Two-dimensional Nonlinear Model Research Articles

Stress–strain state of flexible ring plates of variable stiffness in a magnetic field

A nonlinear two-dimensional model of a magnetoelastic flexible current-carrying ring plate is developed.Asystem of nonlinear equations describing the stress-strain state of flexible current-carrying plates in nonstationary mechanical and electromagnetic fields is derived. The stress state of a flexible plate of variable stiffness in a magnetic field is determined

Derivation of a Hierarchy of Nonlinear Two-dimensional Models For Heterogeneous Plates

This paper is concerned with the asymptotic analysis of plates with periodically rapidly varying heterogeneities. The formal asymptotic procedure is performed when both the periods of changes of the material properties and the thickness of the plate are of the same orders of magnitude. Our approach is based on a sequence of recursive minimization problems. We consider a plate made of Ciarlet—Geymonat type materials. Depending on the order of magnitude of the applied loads, we obtain a nonlinear membrane model and a nonlinear membrane inextensional bending model as announced by Pruchnicki.

Two-dimensional nonlinear models for heterogeneous plates

We consider a formal asymptotic study of plates with periodically rapidly varying heterogeneities. The asymptotic analysis is performed when both the period of change of the material properties and the thickness of the plate are of the same orders of magnitude. We consider a plate made of Ciarlet–Geymonat type materials (P.G. Ciarlet and G. Geymonat (1982)). Depending on the order of magnitude of the applied loads, we obtain a nonlinear membrane model and a nonlinear membrane inextensional-bending model as announced in E. Pruchnicki (2006). Our approach is based on a sequence of recursive minimization problems. To cite this article: E. Pruchnicki, C. R. Mecanique 337 (2009).

Side effect of using response spectral amplification ratios for soil sites—variability and earthquake-magnitude and source-distance dependent amplification ratios for soil sites

We investigate a special type of variability in response spectral amplification ratios computed from numerical “engineering” models for a soft soil site. The engineering models are defined by shallow soil layers over “engineering” bedrock with a shear-wave velocity over 600–700 m/s and the model is subjected to vertical propagating shear waves. The variability, perhaps unique in earthquake engineering, is a result of the “perfectly accurate” computational procedure. For example, an engineering soil site model, subjected to two rock site records or the two horizontal components of a rock site record, produces different response spectral amplification ratios. We use a large number of strong-motion records from “engineering” rock sites, with a reasonably balanced distribution with respect to magnitude and source distance, generated by subduction earthquakes in Japan, to investigate the nature of the variability. In order to avoid any approximation in removing the effect of soil nonlinear response, we use a simple model, a single horizontal soil layer over a bedrock, modelled as elastic. We then demonstrate that a similar type of variability observed in the one- or two-dimensional nonlinear soil models is caused by the nature of response spectral amplification ratios, not a direct result of soil nonlinear response. Examination of variability reveals that the average of response spectral amplification ratios systematically depends on both earthquake magnitude and source distance. We find that, at periods much longer than the site natural periods of the soil sites, the scatter of the amplification ratios decreases with increasing magnitude and source distance. These findings may have a potential impact in establishing design spectra for soft soil sites using strong-motion attenuation models or dynamic numerical modelling.

UNSTEADY RESPONSE OF BLOOD FLOW THROUGH A COUPLE OF IRREGULAR ARTERIAL CONSTRICTIONS TO BODY ACCELERATION

A two-dimensional (2D) nonlinear mathematical model to study the response of the pulsatile flow of blood through a couple of irregular stenoses influenced by externally imposed periodic body acceleration is developed. The model is 2D and axisymmetric with an outline of the stenosis obtained from the three-dimensional (3D) casting of a mildly stenosed artery. The combined influence of an asymmetric shape and surface irregularities of the constrictions is explored in a computational study of blood flow through arterial stenoses with 48% areal occlusion. The arterial wall is treated as an elastic (moving wall) cylindrical tube having a couple of stenoses in its lumen, while the streaming blood is considered to be Newtonian. Solutions of the time-dependent nonlinear Navier–Stokes equations in the cylindrical coordinate system are obtained using a finite difference method based on the nonuniform and nonstaggered grids. The finite difference approximation helps to estimate the effects of body acceleration on the doubly constricted flow phenomena through several graphical representations quantitatively in order to validate the applicability of the present, improved mathematical model.

