Newton Method using Recursive Programming

The following paper presents one way to define and classify the fractional quasi-Newton method through a group of fractional matrix operators, as well as a code written in recursive programming to implement this method, which through minor modifications, can be implemented in any fractional fixed-point method that allows solving nonlinear algebraic equation systems.

Numerical method for determining the coefficient of hydraulic resistance two-phase flow in a gas lift well

The process of stationary movement of a gas-liquid mixture in the lifting pipe of a gas-lift well is considered. To describe this two-phase flow, a mathematical model is proposed that includes the equation of flow motion and the continuity equation for each phase. The presented model is transformed to a single nonlinear ordinary differential equation with respect to pressure. Within the framework of the obtained model, the task is set to determine the hydraulic resistance coefficient of a two-phase flow according to an additionally specified condition with respect to pressure. An additional condition, presented in the form of a nonlinear algebraic equation, is transformed into an ordinary differential equation with respect to an unknown coefficient of hydraulic resistance by applying the method of differentiation by parameter. The solution of the resulting Cauchy problem is determined by the finite difference method. Based on the proposed computational algorithm, numerical experiments were carried out for model data. Keywords: gas lift; two-phase flow; hydraulic resistance coefficient; parameter differentiation method; finite difference method.

Code of a Multidimensional Fractional Quasi-Newton Method with an Order of Convergence at Least Quadratic using Recursive Programming

The following paper presents a way to define and classify a family of fractional iterative methods through a group of fractional matrix operators, as well as a code written in recursive programming to implement a variant of the fractional quasi-Newton method, which through minor modifications, can be implemented in any fractional fixed-point method that allows solving nonlinear algebraic equation systems

A numerical technique to solve a problem of the fluid motion in a straight plane rigid duct with two axisymmetric rectangular constrictions

A second-order numerical technique is developed to study the steady laminar fluid motion in a straight two-dimensional hard-walled duct with two axisymmetric rectangular constrictions. In this technique, the governing relations are solved via deriving their integral analogs, performing a discretization of these analogs, simplifying the obtained (after making the discretization) coupled nonlinear algebraic equations, and the final solution of the resulting (after making the simplification) uncoupled linear ones. The discretization consists of the spatial and temporal parts. The first of them is performed with the use of the TVD-scheme and a two-point scheme of discretization of the spatial derivatives, whereas the second one is made on the basis of the implicit three-point asymmetric backward differencing scheme. The above-noted uncoupled linear algebraic equations are solved by an appropriate iterative method, which uses the deferred correction implementation technique and the technique of conjugate gradients, as well as the solvers ICCG and Bi-CGSTAB.

On solvability of boundary value problem for a nonlinear Fredholm integro-differential equation

The paper proposes a constructive method to solve a nonlinear boundary value problem for a Fredholm integro-differential equation. Using D.S. Dzhumabaev parametrization method, the problem under consideration is transformed into an equivalent boundary value problem for a system of nonlinear integrodifferential equations with parameters on the subintervals. When applying the parametrization method to a nonlinear Fredholm integro-differential equation, the intermediate problem is a special Cauchy problem for a system of nonlinear integro-differential equations with parameters. By substitution the solution to the special Cauchy problem with parameters into the boundary condition and the continuity conditions of the solution to the original problem at the interior partition points, we construct a system of nonlinear algebraic equations in parameters. It is proved that the solvability of this system provides the existence of a solution to the original boundary value problem. The iterative methods are used to solve both the constructed system of algebraic equations in parameters and the special Cauchy problem. An algorithm for solving boundary value problem under consideration is provided.

Certain New Models of the Multi-Space Fractal-Fractional Kuramoto-Sivashinsky and Korteweg-de Vries Equations

The main objective of this paper is to introduce and study the numerical solutions of the multi-space fractal-fractional Kuramoto-Sivashinsky equation (MSFFKS) and the multi-space fractal-fractional Korteweg-de Vries equation (MSFFKDV). These models are obtained by replacing the classical derivative by the fractal-fractional derivative based upon the generalized Mittag-Leffler kernel. In our investigation, we use the spectral collocation method (SCM) involving the shifted Legendre polynomials (SLPs) in order to reduce the new models to a system of algebraic equations. We then use one of the known numerical methods, the Newton-Raphson method (NRM), for solving the resulting system of the nonlinear algebraic equations. The efficiency and accuracy of the numerical results are validated by calculating the absolute error as well as the residual error. We also present several illustrative examples and graphical representations for the various results which we have derived in this paper.