scholarly journals Nonlinear higher-order spectral solution for a two-dimensional moving load on ice

2009 ◽  
Vol 621 ◽  
pp. 215-242 ◽  
Author(s):  
FÉLICIEN BONNEFOY ◽  
MICHAEL H. MEYLAN ◽  
PIERRE FERRANT
Keyword(s):  
Critical Speed ◽  
Moving Boundary ◽  
Moving Load ◽  
Phase Speed ◽  
Nonlinear Effects ◽  
Higher Order ◽  
Two Dimensions ◽  
Linear Solution ◽  
Nonlinear Solution ◽  
Bernoulli’S Equation

We calculate the nonlinear response of an infinite ice sheet to a moving load in the time domain in two dimensions, using a higher-order spectral method. The nonlinearity is due to the moving boundary, as well as the nonlinear term in Bernoulli's equation and the elastic plate equation. We compare the nonlinear solution with the linear solution and with the nonlinear solution found by Parau & Dias (J. Fluid Mech., vol. 460, 2002, pp. 281–305). We find good agreement with both solutions (with the correction of an error in the Parau & Dias 2002 results) in the appropriate regimes. We also derive a solitary wavelike expression for the linear solution – close to but below the critical speed at which the phase speed has a minimum. Our model is carefully validated and used to investigate nonlinear effects. We focus in detail on the solution at a critical speed at which the linear response is infinite, and we show that the nonlinear solution remains bounded. We also establish that the inclusion of nonlinearities leads to significant new behaviour, which is not observed in the linear solution.

Nonlinear effects in the response of a floating ice plate to a moving load

2002 ◽  
Vol 460 ◽  
pp. 281-305 ◽  
Author(s):  
EMILIAN PĂRĂU ◽  
FREDERIC DIAS
Keyword(s):  
Shallow Water ◽  
Deep Water ◽  
Critical Speed ◽  
Moving Load ◽  
Ice Sheet ◽  
Phase Speed ◽  
Nonlinear Effects ◽  
Line Load ◽  
Weakly Nonlinear ◽  
Water Depths

The steady response of an infinite unbroken floating ice sheet to a moving load is considered. It is assumed that the ice sheet is supported below by water of finite uniform depth. For a concentrated line load, earlier studies based on the linearization of the problem have shown that there are two ‘critical’ load speeds near which the steady deflection is unbounded. These two speeds are the speed c0 of gravity waves on shallow water and the minimum phase speed cmin. Since deflections cannot become infinite as the load speed approaches a critical speed, Nevel (1970) suggested nonlinear effects, dissipation or inhomogeneity of the ice, as possible explanations. The present study is restricted to the effects of nonlinearity when the load speed is close to cmin. A weakly nonlinear analysis, based on dynamical systems theory and on normal forms, is performed. The difference between the critical speed cmin and the load speed U is taken as the bifurcation parameter. The resulting normal form reduces at leading order to a forced nonlinear Schrödinger equation, which can be integrated exactly. It is shown that the water depth plays a role in the effects of nonlinearity. For large enough water depths, ice deflections in the form of solitary waves exist for all speeds up to (and including) cmin. For small enough water depths, steady bounded deflections exist only for speeds up to U*, with U* < cmin. The weakly nonlinear results are validated by comparison with numerical results based on the full governing equations. The model is validated by comparison with experimental results in Antarctica (deep water) and in a lake in Japan (relatively shallow water). Finally, nonlinear effects are compared with dissipation effects. Our main conclusion is that nonlinear effects play a role in the response of a floating ice plate to a load moving at a speed slightly smaller than cmin. In deep water, they are a possible explanation for the persistence of bounded ice deflections for load speeds up to cmin. In shallow water, there seems to be an apparent contradiction, since bounded ice deflections have been observed for speeds up to cmin while the theoretical results predict bounded ice deflection only for speeds up to U* < cmin. But in practice the value of U* is so close to the value of cmin that it is difficult to distinguish between these two values.

