Artificial Small Parameter Research Articles

Collapse of corroded pipes under external pressure considering corrosion asymmetry, non-uniformity, and initial ovality

This paper presents an analytical investigation of elastic collapse of corroded pipes considering the corrosion asymmetry, non-uniformity and initial ovality and one novelty is that corrosion asymmetry effect is considered by introducing a misalignment parameter accounting for co-existing internal corrosion and external corrosion. Governing differential equations are derived including the effect of corrosion asymmetry, initial ovality and membrane extensibility. By perturbative techniques, the interaction between corrosion asymmetry and initial ovality is extensively discussed by explicitly calculating the critical initial ovality governing the mode shape changes during initial deformation and collapse deformation phase. One novelty is the introduction of mode shape [Formula: see text] and mode shape 8 characterized by the sign of sum of displacements of two points independent of rigid-body motion. The theoretical solution is agreeing well with the FEA results. Furthermore, for general non-uniform corrosion, unconditionally convergent power series solutions are formulated by Lyapunov artificial small parameter method. By using Holder’s inequality, an explicit estimation method of the needed truncation number is presented for good computation accuracy by a minimization process over Holder’s exponent thus enabling early truncation of the series properly in advance. An improved estimation method is proposed by coupling with Rayleigh quotient techniques and the domain interval partition approach is introduced to accelerate convergence by thoroughly discussing the special uniformly corroded case by mathematical arguments. As a novelty, this paper illustrates why this improvement might be not evident when corrosion is rather narrow theoretically. Finally by using Chebyshev polynomials and the mathematical algebraic ring structure (generated by polynomial and trigonometric functions) closed under addition, multiplication and integration operations, some calculations for rings with corrosion profile in a non-polynomial mode are presented. This paper is intending to enhance the understanding of the effect of corrosion defect on collapse of pipes under external pressure in offshore industry.

The measure of noncompactness in a generalized coupled fixed point theorem and its application to an integro-differential system

This paper discusses the existence of a solution for the system of integro-differential equations. By using a certain measure of noncompactness through the generalized Darbo fixed point theorem, we deduce conditions that guarantee the existence of a solution for the system. The advantage of the Darbo fixed point theorem is it helps us to impose weaker conditions in equations, because the contraction condition reduces to continuous and self-mapping conditions. To illustrate our results, we provide multiple examples and solve them by using the artificial small parameter method, with tables and figures that demonstrate the exactitude of the results.

Hybrid method between Lyapunov’s artificial small parameter and bat algorithm for fractional differential equations

In this paper, a hybrid method between Lyapunov’s artificial small parameter method (LASPM) and the bat algorithm (BA) was proposed to solve fractional differential equations, where the best parameter value (ϵ) for the LASPM was estimated by the BA. The results of the proposed method, BA-LASPM, demonstrated reliability and efficiency compared to the classical method by applying it to some different examples for solving problems of fractional differential equations.

Solving nonlinear differential equations using hybrid method between Lyapunov's artificial small parameter and continuous particle swarm optimization

<p style='text-indent:20px;'>In this paper, Lyapunov's artificial small parameter method (LASPM) with continuous particle swarm optimization (CPSO) is presented and used for solving nonlinear differential equations. The proposed method, LASPM-CPSO, is based on estimating the <inline-formula><tex-math id="M1">\begin{document}$ \varepsilon $\end{document}</tex-math></inline-formula> parameter in LASPM through a PSO algorithm and based on a proposed objective function. Three different examples are used to evaluate the proposed method LASPM-CPSO, and compare it with the classical method LASPM through different intervals of the domain. The results from the maximum absolute error (MAE) and mean squared error (MSE) obtained through the given examples show the reliability and efficiency of the proposed LASPM-CPSO method, compared to the classical method LASPM.</p>

A semi-analytical method applied to turbocharger engine model

In this study, we apply a new version of the Homotopy Analysis Method called decomposition of the homotopy analysis method (DHAM). The DHAM method is based on the decomposition of the right-hand side of a given system of differential equations into a sum of functions. After the decomposition one can apply the HAM method. The physical model that we investigate in this paper is a complex system of equations that contains nonlinear ordinary differential equations of the first order. The system of equations takes into account the important variables such as the pressure, the temperature, the mass flow, the torque due to the turbine turbocharger, the torque from the compressor, the speed of turbocharger, etc. This system is very complex and cannot be solved analytically. The HAM method includes an artificial small parameter that inserts into the physical model and hence it enables one to apply different asymptotic methods. We compared the results of DHAM and HAM to numerical simulations analyses. We concluded that the DHAM results are closer to the numerical simulation results.

