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In a transformation method, the numerical solution of a given boundary value problem is obtained by solving one or more related initial value problems. Therefore, a transformation method, like a shooting method, is an initial value method. The main difference between a transformation and a shooting method is that the former is conceived and derive its formulation from the scaling invariance theory. This paper is concerned with the application of the iterative transformation method to several problems in the boundary layer theory. The iterative method is an extension of the T{��}pfer's non-iterative algorithm developed as a simple way to solve the celebrated Blasius problem. This iterative method provides a simple numerical test for the existence and uniqueness of solutions. Here we show how the method can be applied to problems with a homogeneous boundary conditions at infinity and in particular we solve the Sakiadis problem of boundary layer theory. Moreover, we show how to couple our method with Newton's root-finder. The obtained numerical results compare well with those available in the literature. The main aim here is that any method developed for the Blasius, or the Sakiadis, problem might be extended to more challenging or interesting problems. In this context, the iterative transformation method has been recently applied to compute the normal and reverse flow solutions of Stewartson for the Falkner-Skan model [Comput. \& Fluids, {\bf 73} (2013) pp. 202-209]., 48 pages, 12 figures, 7 tables. arXiv admin note: substantial text overlap with arXiv:1212.5057, arXiv:1410.2043, arXiv:2003.07730

Size-dependent behavior of a bistable microelectromechanical system subjected to electrostatic and piezoelectric actuations is investigated. The system is comprised of an Euler–Bernoulli slightly curved capacitive microbeam which is laminated between piezoelectric layers. Applying DC voltage to piezoelectric layers induces axial force in the beam and modulates its stiffness and consequently, its frequencies. Hamilton’s principle is used to derive the governing equation of motion and the size effect is introduced into the formulation through the so-called strain gradient theory. The proposed model accounts for nonlinearities due to mid-plane stretching, curvature, and the electrostatic loading. The reduced order model is obtained via utilizing the Galerkin approach. A detailed study is carried out to reveal the influences of initial curvature, size effect, and piezoelectric actuation on the behavior of the system. The primary resonance is characterized using shooting method and the stability of the limit cycles is determined using the Floquet theorem. The results are compared in some cases with the method of multiple time scales. The static and dynamic snap-through are identified and the influences of the system parameters on the snap-through bandwidth is investigated. It is shown that by applying appropriate piezoelectric actuation, the bandwidth and center frequency of the dynamic snap-through as well as the static bistability band can be tuned effectively. The results can be used in design and analysis of bistable MEMS-based devices especially in switch and filter design applications.

Nonlinear vibration of a fractional viscoelastic micro-beam is investigated in this paper. The Euler–Bernoulli beam theory and the nonlinear von Karman strain are used to model the beam. The small-scale effects are considered by employing the Modified Couple Stress Theory (MCST). The viscoelastic material of the beam is modeled via the fractional Kelvin–Voigt model. Utilizing the Hamilton’s Principle, a partial fractional differential equation is derived as the governing equation of motion. The Finite Difference Method (FDM) and the Galerkin method are used together for solving the partial fractional differential equation. The FDM is utilized to discretize the time domain, and the Galerkin method is employed to discretize the space domain. In this paper, the FDM and the Shooting method are coupled together to find the periodic solution of the fractional micro-beam and draw the corresponding amplitude–frequency curve. The effects of the order of the fractional derivative, viscoelastic model, and the micro-scale are studied numerically here in this study. Numerical simulations suggest that, the effect of the fractional derivative is very strong and must be considered for modeling the viscoelastic behavior; especially when the amplitude is high. Results also show that the effects of the nonlinear viscoelastic part are considerable when the amplitude is high; this may happen when the excitation frequency is near the natural frequency, at which the maximum amplitude occurs.