Nonlinear wave packets in the Kelvin-Helmholtz instability

1979 ◽  
Vol 290 (1377) ◽  
pp. 639-681 ◽  
Keyword(s):  
Numerical Solutions ◽  
Initial Conditions ◽  
Density Ratio ◽  
Nonlinear Effects ◽  
Amplitude Equation ◽  
Two Dimensions ◽  
Spatial Modulation ◽  
Linear Solution ◽  
Nonlinear Stabilization ◽  
Nonlinear Solutions

The instability of two layers of immiscible inviscid and incompressible fluids in relative motion is studied with allowance for small, but finite, disturbances and for spatial as well as temporal development. By using the method of multiple scaling, a generalized formulation of the amplitude equation is obtained, applicable to both stable and marginally unstable regions of parameter space. Of principal concern is the neighbourhood of the critical point for instability, where weakly nonlinear solutions can be found for arbitrary initial conditions. Among the analytical results, it is shown that (1) the nonlinear effects can be stabilizing or destabilizing depending on the density ratio, (2) the existence of purely spatial instability depends upon the frame of reference, the density ratio, and whether the nonlinear effects are stabilizing, (3) exact nonlinear solutions of the amplitude equation exist representing modulations of permanent form travelling faster than the signal velocity of the linear equation (in particular, a solution is found that represents a solitary wave packet), and (4) the linear solution to the impulsive initial value problem has 'fronts’ which travel with the two (multiple) values of the group velocity (the packet as a whole moves with the mean of the two values). Numerical solutions of the amplitude equation (a nonlinear, unstable Klein-Gordon equation) are also presented for the case of nonlinear stabilization. These show that the development of a localized disturbance, in one or two dimensions, is highly dependent on the precise form of the initial conditions, even when the initial amplitude is very small. The exact solutions mentioned above play an important role in this development. The numerical experiments also show that the familiar uniform solution, an oscillatory function of time only, is unstable to spatial modulation if the amplitude of oscillation is large enough.

Wind-forced linear and nonlinear Kelvin waves along an irregular coastline

1977 ◽  
Vol 83 (2) ◽  
pp. 337-348 ◽  
Author(s):  
Allan J. Clarke
Keyword(s):  
Boundary Layer ◽  
Perturbation Theory ◽  
Phase Speed ◽  
Nonlinear Effects ◽  
Kelvin Waves ◽  
Ekman Transport ◽  
Linear Solution ◽  
Main Effect ◽  
Speed Change ◽  
Linear And Nonlinear

Using a normal and tangential co-ordinate approach, a perturbation theory is developed for wind-forced linear and nonlinear Kelvin waves propagating along an irregular coastline. The theory is valid for coastline curvatures which are non-dimensionally small, the curvature being non-dimensionalized with respect to the reciprocal of the boundary-layer trapping scale, i.e. the reciprocal of the radius of deformation. According to linear theory, the main effect of a coastline of small curvature is to cause a phase-speed change in the wave (from – c to – c(1 – ½k(s)), where k(s) is the non-dimensional curvature a distance s along the coast from the origin) and to make the offshore Ekman transport change more rapidly along the coast, the latter effect implying a more ‘wavelike’ ocean or lake response. Two discernible nonlinear effects were found to be an increase (decrease) in the linear-solution longshore gradients in regions of positive (negative) isopycnal displacement and a tendency for increased (decreased) isopyncal displacement at capes (bays).

scholarly journals Meshless Local Petrov–Galerkin Formulation of Inverse Stefan Problem via Moving Least Squares Approximation

2019 ◽  
Vol 24 (4) ◽  
pp. 101
Author(s):  
A. Karami ◽  
Saeid Abbasbandy ◽  
E. Shivanian
Keyword(s):  
Free Boundary ◽  
Least Squares ◽  
Boundary Problem ◽  
Moving Boundary ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Moving Least Squares ◽  
Heaviside Step Function ◽  
Mlpg Method ◽  
Mls Approximation

In this paper, we study the meshless local Petrov–Galerkin (MLPG) method based on the moving least squares (MLS) approximation for finding a numerical solution to the Stefan free boundary problem. Approximation of this problem, due to the moving boundary, is difficult. To overcome this difficulty, the problem is converted to a fixed boundary problem in which it consists of an inverse and nonlinear problem. In other words, the aim is to determine the temperature distribution and free boundary. The MLPG method using the MLS approximation is formulated to produce the shape functions. The MLS approximation plays an important role in the convergence and stability of the method. Heaviside step function is used as the test function in each local quadrature. For the interior nodes, a meshless Galerkin weak form is used while the meshless collocation method is applied to the the boundary nodes. Since MLPG is a truly meshless method, it does not require any background integration cells. In fact, all integrations are performed locally over small sub-domains (local quadrature domains) of regular shapes, such as intervals in one dimension, circles or squares in two dimensions and spheres or cubes in three dimensions. A two-step time discretization method is used to deal with the time derivatives. It is shown that the proposed method is accurate and stable even under a large measurement noise through several numerical experiments.

