Semi-hyperbolic mappings in Banach spaces

The definition of semi-hyperbolic dynamical systems generated by Lipschitz continuous and not necessarily invertible mappings in Banach spaces is presented in this thesis. Like hyperbolic mappings, they involve a splitting into stable and unstable spaces, but a slight leakage from the strict invariance of the spaces is possible and the unstable subspaces are assumed to be finite dimensional. Bi-shadowing is a combination of the concepts of shadowing and inverse shadowing and is usually used to compare pseudo-trajectories calculated by a computer with the true trajectories. In this thesis, the concept of bi-shadowing in a Banach space is defined and proved for semi-hyperbolic dynamical systems generated by Lipschitz mappings. As an application to the concept of bishadowing, linear delay differential equations are shown to be bi-shadowing with respect to pseudo-trajectories generated by nonlinear small perturbations of the linear delay equation. This shows robustness of solutions of the linear delay equation with respect to small nonlinear perturbations. Complicated dynamical behaviour is often a consequence of the expansivity of a dynamical system. Semi-hyperbolic dynamical systems generated by Lipschitz mappings on a Banach space are shown to be exponentially expansive, and explicit rates of expansion are determined. The result is applied to a nonsmooth noninvertible system generated by delay differential equation. It is shown that semi-hyperbolic mappings are locally φ-contracting, where -0 is the Hausdorff measure of noncompactness, and that a linear operator is semi-hyperbolic if and only if it is φ-contracting and has no spectral values on the unit circle. The definition of φ-bi-shadowing is given and it is shown that semi-hyperbolic mappings in Banach spaces are φ-bi-shadowing with respect to locally condensing continuous comparison mappings. The result is applied to linear delay differential equations of neutral type with nonsmooth perturbations. Finally, it is shown that a small delay perturbation of an ordinary differential equation with a homoclinic trajectory is ‘chaotic’.

High-dimensional and large-scale anomaly detection using a linear one-class SVM with deep learning

High-dimensional problem domains pose significant challenges for anomaly detection. The presence of irrelevant features can conceal the presence of anomalies. This problem, known as the '. curse of dimensionality', is an obstacle for many anomaly detection techniques. Building a robust anomaly detection model for use in high-dimensional spaces requires the combination of an unsupervised feature extractor and an anomaly detector. While one-class support vector machines are effective at producing decision surfaces from well-behaved feature vectors, they can be inefficient at modelling the variation in large, high-dimensional datasets. Architectures such as deep belief networks (DBNs) are a promising technique for learning robust features. We present a hybrid model where an unsupervised DBN is trained to extract generic underlying features, and a one-class SVM is trained from the features learned by the DBN. Since a linear kernel can be substituted for nonlinear ones in our hybrid model without loss of accuracy, our model is scalable and computationally efficient. The experimental results show that our proposed model yields comparable anomaly detection performance with a deep autoencoder, while reducing its training and testing time by a factor of 3 and 1000, respectively.

Reduced order functional observers with application to partial state estimation of linear systems with input delay

Reduced order multi-functional observer design for multi-input multi-utput (MIMO) linear time-invariant (LTI) systems with constant delayed inputs is studied. This research is useful in the input estimation of LTI systems with actuator delay, as well as system monitoring and fault detection of these systems. Two approaches for designing an asymptotically stable functional observer for the system are proposed: delay-dependent and delay-free. The delay-dependent observer is infinite-dimensional, while the delay-free structure is finite-dimensional. Moreover, since the delay-free observer does not require any information on the time delay, it is more practical in real applications. However, the delay-dependent observer contains less restrictive assumptions and covers more variety of systems. The proposed observer design schemes are novel, simple to implement, and have improved numerical features compared to some of the other available approaches to design (unknown-input) functional observers. In addition, the proposed observers usually possess lower order than ordinary Luenberger observers, and the design schemes do not need the observability or detectability requirements of the system. The necessary and sufficient conditions of the existence of an asymptoticobserver in each scenario are explored. The extensions of the proposed observers to systems with multiple delayed-inputs are also discussed. Several numerical examples and simulation results are employed to support our theories.

Three-dimensional Finite Element modelling of laser shock peening process

Laser shock peening (LSP) is an innovative surface treatment technique for metal alloys, with the great improvement of their fatigue, corrosion and wear resistance performance. Finite element method has been widely applied to simulate the LSP to provide the theoretically predictive assessment and optimally parametric design. In the current work, 3-D numerical modelling approaches, combining the explicit dynamic analysis, static equilibrium analysis algorithms and different plasticity models for the high strain rate exceeding 106s-1, are further developed. To verify the proposed methods, 3-D static and dynamic FEA of AA7075-T7351 rods subject to two-sided laser shock peening are performed using the FEA package–ABAQUS. The dynamic and residual stress fields, shock wave propagation and surface deformation of the treated metal from different material modelling approaches have a good agreement.

