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http://dspace.cityu.edu.hk:80/handle/2031/718
2015-07-03T02:42:43ZAnalytical and numerical studies on local and nonlocal elastic bars in tension and neutralizer-based iterative methods
http://dspace.cityu.edu.hk:80/handle/2031/7427
Title: Analytical and numerical studies on local and nonlocal elastic bars in tension and neutralizer-based iterative methods
Authors: Zhu, Xiaowu (朱小武)
Abstract: In this thesis, we study some problems associated with both local and nonlocal elasticity
and, present the so-called neutralizer-based iterative methods for integral equations
of the second kind. The details are as follows.
The local problem is on the strain softening of materials. Strain-softening, i.e.,
the decrease of stress with the increase of strain, which is a common post-peak phenomenon that has been recorded for a variety of materials. Snap-back due to strain
softening may be one of the most interesting and the most common structural instability
phenomena observed in experiments. There have been many efforts in the past
decades to investigate strain-softening with localization experimentally, numerically,
and analytically. However, there is not any analytical study with general nonlinear
constitutive relations in the open literature which explores the role played by the convexity of the constitutive curve of the softening part and the coupling effect between
this convexity and the size. Also, both snap-back and snap-through were observed in
some experiments, but no analytical results are available for explaining the transition
from snap-back to snap-through.
Nonlocal elasticity is a growing direction of continuum mechanics nowadays.
There are many works contributed to this area. Due to the essential difficulty of the
integral equation, analytical approximate solutions are usually prohibited. Thus, many
existing literature devote to apply the approximate differential models suggested by
Eringen et al. However, one weak point of the approximations is that the possible boundary effects, which are present in the integral formulations, are neglected. Thus,
it would be desirable to have proper differential formulations which take into account
such effects. For this purpose, a first step is to know both qualitative and quantitative
behavior near the ends (e.g., the influences of the parameters). Thus, some analytical
solutions are needed to provide convincing results.
In this thesis, firstly for the local problem, we modify an existing model and set up
the stress-strain equations for the structure in the post-peak region, which are nonlinear
as compared with the bilinear case in the literature. After some analysis, we derive the
mathematical conditions for the occurrence of several important curves as frequently
observed in experiments, including the snap-through (which cannot be captured by the
bilinear assumptions). Two examples are also given to illustrate these cases, and the
post-peak curves are consistent with our theoretical predictions.
Secondly, for a static tension problem in nonlocal elasticity (the uniform case), we
apply an existing iterative method that are efficient for a special kind of kernel to handle
the resultant integral equation. By explicitly evaluating the integral in the second
iterative solution, we are able to get a good approximate analytical solution for this
problem. Some features of the nonlocal theory can then be closely examined, especially
the boundary effects. It seems that the analytical results obtained here would
give some insights into nonlocal theory, particularly for its applications in nanomaterials.
Moreover, in view of the boundary effects, we also present a new model for
nonuniform nonlocal bar-a varying volume fraction in the nonlocal phase. The numerical
results of different shapes of materials show that, as compared with a uniform
bar and that in local elasticity, the model herein shows more features, such as stress
concentration.
Thirdly, we present neutralizer-based iterative methods for integral equations of
the second kind. As is known to all, there is an abundant of numerical techniques
for solving integral equations, such as Neumann series, multi-grid method, GMRES and so on. Here, we introduce the concept of neutralizer, which is sometimes used
in dealing with integrals (e.g., the asymptotic expansions of integrals), to obtain the
neutralizer-based iterative method. Some features of such iterative method is numerically
explored. Several meaningful examples are given, showing that the methods
perform well as compared with some related methods.
