Download - Complete Thesis-Final
DYNAMIC SOIL-STRUCTURE INTERACTION FOR LAYERED GROUND
By
HAO SHEN
A thesis in fulfillment of the requirements for the degree of Bachelor of Engineering
School of Civil and Environmental Engineering
The University of New South Wales
Sydney, Australia
October 2013
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Abstract Taking into account dynamic soil-structure interaction in layered soil is quite
essential for some real cases, like earthquakes or machine-induced vibrations.
Both the out-of-plane and in-plane motion of a homogeneous, semi-infinite soil
layer under dynamic load is studied. For this thesis, the main purpose is to explain
and test two different methods of getting the acceleration unit-impulse response
matrix and finding out which method is more reliable and efficient.
First of all, the dynamic stiffness is required, which represents the
relationship between applied forces and the displacements. But, the key point is to
find out the acceleration unit-impulse response matrix . Based on the scaled
boundary finite element method, the scaled boundary finite element equation is
obtained, which can be used to deduce the values of in each time step. In
this thesis, there are mainly two methods to calculate : the constant scheme
and the linear scheme. For the constant scheme, the whole time period is
discretized into many time intervals with finer mesh. In addition, the unit-impulse
response coefficient is assumed to be constant within each time step. While
for the linear scheme, within each time interval, the values of change
linearly.
There are three numerical examples used to test these two schemes in a
Matlab program. For each example, the results will be compared with the
reference solutions both with respect to accuracy and time consumption (CPU
time). Then, we can find out which method is more efficient.
In conclusion, for the constant scheme, the results are usually in high
accuracy. But it costs lots of time. Although there are some errors, the linear
scheme usually requires less time. In addition, the errors can be diminished if we
choose the appropriate parameters.
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Acknowledgements
I would like to express my deep sense of gratitude to Dr. Carolin Birk, my
supervisor, for advising me on this research topic. She is an expert in the structural
dynamics and engineering mechanics fields. Dr Birk continuously and persuasively
inspired me throughout my study here. Without his timely support,
encouragement and advice, this thesis would not have been completed.
I would also like to thank Professor. Chongmin Song and John. P Wolf, who
established and developed the theory of scaled boundary finite element method,
and the basic knowledge about finite element method, and the Matlab codes
created by Dr Song, which helps me a lot on numerical example.
I am grateful to Dr. Wei Gao for exchanging research ideas and some basic
knowledge about material properties, and also Jason Zhao about helping me on
English grammar, format and editing in my thesis writing.
I wish to thank my father Peng Shen, my mother Xi Wang as well as other
members of my family for teaching, raising and supporting me.
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ORIGINALITY STATEMENT
‘I hereby declare that this submission is my own work and to the best
of my knowledge it contains no materials previously published or
written by another person, or substantial proportions of material which
have been accepted for the award of any other degree or diploma at
UNSW or any other educational institution, except where due
acknowledgement is made in the thesis. Any contribution made to the
research by others, with whom I have worked at UNSW or elsewhere,
is explicitly acknowledged in the thesis. I also declare that the
intellectual content of this thesis is the product of my own work, except
to the extent that assistance from others in the project's design and
conception or in style, presentation and linguistic expression is
acknowledged.’
Signed ……………………………………………..............
Date ……………………………………………..............
