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Preface
The story of this book goes back to the author’s studies as an MSc/PhD candidate in the department of engineering
mechanics, Tsinghua University at Beijing China in 1978. At that time, investigations on fluid�solid interaction (FSI)
dynamics were motivated by widely intensive attention in world academic and engineering communities. While search-
ing and reading references on FSI published in the world journals, he realized that the solution of practical FSI pro-
blems involving complex physical conditions must rely on some numerical simulations, so he chose numerical methods
for structural and FSI dynamics as his research direction for his MSc and PhD theses. As the result of 6 years of study,
the MSc thesis for the investigation of the mode synthesis approach based on variational principles and the PhD thesis
on some of the theoretical and computational aspects of the finite element method (FEM) and substructure�subdomain
techniques for dynamic analysis of coupled FSI problems were completed and he passed his viva in 1980 and 1984,
respectively. In these theses, the fundamental principles to construct the substructure methods were revealed, and some
general theorems and variational principles for electrodynamics and FSIs were developed. Then these findings were
used to create the corresponding displacement, equilibrium, and mixed types of finite element (FE)�substructure mod-
els for numerical analysis of FSI problems. With his MSc/PhD degrees awarded, the author moved from the Institute of
Structure and Strength, where he had worked as a research engineer in structure dynamic analysis and testing for about
10 years before his MSc study, and joined academia at Beijing University of Aeronautics and Astronautics (BUAA) in
December of 1984. Later, the author was promoted to professor of theoretical and applied mechanics at BUAA. During
that period, when dealing with a numerical simulation of an aircraft fuel tank�wing interaction system, which aims to
obtain its coupled natural vibration frequencies and modes for more accurate flutter analysis of aircraft, it was further
realized that the occurrence of zero-energy modes in the displacement element model of fluids was a serious difficulty
to overcome for FSI problems involving free surface motions in the lower frequency range. This motived the author to
further tackle FSI problems with different free surface wave expressions. As a result, two variational principles were
developed, which respectively adopt fluid pressure and velocity potential to replace fluid displacement as the variables
describing fluid motions. These two principles have formed the basis for developing effective numerical models, espe-
cially the mixed pressure�displacement type, for linear FSI analysis.
When the author visited Brunel University, United Kingdom, in 1989�90, he met former Professor R.E.D. Bishop
and Professor W.G. Price, known as the father of hydroelasticity. This visit started his joint research with Professor
Price, which has been continued and further developed since his joining the Department of Ship Science at the
University of Southampton in 1993. During more than 25 years of teaching and research in close collaborations with
colleagues and students, the following developments on FSI analysis were made: (1) the variational principles for linear
FSI problems were further modified to include more complex practical cases, such as the boundary conditions for sur-
face tensions, gas�liquid coupling interfaces, floating FSI interfaces, etc.; (2) variational principles for nonlinear FSI
dynamics; (3) further development of the original mixed pressure�displacement model for linear FSI problems to con-
sider more complex boundary and coupling conditions and demonstrations dealing with many practical problems in
maritime engineering, such as water�very large floating structure (VLFS) interactions, liquefied natural gas (LNG)�
ship�water couplings, and three-phase interaction for air�water-shell systems, etc.; (4) development of the mixed
FE�boundary element (BE) model to deal with linear FSI problems involving infinite fluid or solid domains, so that
the number of degrees of freedom for infinite domains can be largely reduced and the efficiency of computations
increased; (5) creation of the mixed FE�computational fluid dynamics (CFD) model, allowing nonlinear FSI problems
to be simulated using available commercial solid FE and fluid CFD computer codes in association with a partitioned
iteration approach; (6) proposal of the mixed FE�smoothed particle method (SPM), which is convenient in simulating
nonlinear FSI problems suffering from fluid�solid separations and breaking waves, such as green water, and missiles
moving into and out of water. These theoretical results and numerical methods, developed and practiced by the author
and his colleagues, are presented in this book.
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Parts of chapters of the original draft of this book were used as lecture notes of linear and nonlinear numerical meth-
ods for FSI dynamics twice given at BUAA, for graduate students in the fields of mechanics and mathematics and three
times at Zhejiang University for students in the field of ocean engineering. Considering the requirements of teaching
and textbooks, when presenting each numerical method for a specified FSI problem, the author has always kept in mind
that the main aim of a book is for readers, especially students and new researchers, to learn fundamental theoretical
knowledge, essential techniques, and reliable methods, and then to apply their independent research ability to model
and numerically solve practical complex FSI problems. Toward this end, for each numerical method described through-
out the book, the author always tries to address the key steps to solve an engineering FSI problem (shown in Fig. 1.9).