Notes: CityU Call Number: QA931 .Z45 2012; vi, 102 leaves : ill. 30 cm.; Thesis (Ph.D.)--City University of Hong Kong, 2012.; Includes bibliographical references (leaves [95]-102)2012-01-01T00:00:00ZTheoretical investigation and numerical computation on meshless collocation method for solving partial differential equations
http://dspace.cityu.edu.hk:80/handle/2031/7426
Title: Theoretical investigation and numerical computation on meshless collocation method for solving partial differential equations
Authors: Zheng, Yanjun (鄭豔君)
Abstract: In the last decades, the use of radial basis functions (RBFs) has proven to be efficient
and robust in multivariate interpolation and solving partial differential equations
(PDEs). In this thesis we focus on the stability analysis using meshless collocation
method by RBFs for solving partial differential equations. The original umsymmetric
meshless collocation method was firstly introduced by E. Kansa in 1986. Hon and his
collaborators later extended the method to solve various nonlinear initial and boundary
value problems. In the first part of the thesis, we investigate the stability and convergence
of unsymmetric meshless collocation methods. Some theoretical results are
obtained based on the work of R. Schaback who gave a general framework for obtaining
error bounds and convergence of a large class of unsymmetric meshless numerical
methods in solving well-posed linear operator equations. For simplicity, we consider
in this thesis the standard Poisson boundary value problem (PBVP). Using the works
of F. J. Ward, H. Wendland and R. Arcangeli et al., we give in this thesis a stability
condition of meshless collocation methods in solving the PBVP. Based on the theoretical
stability result, in the second part of the thesis, we devise a meshless computational
algorithm for solving a real application problem arisen from financial option pricing
model. This involves the numerical techniques for solving a partial integro-differential
governing equation under some initial and boundary condition problem with unknown
free boundary. Numerical examples are given to verify the effectiveness, accuracy, and
robustness of the meshless collocation method.
Notes: CityU Call Number: QA377 .Z4427 2010; iv, 93 leaves : ill. 30 cm.; Thesis (Ph.D.)--City University of Hong Kong, 2010.; Includes bibliographical references (leaves 61-68)2010-01-01T00:00:00ZMathematical modeling and numerical study of multi-phase and multi-component flows in complex porous media
http://dspace.cityu.edu.hk:80/handle/2031/7425
Title: Mathematical modeling and numerical study of multi-phase and multi-component flows in complex porous media
Authors: Zhang, Qian (张谦)
Abstract: In this thesis, we study two multi-phase and multi-component fluid flows in clothing
assemblies and concrete materials, respectively. The former arises from the design
of functional clothing in textile industries and the latter can be found in construction
industries and structure engineering.
A typical clothing assembly consists of a thick fibrous batting sandwiched by two
thin covering fabrics. The outside cover is exposed to a cold environment with a lower
temperature and relative humidity while the inside cover is exposed to a gaseous mixture
of air and vapor with a higher temperature and relative humidity. In this application,
the heat and moisture transport is coupled in rather complicated mechanisms, the
phase changes, condensation/evaporation and fiber absorption, play an important role.
Firstly, we present some more precise formulations on condensation/evaporation, fiber
absorption and heat capacity to maintain the physical conservation of mass and energy.
Numerical results show that the proposed formulations are more realistic to describe
the phase changes. A human sweating model is simulated with a normal sweating rate
and a profuse sweating rate during exercise, respectively.
Secondly, since the environment temperature is usually lower than the freezing
point, there is an interface arising in the batting area of clothing assemblies. A Dirichlet
to Neumann map type interface method (DNMIM) is proposed for this problem
such that the moving interface can be captured implicitly and no extra iterations are
needed. An uncoupled semi-implicit finite volume method is applied for the system of
nonlinear parabolic equations. Two types of clothing assemblies are investigated
numerically and an artificial example is also presented to show the accuracy of our
algorithm.
Thirdly, we propose an uncoupled leap-frog finite difference method for a single component
model in porous textile materials. In this method, the leap-frog scheme is
applied in the temporal direction and a central finite difference approximation is used
in the spatial direction. We prove existence and uniqueness of the solution to the finite
difference system with optimal error estimates in both L2 and H1 norms. Numerical results
are presented to verify the theoretical accuracy. The numerical method presented
here can be easily extended to multi-component models.
