The nonlinear waves described by the nonlinear equation are numerically investigated. With the aid of atomistic multiscale modelling and analytical approaches, buckling strength has been determined for carbon nanofibres/epoxy composite systems. Various nanofibres configurations considered are single walled carbon nano tube and single layer graphene sheet and SLGS/SWCNT hybrid systems. Computationally, both eigen-value and non-linear large deformation-based methods have been employed to calculate the buckling strength.

What is multiple scale method

This is done by introducing fast-scale and slow-scale variables for an independent variable, and subsequently treating these variables, fast and slow, as if they are independent. In the solution process of the perturbation problem thereafter, the resulting additional freedom – introduced by the new independent variables – is used to remove secular terms. The latter puts constraints on the approximate solution, which are called solvability conditions. It is observed in Figure 7 that the mode in the two junction types does not oscillate with an unbounded or growing amplitude. After a while, there is a balance of energy input into the breathing mode due to the external drive and the radiative damping. This result shows the regular oscillation of the mode that the junction voltage disappears even at the condition when the driving frequency is same as the eigen-frequency.

Multiscale Modeling of Hybrid Machining Processes

Further, we also describe the breathing modes for various order of perturbation. At the end, we compare the solutions obtained via perturbation and numerical methods of parametric-driven sine-Gordon equation with phase shifts. Finally, we concluded that the modes of the breathing decay to a constant in both cases. Beginning with new material on the development of cutting-edge asymptotic methods and multiple scale methods, the book introduces this method in time domain and provides examples of vibrations of systems. Clearly written throughout, it uses innovative graphics to exemplify complex concepts such as nonlinear stationary and nonstationary processes, various resonances and jump pull-in phenomena. It also demonstrates the simplification of problems through using mathematical modelling, by employing the use of limiting phase trajectories to quantify nonlinear phenomena.

Following the review, we highlight a new multiple-scale method, the bridging scale, and compare it to existing multiple-scale methods. Next, we show example problems in which the bridging scale is applied to fully non-linear problems. Concluding remarks address the research needs for multiple-scale methods in general, the bridging scale method in particular, and potential applications for the bridging scale. We develop two new equations which describe propagation of two different wave modes simultaneously.

An atomistic-based finite deformation membrane for single layer crystalline films

A new phenomenon was revealed for long-time resonant energy exchange in carbon nanotubes with radial breathing mode. The modified nonlinear Schrödinger equation describes the nonlinear dynamics of CNTs given in Smirnov and Manevitch . An initial value problem of Hamilton’s principle applied to nonconservative systems, was proposed for complex partial differential equations of the NLS type equation. In the study by Rossi et al. , dynamics of coherent solitary wave structures of NCVA was examined, and also it is proved that the NLSr and linear perturbative variation equations are equivalent through nonconservative variation methods.

What is multiple scale method

The mode shapes of the microbeam are found to be generally three-dimensional spatial in the presence of the longitudinal magnetic field. It is interesting that buckling instability would concurrently occur in the first mode or in the higher-order modes when the magnetic field parameter becomes sufficiently large. We present a detailed introduction to the available technologies in the field of multiple-scale analysis. In particular, our review centers on methods which aim to couple molecular-level simulations to continuum level simulations . Using this definition, we first review existing multiple-scale technology, and explain the pertinent issues in creating an efficient yet accurate multiple-scale method.

Fast homogenization through clustering-based reduced-order modeling

Non-linear wave processes on the surface of shallow water under a layer of ice are considered taking bending deformations and tension compression into account. A closed system of equations in the water level perturbations and the velocity potential is derived to describe them. A periodic solution of the equation obtained is constructed, expressed in terms of Weierstrass elliptic functions.

  • We will use the Cole-Hopf transformation combined with the simplified Hirota’s method to conduct this analysis.
  • Here, it is extended to study the Heisenberg operator equations of motion and the Schrödinger equation for the quantum anharmonic oscillator.
  • Due to this, the driving frequency appears in the higher order and oscillate the system rapidly.
  • It is shown that, for periodic and solitary waves, two forms of wave profiles exist depending on the parameters of the mathematical model.
  • The compensation of radiation loses along with additional dissipative loses were studied by resonant drive of kink.
  • This is firstly demonstrated for a symmetric system, where an isola envelops the secondary backbone curves, which emerge from a bifurcation.

The material properties of coarse-scale are modeled with the nonlinear finite element method, in which the stress tensor and tangent modulus are computed using the Hill-Mandel principal through the atomistic RVE. In order to clearly represent the mechanical behavior of the fine-scale, the stress-strain curves of the atomistic RVE undergoing distinct type of deformation modes are delineated. These results are then assessed to obtain the proper fine-scale parameters for the multi-scale analysis. Finally, several numerical examples are solved to illustrate the capability of the proposed computational algorithm.

Pulse propagation in a nonlinear viscoelastic rod of finite length

Are nonzero constants of integration that are also determined by the continuity conditions at the discontinuous points. Are determined by applying the continuity conditions at the discontinuity points. Sensitivity analysis of the frequency response of a piecewise linear system in a frequency island.

What is multiple scale method

Multiple-scale analysis is employed for the study of nonlinear wave propagation in periodic layered media. In a first step, wave propagation in each individual layer is modeled by a corresponding equivalent nonlinear transmission line. The multiple-scale analysis is then employed to establish a system of nonlinear equations for the amplitudes of the forward and backward waves in the transmission lines of the model mentioned. This system of nonlinear equations is solved with the aid of a continuation technique for derivation of transmission characteristics of the periodic structure. To study optical bistability, a parameter-switching algorithm is utilized for obtaining the solutions of the system including both stable and unstable ones. For the sake of verification, we have also utilized a nonlinear finite-difference time-domain method to analyze the wave propagation in the aforementioned structure.

Soliton models of long internal waves

The variations of these parameters with the wave number k and the other physical parameters are discussed and the possibility of occurence of modulational instability is indicated. Multiple-scale analysis is a global perturbation scheme that is useful in systems characterized by disparate time scales, such as weak dissipation in an oscillator. These effects could be insignificant on short time scales but become important on long time scales. Classical perturbation methods generally break down because of resonances that lead to what are called secular terms.

What is multiple scale method

Here, it is extended to study the Heisenberg operator equations of motion and the Schrödinger equation for the quantum anharmonic oscillator. In the former case, it leads to a system of coupled operator differential equations, which is solved exactly. In the latter case, multiple-scale analysis elucidates the connection between weak-coupling perturbative and semiclassical nonperturbative aspects of the wave function. In this paper, a hierarchical RVE-based continuum-atomistic multi-scale procedure is developed to model the nonlinear behavior of nano-materials. The atomistic RVE is accomplished in consonance with the underlying atomistic structure, and the inter-scale consistency principals, i.e. kinematic and energetic consistency principals, are exploited. To ensure the kinematic compatibility between the fine- and coarse-scales, the implementation of periodic boundary conditions is elucidated for the fully atomistic method.