The first is to define the physical FSI problem of concern, from which the purpose, characteristics, and important and
neglectable factors affecting the solution can be clarified to facilitate the formulation of a suitable mathematical model.
For example, when dealing with a water�structure interaction problem, the compressibility of water can be neglected
for problems aiming to obtain the fluid loads on a structure caused by surface waves, but it has to be considered for a
problem involving an underwater explosion wave problem. The second is to formulate the physical problem into a
mathematical model consisting of a set of partial differential equations with boundary conditions defined in the contin-
uum of the FSI system. This is completed by using assumptions to keep the essential factors and to neglect nonessential
ones explored in the first step, based on the fundamental mechanical conservative laws governing the given FSI prob-
lem. The resultant formulation constructs a mathematical base to solve the problem. The third step is to convert the
mathematical formulation derived in the second step into its numerical form for solving. To develop a practical and use-
ful numerical scheme, the essential discretization approach, with its consistency, stability, and convergence perfor-
mance, has to be considered. The last step is to illustrate and demonstrate the developed methods by the examples
supported by available experiments or other reported results.
To complete these basic tasks, this book provides the fundamental theories, methodologies, and results developed in
different applications of FSI dynamics in a coherent format for readers to learn, understand, and use them to accomplish
the four tasks to solve their physical FSI problems in engineering: (1) establish the fundamental principles, equations,
and different types of boundary conditions in continuum dynamics, as well as their simplified forms, by introducing
some assumptions, giving the necessary knowledge and approaches for readers to define their physical FSI problems
and to formulate them in reasonable and effective mathematical models; (2)develop the variational principles for linear
and nonlinear FSI systems, in which the conditions for free surface waves, surface tensions, floating FSI interfaces, and
two-phase gas�liquid coupling interfaces are included (these variational principles establish the basis for developing
new numerical methods); (3) determine the related numerical discretization theories, approaches, their formulations and
performance analysis, such as for the FE, BE, finite difference (FD), and SPM models described in each solution
method for different types of FSI problems; (4) examine the simultaneous numerical solution approaches for linear FSI
equations and the partitioned iteration approach for nonlinear problems, which enables the solid and the fluid equations
for nonlinear FSI problems to be separately solved in time steps using commercial codes, and then arriving at conver-
gence through coupling iteration until the time step of interest is reached. The book, to some extent, is of very practical
use with its complete and comprehensive knowledge covering theories, numerical methods, and their solutions to deal
with various FSI problems in engineering applications. No doubt that the number of pages in any book aiming to cover
FSI dynamics will soon run out, as it is not possible—and actually there is no need—to give very detailed information
on the related numerical methods that have been fully theoretically and practically demonstrated elsewhere. To amend
this and to satisfy the further reading requirements for some readers, when describing related knowledge and numerical
methods, the author always provides the references to world-influential books on the methods involved for more infor-
mation if it is needed.
Throughout the book, the author continually returns to some examples for each method and has tried to illustrate
even the most abstract results in order to demonstrate the developed theory and approaches, as well as applications. The
simpler examples can be solved by hand, and doing so can clearly enable readers, especially students, to better under-
stand the related FSI mechanisms. The selected practical application examples include air�liquid-shell three-phase
interactions, LNG ship�water sloshing, acoustic analysis of an air-building interaction system excited by human foot
impacts, transient dynamic response of an airplane�VLFS�water interaction system excited by airplane landing
impacts, turbulent flow�body interactions, a structure dropping down on the water surface with breaking waves, etc.
The numerical results are compared with available experiments or numerical data to demonstrate the accuracy of the
various approaches and their value for engineering applications. Based on FSI analysis of integrated wave harvesting
systems, it is revealed that the energy harvesting device acts as a damping mechanism in the system, which can keep
the resonance of the linear system and the periodical oscillation, such as flutter, of the nonlinear system in their
xii Preface
stable states. As a result, these two harmful phenomena—resonance and flutter—having to be avoided in traditional
designs conversely are used to effectively harvest wave energies.
