DSpace Community:
http://dspace.cityu.edu.hk:80/handle/2031/718
2017-02-17T02:13:47ZNew analysis of linearized numerical schemes for several nonlinear convection-diffusion equations and Schrödinger type system
http://dspace.cityu.edu.hk:80/handle/2031/8627
Title: New analysis of linearized numerical schemes for several nonlinear convection-diffusion equations and Schrödinger type system
Authors: Wang, Jilu (王冀魯)
Abstract: The thesis is concerned with a new analysis of linearized numerical schemes for several
nonlinear convection-diffusion equations and Schrödinger type system.
The method of characteristics is especially effective for convection-dominated diffusion
problems. Due to the nature of characteristic temporal discretization, the method allows one
to use a large time step in many practical computations, while all previous theoretical analyses always required certain restrictions on the time stepsize. The first part of the thesis is to present a new analysis to establish unconditionally optimal error estimates for a modified
method of characteristics with finite element approximation. Here, we consider two physical
models: incompressible miscible flow in porous media and the time-dependent Navier-Stokes
equations. For this purpose, we introduce a characteristic time-discrete system. We prove that the L2 error bound of the fully discrete method of characteristics to the time-discrete system is τ-independent and the numerical solution is bounded in W1,∞-norm unconditionally,
where τ denotes the time stepsize. With the boundedness, optimal error estimates are established in a traditional manner.
Secondly, we study linearized Crank-Nicolson finite element methods (FEMs) and finite
difference methods (FDMs) for Schrödinger type system. We obtain optimal L2 error estimates
without any time-step restrictions. Our approaches are based on an error splitting technique
for FEMs, and a rigorous analysis in both real and imaginary parts of the error functions
for FDMs.
The third part of this thesis is concerned with mathematical modeling, analysis and computations
of heat and sweat transport in fibrous media with a non-local thermal radiation and
phase change. The model, based on a combination of these classical heat transfer mechanisms
(convection, conduction and radiation), is governed by a nonlinear, degenerate and strongly coupled parabolic system. We prove the global existence of positive/non-negative weak
solutions of this nonlinear parabolic system. A typical clothing assembly with a polyester
batting material sandwiched in two laminated covers is investigated numerically. Numerical
results show that the contribution of radiative heat transfer is comparable with that of conduction/
convection in the sweating system.
Notes: CityU Call Number: QA377 .W34 2015; ix, 157 pages : illustrations 30 cm; Thesis (Ph.D.)--City University of Hong Kong, 2015.; Includes bibliographical references (pages 141-157)2015-01-01T00:00:00ZVertical mode expansion method for scattering of light by multiply layered photonic structures
http://dspace.cityu.edu.hk:80/handle/2031/8626
Title: Vertical mode expansion method for scattering of light by multiply layered photonic structures
Authors: Shi, Hualiang (時華良)
Abstract: A layered structure is a one-dimensional (1D) structure for which the material
properties depend only on one spatial variable z. A multiply layered structure
is a three-dimensional (3D) structure consisting of different cylindrical 1D
layered structures in different regions. In photonics, the study of multiply layered
structures is of great practical importance due to the existing fabrication
techniques. Some important examples of multiply layered structures include a
metallic film with cylindrical apertures, cylindrical metallic nanoparticles on a
substrate, and a photonic crystal slab with cylindrical holes. A fundamental
problem is to analyze the scattering of light by 3D multiply layered structures.
Numerical methods, such as the finite-difference time-domain method, the finite
element method, the volume and surface integral equation methods, can be used
to solve these scattering problems. However, it is desirable and often possible to
develop special numerical or semi-analytic methods that are more efficient and accurate
than the general methods. The usual mode matching method (also called
modal method or mode expansion method) is applicable to piecewise z-invariant
structures, and it expands the electromagnetic field in each z-invariant segment
using the eigenmodes of that segment. These eigenmodes are functions of two
transverse variables. The method is not very efficient since a large number of
eigenmodes are required, and they are full vectorial and expensive to calculate.
In this thesis, we develop a vertical mode expansion method (VMEM) for
analyzing the scattering of light by multiply layered structures. Our starting
point is a mode expansion technique in general 1D layered structures. The
electromagnetic field is expanded in 1D vertical modes which depend on z, where the
“expansion coefficients” are functions of the two transverse variables and satisfy
two-dimensional (2D) Helmholtz equations. Our VMEM requires the so-called
Dirichlet-to-Neumann (DtN) or Neumann-to-Dirichlet (NtD) maps for the related
2D Helmholtz equations. These operators provide relations between the
solutions and their normal derivatives on the boundaries of 2D domains in the xy
plane. With the help of DtN or NtD maps, VMEM establishes a linear system by
matching the tangential components of the electromagnetic filed on the vertical
boundaries of the different regions. The VMEM gives a 2D formulation for the
original 3D problem. It is relatively simple to implement and relatively efficient.