3D guidance for hypersonic reentry gliders based on analytical prediction

In order to give full play to the potential of hypersonic gliders to strike a longrange of targets and perform large lateral maneuvering missions, a three-dimensional guidance method is proposed on the basis of analytical predictions. We first construct an entry guidance model for a glider and derive the analytical solution for the trajectory based on Lyapunov's artificial small parameter method. Such an approach ensures high accuracy in the theoretical prediction of flight trajectory. Subsequently, we construct a three-dimensional (3D) flight corridor that satisfies multiple constraints including heat rate, dynamic pressure, overload, angle of attack, and bank angle. Furthermore, the flight profile and bank reversal strategy in the 3D flight corridor model are designed on the basis of the analytical solution of the flight trajectory considering both longitudinal flight range and lateral flight mobility requirements of the glider. In addition, the control commands of the angle of attack and bank angle are generated by tracking the profile of the flight. The method has been tested in three cases of guidance under short-, medium-, and long-range flights. The results show that it provides high prediction accuracy and strong self-adaptive capability for all the test cases. Our proposed method targets the design of a 3D flight profile in a 3D flight corridor. Furthermore, the method considers flight requirements in both the longitudinal and lateral directions, thus maximising the aircraft's ability to reach the target while maintaining high levels of flexibility and self-adaptive capability during the mission.

A modified perturbation solution to the one-dimensional Bratu problem

An approximate analytical solution to the one-dimensional Bratu boundary value problem is introduced in this paper. The solution is based on some perturbation expansion methods. The first step is taken from the linearized perturbation technique which was developed to solve initial value problem with a nonlinear term and no small parameter. An artificial small parameter is embedded and the dependent variable can then be expanded in terms of this parameter. However, necessary modifications are introduced to implement some techniques to solve a boundary value problem in order to allow for different nonlinearities. The current solution showed good convergence at any value of the Bratu constant, when compared with the exact lower branch of the solution to the one-dimensional Bratu problem. Also its recursive nature allows for more iterations and adding more correction terms to the final approximate solution which increases the accuracy.

Simple analysis of scattering data with the Ornstein-Zernike equation.

In this paper we propose and explore a method of analysis of the scattering experimental data for uniform liquidlike systems. In our pragmatic approach we are not trying to introduce by hands an artificial small parameter to work out a perturbation theory with respect to the known results, e.g., for hard spheres or sticky hard spheres (all the more that in the agreement with the notorious Landau statement, there is no physical small parameter for liquids). Instead of it being guided by the experimental data we are solving the Ornstein-Zernike equation with a trial (variational) form of the interparticle interaction potential. To find all needed correlation functions this variational input is iterated numerically to satisfy the Ornstein-Zernike equation supplemented by a closure relation. Our method is developed for spherically symmetric scattering objects, and our numeric code is written for such a case. However, it can be extended (at the expense of more involved computations and a larger amount of required experimental input information) for nonspherical particles. What is important for our approach is that it is sufficient to know experimental data in a relatively narrow range of the scattering wave vectors (q) to compute the static structure factor in a much broader range of q. We illustrate by a few model and real experimental examples of the x-ray and neutron scattering data how the approach works.

Fatigue crack growth modeling and prediction with uncertainties via stochastic perturbation series expansion method

The fatigue crack growth process is characterized by uncertainties inherent to the variations in geometrical properties, material properties, and operating conditions. This paper proposes a novel stochastic perturbation series expansion method for predicting fatigue crack growth evolution which accounts for uncertain parameters. Unlike the deterministic method, the parameters of the crack growth model are considered to be stochastic with perturbation terms. The solution to the crack growth equation is then transformed into a perturbation series after introducing said small parameters. A series of modified differential equations for predicting the stochastic characteristics of the crack length history are derived via first-order Taylor series expansion method. The initial crack length under perturbation can be expressed via the artificial small parameter method per the measurement errors. Further, by substituting the initial conditions into the modified stochastic equations, variations in the mean value and standard deviation as well as the probabilistic region of crack length history can be successfully obtained. The performance of the proposed method is evaluated through two examples. Comparison against experimental and Monte-Carlo simulation data confirm that the proposed method is indeed feasible and effective for predicting fatigue crack growth evolution with consideration of stochastic uncertainties.

Asymptotics of eigenvalues of the nonlinear eigenvalue problem arising from the near mixed-mode crack-tip stress-strain field problems

In the present paper, approximate analytical and numerical solutions to nonlinear eigenvalue problems arising in nonlinear fracture mechanics in studying stress-strain fields near a crack tip under mixed-mode loading are presented. Asymptotic solutions are obtained by the perturbation method (the artificial small parameter method). The artificial small parameter is the difference between the eigenvalue corresponding to the nonlinear eigenvalue problem and the eigenvalue related to the linear “undisturbed” problem. It is shown that the perturbation technique is an effective method of solving nonlinear eigenvalue problems in nonlinear fracture mechanics. A comparison of numerical and asymptotic results for different values of the mixity parameter and hardening exponent shows good agreement. Thus, the perturbation theory technique for studying nonlinear eigenvalue problems is offered and applied to eigenvalue problems arising in fracture mechanics analysis in the case of mixed-mode loading.