The book consists of 12 chapters. The first chapter gives an introduction of FSI dynamics. After a general discussion
on its position and characteristics in mechanics, some FSI problems in engineering are listed. Solution approaches for
FSI problems include no coupling approximation, the quasicoupling approach, and integrated coupling methods. As a
preliminary introduction, a simple example is given to show how to construct a numerical model in order to solve com-
plex engineering problems. These include the Rayleigh�Ritz method based on variational formulations, FE models,
weighted residual approaches, and FD approximations, which are adopted to solve FSI problems in the book. Following
this introduction, a short review of FSI dynamics is given, which covers the subdisciplines of the field, some historical
events affecting the development and progress of research around the world, and important conferences, review papers,
and books on FSI. This short review provides fundamental information and literature resources for readers who wish to
engage in investigations of FSI problems in the future. Following this short review, the main aim of this book is given
and explained in the last subsection of the chapter.
In Chapter 2, Cartesian tensor and matrix calculus, some preliminary knowledge of Cartesian tensor analysis and
matrix calculus, used throughout the book, is briefly discussed, and further reference books are listed. Readers who are
familiar with tensor analysis and matrix calculus may not need to read this chapter.
Chapter 3, Fundamentals of continuum mechanics, presents fundamental knowledge on continuum dynamics,
which includes the different reference systems, deformation and stress analysis, conservative laws, constitutive equa-
tions, and various types of boundary conditions, as well as their corresponding simplifications by using some assump-
tions. This knowledge is necessary to formulate a FSI problem.
Chapter 4, Variational principles of linear fluid�solid interaction systems, presents the variational principles for
linear FSI dynamic systems. Following a short introduction on the history of variational principles for linear FSI
dynamics, it discusses the two new, further developed, variational principles by the author and his colleagues, in which
the floating boundary conditions on the wet interface, the interaction conditions between two different fluids, and the
surface tension conditions on the interfaces of liquids and gases are modeled. The first is a pressure�acceleration form,
an equilibrium or complementary energy form, in which the fluid pressure and the solid acceleration are taken as varia-
tional variables to describe the dynamics of the system. Considering the same conditions, the second example is its
kinematic or potential energy form, in which the velocity potential of fluids and the displacement of solids are chosen
as variables to describe the dynamic motions of the coupling system. After the detailed mathematical proofs on these
two principal variational principles, some varieties are derived by replacing the involved variables or releasing some
variational constraints by means of the Lagrangian multiplier approach. These modified variational formulations include
the two mixed energy forms—a displacement�pressure form and an acceleration�velocity potential form—as well as
two three-field forms: a displacement�pressure�velocity potential one and a displacement�acceleration�pressure one.
To deal with FSI systems with damping, for which the integrated variational in real form cannot be derived, the virtual
forms of two fundamental variational formulations are given, including the contributions from various types of damping
in FSI systems. A comprehensive discussion on the complex form of the integral variational formulations based on
adjoint variables for damped dynamic systems is also given in this chapter. As discussed in Chapter 1, Introduction,
variational formulations provide a powerful approach to converting the governing equations formulating physical FSI
problems in the continuum system into their corresponding numerical forms and then to construct effective numerical
models or to find their approximate solutions for very complex engineering problems that are difficult to solve analyti-
cally. The variational principles presented in this chapter play an important role as the “bridges” used in Chapter 5,
Solutions of some linear fluid�solid interaction problems, to derive approximate solutions of some simple FSI pro-
blems and in Chapter 6, Preliminaries of waves, to develop FE models for linear FSI systems.
Chapter 5, Solutions of some linear fluid�solid interaction problems, deals with some simple FSI problems, whose
theoretical solutions or the approximate solutions,based on the variational principles in association with the
Rayleigh�Ritz method, can be obtained. These selected FSI problems include one-dimensional (1D), 2D, and simpli-
fied 3D problems, such as axis-symmetrical problems and center-symmetrical problems. In Section 5.1.1, the dynamics
of a 1D FSI system subjected to a pressure wave is discussed, in which the natural vibrations and the dynamic response
are given. Sections 5.1.2 and 5.1.3 investigate 1D Sommerfeld systems to show that their natural vibrations rely on a
complex eigenvalue problem and how to use the dry solid natural modes to seek the dynamic response of these systems.
Section 5.1.4 discusses the effect of free surface waves based on Rayleigh�Ritz approximations, while Section 5.1.5
studies the vibration of an FSI system with a floating boundary, in which the gravity potential needs to be considered.