In Chapter 3, we present a VMEM for multiply layered structures with
an elliptic cylindrical region. The method is developed based on a numerical
separation of variables in the elliptic coordinates. The key step is to calculate the
DtN maps for 2D Helmholtz equations inside or outside an ellipse. For numerical
stability reasons, we avoid the analytic solutions of the Helmholtz equations in
terms of the angular and radial Mathieu functions, and construct the DtN maps
by a fully numerical method. The method is used to analyze the transmission of
light through an elliptic aperture in a metallic film, and the scattering of light by
elliptic gold cylinders on a substrate.
In Chapter 4, we develop a more general VMEM for layered cylindrical structures
with arbitrary cross sections in a layered background. A boundary integral
equation (BIE) is used to construct the NtD maps for 2D Helmholtz equations
that appear in the mode expansion process. The method is applied to analyze
subwavelength apertures in metallic films and nanoparticles on substrates.
In Chapter 5, we further extend the VMEM to multiply layered periodic
structures, such as a photonic crystal slab with a square lattice of holes, and a
periodic array of metallic nanoparticles on a substrate. For a multiply layered
periodic structure, a unit cell consists of different 1D layered cylindrical regions.
A BIE is again used to construct the NtD maps, but the quasi-periodic boundary
conditions must be incorporated in the process, and a graded mesh technique is
used to handle the corner singularities. Using the VMEM for periodic structures,
we calculate the transmission and extinction spectra for photonic crystal slabs
and other plasmonic structures
Notes: CityU Call Number: QC427.4 .S54 2015; xii, 99 pages : illustrations 30 cm; Thesis (Ph.D.)--City University of Hong Kong, 2015.; Includes bibliographical references (pages 89-96)2015-01-01T00:00:00ZA study on dynamical behavior for distributed networked systems under fault
http://dspace.cityu.edu.hk:80/handle/2031/8625
Title: A study on dynamical behavior for distributed networked systems under fault
Authors: Chen, Shun (陳舜)
Abstract: Recently, distributed networked systems, such as distributed robots and mobile sensor
networks, have been widely considered due to their broad applications. The distributed
networked system consists of a large number of small, inexpensive systems deployed
over a vast region in a distributed way, in which each small system is capable of collecting,
processing information and communicating with neighboring systems. There exist
many real-world engineering systems which are well described by distributed networked
systems, such as sensor networks, multi-agents systems, and autonomous underwater vehicles.
For distributed networked systems, the collaboration among agents is the key
factor for achieving the desire tasks. However, the communication among various agents
always suffer from different types of physical limitations, for instance, communication
range, power, communication quantization, and processing ability. In addition, in networked
systems, the distributed behavior brings plenty of essential difficulties in theoretical
research. In particular, due to the security requirements in military applications,
or source limitations in power systems, the risk of fault in distributed networked system
becomes more of a concern. Unfortunately, different kinds of faults emerged due to the
unexpected environment effects which can lead to undesirable stability and performance
analysis of distributed networked systems. Thus, due to the increasing complexity and
safety demand of real-world applications, developing dynamical analysis techniques for
distributed networked systems under fault are important. It is highly desirable to exploit
effective protocols and methods for preserving the stability and performance of the distributed
networked systems under fault.
The following issues will be presented in this thesis in detail: (a) fault tolerant multiagents
consensus; (b) fault tolerant coordination with quantization due to limited infor
mation transmission requirements; (c) state estimation for heterogeneous distributed networks
under fault effect. (d) fault estimation for multiple distributed sensor networks;
The main contributions of this thesis are listed as follows:
• Fault tolerant consensus in multi-agents system using distributed adaptive protocol
is investigated. Distributed adaptive online updating strategies for some parameters
are proposed based on local information of the network structure. Based on the
online updating parameters, distributed adaptive protocols are developed to compensate
the fault effects and the uncertainty effects in both leaderless multi-agent
system and leader-follower multi-agent system.
• The coordination control under fault due to attacks is considered on the security aspect.
Passivity based fault tolerant controls for coordination using both logarithmic
quantizers and uniform quantizers are investigated. Based on the nonsmooth analytical
technique, the effect of the quantization and fault on the coordination results
is examined.
• Two sets of estimator designs for distributed sensor networks in multi-targets tracking
under signal transmission faults due to the uncertain environments are presented.
Two-targets tracking distributed sensor networks are firstly proposed to
simplify the illustration of complicated mathematics. The estimation approach in
two-targets tracking sensor networks is to construct fault estimators for the signal
transmission faults. Then, estimators for both fault and state are designed for
multi-targets tracking sensor networks. Furthermore, two applications are used to
demonstrated the effectiveness of the proposed theoretical results.