Finally, we study a mathematical model of carbonation process in porous concrete
materials. Based on the physio-chemical mechanisms, the whole process can be
viewed as the multi-component flow coupled the carbon dioxide-moisture-calcium ion
transport. Since dissolution of calcium hydroxide plays a crucial role, we present a
modified formulation for the dissolution rate and more precise mathematical descriptions
on physio-chemical conservation. An uncoupled finite volume method with a
semi-implicit Euler scheme is proposed to solve the nonlinear parabolic equations.
Numerical simulations under the normal carbonation condition and the accelerated
carbonation condition are presented and analyzed, respectively.
Notes: CityU Call Number: QC173.4.P67 Z45 2012; vi, 115 leaves : ill. 30 cm.; Thesis (Ph.D.)--City University of Hong Kong, 2012.; Includes bibliographical references (leaves [104]-115)2012-01-01T00:00:00ZThe perturbation-incremental method for nonlinear dynamical systems
http://dspace.cityu.edu.hk:80/handle/2031/7424
Title: The perturbation-incremental method for nonlinear dynamical systems
Authors: Cao, Yingying (曹瑩瑩)
Abstract: In this thesis, we will study the application of the Perturbation-Incremental (PI)
method in nonlinear dynamical systems. The thesis consists of three parts.
The first part is about a novel construction of homoclinic/heteroclinic orbits (HOs)
in nonlinear oscillators by the Perturbation-Incremental method. Consider strongly
nonlinear oscillators of the form x + g(x) = Ɛf(x, x, μ), (0.1)
where g and f are arbitrary nonlinear functions of their arguments, Ɛ and μ are parameters
of arbitrary magnitude. Accurate analytical solution of a HO for small perturbation
can be obtained in terms of trigonometric functions. An advantage of the
present construction is that it gives an accurate approximate solution of a HO for large
parametric value in relatively few harmonic terms while other analytical methods such
as the Lindstedt-Poincare method and the multiple scales method fail to do so.
In the second part, we describe the application of the PI method to one-dimensional
complex Ginzburg-Landau equation. We first consider the cubic complex Ginzburg-Landau
equation as the form ∂tA=μA+β|A|2A+D∂xxA, (0.2)
where β=βr+iβi,D=Dr+iDi Є C and μ Є R. Stationary pulse solution
and hole solution are expressed in the form of A = ei(0(ᶓ)+ωt) μ(ᶓ) where μ(ᶓ) and
θ(ᶓ) are real functions with ᶓ = x - vt, ω Є R. From the harmonic balance (HB) method with a nonlinear time transform φ, we obtain some exact stationary coherent
structures including pulse and hole solutions, as well as traveling solutions. Then
we consider the cubic-quintic complex Ginzburg-Landau equation (QCGLE) without
regard to nonlinear gradient terms ∂tA=μA+β|A|2A+γ|A|4A+D∂xxA, (0.3)
where β = βr + iβi, γ = γr + iγi,D=Dr+iDi Є C and μ Є R. Exact stationary
hole solutions are found by the HB method with a nonlinear time transform φ. Some
numerical solutions are studied by the PI method.
In the third part, a novel approach of using HB method with a nonlinear time
transform is presented to find front, soliton and hole solutions of a modified complex
Ginzburg-Landau equation in the form of given by iut+1/2uxx+1/2(β-iF)uyy+(1-iδ)|u|2u=iγu, (0.4)
where β, F, δ and γ are real constants. Exact stationary and traveling solutions in the
form of u = ei(θ(ᶓ)+ωt)V(ᶓ) are studied, where v(ᶓ) and θ(ᶓ) are real functions with
ᶓ= p1x + p2y + p3t, and p1, p2, p3 are constants to be determined. Three families
of exact solutions are obtained, one of which contains two parameters while the others
one parameter. The HB method is an efficient technique in finding limit cycles of
dynamical systems. In this thesis, the method is extended to obtain HOs and then
coherent structures. It provides a systematic approach in the computation as various
methods may be needed to obtain the same families of solutions.
Notes: CityU Call Number: QA845 .C36 2012; v, 116 leaves 30 cm.; Thesis (Ph.D.)--City University of Hong Kong, 2012.; Includes bibliographical references (leaves [106]-116)2012-01-01T00:00:00Z