Lingnan College Hong Kong : prospectus 1974-1977

https://commons.ln.edu.hk/lingnan_calendar/1001/thumbnail.jp

Lingnan College Hong Kong : prospectus 1972-1974

https://commons.ln.edu.hk/lingnan_calendar/1000/thumbnail.jp

The method of fundamental solutions for some direct and inverse problems

We propose and investigate applications of the method of fundamental solutions (MFS) to several parabolic time-dependent direct and inverse heat conduction problems (IHCP). In particular, the two-dimensional heat conduction problem, the backward heat conduction problem (BHCP), the two-dimensional Cauchy problem, radially symmetric and axisymmetric BHCPs, the radially symmetric IHCP, inverse one and two-phase linear Stefan problems, the inverse Cauchy-Stefan problem, and the inverse two-phase one-dimensional nonlinear Stefan problem. The MFS is a collocation method therefore it does not require mesh generation or integration over the solution boundary, making it suitable for solving inverse problems, like the BHCP, an ill-posed problem. We extend the MFS proposed in Johansson and Lesnic (2008) for the direct one-dimensional heat equation, and Johansson and Lesnic (2009) for the direct one-phase one-dimensional Stefan problem, with source points placed outside the space domain of interest and in time. Theoretical properties, including linear independence and denseness, the placement of source points, and numerical investigations are included showing that accurate results can be efficiently obtained with small computational cost. Regularization techniques, in particular, Tikhonov regularization, in conjunction with the L-curve criterion, are used to solve the illconditioned systems generated by this method. In Chapters 6 and 8, investigating the linear and nonlinear Stefan problems, the MATLAB toolbox lsqnonlin, which is designed to minimize a sum of squares, is used

Topological Photonics

Topological photonics is a rapidly emerging field of research in which geometrical and topological ideas are exploited to design and control the behavior of light. Drawing inspiration from the discovery of the quantum Hall effects and topological insulators in condensed matter, recent advances have shown how to engineer analogous effects also for photons, leading to remarkable phenomena such as the robust unidirectional propagation of light, which hold great promise for applications. Thanks to the flexibility and diversity of photonics systems, this field is also opening up new opportunities to realize exotic topological models and to probe and exploit topological effects in new ways. This article reviews experimental and theoretical developments in topological photonics across a wide range of experimental platforms, including photonic crystals, waveguides, metamaterials, cavities, optomechanics, silicon photonics, and circuit QED. A discussion of how changing the dimensionality and symmetries of photonics systems has allowed for the realization of different topological phases is offered, and progress in understanding the interplay of topology with non-Hermitian effects, such as dissipation, is reviewed. As an exciting perspective, topological photonics can be combined with optical nonlinearities, leading toward new collective phenomena and novel strongly correlated states of light, such as an analog of the fractional quantum Hall effect.Comment: 87 pages, 30 figures, published versio

Recent Advances in Theoretical and Computational Modeling of Composite Materials and Structures

The advancement in manufacturing technology and scientific research has improved the development of enhanced composite materials with tailored properties depending on their design requirements in many engineering fields, as well as in thermal and energy management. Some representative examples of advanced materials in many smart applications and complex structures rely on laminated composites, functionally graded materials (FGMs), and carbon-based constituents, primarily carbon nanotubes (CNTs), and graphene sheets or nanoplatelets, because of their remarkable mechanical properties, electrical conductivity and high permeability. For such materials, experimental tests usually require a large economical effort because of the complex nature of each constituent, together with many environmental, geometrical and or mechanical uncertainties of non-conventional specimens. At the same time, the theoretical and/or computational approaches represent a valid alternative for designing complex manufacts with more flexibility. In such a context, the development of advanced theoretical and computational models for composite materials and structures is a subject of active research, as explored here for a large variety of structural members, involving the static, dynamic, buckling, and damage/fracturing problems at different scales

Reconstruction based error detection for robust approximation of partial differential equations

In this work we present a framework for the construction of robust a posteriori estimates for classes of finite difference schemes. We are motivated by the relative lack of such frameworks compared to existing ones for other numerical discretisation methods, such as finite elements and finite volumes. The framework we propose is based on the use of reconstructions, which are obtained by post-processing the finite difference solution. The post-processed object is a key ingredient in obtaining a posteriori bounds using the relevant stability framework of the problem. The resulting bounds are fully computable and allow us to establish a posteriori error control over the problem at hand. In the first part of the thesis we motivate and investigate the behaviour of our framework using model ODE, elliptic and hyperbolic problems. We use our framework to obtain reconstructions which are used to compute a posteriori error estimates. We validate the numerical behaviour of these estimates using solutions of varying regularity. In the second part of the thesis we focus on hyperbolic conservation laws in one spatial dimension and we deal with scalar problems as well as systems. Hyperbolic conservation laws are widely used in the modelling of physical phenomena. The numerical modelling of conservation laws, which arises due to the frequent lack of explicit solutions, is challenging, largely due to the complex behaviour these problems exhibit, such as shock formation even with smooth initial conditions. In this setting, we present a framework which is applicable to general non-linear conservation laws. We investigate its numerical behaviour and showcase our results by using popular finite difference discretisations for a range of problems. We demonstrate that the the framework can produce optimal estimates, capable of tracking features of interest and act as refinement/coarsening indicators