Stability theory and numerical analysis of non-autonomous dynamical systems

The development and use of cocycles for analysis of non-autonomous behaviour is a technique that has been known for several years. Initially developed as an extension to semi-group theory for studying rion-autonornous behaviour, it was extensively used in analysing random dynamical systems [2, 9, 10, 12]. Many of the results regarding asymptotic behaviour developed for random dynamical systems, including the concept of cocycle attractors were successfully transferred and reinterpreted for deterministic non-autonomous systems primarily by P. Kloeden and B. Schmalfuss [20, 21, 28, 29]. The theory concerning cocycle attractors was later developed in various contexts specific to particular classes of dynamical systems [6, 7, 13], although a comprehensive understanding of cocycle attractors (redefined as pullback attractors within this thesis) and their role in the stability of non-autonomous dynamical systems was still at this stage incomplete. It was this purpose that motivated Chapters 1-3 to define and formalise the concept of stability within non-autonomous dynamical systems. The approach taken incorporates the elements of classical asymptotic theory, and refines the notion of pullback attraction with further development towards a study of pull-back stability arid pullback asymptotic stability. In a comprehensive manner, it clearly establishes both pullback and forward (classical) stability theory as fundamentally unique and essential components of non-autonomous stability. Many of the introductory theorems and examples highlight the key properties arid differences between pullback and forward stability. The theory also cohesively retains all the properties of classical asymptotic stability theory in an autonomous environment. These chapters are intended as a fundamental framework from which further research in the various fields of non-autonomous dynamical systems may be extended. A preliminary version of a Lyapunov-like theory that characterises pullback attraction is created as a tool for examining non-autonomous behaviour in Chapter 5. The nature of its usefulness however is at this stage restricted to the converse theorem of asymptotic stability. Chapter 7 introduces the theory of Loci Dynamics. A transformation is made to an alternative dynamical system where forward asymptotic (classical asymptotic) behaviour characterises pullback attraction to a particular point in the original dynamical system. This has the advantage in that certain conventional techniques for a forward analysis may be applied. The remainder of the thesis, Chapters 4, 6 and Section 7.3, investigates the effects of perturbations and discretisations on non-autonomous dynamical systems known to possess structures that exhibit some form of stability or attraction. Chapter 4 investigates autonomous systems with semi-group attractors, that have been non-autonomously perturbed, whilst Chapter 6 observes the effects of discretisation on non-autonomous dynamical systems that exhibit properties of forward asymptotic stability. Chapter 7 explores the same problem of discretisation, but for pullback asymptotically stable systems. The theory of Loci Dynamics is used to analyse the nature of the discretisation, but establishment of results directly analogous to those discovered in Chapter 6 is shown to be unachievable. Instead a case by case analysis is provided for specific classes of dynamical systems, for which the results generate a numerical approximation of the pullback attraction in the original continuous dynamical system. The nature of the results regarding discretisation provide a non-autonomous extension to the work initiated by A. Stuart and J. Humphries [34, 35] for the numerical approximation of semi-group attractors within autonomous systems. . Of particular importance is the effect on the system's asymptotic behaviour over non-finite intervals of discretisation.

Experimental and numerical analysis of the deformed process of cracks/cavities

In this paper, the three-point bending tests of beams with a small single-edge crack are conducted to demonstrate the reliability and accuracy of the laser diffraction technique in detecting its deformation and evolution. As a comparison, the quasi-static analysis of the beam subjected to 3-point bending is further proceeded using the finite element method and the related results have been in good agreement with those from the tests.

Numerical analysis of heat transfer and the optimisation of regenerators

To optimize a regenerator’s structure and its operation parameters and, consequently, to increase the efficiency of heat recovery and to save energy, a computational approach is used to study the unsteady three-dimensional flow and heat transfer. The simulation is performed in two steps. In the first step, the gas flow and heat transfer in a typical sphere-bed unit is simulated to deduce a dimensionless equation of heat transfer between gas and sphere. In the second step, a model is developed to simplify the prototype and to simulate the gas flow and heat transfer in the whole regenerator. The heat exchange process in regenerators and the effects of the regenerator’s structure and operation parameters, such as gas mass flux, reversal time, regenerator height, sphere diameter, and thermophysical properties of the spheres, are studied with the model to determine efficiency of heat recovery.