The longest reach problem of an elastic cantilevered rod subjected to a tip load is here revisited. It has been investigated for the first time recently (Wang, 2016). In this paper, two extensions of the problem are covered: the case of rods of extensible axis and of variable cross sectional area distribution. The problem is stated and solved numerically by means of the shooting method. Further distances with respect to those obtained in the literature can be reached employing decreasing bending rigidity along the rod axis.

This paper aims to solve the equations of geometrically exact 3D nonlinear Cosserat static rods under large displacements with a mesh free alternative method to finite elements: the shooting method. One of the main goal of the work presented is to introduce a new multiple shooting method to handle geometrical discontinuities as well as beams assembled through mechanical linkages in a multi-body framework. An additional purpose is to propose a new shooting method algorithm in which the dual set of Modified Rodrigues Parameters (MRP) and Shadow Modified Rodrigues Parameters (SMRP) are used for rotations parameterization along the length of the beam in order to model extremely large rotations. Several numerical examples demonstrate the validity, the performance and accuracy of the proposed approach.

In this paper, the buckling and post-buckling behavior of an elastic lattice system referred to as the discrete elastica problem is investigated using an equivalent non-local continuum approach. The geometrically exact post-buckling analysis of the elastic chain, also called Hencky system, is first numerically solved using the shooting method. This discrete physical model is also mathematically equivalent to a finite difference formulation of the continuum elastica. Starting from the exact difference equations of the discrete problem, a continualization method is applied for approximating the difference operators by differential ones, in order to better characterize the discrete system by an enriched continuous one. It is shown that the new continuum associated with the discrete system exactly fits the discrete elastica post-buckling problem, where the non-locality is of Eringen׳s type (also called stress gradient non-local model). An asymptotic expansion is performed for both the discrete and the non-local continuum models, in order to approximate the post-buckling branches of the discrete system. Some numerical investigations show the efficiency of the non-local approach, especially for capturing the scale effects inherent to the cell size of the lattice model.

Large-amplitude forced vibration, before damage onset, of variable stiffness composite laminated plates with curvilinear fibres are studied. The fibre paths considered change linearly in relation to one Cartesian coordinate. The plates are rectangular and with clamped edges. The displacement field is modelled by a third order shear deformation theory and the equations of motion, in the time domain, are obtained using a p -version finite element method. The in-plane inertia is neglected, still taking into consideration the in-plane displacements, and the model is statically condensed. The condensed model is transformed to modal coordinates in order to have a reduced model with a smaller number of degrees-of-freedom. A shooting method using fifth-order Runge–Kutta method, as well as adaptive stepsize control, is used to find periodic solutions of the equations of motion. Frequency-response curves of composite laminates with different curvilinear fibre angles and various thicknesses are plotted and compared. Tsai–Wu criterion is employed in order to predict the damage onset. When it is detected that damaged started, the continuation method is interrupted and no further points of the response curve are computed. The reason behind this interruption is that the model does not include the effects of damage. Examples of bifurcations are presented and studied in detail, using projections of trajectories in a phase plane and Fourier spectra. The time histories and frequency spectra of steady-state stresses are plotted for VSCL plates with different fibre angles. The steady-state stresses are also displayed for bifurcated branches of the solutions.

In this study, the early stages of a Soret-driven convection of a nanoparticles suspension with large negative separation ratio ψ confined within a Hele-Shaw cell or in a porous media, heated from above is investigated using linear and nonlinear analyses. The new stability equations are formulated in a similarity transformation ( τ , ζ ) -domain as well as a ( τ , z ) -domain by introducing a new characteristic dimensionless parameter R s H . With and without the quasi-steadiness assumptions, the resulting stability equations are solved analytically by expanding the disturbances as a series of orthogonal functions, and in addition, the numerical shooting method is used. The critical time of the onset of convection and the corresponding wave number are obtained as a function of R s H . It is found that the onset time of convective instability decreases with increasing R s H . The linear stability limits are independent of the solution methods and the coordinate system, if the trial functions for the disturbance quantities are properly chosen. Based on the results of the linear stability analysis, the nonlinear analyses are also tried by employing a finite volume method. The present nonlinear theory explained the linear stability theory well and also predicted the existing experimental results reasonably.

