Numerical methods in Nonlinear Analysis of shell structures

The design of shell and spatial structures represents an important challenge even with the use of the modern computer technology.If we concentrate in the concrete shell structures many problems must be faced,such as the conceptual and structural disposition, optimal shape design, analysis, construction methods, details etc. and all these problems are interconnected among them. As an example the shape optimization requires the use of several disciplines like structural analysis, sensitivity analysis, optimization strategies and geometrical design concepts. Similar comments can be applied to other space structures such as steel trusses with single or double shape and tension structures. In relation to the analysis the Finite Element Method appears to be the most extended and versatile technique used in the practice. In the application of this method several issues arise. First the derivation of the pertinent shell theory or alternatively the degenerated 3-D solid approach should be chosen. According to the previous election the suitable FE model has to be adopted i.e. the displacement,stress or mixed formulated element. The good behavior of the shell structures under dead loads that are carried out towards the supports by mainly compressive stresses is impaired by the high imperfection sensitivity usually exhibited by these structures. This last effect is important particularly if large deformation and material nonlinearities of the shell may interact unfavorably, as can be the case for thin reinforced shells. In this respect the study of the stability of the shell represents a compulsory step in the analysis. Therefore there are currently very active fields of research such as the different descriptions of consistent nonlinear shell models given by Simo, Fox and Rifai, Mantzenmiller and Buchter and Ramm among others, the consistent formulation of efficient tangent stiffness as the one presented by Ortiz and Schweizerhof and Wringgers, with application to concrete shells exhibiting creep behavior given by Scordelis and coworkers; and finally the development of numerical techniques needed to trace the nonlinear response of the structure. The objective of this paper is concentrated in the last research aspect i.e. in the presentation of a state-of-the-art on the existing solution techniques for nonlinear analysis of structures. In this presentation the following excellent reviews on this subject will be mainly used.

Energy-consistent time integration for nonlinear viscoelasticity

This work is concerned with the numerical solution of the evolution equations of thermomechanical systems, in such a way that the scheme itself satisfies the laws of thermodynamics. Within this framework, we present a novel integration scheme for the dynamics of viscoelastic continuum bodies in isothermal conditions. This method intrinsically satisfies the laws of thermodynamics arising from the continuum, as well as the possible additional symmetries. The resulting solutions are physically accurate since they preserve the fundamental physical properties of the model. Furthermore, the method gives an excellent performance with respect to robustness and stability. Proof for these claims as well as numerical examples that illustrate the performance of the novel scheme are provided

Some aspects of lubrication in heavy regimes: thermal effects, stability and turbulence

In this work, a combination of numerical methods applied to thermohydrodynamic lubrication problems with cavitation is presented. It should be emphasized the difficulty of the nonlinear mathematical coupled model involving a free boundary problem, but also the simplicity of the algorithms employed to solve it. So, finite element discretizations for the hydrodynamic and thermal equations combined with upwind techniques for the convection terms and duality methods for nonlinear features are proposed. Additionally, a model describing the movement of the shaft is provided. Considering the shaft as a rigid body this model will consist of an ODE system relating acceleration of the center of gravity and external and pressure loads. The numerical experiments of mechanical stability try to clarify the position of the neutral stability curve. Finally, a rotating machine for ship propulsion involving both axial and radial bearings operating with nonconventional lubricants (seawater to avoid environmental pollution) is analyzed by using laminar and turbulent inertial flows.

Nonlinear analysis of a simple model of temperature evolution in a satellite

We analyze a simple model of the heat transfer to and from a small satellite orbiting round a solar system planet. Our approach considers the satellite isothermal, with external heat input from the environment and from internal energy dissipation, and output to the environment as black-body radiation. The resulting nonlinear ordinary differential equation for the satellite’s temperature is analyzed by qualitative, perturbation and numerical methods, which prove that the temperature approaches a periodic pattern (attracting limit cycle). This approach can occur in two ways, according to the values of the parameters: (i) a slow decay towards the limit cycle over a time longer than the period, or (ii) a fast decay towards the limit cycle over a time shorter than the period. In the first case, an exactly soluble average equation is valid. We discuss the consequences of our model for the thermal stability of satellites.

Kink stability, propagation, and length-scale competition in the periodically modulated sine-gordon equation

We have examined the dynamical behavior of the kink solutions of the one-dimensional sine-Gordon equation in the presence of a spatially periodic parametric perturbation. Our study clarifies and extends the currently available knowledge on this and related nonlinear problems in four directions. First, we present the results of a numerical simulation program that are not compatible with the existence of a radiative threshold predicted by earlier calculations. Second, we carry out a perturbative calculation that helps interpret those previous predictions, enabling us to understand in depth our numerical results. Third, we apply the collective coordinate formalism to this system and demonstrate numerically that it reproduces accurately the observed kink dynamics. Fourth, we report on the occurrence of length-scale competition in this system and show how it can be understood by means of linear stability analysis. Finally, we conclude by summarizing the general physical framework that arises from our study.