Integrated fluid-thermal-structural numerical analysis for the quenching of metallic components

The quenching of a metal component with a channel section in a water tank is numerically simulated. Computational fluid dynamics (CFD) is used to model the multiphase flow and the heat transfer in film boiling, nucleate boiling and convective cooling processes to calculate the difference in heat transfer rate around the component and then combining with the thermal simulation and structure analysis of the component to study the effect of heat transfer rate on the distortion of the U-channel component. A model is also established to calculate the residual stress produced by quenching. The coupling fluid-thermal-structural simulation provides an insight into the deformation of the component and can be used to perform parameter analysis to reduce the distortion of the component. © 2011 Shanghai Jiaotong University and Springer-Verlag Berlin Heidelberg.

Numerical modeling and analysis for forming process of dual-phase 980 steel exposed to infrared local heating

A recent experiment confirmed that the infrared (IR) local heating method drastically reduces springback of dual-phase (DP) 980 sheets. In the experiment, only the plastic deformation zone of the sheets was locally heated using condensed IR heating. The heated sheets were then deformed by V-bending or 2D-draw bending. Although the experimental observation proved the merit of using the IR local heating to reduce springback, numerical modeling has not been reported. Numerical modeling has been required to predict springback and improve the understanding of the forming process. This paper presents a numerical modeling for V-bending and 2D-draw bending of DP 980 sheets exposed to the IR local heating with the finite element method (FEM). For describing the thermo-mechanical behavior of the DP 980 sheet, a flow stress model which includes a function of temperature and effective plastic strain was newly implemented into Euler-backward stress integration method. The numerical analysis shows that the IR local heating reduces the level of stress in the deformation zone, although it heats only the limited areas, and then it reduces the springback. The simulation also provides a support that the local heating method has an advantage of shape accuracy over the method to heat the material as a whole in V-bending. The simulated results of the springback in both V-bending and 2D-draw bending also show good predictions.

Experimental and numerical investigation of low pressure tube hydroforming on 409 stainless steel

Tube hydroforming has been widely used to produce automotive structural components due to the superior properties of the hydroformed parts in terms of their light weight and structural rigidity. Compared to the traditional manufacturing process for a closed-section member including stamping and followed by welding, tube-hydro forming leads to cost savings due to reduced tooling and material handling. However, the high pressure pumps and high tonnage press required in hydroforming, lead to increased capital investment reducing the cost benefits. This study explores low pressure tube hydro forming which reduces the internal fluid pressure and die closing force required to produce the hydroformed part. The experimental and numerical analysis was for low pressure hydro formed stainless steel tubes. Die filling conditions and thickness distributions are measured and critically analysed.

Numerical investigation of a grating and graphene-based multilayer surface plasmon resonance biosensor

A multilayer surface plasmon resonance biosensor (SPRB) incorporating a grating-graphene configuration is investigated for enhanced sensitivity. The numerical analysis of the impact of integrating a periodic array of subwavelength grating on top of a layer of graphene sheet for improving sensitivity is presented. The result of monitoring the biomolecular interactions of DNA hybridization is compared against the outcome of the conventional SPRB, a graphene-based multilayer SPRB, and a multilayer layer grating SPRB, and is mathematically validated. It is demonstrated that the inclusion of a grating and graphene layer on top of the gold thin film is an excellent candidate for a highly sensitive SPRB. To achieve further enhancement of sensitivity, the subwavelength grating is numerically optimized against its geometry including grating configurations (rectangular, sinusoidal, and triangular), grating depth, volume factor, and grating period. © 2014 Taylor & Francis.