For 2D problems, Section 5.2.1 discusses the sloshing problem of a 2D incompressible water tank to further explain
how the variable separation method works to find solutions. Section 5.2.2 investigates a 2D Sommerfeld system
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involving both free surface waves and compressible waves aiming to explain the new radiation parameter proposed by
the author. Section 5.2.3 studies a dam-water system subjected to earthquake excitations, of which the solutions may be
useful in dam design, while Section 5.2.4 investigates a beam�water interaction system that is an often used as a model
to tackle offshore structure�water interactions. For 3D problems, an axis-symmetrical problem involving Bessel func-
tions is discussed in Section 5.3.1, while a center-symmetrical problem on underwater noise reduction is solved in
Section 5.3.2. It should be emphasized that, although the problems discussed in this chapter are simple, the solution
approaches used in the examples are carefully selected to explain the methods and concepts mentioned in previous
chapters, as well as to understand and reveal some FSI mechanisms more directly and obviously. For those researchers
and students who are just starting to deal with FSI dynamics, these simple examples provide very handy practices.
Chapter 6, Preliminaries of waves, presents some fundamental knowledge and concepts on waves. The concept of
wave propagation is developed from the simplest model for 1D motion, from which d’Alembert’s solution, dispersive
waves, and dissipation waves are introduced. Some nonlinear wave equations, such as Burgers, KdV, and Boussinesq
equations with their solitary wave solutions, are discussed based on the Jacobian elliptic function expansion method.
Other content concerns linear plane waves in order to provide knowledge on the method of variable separation, deep
and shallow water waves, standing and traveling waves, as well as the wave field and its energy transmission. This
knowledge can be used when dealing with wave�structure interactions.
Chapter 7, Finite element models for linear fluid�structure interaction problems, presents the two FE models for
linear FSI problems. The motions of solid structure are governed by the equations in linear elasticity theory, while the
fluid motions satisfy a linear pressure wave equation with possible linear gravity wave conditions on the free surface, if
it exists. Also, the surface tension can be considered if this is necessary. On the FSI boundaries, the conditions for the
force equilibrium and the motion consistency, represented by the corresponding variables adopted in each model, are
required. If a floating boundary is involved, the change of gravity potential caused by the flotation of the boundary is
considered. The forms of conditions on the solid and fluid boundaries depend on the variables used in the two models,
respectively. The first model is a displacement�velocity potential model in which the displacement in the solid and the
potential of velocity in the fluid are used as the variables to investigate the FSI. On the boundaries of the solid, the trac-
tion or the displacement condition can be defined, while the boundary conditions of the fluid are defined by the pre-
scribed potential of velocity or its spatial derivatives (fluid velocity). The second model is a mixed one in which the
displacement of the solid and the dynamic pressure of the fluid are chosen as the variables. Traction or acceleration is
required on the boundary of solid, while on the fluid boundaries, the pressure or the acceleration is defined. To improve
the efficiency of solving large complex FSI problems, the substructure�subdomain methods are introduced. Since the
presented numerical methods are based on FE models in both the solid and the fluid, any complex geometries of the
problem can be modeled by choosing suitable elements, and the traditional FE procedures can be directly used in the cal-
culations. For example, the subspace iteration program to solve the natural frequencies with the corresponding natural
modes of FSI systems and the available time integral approaches can be adopted to solve their dynamic responses. The
second displacement�pressure model is more convenient to simulate many engineering problems involving defined
pressure boundary conditions and obtaining dynamic pressures on structures, for example, the dynamic pressure on the
tank wall caused by fluid sloshing, the sound pressure in structure�acoustic volume interactions, the dynamic responses
of FSI systems subject to various excitations: explosion pressure wave, dynamic impacts, earthquakes, human foot
impacts, etc.
Chapter 8, Mixed finite element�boundary element model for linear water�structure interactions, deals with lin-
ear structure�water dynamic interactions based on a mixed FE�BE model. The water is treated as an ideal incompress-
ible fluid with its flow irrotational in the Euler coordinate system. Therefore its motions are governed by the theory of
potential flows presented in Section 3.6.3. The structures are considered as elastic bodies satisfying the equations in the
Lagrange coordinate system given in Chapter 3, Fundamentals of continuum mechanics. Following a description of
the fundamentals of the BE method, an important water�structure interaction problem is discussed: an integrated cou-
pling system of a very large floating structure (VLFS) subject to aircraft landing impacts. The problem involves
aircraft�VLFS�water interactions, for which a detailed mathematical description, the numerical solution method, as
well as validation and examples are given.