Determining the stress state of flexible orthotropic shells of revolution in magnetic field

A two-dimensional nonlinear magnetoelastic model of a current-carrying orthotropic shell of revolution is constructed taking into account finite orthotropic conductivity, permeability, and permittivity. It is assumed that the principal axes of orthotropy are aligned with the coordinate axes and that the orthotropic body is magnetically and electrically linear. The coupled nonlinear differential equations derived describe the stress-strain state of flexible current-carrying orthotropic shells of revolution that have an arbitrarily shaped meridian and orthotropic conductivity and are in nonstationary mechanical and electromagnetic fields. A method to solve this class of problems is proposed

Flutter margin with non-linearities: Real-time prediction of flutter onset speed

The flight envelope states the conditions where an aircraft is safe to fly and those conditions must be carefully determined for every configuration. That is why, even today, the numerical tools have become more and more precise, flight test is the only reliable method to define the flight envelope, especially in the conditions where the numerical model reaches a great degree of complexity (non-linearities). However, flight test is expensive and different procedures must arise in order to match reliability (safety) and cost (flight hours). This dilemma is even worse when free-flutter conditions must be provided for a new aircraft or a new external configuration, because the envelope expansion is done by increasing the flight test hours. The present article shows a procedure to predict the flutter speed based on real-time tuning of a quasi non-linear aeroelastic model. A two-dimensional non-linear (freeplay) aeroeslastic model is implemented in MatLab/Simulink with incompressible aerodynamic conditions. A comparison with real compressible conditions is provided. Once the numerical validation is accomplished, a parametric aeroelastic model is built in order to describe the proposed procedure and contribute to reduce the number of flight hours needed to expand the flutter envelope.

Numerical Simulation of a Flapping Foil with Chordwise or Spanwise Flexibility

Motivated by the discovery that living creatures may be able to enhance their locomotion capability through passive or active deformations of their wings or fins, we carry out a fully coupled fluid-structure interaction study to investigate the flapping motion of a foil with either chordwise or spanwise flexibility. We employ a fluid-structure interaction model, which accounts for fluid dynamics using the boundary-element method and structural dynamics using a two-dimensional nonlinear thin-plate model. With this approach, we numerically investigate the effect of structural deformation on the performance of the foil when it is immersed in two different fluids, a low-density fluid (e.g., air), in which the deformation is determined primarily by the inertia of the foil, and a high-density fluid (e.g., water), in which the fluid loading has a significant impact. In the first scenario, we find that the chordwise flexibility reduces both the thrust and the propulsion efficiency and the spanwise flexibility increases the thrust without efficiency reduction within a small range of structural parameters. In the second scenario, we find that the chordwise flexibility increases the efficiency. The spanwise flexibility, on the other hand, compromises the performance of the foil by diminishing both the thrust and the efficiency. Possible applications of these findings and their relation to designs in aquatic animals are discussed.

Comparison of Residual Stresses in the Mechanical Roll Expansion of HX Tubes Into TEMA Grooves

The mechanical roll expansion of heat exchanger tubes into tubesheets containing TEMA grooves, which are thought to aid in the mechanical integrity of the tube-to-tubesheet (TTT) joint, has for many years provided an acceptable means of completing a TTT joint. Inherent with the intentional roll expansion of the tube is the creation of a tensile residual stress field within the tube that is greatest in the transition region between the expanded and unexpanded zones of the tube. An additional complicating factor in the tube-to-tubesheet joint design is the choice of utilizing a seal weld or a “full strength” weld at the tube end in conjunction with a level of roll expansion quantified by the degree of tube wall reduction. This paper presents the results of an initial study of the mechanical roll expansion of 1in. diameter tubes into a typical TEMA-R designed tubesheet, utilizing two grooves in the tubesheet hole. Two combinations of tube and tubesheet materials are studied that include duplex stainless steel tubes and tubesheet, while the second combination includes type 321 tubes roll expanded into a 2‐1∕4 Cr-1 Mo tubesheet, clad with 321 SS overlay. The residual stress fields are calculated by the finite element method and employ a simplified two-dimensional nonlinear axisymmetric model.

Effect of hydrodynamic properties of the sea bottom on the tidal dynamics in a rectangular basin

Expressions derived for the friction coefficient in an oscillatory rotating turbulent bottom boundary layer (BBL) over rough, incompletely rough (smoothly rough), and smooth underlying surfaces are incorporated as an individual module into a two-dimensional nonlinear tidal model, and the standard version of the model and its modified analogue are used to discuss the titular subject. It is established that the dynamics of tides in the Taylor basin can change noticeably under the effect of hydrodynamic properties of the sea bottom. Such changes occur mainly in the influence domains of amphidromies. In the remaining parts of the basin, relative changes in the amplitudes and phases of tidal sea-surface level elevations do not exceed ±10% and ±10°, respectively. The largest discrepancies of tidal characteristics take place in the cases of the incompletely rough and smooth sea bottoms; the smallest discrepancies, in the case of the rough sea bottom. Estimates for the changes in tidal characteristics that are caused by the usually neglected effects of rotation and phase difference between the bottom friction and the tidal velocity at the upper BBL boundary are presented as well.