Analysis of backward congestion notification with delay for enhanced Ethernet networks

At present, companies and standards organizations are enhancing Ethernet as the unified switch fabric for all of the TCP/IP traffic, the storage traffic and the high performance computing traffic in data centers. Backward congestion notification (BCN) is the basic mechanism for the end-to-end congestion management enhancement of Ethernet. To fulfill the special requirements of the unified switch fabric, i.e., losslessness and low transmission delay, BCN should hold the buffer occupancy around a target point tightly. Thus, the stability of the control loop and the buffer size are critical to BCN. Currently, the impacts of delay on the performance of BCN are unidentified. When the speed of Ethernet increases to 40 Gbps or 100 Gbps in the near future, the number of on-the-fly packets becomes the same order with the buffer size of switch. Accordingly, the impacts of delay will become significant. In this paper, we analyze BCN, paying special attention on the delay. We model the BCN system with a set of segmented delayed differential equations, and then deduce sufficient condition for the uniformly asymptotic stability of BCN. Subsequently, the bounds of buffer occupancy are estimated, which provides direct guidelines on setting buffer size. Finally, numerical analysis and experiments on the NetFPGA platform verify our theoretical analysis.

Evaluation of numerical procedures to determine seismic response of structures under influence of soil-structure interaction

In this study, the accuracy and reliability of fully nonlinear method against equivalent linear method for dynamic analysis of soil-structure interaction is investigated comparing the predicted results of both numerical procedures with the results of experimental shaking table tests. An enhanced numerical soil-structure model has been developed which treats the behaviour of the soil and the structure with equal rigour. The soil-structural model comprises a 15 storey structural model resting on a soft soil inside a laminar soil container. The structural model was analysed under three different conditions: (i) fixed base model performing conventional time history dynamic analysis, (ii) flexible base model (considering full soil-structure interaction) conducting equivalent linear dynamic analysis, and (iii) flexible base model performing fully nonlinear dynamic analysis. The results of the above mentioned three cases in terms of lateral storey deflections and inter-storey drifts are determined and compared with the experimental results of shaking table tests. Comparing the experimental results with the numerical analysis predictions, it is noted that equivalent linear method of dynamic analysis underestimates the inelastic seismic response of mid-rise moment resisting building frames resting on soft soils in comparison to the fully nonlinear dynamic analysis method. Thus, inelastic design procedure, using equivalent linear method, cannot adequately guarantee the structural safety for mid-rise building frames resting on soft soils. However, results obtained from the fully nonlinear method of analysis fit the experimental results reasonably well. Therefore, this method is recommended to be used by practicing engineers.

The effect of laser power density on the fatigue life of laser-shock-peened 7050 aluminium alloy

Laser shock peening (LSP) is an innovative surface treatment method that can result in significant improvement in the fatigue life of many metallic components. The process produces very little or no surface profile modification while producing a considerably deeper compressive residual stress layer than traditional shot peening operations. The work discussed here was designed to: (a) quantify the fatigue life improvement achieved by LSP in a typical high strength aircraft aluminium alloy and (b) identify any technological risks associated with its use. It is shown that when LSP conditions are optimal for the material and specimen configuration, a —three to four times increase in fatigue life over the as-machined specimens could be achieved for a representative fighter aircraft loading spectrum when applied at a representative load level. However, if the process parameters are not optimal for the material investigated here, fatigue lives of LSP treated specimens may be reduced instead of increased due to the occurrence of internal cracking. This paper details the effect of laser power density on fatigue life of 7050-T7451 aluminium alloy by experimental and numerical analysis.

Data quality issues in practice and theory : a cross-cultural example

Practical considerations and traditions play a substantial role in data collection exercises, often limiting the focus of study to either qualitative or quantitative issues. An industry with a particularly strong quantitative emphasis is the insurance and reinsurance industry, where actuarial decisions are based on detailed and exacting numerical analysis of data that are assumed to be reliable and valid. However, the qualitative investigation of the quality of data in one reinsurance setting reported in this paper shows that where the meanings of the questions asked and of the answers provided are subject to interpretation, the quality of data collected for entry to databases can be poor. While this can be exacerbated in cross-cultural contexts, it is also generally true. Due to the constrained nature of insurance practice, the existence of a range of techniques combining qualitative and quantitative methods is somewhat academic. Therefore, because researchers have the latitude to investigate both qualitative and quantitative factors in the industrial context, a call is made for researchers and industry to work more closely together.