Chapter 9, Hydroelasticity theory of ship�water interactions, discusses hydroelasticity theory. Section 9.1 presents
the fundamentals of ship�water interaction dynamics, including the definitions of the three reference coordinate sys-
tems, generalized nonlinear governing equations, the static equilibrium solution, and the steady motion solution of the
system. The moving reference frame with the ship’s forward speed is used as a convenience in modeling the dynamics
of moving ships. Section 9.2 summarizes some concepts on incident water waves. Following the knowledge described
in Sections 9.1 and 9.2, Section 9.3 investigates the dynamic response of the linear integrated ship�water interaction
xiv Preface
system excited by incident waves using the traditional approach in the hydroelasticity theory of ships. Since the mode
summation approach is used in traditional hydroelasticity theory, which is not valid for nonlinear systems, only linear
problems are discussed in this chapter, leaving nonlinear ones to be tackled in other related chapters for nonlinear FSI
problems. Some differences between the mixed FE�BE model for water�structure interactions presented in Chapter 8,
Mixed finite element�boundary element model for linear water�structure interactions, and the traditional hydroelas-
ticity method are mentioned for readers to choose a suitable approach in solving their structure�water interaction pro-
blems of interest.
Chapter 10, Variational principlesfor nonlinear fluid�solid interactions, presents the variational principles for
nonlinear FSI dynamics developed by the author and his colleague. It is most important to find the main differences in
deriving variations of a variational principle for nonlinear FSI compared with linear cases. For linear theory, we assume
that the motions of the fluid and the solid are small, so that its original configuration is taken as our reference state, and
there is no need to distinguish Lagrange and Euler coordinates, as well as the variations involved. For example, when
we take the variation of a quantity defined in a fluid domain, we consider its boundary fixed at the original position and
neglect the effect caused by boundary motion. Also, as used widely, we always freely exchange the order of time and
space integrations in the linear variation process without any considerations. However, for nonlinear cases, these
operations are no longer valid, since large motions cause boundary motions, which have to be considered in the
variation process. After a short review of historical studies on the variations of nonlinear dynamical systems in
Section 10.1, the fundamental concepts of the variational process valid for nonlinear FSI systems are detailed, as
discussed in Section 10.2, for readers to learn these mathematical tools and to derive the variational principles for
nonlinear systems. Based on this knowledge and these methods, the variational principles for nonlinear FSI systems are
derived, and some selected application examples are presented in order to obtain their approximate solutions.
Chapter 11, Mixed finite element�computational fluid dynamics method for nonlinear fluid�solid interactions,
presents the mixed FE�CFD method for nonlinear FSI systems, which has been developed and practiced by the author
and his colleagues. In this mixed method, the well developed, both theoretically and numerically, FEMs in structural
analysis and CFD for fluid flows are combined to deal with nonlinear FSI problems, the benefit of which is effectively
using the available powerful commercial software for FEM and CFD to simulate complex nonlinear FSI problems in
engineering. Generally, we assume that the structure undergoes large rigid motions as well as large elastic deformation,
while the fluid flow is governed by nonlinear viscous or nonviscous field equations with nonlinear boundary conditions
applied to the free surface and FSI interfaces. The updated Lagrangian description in FEM analysis of solids and the
updated Arbitrary-Lagrangian�Eulerian mesh description in CFD, discussed in Chapter 3, Fundamentals of continuum
mechanics, are used to overcome the difficulty caused by large motions of coupling interfaces to solve the FSI equa-
tions in association with the partitioned iteration approach. The detailed numerical implication process and application
examples are presented after the fundamental theory is described.
Chapter 12, Mixed finite element�smoothed particle methods for nonlinear fluid�solid interactions, presents a
mixed FE�SPM, in which the solid structure is modeled by a powerful nonlinear FE model, while the fluid motions are
modeled by the SPM, to deal with nonlinear FSI problems involving violent fluid flows, such as fluid�solid separation
and breaking waves. Since both FE and SPM are based on the Lagrange description of motion, this mixed approach
easily traces fluid particle large motions occurring with breaking waves, compared with the FE�CFD method given in
Chapter 11, Mixed finite element�computational fluid dynamics method for nonlinear fluid�solid interactions. In
order to for readers to fully understand the SPM involved in this mixed scheme, a briefing on the history, characteris-
tics, developments with applications, and fundamental theory within the influential books are summarized in Sections
12.1�12.3 before discussing the proposed mixed FE�SPM in Section 12.4. To demonstrate the proposed mixed
method, selected examples include beam�water interactions, the rigid and flexible wedges dropping on the water,
flow-induced vibrations of a 2D cylinder, etc., whose results are compared with available experiments and other
reports.