Nonlinear shallow-water solutions using the weak temperature gradient approximation

A form of the weak temperature gradient (WTG) approximation, in which the temperature tendency and advection terms are neglected in the temperature equation so that the equation reduces to a diagnostic balance between heating and vertical motion, is applied to a two-dimensional nonlinear shallow-water model with the heating (mass source) parameterized as a Newtonian relaxation on the temperature (layer thickness) towards a prescribed function of latitude and longitude, containing an isolated maximum or minimum, as in the classic linear Gill problem. In this model, temperature variations are retained in the Newtonian heating term, so that it is not a pure WTG system. It contains no free unbalanced modes, but reduces to the Gill model in the steady linear limit, so that steady solutions may be thought of as containing components corresponding to unbalanced modes in the same sense as the latter. The equations are solved numerically and are compared with full shallow-water solutions in which the WTG approximation is not made. Several external parameters are varied, including the strength, location, sign, and horizontal scale of the mass source, the Rayleigh friction coefficient, and the time scale for the relaxation on the mass field. Indices of the Walker and Hadley circulations are examined as functions of these external parameters. Differences between the WTG solutions and those from the full shallow-water system are small over most of the parameter regime studied, which includes time-dependent as well as steady solutions.

One-sided invariant manifolds, recursive folding, and curvature singularity in area-preserving nonlinear maps with nonuniform hyperbolic behavior

Two-dimensional nonlinear models of conservative dynamics are typically nonuniformly hyperbolic in that there are nonhyperbolic trajectories that coexist with a “massive” hyperbolic region. We investigate the influence of nonhyperbolic points on the global geometric structure of a invariant manifolds associated with points of the hyperbolic region. As a case study, we consider a transformation of the Standard Map family and analyze the structure of invariant manifolds in the neighborhood of an isolated parabolic (fixed) point xp. This analysis shows the existence of lobes enclosing the parabolic point, that is, of simply connected regions containing xp whose boundary is formed by two continuous arcs of stable and unstable manifolds that intersect only at two points. From the existence of such regions, we derive that (i) there are points of the hyperbolic region where the local curvature of invariant manifolds is arbitrarily large and (ii) manifolds possess the recursively folding property. Property (ii) means that given an invariant manifold W and established an orientation on it, in the neighborhood of any point of the chaotic region there are nearby arcs of W that are traveled in opposite directions. We propose an archetypal model for which the existence of lobes and the recursive folding property can be derived analytically. The impact of nonuniform hyperbolicity on the evolution of physical processes that occur along with phase space mixing is also addressed.

Minimizing manoeuvre advantage requirements for a hit-to-kill interceptor

Hit-to-kill requires extremely high intercept accuracies. Such performance places severe demands on interceptor manoeuvrability and flight control response time constant when engaging targets that are intentionally or unintentionally manoeuvring. A well-known guidance law for use against a manoeuvring target is augmented proportional navigation (APN). It is an optimal guidance law under various assumptions including a constant-lateral-acceleration target. It is also effective when used against other target manoeuvres provided that the lateral acceleration capabilities of the interceptor is significantly greater than the lateral acceleration capabilities of the target, and if the time constant of the interceptor flight control system is small. The advantage of using a new guidance algorithm for a hit-to-kill interceptor against a manoeuvring target in two dimensions is demonstrated in this paper. A large manoeuvre advantage is not required, and it is effective with time constants that would otherwise degrade the performance of APN. Results are based on a two-dimensional non-linear model with the target performing a weave manoeuvre. The new algorithm can achieve increased intercept accuracy under minimal manoeuvre advantage requirements over a wide range of initial conditions with reasonable levels of angle noise and target manoeuvre estimation error.

Non-linear two-dimensional model of melt flows and interface instability in aluminum reduction cells

We derive a new two-dimensional model for the melt flows and interface instability in aluminum reduction cells. The model is based on the de St. Venant shallow water equations and incorporates the essential features of the system such as the magnetohydrodynamic instability mechanism and non-linear coupling between the flows and interfacial waves. The model is applied to investigate the impact of background melt flows and magnetic field perturbations on the instability.