In book II of Newton's "Principia Mathematica" of 1687 several applicative problems are introduced and solved. There, we can find the formulation of the first calculus of variations problem that leads to the first free boundary problem of history. The general calculus of variations problem is concerned with the optimal shape design for the motion of projectiles subject to air resistance. Here, for Newton's optimal nose cone free boundary problem, we define a non-iterative initial value method which is referred in literature as transformation method. To define this method we apply invariance properties of Newton's free boundary problem under a scaling group of point transformations. Finally, we compare our non-iterative numerical results with those available in literature and obtained via an iterative shooting method. We emphasize that our non-iterative method is faster than shooting or collocation methods and does not need any preliminary computation to test the target function as the iterative method or even provide any initial iterate. Moreover, applying Buckingham Pi-Theorem we get the functional relation between the unknown free boundary and the nose cone radius and height., 17 pages, 4 figures, 2 tables

An approach is presented to study the nonlinear dynamic responses to rotating drill shaft with multi-span supports. Based on the finite element method, the drilling shaft is modeled as lots of 2-node Timoshenko shaft element model with free-interface that can take the effect of inertia and shear into consideration. In these cases, the governing equations of drilling shaft system consist of the coupled linear and nonlinear components. According to the feature of such systems, a modified transformation is introduced, by means of which the linear degrees of freedom of the drilling shaft system are reduced significantly whereas nonlinear degrees of freedom of the system are retained in the physical space. A modified Newton shooting method is used to obtain the periodic trajectories of the dynamic system. The advantage of this method has reduced much of the computational cost in the past, and the hydrodynamic forces of cutting fluid, cutting forces and unbalance forces can easily be added to the system equations. Further, the numerical schemes of this study are applied to a large-scale deep-hole drill machine with two intermediate supports. The periodic dynamic behaviors of the drilling shaft system and the region of unstable rotation are investigated numerically, whereby revealing some interesting phenomena.

The non-linear response of a buckled beam to a primary resonance of its first vibration mode in the presence of internal resonances is investigated. We consider a one-to-one internal resonance between the first and second vibration modes and a three-to-one internal resonance between the first and third vibration modes. The method of multiple scales is used to directly attack the governing integral–partial–differential equation and associated boundary conditions and obtain four first-order ordinary-differential equations (ODEs) governing modulation of the amplitudes and phase angles of the interacting modes involved via internal resonance. The modulation equations show that the interacting modes are non-linearly coupled. An approximate second-order solution for the response is obtained. The equilibrium solutions of the modulation equations are obtained and their stability is investigated. Frequency–response curves are presented when one of the interacting modes is directly excited by a primary excitation. To investigate the global dynamics of the system, we use the Galerkin procedure and develop a multi-mode reduced-order model that consists of temporal non-linearly coupled ODEs. The reduced-order model is then numerically integrated using long-time integration and a shooting method. Time history, fast Fourier transforms (FFT), and Poincare sections are presented. We show period doubling bifurcations leading to chaos and a chaotically amplitude-modulated response.

For power-law fluids we propose a Lie-group shooting method to tackle the boundary-layer problems under a suction/injection as well as a reverse flow boundary conditions. The Crocco transformation is employed to reduce the third-order equation to a second-order ordinary differential equation, and then through a symmetric extension to a boundary value problem with constant boundary conditions, which can be solved numerically by the Lie-group shooting method. However, the resulting equation is singular, which might be difficult to solve, and we propose a technique to overcome the initial impulse caused by the singularity using a very small time-step integration at the first few time steps. Because we can express the missing initial condition through a closed-form formula in terms of the weighting factor r ∈ ( 0 , 1 ) , the Lie-group shooting method is very effective for searching the multiple-solutions of a reverse flow boundary condition.