Asymptotics of the stress field near acrack tip under mixed - mode loading: small parameter method

In the present paper approximate analytical and numeric solutions to nonlinear eigenvalue problems arising in nonlinear fracture mechanics in analysis of stress — strain fields near a crack tip under mixed mode loading are presented. Asymptotic solutions are obtained via perturbation method technique (small parameter method). The artificial small parameter is the difference between the eigenvalue corresponding to the nonlinear eigenvalue problem and the eigenvalue related to the linear ”undisturbed” problem. It is shown that the perturbation technique gives an effective method of solving nonlinear eigenvalue problems in nonlinear fracture mechanics. Comparison results of numeric and asymptotic results for different value of the mixity parameter and hardening exponent shows good agreement. Thus the perturbation theory technique for study of nonlinear eigenvalue problems is offered and applied for eigenvalue problems arising from fracture mechanics analysis in the case of mixed mode loading.

Adomian decomposition method is a special case of Lyapunov’s artificial small parameter method

The Adomian decomposition method and Lyapunov’s artificial small parameter method are two popular analytic methods for solving nonlinear differential equations. In this short paper, we show that the Adomian decomposition method can be derived from the more general Lyapunov’s artificial small parameter method.

Asymptotics of the near-crack-tip stress field of a growing fatigue crack in damaged materials: Numerical experiment and analytical solution

In this paper, an asymptotic analysis of growing fatigue near-crack-tip fields in a damaged material is performed. The integrity parameter describing the damage accumulation process in the vicinity of the crack tip is incorporated into the constitutive law of an isotropic linear elastic material. An asymptotic solution based on the eigenfunction expansion method is obtained. It is shown that the problem is reduced to a nonlinear eigenvalue problem. An analytical solution of the nonlinear eigenvalue problem is found by the artificial small parameter method. The perturbation theory approach allows us to derive the analytical presentation of the stress and integrity distributions near the crack tip. The technique proposed permits us to find higher-order terms of the asymptotic expansions of the stress components and the integrity parameter.

Acoustic scattering by a cluster of small sound-soft obstacles

The scattering problem of an incident plane sound wave by a finite number of small sound-soft arbitrarily shaped obstacles is considered. First, we study the case of multiple scattering in the long wave limit. By analogy with Greenwood's approximation for the electrical constriction resistance of a circular cluster of microcontacts, we obtain an approximation for the harmonic capacity of a system of a large number of small sound-soft obstacles grouped into a spherical cluster. We generalize Greenwood's approach for the case of an arbitrary convex cluster and, as a by-product of our analysis, we elicit an approximate formula for the harmonic capacity of a convex solid. The study of the general case of multiple scattering by a cluster of small soft-sound obstacles is based on the first order asymptotic model obtained in our previous paper. We use the method of artificial small parameter for constructing Padé approximants of the scattering amplitude. Explicit formulas are derived and illustrated on several examples.

An artificial small perturbation parameter and nonlinear plate vibrations

Nonlinear natural in-plane vibrations of a rectangular plate are studied using three small parameters. Firstly, the nonlinearity is assumed to be small. Then, a solution to a problem of the zeroth order (linear) is sought in the form of an asymptotic series with respect to the ratio of stiffness characteristics. For internal resonance, vibration modes are coupled via an infinite system of nonlinear algebraic equations, and the artificial small parameter approach is proposed to solve the obtained system. Analytical formulas for the amplitude–frequency characteristics are derived and the solutions are compared with numerical results.

ASYMPTOTIC APPROACH FOR NON-LINEAR PERIODICAL VIBRATIONS OF CONTINUOUS STRUCTURES

An asymptotic approach for determining periodic solutions of non-linear vibration problems of continuous structures (such as rods, beams, plates, etc.) is proposed. Starting with the well-known perturbation technique, the independent displacement and frequency is expanded in a power series of a natural small parameter. It leads to infinite systems of interconnected non-linear algebraic equations governing the relationships between modes, amplitudes and frequencies. A non-trivial asymptotic technique, based on the introduction of an artificial small parameter is used to solve the equations. An advantage of the procedure is the possibility to take into account a number of vibration modes. As examples, free longitudinal vibrations of a rod and lateral vibrations of a beam under cubically non-linear restoring force are considered. Resonance interactions between different modes are investigated and asymptotic formulae for corresponding backbone curves are derived.

A New Perturbation Approach to the Laminar Fluid Flow Behind a Two-Dimensional Solid Body

In the present paper, the asymptotic behavior of the Navier–Stokes solutions at a large distance from an arbitrary cylindrical obstacle immersed in an otherwise uniform fluid flow is considered by means of a new perturbation approach. Oseen or boundary-layer differential equations have been used by many investigators for this classical problem, under the assumption that the vorticity decays exponentially far downstream. An integral formulation has the advantage of being less likely to omit terms in the perturbation expansion; however, previously the asymptotic solutions have not been derived from such integral representations. The governing integral representations for this problem are first derived by considering the boundary conditions on the surface of the obstacle and at infinity. Second an artificial small parameter, $\varepsilon $, introduced by Chang (1961), is used and the method of matched asymptotic expansions proposed by Kida and Miyai (1972) is applied to these integral equations. The present approach shows that the principle of rapid decay of vorticity is rational, supposing that an asymptotic expansion with respect to $\varepsilon $ exists, and it is shown that the same asymptotic solutions as Chang’s are obtained without using the integral theorem.