The effect of clamping sequence on dimensional variability in sheet metal assembly

Finite element (FE) modelling techniques have become a popular tool for exploring welding and clamping sequence dependence in sheet metal assemblies. In the current paper, the dimensional variability associated with different assembly clamping sequences is investigated with a FE contact modelling approach implemented in the commercial code Abaqus. A simplified channel section assembly consisting of a top hat and bottom plate is the case study investigated. Expected variation modes of bow and twist were used to simulate key variability sources in the main structural component under investigation; the top hat of the channel section. It was found that final assembly variability can change considerably depending on clamp sequence selection. It was also found that different clamp sequences can control particular modes of variation better than others, and that there is not one particular clamping sequence that is the best for containing all variation modes. An adaptable assembly process is therefore suggested, where given the shape of input components the best available clamping sequence is selected. Comparison of the performance of the proposed adaptable clamping sequence to traditional fixed clamping sequences shows improvements for the dimensional control of variability in non-rigid components. While introducing such a method in production would require inspection of each component being assembled and investigation of the alternative clamping sequences, given access to fast and detailed dimensional inspection technology such as optical coordinate measuring machines (OCMM's), the approach shows promise for future application.

Scope for further analytical solutions for constant flux infiltration into a semi-infinite soil profile or redistribution in a finite soil profile.

We attempt to generate new solutions for the moisture content form of the one-dimensional Richards' [1931] equation using the Lisle [1992] equivalence mapping. This mapping is used as no more general set of transformations exists for mapping the one-dimensional Richards' equation into itself. Starting from a given solution, the mapping has the potential to generate an infinite number of new solutions for a series of nonlinear diffusivity and hydraulic conductivity functions. We first seek new analytical solutions satisfying Richards' equation subject to a constant flux surface boundary condition for a semi-infinite dry soil, starting with the Burgers model. The first iteration produces an existing solution, while subsequent iterations are shown to endlessly reproduce this same solution. Next, we briefly consider the problem of redistribution in a finite-length soil. In this case, Lisle's equivalence mapping is generalized to account for arbitrary initial conditions. As was the case for infiltration, however, it is found that new analytical solutions are not generated using the equivalence mapping, although existing solutions are recovered.

Post processing of the hot torsion test results using a multi-dimensional modelling approach

A model selection scheme was extended to a multi-dimensional representation of the hot torsion test torque, twist and twist rate data to calculate partial derivatives of the torque data with respect to twist and twist rate. These enabled calculation of the instantaneous strain and strain rate hardening indices in the Fields and Backofen method. The concept of an iso-parametric shape function has been borrowed from the finite element method for adding twist rate as a dependant variable to the torque-twist models identified by the model selection scheme. Expressions to calculate the hardening indices, when employing a rational model of torsion data, were derived and presented. Subsequently, the models were used for post processing the data and determining hot strength behaviour, taking into account variations of strain and strain rate hardening indices during the deformation. To substantiate the technique, the hot flow behaviour of API-X70 micro-alloyed steel was determined using a range of hot torsion test data for the material. The flow stress obtained using the instantaneous hardening indices were compared with that obtained by the orthodox technique. For the investigated cases, the onset of dynamic recrystallization (DRX) predicted by the presented technique deviated considerably from those obtained when the average indices were used.

Three dimensional stability charts for slopes based on limit analysis methods

This paper uses finite element upper and lower bound limit analysis to produce chart solutions for three-dimensional (3D) natural slopes for both short- and long-term stability. The presented chart solutions are convenient tools that can be used for preliminary design purposes. The rigorous limit analysis results in this paper were found to bracket the true factor of safety within ±10% or better, which can be used as a benchmark for the solutions from other methods. The depth of the slip surfaces is observed to be generally shallow for most analyzed cases, particularly for the long-term slope stability problem. In addition, it was found that using a two-dimensional (2D) analysis may lead to significant differences in estimating safety factors, which can differ by 2%–60% depending on the slope geometry and soil properties. Therefore, great care and judgement are required when applying 2D analyses to 3D slope problems.

Finite element analysis of an axi-symmetric forward spiral extrusion of Mg-1.75Mn alloy

A potential severe plastic deformation process known as axi-symmetrical forward spiral extrusion (AFSE) has been studied numerically and experimentally. The process is based on the extrusion of cylindrical samples through a die with engraved spiral grooves in a near zero shape change manner. The process was simulated using a three dimensional finite element (FE) model that has been developed using commercial software, ABAQUS. In order to verify the finite element results, hot rolled and annealed samples of the alloy were experimentally processed by AFSE. The required extrusion forces during the process were estimated using the FE model and compared with the experimental values. The reasonable agreement between the FE results and experimental data verified the accuracy of the FE model. The numerical results indicate the linear strain distribution in the AFSE sample is only valid for a core concentric while the strain distribution in the vicinity of the grooves is non axi-symmetric. The FE simulation results from this research allows a better understanding of AFSE kinematics especially near the grooves, the required extrusion force and the resultant induced strain distribution in the sample. To compare the mechanical properties of the Mg-1.75Mn alloy before and after the process, a micro shear punch test was used. The tests were performed on samples undergoing one and four passes of AFSE. After four passes of AFSE, it was observed that the average shear strength of the alloy has improved by about 21%. The developedfinite element model enables tool design and material flow simulation during the process.