International audience; Large-amplitude non-linear vibrations of micro- and nano-electromechanical resonant sensors around their primary resonance are investigated. A comprehensive multiphysics model based on the Galerkin decomposition method coupled with the averaging method is developed in the case of electrostatically actuated clamped-clamped resonators. The model is purely analytical and includes the main sources of non-linearities as well as fringing field effects. The influence of the higher modes and the validation of the model is demonstrated with respect to the shooting method as well as the harmonic balance coupled with the asymptotic numerical method. This model allows designers to investigate the sensitivity variation of resonant sensors in the non-linear regime with respect to the electrostatic forcing.

The quasi-static response and the stored and dissipated energies due to large deflections of a slender inextensible beam made of a linear viscoelastic material and subjected to a time-dependent inclined concentrated load at the free end are investigated. The beam cross-section is considered prismatic, the self-weight is disregarded and the material is initially stress free. The set of four first-order non-linear partial integro-differential equations obtained from geometrical compatibility, equilibrium of forces and moments, and linear viscoelastic constitutive relation is numerically solved using a one-parameter shooting method combined with a fourth-order Runge–Kutta algorithm. An analytical expression is derived to divide the energy supplied by the external load into conserved and dissipated parts. For the case study presented, a three-parameter solid linear viscoelastic constitutive model is employed and a step load is applied. The variables are made non-dimensional, so four parameters govern the problem: the ratio between the final and initial relaxation moduli, the load magnitude, the angle of inclination and the unloading time. A finite-element model is also performed to compare and validate the analytical and numerical formulations. Results are presented for encastre curvature and tip displacement versus time, geometrical configuration, load versus tip displacement, total work done by the external force, stored and dissipated energies versus time, energy per unit length along arc length for three times and versus time for two beam sections.

For the Falkner–Skan equation, including the Blasius equation as a special case, we develop a new numerical technique, transforming the governing equation into a non-linear second-order boundary value problem by a new transformation technique, and then solve it by the Lie-group shooting method. The second-order ordinary differential equation is singular, which is, however, much saving computational cost than the original third-order equation defined in a semi-infinite range. In order to overcome the singularity we consider a perturbed equation. The newly developed Lie-group shooting method allows us to search a missing initial slope at the left-end in a compact space of t ∈ [ 0 , 1 ] , and moreover, the initial slope can be expressed as a closed-form function of r ∈ ( 0 , 1 ) , where the best r is determined by matching the right-end boundary condition. All that makes the new method much superior than the conventional shooting method used in the boundary layer equation under imposed boundary conditions. When the initial slope is available we can apply the fourth-order Runge–Kutta method to calculate the solution, which is highly accurate. The present method is very effective for searching the multiple-solutions under very complex boundary conditions of suction or injection, and also allowing the motion of plate.

Non-linear shooting and Adomian decomposition methods have been proposed to determine the large deflection of a cantilever beam under arbitrary loading conditions. Results obtained only due to end loading are validated using elliptic integral solutions. The non-linear shooting method gives accurate numerical results while the Adomian decomposition method yields polynomial expressions for the beam configuration. With high load parameters, occurrence of multiple solutions is discussed with reference to possible buckling of the beam-column. An example of concentrated intermediate loading (cantilever beam subjected to two concentrated self-balanced moments), for which no closed form solution can be obtained, is solved using these two methods. Some of the limitations and recipes to obviate these are included. The methods will be useful toward the design of compliant mechanisms driven by smart actuators.

This article presents the behavior of slender elastic rods subjected to axial terminal forces and self-weight. The mathematical formulation is presented, a solution is sought for a double-hinged boundary condition and the analysis is carried out for different values of non-dimensional weight. The formulation derives from geometrical compatibility, equilibrium of forces and moments and constitutive relations yielding a set of six first order non-linear ordinary differential equations with boundary conditions specified at both ends, which characterizes a complex two-point boundary value problem. Furthermore, a perturbation method is used to find the critical buckling loads and initial post-buckling solutions. A numerical integration scheme based on a three parameter shooting method is employed in the post-buckling solutions.