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Table of Contents
ABSTRACT ................................................................................................................................................ I
ACKNOWLEDGEMENTS ....................................................................................................................... II
ORIGINALITY STATEMENT .............................................................................................................. III
1 INTRODUCTION ................................................................................................................................. 1 1.1 STATEMENT OF PROBLEM ........................................................................................................................ 1 1.2 DIRECT METHOD ....................................................................................................................................... 2 1.3 SUBSTRUCTURE METHOD ......................................................................................................................... 3 1.4 THESIS OUTLINE ........................................................................................................................................ 6
2 LITERATURE REVIEW ...................................................................................................................... 8 2.1 INTRODUCTION .......................................................................................................................................... 8 2.2 FINITE ELEMENT METHOD (FEM) ........................................................................................................ 9 2.3 BOUNDARY ELEMENT METHOD .......................................................................................................... 10 2.4 INFINITE ELEMENTS .............................................................................................................................. 11 2.5 ABSORBING BOUNDARY CONDITIONS................................................................................................. 12 2.6 SCALED BOUNDARY FINITE ELEMENT METHOD ................................................................................ 14
3 THEORETICAL CONCEPTS ............................................................................................................17 3.1 INTRODUCTION OF HOMOGENEOUS, SEMI-INFINITE SOIL LAYER PROBLEM ................................. 17 3.2 TRANSFORMING CARTESIAN COORDINATES INTO LOCAL COORDINATES ..................................... 21 3.3 DERIVATION OF THE SCALED BOUNDARY FINITE ELEMENT EQUATION IN FREQUENCY DOMAIN
FOR OUT-OF-PLANE MOTION IN A 2D LAYER ............................................................................................ 26 3.4 DERIVATION OF SCALED BOUNDARY FINITE ELEMENT EQUATION IN TIME- DOMAIN .............. 32 3.5 CONSTANT SCHEME OF DERIVING THE ACCELERATION UNIT-IMPULSE RESPONSE MATRIX ... 37 3.6 LINEAR SCHEME OF THE OBTAINING ACCELERATION UNIT-IMPULSE RESPONSE MATRIX ....... 39 3.7 TIME-DOMAIN SOLUTION FOR ................................................................................................... 43
4 NUMERICAL EXAMPLES ................................................................................................................49 4.1 OUT-OF-PLANE MOTION OF HOMOGENEOUS-SOIL LAYER ............................................................... 49 4.2 IN-PLANE MOTION OF HOMOGENEOUS-SOIL LAYER ......................................................................... 67 4.3 HOMOGENEOUS SEMI-INFINITE SOIL LAYER WITH TRENCH ............................................................ 80
5 CONCLUSION AND FUTURE WORK ............................................................................................96 5.1 SUMMARY ................................................................................................................................................ 96 5.2 RECOMMENDATIONS FOR FUTURE RESEARCH................................................................................... 98
APPENDIX A ..........................................................................................................................................99
APPENDIX B ....................................................................................................................................... 100
APPENDIX C ....................................................................................................................................... 101
APPENDIX D ....................................................................................................................................... 102
BIBLIOGRAPHY ................................................................................................................................. 103
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Chapter 1
Introduction
1.1 Statement of problem
To civil engineers, the analysis of structures on soils or fluids is one of the most
difficult technical problems need to be solved. For both the design and
construction, the bearing capacity of static loads of structures is one of the major
problems to be dealt with. At present, a lot of proven techniques can be used to
solve that problem, like ground improvement, foundation work and reinforced
concrete. Not just the static load, dynamic soil-structure interactions, like
earthquake, waves from moving train underground and the machine induced
vibration, are also major factors of the collapse of the buildings and structures.
Because of overlooking and lacking of techniques, a large amount of buildings are
damaged from dynamic loads. In the past few years, many experts and engineers
studied in this area and try to find a method to solve the problems.
In order to study how the dynamic action affects structures, we need to
determine the actual response of the structures and surrounding soil materials,
like the displacements. A so-called dynamic soil-structure interaction analysis
should be performed. Before that, a numerical model of structure and soil domains
needs to be established. In this chapter, some previous strategies of modeling
media of infinite extent and their shortages will be introduced.
According to Trinks (2005), there is a strong relationship between response
of structure and the condition of the underlying ground. Waves can propagate
through the soil and the structure interacts dynamically with the unbounded
medium. In that case, those numerical models should include both the bounded
structure and the unbounded medium. From Wolf and Song (1996), the most
difficult problem remained is how to model and analyze the unbounded soils.
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For statics analysis, the displacement of unbounded domain tends to zero at
some distance from the loading position. Hence, a fixed boundary can be created
with specific boundary condition in the model. Thus, finite element method can be
used for statics load problems. But it is totally different for the dynamic analysis of
the unbounded domain.
Nowadays, there are mainly two methods to analyze the dynamic
unbounded medium-structure-interaction, Direct Method and Substructure Method.
Both of them are summarized in the next few paragraphs.
1.2 Direct method
In direct method, only a part of soil adjacent to structure will be analyzed together
with the structure. In order to satisfy the radiation conditions, an artificial
boundary need to be created in a suitable place to enclose the finite part. The finite
part or bounded domain is the combination of structure and near-field medium.
Finite Element Method can be used to analysis this coupled system. Finite
Element Method (FEM) is an efficient and well-developed numerical method to
obtain approximate solutions of vibration systems. But the problem of this method
is the reflecting of waves on artificial boundary, which may require additional
measures. So, finite element method is not suitable for the dynamic case here.