Fracture modelling of DP780 sheets using a hybrid experimental-numerical method and two-dimensional digital image correlation

This paper is concerned with the construction of fracture envelopes of DP780 sheets using two methods: a hybrid experimental-numerical method; two-dimensional digital image correlation (2D-DIC). For the hybrid method, four types of ductile fracture tests were carried out covering a wide range of stress states on specimens: with a central hole; two symmetric circular notches; flat grooved; and diagonally double-notched. Based on the fracture strain and loading paths identified with finite element simulation, a fracture envelope was obtained by employing the three-parameter modified Mohr-Coulomb fracture model. In addition, the fracture surface strain was directly measured using 2D-DIC. Loading histories of each test were extracted from a surface element of a three dimensional finite element model. The comparison of fracture envelopes constructed by the two methods reveals that there is little difference. Thus, it can be concluded that 2D-DIC is applicable to fracture modelling of DP780 sheets despite the assumption of the plane stress condition even after necking

Use of artificial neural networks as explicit finite difference operators

Results of a numerical exercise, substituting a numerical operator by an artificial neural network (ANN) are presented in this paper. The numerical operator used is the explicit form of the finite difference (FD) scheme. The FD scheme was used to discretize the one-dimensional transport equation, which included both the advection and dispersion terms. Inputs to the ANN are the FD representation of the transport equation, and the concentration was designated as the output. Concentration values used for training the ANN were obtained from analytical solutions. The numerical operator was reconstructed from a back calculation of the weights of the ANN. Linear transfer functions were used for this purpose. The ANN was able to accurately recover the velocity used in the training data, but not the dispersion coefficient. This capability was improved when numerical dispersion was taken into account; however, it is limited to the condition: C/P<0.5 , where C is the Courant number and P , the Peclet number (i.e., the restriction imposed by the Neumann stability condition).

Continuous finite-time state feedback stabilizers for some nonlinear stochastic systems

This paper is concerned with the problem of finite-time stabilization for some nonlinear stochastic systems. Based on the stochastic Lyapunov theorem on finite-time stability that has been established by the authors in the paper, it is proven that Euler-type stochastic nonlinear systems can be finite-time stabilized via a family of continuous feedback controllers. Using the technique of adding a power integrator, a continuous, global state feedback controller is constructed to stabilize in finite time a large class of two-dimensional lower-triangular stochastic nonlinear systems. Also, for a class of three-dimensional lower-triangular stochastic nonlinear systems, a recursive design scheme of finite-time stabilization is given by developing the technique of adding a power integrator and constructing a continuous feedback controller. Finally, a simulation example is given to illustrate the theoretical results. © 2014 John Wiley & Sons, Ltd.

Slope stability charts for two-layered purely cohesive soils based on finite-element limit analysis methods

Stability charts for soil slopes, first produced in the first half of the twentieth century, continue to be used extensively as design tools, and draw the attention of many investigators. This paper uses finite-element upper and lower bound limit analysis to assess the short-term stability of slopes in which the slopematerial and subgrade foundation material have two distinctly different undrained strengths. The stability charts are proposed, and the exact theoretical solutions are bracketed to within 4.2% or better. In addition, results from the limit-equilibrium method (LEM) have been used for comparison. Differences of up to 20% were found between the numerical limit analysis and LEM solutions. It also shown that the LEM sometimes leads to errors, although it is widely used in practice for slope stability assessments.

Three-dimensional slope stability assessment of two-layered undrained clay

This paper uses the finite element upper and lower bound limit analysis methods to investigate the three-dimensional (3D) slope stability of two-layered undrained clay slopes. The solutions obtained from the slope stability analyses are bracketed to within ±10% or better. For comparison purposes, results from two-dimensional (2D) analyses based on the numerical limit analysis methods and the conventional limit equilibrium method (LEM) are also discussed. This study shows that 3D boundary of a slope can have significant effects on the slope stability. In addition, the results are presented in the form of stability charts which can be convenient tools for practicing engineers.

Slope stability analysis for fill slopes using finite element limit analysis

This paper investigates the stability of fill slopes often found in embankment cases where frictional fill materials are placed on purely cohesive undrained clay with increasing strength. By using finite element upper and lower bound limit analysis for this investigation, the limit load can be truly bounded. It is known that two-dimensional analysis yields a more conservative result due to plain strain condition when compared to three-dimensional analysis. Therefore, this paper will focus on three-dimensional (3D) slope stability analysis and for comparison purposes two-dimensional analysis results will be employed. In fact, the final results are presented in the form of comprehensive chart solutions for the convenience of practicing engineers during preliminary slope design. The failure mechanism will also be discussed in order to further illustrate the situation during failure. It should be highlighted that the failure mechanisms are obtained through the numerical method itself and no prior assumptions are required, therefore, are more realistic and able to provide a better understanding for the slope failure surfaces.