Heat and mass transfer along a semi-infinite vertical flat plate under the combined buoyancy force effects of thermal and species diffusion is investigated in the presence of a strong non-uniform magnetic field and the Hall currents are taken into account. The induced magnetic field due to the motion of the electrically conducting fluid is negligible. This assumption is valid for a small magnetic Reynolds number. The similarity solutions are obtained using the scale group of transformations. These are the only symmetry transformations admitted by the field equations. The non-linear boundary layer equations with the boundary conditions are transferred to a system of non-linear ordinary differential equations with the appropriate boundary conditions. Furthermore, the similarity equations are solved numerically by using a fourth order Runge–Kutta scheme with the shooting method. Numerical results for the velocity profiles, the temperature profiles and the concentration profiles are presented graphically for various values of the magnetic parameter M in the range of 0–1 with the Hall parameter m taking the values 0.5, 1, 2, and 3.

The inextensional, spacial deformation of a slender beam subjected to end loads and/or distributed loads can be characterized by a system of 12 coupled, non-linear ordinary differential equations, resulting in a complicated two-point boundary value problem. However, by an iterative shooting method, this problem is converted to an initial value problem, and a variety of existing computer programs can then be used to integrate the 12 equations. Two numerical examples are presented to demonstrate the method. Example 1 considers the finite deformation of a cantilever beam subjected to a concentrated end load. Example 2 considers the same beam subjected to a distributed load. A connection (and numerical check) is established with the corresponding linearized problems. In addition, each solution of the non-linear problem provides a quantitative assessment of the important effect of load misalignment.

Cylindrical shells of arbitrary wall thickness subjected to uniform radial tensile or compressive dead-load traction are investigated. The material of the shell is assumed to be homogeneous, isotropic, compressible and hyperelastic. The stability of the finitely deformed state and small, free, radial vibrations about this state are investigated using the theory of small deformations superposed on large elastic deformations. The governing equations are solved numerically using both the multiple shooting method and the finite element method. For the finite element method the commercial program ABAQUS is used. 1 The loss of stability occurs when the motions cease to be periodic. The effects of several geometric and material properties on the stress and the deformation fields are investigated.

The steady two-dimensional flow of an incompressible viscous and electrically conducting fluid over a non-linearly semi-infinite stretching sheet in the presence of a chemical reaction and under the influence of a magnetic field is analyzed. The equations governing the flow and concentration field are reduced to a system of coupled non-linear ordinary differential equations. These non-linear differential equations are solved numerically by using the shooting method. The numerical results for the concentration field are presented through graphs. (c) 2006 Elsevier Ltd. All rights reserved. International Journal of Non-Linear Mechanics

The effects of fluid viscoelasticity on the temperature and flow fields of the flow past a wedge were studied. The combined effects of the shape factor, suction/injection rates and viscoelasticity were analysed by the series expansion method, similarity transformation, fourthorder Runge-Kutta integration and the shooting method. The stream function was introduced into the momentum boundary layer equations of a second-grade fluid to eliminate the pressure terms. The energy equation is also analysed by the similarity transformation. The results indicated that the velocity profiles, temperature distributions, surface friction and heat transfer are greatly affected by the elastic parameter, the shape factor, the Prandtl number and the suction/injection rates.

Some results of qualitative investigations and of numerical solutions to the problem of axisymmetrical deformations of circular, geometrically non-linear, rotationally orthotropic plates are presented. The qualitative studies reveal certain general characteristics of behavior and proofs are obtained only on the basis of form, in analogy to the qualitative theory of differential equations. For numerical solutions we employ the shooting method in combination with the ‘deformation map’, which is similar to Poincare's phase plane.

It is shown that a general non-linear boundary value problem can be reduced to an initial value problem. This is important both conceptually and computationally.

A numerical method for computing stability boundaries for Hill's equation is presented.