In order to solve that problem, some applicable strategies were created to
simulate the boundary condition. One of such methods is called Absorbing
Boundaries. The key point of Absorbing Boundaries Strategy is assuming that the
artificial boundaries can absorb the wave propagating through the interface to the
unbounded domain without any reflection. In that case, no additional measures
need to be done and this makes FEM possible here. But it is still unsuitable because
of the low level of accuracy and efficiency. These strategies are also referred to as
Transmitting Boundaries and non-reflecting boundaries. Comprehensive reviews
about those methods can be founded in the literature review part.
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1.3 Substructure method
Compare to Direct Method, Substructure Method is more efficient and accurate for
dynamic interaction analysis. For this kind of method, the problem domain is
divided into two parts. First part is the combination of structure and the soil
surrounding structure, which is called bounded domain. The rest of the infinite
medium is the second part, called unbounded domain.
For the bounded domain only, an external load { } applies on the
structure. Along the artificial boundary, there is an internal load { }. In
addition, there are also nodal displacement responses to the dynamic load along
the boundary. The near field part is modeled by finite element method.
Figure 1.2. 1, Model of interaction problem in near-field part
Equation of motion in time-domain is shown below:
[ ] { } [ ] { } { } { } 1.2. 1
The purpose of this formulation is finding out the displacements { }. In
this equation, term { } is the static stiffness matrix of nodes for the bounded
domain, which can be calculated out. The amount of { }can be monitored from
equipment. The sum total mass of the structure and surrounded soil { }is also
given. In that case, the only unknown is the internal force { }.
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In order to find { } , we need another equation to analysis the
unbounded domain.
Figure 1.2. 2, Model of dynamic interaction problem in unbounded domain
For substructure method, the discretized boundary, which coincides with
the soil-structure interface, encloses the bounded domain. Only the nodes along
the discretized boundary will be analyzed.
There are several different numerical methods which are suitable to model
unbounded domain. Typically, all these methods assume that the internal force
and displacements are harmonic. The equations look like:
{ } { } 1.2. 2
{ } { } 1.2. 3
Equation for unbounded domain is in frequency domain at first,
{ } [ ] { } 1.2. 4
In equation 1.2.4, term { }is the displacement vector of the discretized
nodes along the boundary in frequency domain. The symbol [ ]is the dynamic
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stiffness matrix of the soil of the interface, and it is also in frequency domain. One
of the major problems of substructure method is to find out the values of [ ].
The matrix [ ] can be found using a numerical method, like Boundary
Element Method and Scaled Boundary Finite Element Method. Compare to the
Boundary Element Method, Scaled Boundary Finite Element Method (SBFEM) are
more suitable to model the far-field part. From Wolf and Song (1996), this method
combines advantages of finite element method and boundary element method. It is
also an accurate semi-analytical model of the radiation damping effect. The details
of this method will be introduced in literature review part.
We cannot just replace { }in equation 1.2.1 into equation 1.2.4, since
they are in different domain. So, first of all, equation 1.2.4 should be transferred
from frequency domain into time domain. A method called inverse Fourier
Transformation (James, 2011) is employed. The basic equations of Fourier
Transformation Method are:
∫
1.2. 5
∫
1.2. 6
Equation 1.2.5 is used to transfer the interaction force in frequency domain
into time domain. Equation 1.2.6 is the inverse form of 1.2.5. After transforming,
equation 1.2.4 becomes
{ } ∫ [ ] { }
1.2. 7
Here, the matrix term [ ] is called the displacement unit-impulse
response matrix. According to Prempramote (2011), there is a force applied at a
point on discretized boundary and displacements take place at all degree of
freedom along this boundary. The displacements in each degree of freedom not
just only depend on the present applied force, but also the previous applied force.
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In that case, a convolution integral needs to be calculated at each time step, which
leads to a huge computational effort. Although this method is quite accurate,
because of the using of inverse Fourier transformation, it is still inefficient.
Recently, a more efficient method has been developed to calculate the unit-
impulse response matrix [ ] and evaluate the convolution integral (Radmanovic
& Katz, 2010). It is based on approximating the displacement unit-impulse
response matrix by a piece-wise linear function and truncating the convolution
integral after a certain point in time, when the displacement unit-impulse
responses close to zero. It is named as the new scheme or linear scheme to find out
the acceleration unit-impulse response matrix, which is employed in later study.