Appendix A provides some numerical methods with its FORTRAN program to solve dynamic FE equations
developed by author. The principles, solution processes, example, and comparison of five time integration methods
often used in FEM are described. The computer code provides a generalized functional module for the five algorithms
to be incorporated into any computer program to investigate complex dynamics problems in engineering.
The book ends with an extensive list of more than 800 references. For the listed references, the author makes no
claims for the completeness of the listed publications but has tried to include the bulk of the papers, monographs, and
books that have proved useful to the author, his colleagues, and his students, but he recognizes that his bias
probably makes this a rather eclectic selection, especially the papers by the author, his colleagues, and students in their
collaborated researche on FSI problems.
Preface xv
The book can be used as an undergraduate or graduate textbook or a comprehensive source for scientists, academics,
researchers, and engineers, providing the state of the art on the theory, variational principles, numerical modeling, and
applications of FSI dynamics.
The author would like to thank his supervisors, former Professor Q.H. Du and former Professor Z.C. Zheng, as well
as the teachers, especially Professor K.Z. Huang, in the department of engineering mechanics at Tsinghua University,
who steered the author into engaging this research and who transferred his solid knowledge in mathematics and
mechanics to the author, which has created the lifetime benefit of being able to deal with complex FSI problems. The
author would like to deeply thank the Faculty of Engineering and Physical Sciences at the University of Southampton
for awarding an Emeritus Professor position and for allowing the university’s office and facilities to be used in writing
this book. The author’s colleagues in the Fluid�Structure Interaction Research Group and the Institute of Sound and
Vibration of the university, especially Professor W.G. Price, Fellow of Royal Society (FRS), deserve more thanks than
anyone can give for their important help and support lasting more than 20 years of working together.
The author extends his sincere thanks to his colleagues and friends who have helped him or jointly engaged in FSI
research problems. In particular, the author thanks Professor R.A. Shenoi, who led a team including the author to
manage the UK EPSRC and EU MASTRUC projects in which the involved FSI problems in marine engineering
were further practiced and demonstrated; Professor G.E. Hearn, who helped to check the PhD thesis
on aircraft�VLFS�water interactions completed by one of the author’s students; Dr. Y.P. Xiong, Dr. M. Tan, and
Dr. K. Djidjeli, who have jointly completed FSI research projects, supervised graduate students, and published a number
of FSI research papers with the author for many years. The author’s great thanks should be given to his former research
assistant and PhD candidates for their dedication in FSI research, particularly Dr. Y. Chen for the mixed FE�FD
method, Dr. S. Zhao for beam�water interactions, Dr. Z. Jin for aircraft�VLFS�water interactions, Dr. J. Yang for
the nonlinear airfoil flutter system to harvest fluid energy, Dr. F. Sun and Dr. Z. Sun for the mixed FE�SPM, and
Dr. A. Javed for the flow-excited vibration with the mixed FD�smoothed particle approach. The author also wishes to
thank Dr. M. Toyoda at Ishikawajima-Harima Heavy Industries Co. Ltd. (IHI), Japan, for the joint research in the
experiment on a water-spherical shell tank interaction vibration. Without the hard works of these RA and PhD students
on the theoretical analysis and numerical simulations for the related FSIproblems, this book could not have been pre-
sented in its current form.
The author expresses his deep thanks to Professor Bohua Sun, member of the Academy of Science of South Africa,
Cape Peninsula University of Technology, South Africa; Professor David Yang Gao, Alex Rubinov Professor of
Mathematics, Federation University Australia; and Professor Pihua Wen, reader in Computational Mechanics, School of
Engineering and Materials Science, Queen Mary University of London, for agreeing to let me nominate them as possi-
ble referees of this book, as required by the publisher when this book proposal was provided.
Finally, the author would like to acknowledge the encouragement, advice, and gentle criticisms of editors, whose
careful readings of the manuscripts enabled him to make corrections and improvements.
At last, the author thanks his wife and children for their understanding, patience, and supports during the production
of this addition to his family in the author’s retired life.
Jing Tang Xing
Southampton
September 2018
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