Stability in Linear Optimization Under Perturbations of the Left-Hand Side Coefficients

This paper studies stability properties of linear optimization problems with finitely many variables and an arbitrary number of constraints, when only left hand side coefficients can be perturbed. The coefficients of the constraints are assumed to be continuous functions with respect to an index which ranges on certain compact Hausdorff topological space, and these properties are preserved by the admissible perturbations. More in detail, the paper analyzes the continuity properties of the feasible set, the optimal set and the optimal value, as well as the preservation of desirable properties (boundedness, uniqueness) of the feasible and of the optimal sets, under sufficiently small perturbations.

An investigation of the effectiveness of existing bridge design methodology in providing adequate structural resistance to seismic disturbances : Phase III, Nonlinear soil-structure interaction of skew highway bridges : final report : prepared for Federal Highway Administration, Offices of Research and Development /

Bibliography: p. 46-48.

Study of stability for non-linear ordinary differential equations.

"C00-1469-0118."

Future developments in stiff integration techniques : stability of methods that do not use an exact Jacobian /

"COO-2383-0035."

Simulation of guiding-centre soliton transmission system stability in the presence of polarisation mode dispersion

Results of full numerical simulations of a guiding-centre soliton system with randomly birefringent SMF fibre are shown and analysed. It emerges that the soliton system becomes unstable even for small amounts of PMD.

Simultaneous clock recovery and data regeneration using a nonlinear optical loop mirror as an all-optical mixer

High-speed optical clock recovery, demultiplexing and data regeneration will be integral parts of any future photonic network based on high bit-rate OTDM. Much research has been conducted on devices that perform these functions, however to date each process has been demonstrated independently. A very promising method of all-optical switching is that of a semiconductor optical amplifier-based nonlinear optical loop mirror (SOA-NOLM). This has various advantages compared with the standard fiber NOLM, most notably low switching power, compact size and stability. We use the SOA-NOLM as an all-optical mixer in a classical phase-locked loop arrangement to achieve optical clock recovery, while at the same time achieving data regeneration in a single compact device

Nonlinear dynamics of a periodically driven Duffing resonator coupled to a Van der Pol oscillator

We explore the dynamics of a periodically driven Duffing resonator coupled elastically to a van der Pol oscillator in the case of 1?:?1 internal resonance in the cases of weak and strong coupling. Whilst strong coupling leads to dominating synchronization, the weak coupling case leads to a multitude of complex behaviours. A two-time scales method is used to obtain the frequency-amplitude modulation. The internal resonance leads to an antiresonance response of the Duffing resonator and a stagnant response (a small shoulder in the curve) of the van der Pol oscillator. The stability of the dynamic motions is also analyzed. The coupled system shows a hysteretic response pattern and symmetry-breaking facets. Chaotic behaviour of the coupled system is also observed and the dependence of the system dynamics on the parameters are also studied using bifurcation analysis.

A preliminary study into emergent behaviours in a lattice of interacting nonlinear resonators and oscillators

Future sensor arrays will be composed of interacting nonlinear components with complex behaviours with no known analytic solutions. This paper provides a preliminary insight into the expected behaviour through numerical and analytical analysis. Specically, the complex behaviour of a periodically driven nonlinear Duffing resonator coupled elastically to a van der Pol oscillator is investigated as a building block in a 2D lattice of such units with local connectivity. An analytic treatment of the 2-device unit is provided through a two-time-scales approach and the stability of the complex dynamic motion is analysed. The pattern formation characteristics of a 2D lattice composed of these units coupled together through nearest neighbour interactions is analysed numerically for parameters appropriate to a physical realisation through MEMS devices. The emergent patterns of global and cluster synchronisation are investigated with respect to system parameters and lattice size.

Nonlinear spectral shaping and optical rogue events in fiber-based systems

We provide an overview of our recent work on the shaping and stability of optical continua in the long pulse regime. Fibers with normal group-velocity dispersion at all-wavelengths are shown to allow for highly coherent continua that can be nonlinearly shaped using appropriate initial conditions. In contrast, supercontinua generated in the anomalous dispersion regime are shown to exhibit large fluctuations in the temporal and spectral domains that can be controlled using a carefully chosen seed. A particular example of this is the first experimental observation of the Peregrine soliton which constitutes a prototype of optical rogue-waves.

Nonlinear spectral shaping and optical rogue events in fiber-based systems

We provide an overview of our recent work on the shaping and stability of optical continua in the long pulse regime. Fibers with normal group-velocity dispersion at all-wavelengths are shown to allow for highly coherent continua that can be nonlinearly shaped using appropriate initial conditions. In contrast, supercontinua generated in the anomalous dispersion regime are shown to exhibit large fluctuations in the temporal and spectral domains that can be controlled using a carefully chosen seed. A particular example of this is the first experimental observation of the Peregrine soliton which constitutes a prototype of optical rogue-waves. © 2012 Elsevier Inc. All rights reserved.