In this research, this idea will be extended to layered ground. I will focus on
the study about dynamic soil-structure interaction in a homogeneous semi-infinite
soil layer, which will be described later. The scaled boundary finite element
method will be used to find the dynamic stiffness matrix [ ]. The details about
SBFEM will be discussed in chapter 3 after the literature review.
1.4 Thesis outline
The further outline of this thesis is as follows:
In Chapter 2, the literature review of the existing methods of solving
dynamic interaction problem in unbounded domain is summarized. Those
methods are divided into two groups, global procedures and local procedures. The
global procedures include boundary element method and scaled boundary finite
element method. And the local procedures include most absorbing boundary
methods and finite element method. Each method is introduced in details and their
advantages and disadvantage will also be discussed. Then, a suitable method is
chosen as the theoretical framework of this research.
In Chapter 3, the project about out-of-plane motion of a homogeneous,
semi- finite soil layer is introduced at first, which includes the major difficult
confronted, some figures and governing equations. Then, the governing equation is
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derived, which is called the scaled boundary finite element method in this research.
The equation is in frequency-domain at first. A method of transforming the
equation into time-domain is also introduced in detail. The key step is to calculate
values of the acceleration unit-impulse response matrix. Two different methods
are introduced here.
In Chapter 4, studies about three different numerical examples are
represented. First one is about the out-of-plane motion of homogeneous semi-
infinite soil layer and the second example is the study about the in-plane motion of
the same soil layer. Two methods of obtaining the acceleration unit-impulse
response matrix are employed here, which has been talked about in chapter 3.
After comparing the results in terms of accuracy and time consumption, there is a
conclusion about which method is more efficiency and reliable. For the third
example, a trench is added in the same soil layer near the structure. There is a
study about how the trench influences the propagation of dynamic waves and
displacements of soils.
In Chapter 5, all the works in this research are summarized, and the
possible work in the future research are recommended.
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Chapter 2
Literature Review
In this chapter, some methods of solving the dynamic interaction problem in
unbounded domains will be introduced. Those methods can be clarified into two
groups, global procedures and local procedures. Global procedures include
boundary element method and scaled boundary finite element method, while most
absorbing boundary methods and finite elements belong to local procedures. For
each method, their principles and the process of developments will be talked about.
Their advantages and disadvantages will also be discussed in detail and then a
suitable method will be chosen as the theoretical framework in the soil-structure
interaction problems in layered soil.
2.1 Introduction
The laying of soils can importantly influence the dynamic behavior of the structure.
In this thesis, the problems of in-plane and out-of-plane motion of a homogeneous,
semi-infinite soil layer will be discussed in detail. In the past few decades, a lot of
researches had been undertaken to deal with this kind of problems and some
methods were also developed, for example, the Green’s functions, created by Luco,
Apsel (1983) and Bouchon (1981). But this kind of model is quite complicated and
expensive. Compared to Green’s function, Cone models are much simpler. Meek
and Wolf used this method to calculate the dynamic stiffness coefficients of a disc
of a single layer on rigid rock (Wolf & Meek, 1994). Lysmer and Waas derived the
thin-layer method to solve dynamic interaction problems in layered media
(Lysmer & Waas, 1975). It was developed between 1970s and 1990s. This method
has been considered as the most sophisticate one and mainly used in plane and
cylindrical problems. At recent, the scaled boundary finite element method
(SBFEM) has been widely used in the dynamic interaction problems. Wolf and
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Song developed it in 1990s (1996). And it is usually in high efficiency and accuracy.
The basic theory of SBFEM is introduced in sub-section 2.6.
For the problem in this article is about the dynamic interaction in layered
soils. After discussion and comparing, scaled boundary finite element method
(SBFEM) is chosen to model the far field part. All the calculations and formula
derivations are based on this method, for example the obtaining of the governing
equation and the dynamic stiffness matrix.
2.2 Finite Element Method (FEM)
The finite element method is a numerical technique to solve some field problems,
like temperature distribution and displacement distribution in objects. Those
structures are modeled by small-interconnected elements, called finite elements,
and connect with each other by nodes, then solved by approximating solutions to
partial differential equations. In most case, it is in high efficiency with ideal
accuracy. This kind of analysis is a fairly recent discipline crossing the boundaries
of mathematics, physics, and engineering and computer science. It is widely used
in the structural, thermal and fluid analysis areas (Nikishkov, 2001).