• The state estimation performance for heterogeneous distributed system with fault
based on sampled-data measurement is considered. A performance index for distributed
state estimation which augmenting the effect of the amplitude and frequency
of the fault is proposed. In addition, distributed state estimators are constructed
based on the sampled-data measurement of the heterogeneous distributed system.
Notes: CityU Call Number: QA76.9.F38 C45 2015; x, 140 pages : illustrations 30 cm; Thesis (Ph.D.)--City University of Hong Kong, 2015.; Includes bibliographical references (pages 121-140)2015-01-01T00:00:00ZAnalytic studies on bifurcations of a hyperelastic layer-substrate structure under uniaxial compression
http://dspace.cityu.edu.hk:80/handle/2031/8567
Title: Analytic studies on bifurcations of a hyperelastic layer-substrate structure under uniaxial compression
Authors: Liu, Yang (劉洋)
Abstract: This thesis focuses on the bifurcations of a two-dimensional layer-substrate structure
under uniaxial compression and its applications to buckled and wrinkled fruits and
vegetables. A layer coated to a substrate can be used to model the skin of human beings,
fruits and vegetables with exocarp and sarcocarp, the evolution of thin membrane
in some biological situations, etc. So it is necessary to study such a structure both
mechanically and mathematically due to its widespread applications.
In the first part, we focus on the bifurcation behavior of a compressible hyperelastic
layer bonded to another compressible hyperelastic substrate. A linear bifurcation
analysis is carried out for obtaining the bifurcation condition in the framework of exact
theory of nonlinear elasticity. From the critical stretch curves, it is found that there are
two mode types for the layer: buckling mode and wrinkling mode. By further considering
the eigenfunction, three types of modes for the substrate are identified, including
buckling mode, buckling-surface mode and wrinkling-surface mode. Through a careful
analysis, we manage to classify the plane of the aspect ratio of the layer and the
thickness ratio into six domains for different mode types and whose boundaries determine
where the transitions of mode types take place. Finally, an asymptotic analysis
with double expansions for each unknown is carried out to give the explicit formulas
for the critical mode number and the critical stretch (which also give an improvement
on the existing results for a layer coated to a half-space) . Also, simplified relations for
those critical thickness ratios and aspect ratios are derived.
In the second part, we adopt a generalized plane-strain model to establish the geometrical
constraint for buckled and wrinkled shapes by modeling a fruit/vegetable
with exocarp and sarcocarp as a hyperelastic layer-substrate structure subjected to uniaxial
compression. Our point is that there is a critical thickness ratio which separates
the buckling and wrinkling modes, independently of the material stiffnesses.
More specifically, it is found that if the thickness ratio is smaller than this critical
value a fruit/vegetable should be in a buckled shape (under a sufficient stress); if a
fruit/vegetable is in a wrinkled shape the thickness ratio is always larger than this critical
value. To verify the theoretical prediction, we consider four types of buckled fruits/
vegetables and four types of wrinkled fruits/vegetables with three samples in each
type. The geometrical parameters for the 24 samples are measured and it is found that
indeed all the data fall into the theoretically predicted buckling or wrinkling domains.
In the third part, we revisit the same problem in part one and study the postbifurcation
solutions for buckling mode. By specifying the geometrical and material
parameters within the buckling mode domains and adopting the combined seriesasymptotic
expansions method, two coupled nonlinear ODEs governing the leading
order axial strain and shear strain for the upper surface of layer are obtained. At nearcritical
loads, a perturbation method is used to derive the amplitude equation by utilizing
the solvability condition. Two cases are considered, that is the ratio of Young’s
modulus between layer and substrate is of O(1) or large. The two-terms asymptotic
solutions and leading order solutions are obtained. It is found that there is a transition
between a supercritical bifurcation to a subcritical one when the modulus ratio is increasing.
The validity of our results is examined by comparing the analytical solutions
with the numerical ones for both cases and good agreements are found. The simple
analytical formulas for the deflection, the displacements for both layer and substrate
are also obtained, which can give us the insights for how the geometrical and material
parameters affect the post-bifurcation states. Especially, the relations of deflection, the
critical bifurcation stretch and modulus ratio are non-monotone, which implies there
exists the critical modulus ratios such that the deflection and critical stretch attains a
minimum value. This suggest a possible way for controlling the deflection amplitude
and stability of the structure.
Notes: CityU Call Number: QA380 .L59 2015; 2, x, 170 pages : illustrations 30 cm; Thesis (Ph.D.)--City University of Hong Kong, 2015.; Includes bibliographical references (pages 117-127)2015-01-01T00:00:00Z