Flow over Small Heat Islands: A Numerical Sensitivity Study

Abstract A two-dimensional nonlinear model with physical parameterizations was applied to simulate the observed diurnal variation on the 5-km-wide flat tropical island of Nauru in the trade wind zone. Both the model and Atmospheric Radiation Measurement (ARM) campaign aircraft observations indicate vigorous mixing in the typical sunny daytime conditions, leading to a warm plume downstream of the island. The model's afternoon wind field displayed rising motion downstream and downwash ahead of the island with gravity wave structure, in accordance with linear models of steady flow over a heated island. The roughness difference between sea and land added local rising motion above the windward coast and sinking motion above the lee. Without large-scale wind U, a weakish sea-breeze (SB) pattern develops during the day in this model over the 5-km-wide island/peninsula. This pure SB circulation intensifies with increasing island width up to 40 km. When large-scale wind is present, the morning leeside SB cell is a...

Modulation instability in two-dimensional nonlinear Schrödinger lattice models with dispersion and long-range interactions

The problem of modulation instability of continuous wave and array soliton solutions within the framework of a two-dimensional continuum-discrete nonlinear Schrodinger lattice model, which accounts for dispersion and long-range interactions between elements, is investigated. The linear stability analysis based on an energy principle and a variational approach, which were originally developed for the continuum nonlinear Schrodinger model, is proposed. Regions of instability are identified and analytical expressions for the corresponding thresholds and the growth rate spectra are calculated.

An investigation of spatial and temporal weighting schemes for use in unstructured mesh control volume finite element methods

In this work the control volume finite element (CVFE) method is used to discretise a two-dimensional non-linear transport model on an unstructured mesh. First and second order temporal weighting, combined with various flux limiting techniques (spatial weighting) are analysed in order to identify the most accurate and efficient numerical scheme. An inexact Newton method is used to resolve the underlying discrete non-linear system. In computing the Newton step the performance of the preconditioned iterative solvers BiCGSTAB and GMRES, in conjunction with a two-node Jacobian approximation is also examined. A linear benchmark problem that admits an analytical solution is used to assess the accuracy and computational efficiency of the numerical model. The results show that the flux limited second order temporal scheme substantially reduce numerical diffusion on relatively coarse meshes. The low temperature wood drying non-linear case study highlights that the first order temporal scheme combined with flux limiting achieves good accuracy on a relatively coarse mesh and improves the overall computational efficiency.

Two-dimensional non-linear model of reversed field pinch plasma evolution

A two-dimensional non-linear equilibrium code, able to simulate reversed field pinch (RFP) discharges, has been developed. It is based on the two-dimensional MHD equilibrium code MAXFEA, which has been properly modified by introducing RFP profiles of the poloidal current density flux function in accordance with the μ&p model. In particular, equation parameters consistent with available data from the RFX machine have been used. A complete mesh of the RFX poloidal field system has been prepared and the whole poloidal electrical circuit has been implemented into the code. The RFP equilibrium code has been successfully validated against some RFX experiments, with and without plasma.

Nonlinearly elastic membrane model for heterogeneous plates: a formal asymptotic approach by using a new double scale variational formulation

This paper is concerned with the asymptotic analysis of plates with periodically rapidly varying heterogeneities. The asymptotic analysis is performed when both the material properties and the thickness of the plate are of the same orders of magnitude. We consider a plate made of Saint Venant–Kirchhoff type materials then we justify a new two-scale variational formulation. We assume that both the data and the displacement field admit a formal asymptotic expansion with a negative order of the leading term. We then prove that the lowest order term of the displacement field must be of order zero. Finally, we consider the particular case of a laminated plate clamped along its lateral boundary and we show that this term satisfies a two-dimensional nonlinear membrane model.

Nonlinear two-dimensional model for thermoacoustic engines.

A two-dimensional model and efficient solution algorithm are developed for studying nonlinear effects in thermoacoustic engines. There is no restriction on the length or location of the stack, and the cross-sectional area of the resonator may vary with position along its axis. Reduced model equations are obtained by ordering spatial derivatives in terms of rapid variations across the pores in the stack, versus slow variations along the resonator axis. High efficiency is achieved with the solution algorithm because the stability condition for numerical integration of the model equations is connected with resonator length rather than pore diameter. Computation time is reduced accordingly, by several orders of magnitude, without sacrificing spatial resolution. The solution algorithm is described in detail, and the results are verified by comparison with established linear theory. Two examples of nonlinear effects are investigated briefly, the onset of instability through to saturation and steady state, and nonlinear waveform distortion as a function of resonator shape.