There are mainly two methods of transforming the physical formulation of
the problem into finite element discrete analogue; they are Galerkin method and
vibrational formulation (Nikishkov, 2001). Galerkin method is suitable for
differential equation problem, while vibrational formulation can be used when
physical problem can be formulated as the minimization of a functional.
From the above introduce, it can be find that finite element method is a
widely used method in many fields. In that case, a lot of commercial software about
this method are developed and put into use. So, it is really convenient for us to
solve some questions. Another advantage is that this method can be used in the
arbitrary geometry problems, which means it is suitable for some actual issue with
irregular shaped domain. Furthermore, it can also be used in anisotropic materials
and nonlinear domain problems. However, there are also some limitations. This
method only offers approximate solution and the accuracy of solutions is usually
10
affected by the shape quality and the density of element. Moreover, it is not
suitable for dynamic problems in unbounded domain, which has been explained in
the introduction part. If there are some cracks in specified medium, a large amount
of fine meshes have to be created around those cracks, which may increase the
computational effort a lot and the accuracy of results can be much lower.
2.3 Boundary Element Method
Boundary element method is an important numerical method in the computational
solution of a number of physical problems. In the past few years, the work
presented by Green (1828)really contributes to the development of BEM.
The first step of boundary element method is representing the boundary
value problem in the form of integral formulation, but not the equations of motion
and boundary conditions (Trinks, 2005). There are two methods of doing that,
weighted residuals and Green’s reciprocal theorem. After that, there is a spatial
discretization on the boundary of specific domain and a boundary integral
equation is created. Then, for all nodes of the surface, a global system of algebraic
equations for displacements and forces’ values can be obtained. Thus, the solutions
of the nodes on that boundary can be obtained.
There are many advantages of using boundary element method. Among
them, the most important one is that the spatial dimension of problem can be
reduced by one. This is because the volume integrals of problem can be
transformed into surface integrals, or surface integrals can be transformed into
line integrals. Therefore, the computational process is much simpler. Furthermore,
this method can be used in unbounded medium. Not like finite element method,
boundary element method is applicable when some cracks appear in the medium.
Only some nodes along the edge of crack will be included in the computing.
But there are also some shortages. BEM is very inefficient for the transient
load problems (Trinks, 2005), like earthquake and waves created by travelling
subway. Moreover, an important feature of BEM is the using of fundamental
solution, which increases the inapplicable of this method. For medium with
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anisotropic materials, the fundamental solutions are not easy to obtain. In order to
get a time-domain solution, the Fourier transformation method needed to be
utilized in, which leads to more computational effort. For time-dependent problem,
like dynamic interaction between structure and soil, the observation period also
needs to be discretized into time increment, which is quite expensive in terms of
numerical effort.
2.4 Infinite Elements
The infinite element method is developed from the finite element method and is
quite similar to it. The key step of infinite element is extending the shape function
of elements to infinite. Bettess (1977) did the original work of infinite elements.
The accuracy of the infinite element is controlled by the choice of the shape
function and the order of approximation (Prempramote, 2011). According to Trink
(2005), there are mainly three ways of extending the finite element domain to the
infinity: decay function, mapped infinite elements and wave envelope element.
In order to incorporate the correct decay in the infinite element, Bettess
(1977) and Zienkiewicz (Zienkiewicz, Taylor, & Zhu, 2005) developed the infinite
element with the help of a finite to infinite mapping. The formula of this method
looks like:
∑
∑
2.4. 1
The wave envelope concept was first used in finite elements. Then, this
method developed and transforms into the time-domain with the help of inverse
Fourier transformation.
There are also some disadvantages of infinite element. In order to improve
the accuracy of the results, the order of elements needs to be increase and this may
cause the ill-conditioning problems (Prempramote, 2011). Moreover, the geometry
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of the infinite-element mesh has to be conformed to a separable coordinate system
for the wave equation.
2.5 Absorbing Boundary Conditions
From the problem statement in introduction part, the major problem of direct
method is the reflection of outgoing waves at artificial boundaries. In order to
avoid the errors caused by reflecting waves from artificial boundary, a special
boundary condition can be created here. In the literature, such method is called
absorbing boundary conditions.
From Trink PhD thesis (Trinks, 2005), absorbing boundaries can be divided
into two categories, local and nonlocal. The differential operators with respect to
time and space are used in local absorbing boundaries. But differential and integral
operators are both used in nonlocal-absorbing boundaries.