Impact of higher order nonlinear effects on modulational instability and pulse train generation in birefringent Lakshmanan–Porsezian–Daniel model

Optik ◽  
2021 ◽  
pp. 168462
Author(s):  
Gawarai Dieu-donne ◽  
C.G. Latchio Tiofack ◽  
Malwe Boudoue Hubert ◽  
Gambo Betchewe ◽  
Doka Yamigno Serge ◽  
...  
Keyword(s):  
Pulse Train ◽  
Modulational Instability ◽  
Nonlinear Effects ◽  
Higher Order ◽  
Pulse Train Generation

scholarly journals Nonperturbative Kinetic Description of Electron-Hole Excitations in Graphene in a Time Dependent Electric Field of Arbitrary Polarization

Particles ◽  
2019 ◽  
Vol 2 (2) ◽  
pp. 208-230 ◽  
Author(s):  
Stanislav A. Smolyansky ◽  
Anatolii D. Panferov ◽  
David B. Blaschke ◽  
Narine T. Gevorgyan
Keyword(s):  
Electric Field ◽  
Electric Fields ◽  
Kinetic Equations ◽  
Nonlinear Effects ◽  
Two Dimensions ◽  
Time Dependent ◽  
Polarization Current ◽  
Kinetic Description ◽  
Orthogonal Direction ◽  
Electron Hole

On the basis of the well-known kinetic description of e − e + vacuum pair creation in strong electromagnetic fields in D = 3 + 1 QED we construct a nonperturbative kinetic approach to electron-hole excitations in graphene under the action of strong, time-dependent electric fields. We start from the simplest model of low-energy excitations around the Dirac points in the Brillouin zone. The corresponding kinetic equations are analyzed by nonperturbative analytical and numerical methods that allow to avoid difficulties characteristic for the perturbation theory. We consider different models for external fields acting in both, one and two dimensions. In the latter case we discuss the nonlinear interaction of the orthogonal currents in graphene which plays the role of an active nonlinear medium. In particular, this allows to govern the current in one direction by means of the electric field acting in the orthogonal direction. Investigating the polarization current we detected the existence of high frequency damped oscillations in a constant external electric field. When the electric field is abruptly turned off residual inertial oscillations of the polarization current are obtained. Further nonlinear effects are discussed.

Effect of a Nonlinear Term in a Strain-Displacement Relationship on the Load-Displacement Path of Von Mises Truss

2015 ◽  
Vol 769 ◽  
pp. 85-90
Author(s):  
Jozef Havran ◽  
Martin Psotny
Keyword(s):  
Load Level ◽  
Iteration Algorithm ◽  
Arc Length ◽  
Linear Solution ◽  
Nonlinear Solution ◽  
Total Potential ◽  
Theoretical Approaches ◽  
Newton Raphson ◽  
Von Mises ◽  
Ansys System

Von Misses truss is one of the best examples to explain different theoretical approaches, nature of non-linear solution, define the snap-through, illustrate interactive buckling, etc. The presented paper compares two nonlinear approaches to the problem. Effect of nonlinear terms in strain-displacement relationship on the load level in critical point of nonlinear solution is analyzed. To obtain the nonlinear equilibrium paths, the Newton-Raphson iteration algorithm is used. Corresponding levels of the total potential energy are defined. The peculiarities of the effects of the initial imperfections are investigated. Custom FEM computer program has been used for analysis. Full Newton-Raphson procedure, in which the stiffness matrix is updated at every equilibrium iteration, has been applied. Obtained results are compared with results of the nonlinear analysis using ANSYS system, element type BEAM3 is used. The arc-length method is chosen for analysis, the reference arc-length radius is calculated from the load increment. Only fundamental path of nonlinear solution has been presented.

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