For nonlocal boundary conditions, the relationship between forces and
displacements [ ] at artificial boundaries needs to be founded to describe it.
The DtN method (Bikri, Guenther, & Thomann, 2010) applies here to find this
relationship in a formula form, which is:
{ } [ ]{ } 2.5. 1
Moreover, the force-displacements relationship can be obtained from
approximate solutions of wave transmitting problems in a semi-infinite far field.
The formulation about this relationship is usually derived from frequency domain.
In order to find out the formulation in time domain, the inverse Fourier
transformation need to be used. To prevent the big computational work, some
other strategies are founded to replace Fourier transform, for example
approximating the exact DtN map by a frequency-dependent rational function. A
least-squares method (Wolf J. , 1994) also can be used to calculate the coefficients
of a scalar rational dynamic stiffness approximation. Another method is called
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lumped-parameter model, which were assigned to the corresponding partial
fractions (Wolf J. , 1994).
The major benefit of using global boundary conditions is the result is in
high accuracy and rigorous. For the formulation needs to be transformed from
frequency domain into time domain, the computational effort is very large and
time spending. Compared to the local boundaries condition, it will be more
expensive.
Some absorbing boundaries conditions are a kind of local procedures to
model the wave propagation in unbounded domain. Not like some global
procedures, this method is simpler and approximate. This method first appeared
on 1969, which is called “viscous boundary”, proposed by Lysmer and Kuhlemeyer
(Lysmer & Kuhlemeyer, 1969). For two-dimensional cases, the formulas of viscous
boundary condition are shown here:
2.5. 2
2.5. 3
where a and b are dimensionless parameters, is the mass density, is the
velocity of P-waves, is the velocity of S-waves, is a normal stress, is a shear
stress, is a normal velocity and is a tangential velocity. The ability of absorbing
the waves by viscous boundary depends on the ratio of the transmitting energy
from reflected waves and that from incident wave (Prempramote, 2011). After
research, Lysmer and Kuhlemeyer made a conclusion that the absorption
condition is good when a=b=1 (Lysmer & Kuhlemeyer, 1969).
From the above introduction, it can be easily found that local boundary
conditions is simple to implement and very efficient, when compared to nonlocal
boundaries conditions. The limitation of this method is the results are usually in
low accuracy. Accuracy can be improved by placing the boundary farther away,
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which means enlarge the near field. But this method may also increase the
numerical effort.
2.6 Scaled boundary finite element method
Scaled boundary finite element method (SBFEM) is a fundamental solution-less
boundary-element method. It is also a semi-analytical approach, which combines
the advantages of finite element method and boundary element method. This
method was initially developed by Song and Wolf (1997) and successfully utilized
in solving the elasto-dynamic and allied problems in civil engineering. For the
problems about dynamic interaction within unbounded medium, SBFEM will be
applied here as the theoretical framework.
At first, this method is called consistent infinitesimal finite-element cell
method, which developed for two-dimensional scalar waves in unbounded
domains. The term, called ‘scaled boundary finite element method’, first appeared
in 1997 created by Song and Wolf (1997). After extending this method, it can be
used to solve problems in diffusion (Song & Wolf, 1999), dynamic fluid-structure
interaction (Fan, Li, & Yu, 2005) and acoustics (Lehmann , Langer, & Clasen, 2006).
The scaled boundary finite element method can also apply to analyze problems
with stress singularity. Orders of singularity and stress intensity factors for multi-
material plates can be evaluated by it (Song & Wolf, 2002). Then, it is extended to
model the power-logarithmic singularities and to calculate the T-stresses, high
order terms, angular distributions of stresses under mechanical loading (Song C. ,
2005).
According to Wolf and Song (1999), there are two derivations of the scaled
boundary finite element equations in displacement and dynamic stiffness. One is
called the scaled-boundary-transformation-based derivation and another one is
called mechanically based derivation. For mechanically based derivation, the
finite-element cell is constructed between a similar fictitious boundary and an
artificial boundary. After limiting the cell width to zero, scaled boundary finite-
element equations can be obtained.
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For transformation-based derivation, with the help of Galerkin weighted-
residual technique, scaled boundary finite-element equation in displacement can
be obtained from governing partial differential equation. Like boundary element
method, only the boundary needs to be discretized. So, the spatial dimension can
be reduced by one. In that case, the computational effort can be reduced, which is
much less than that of finite element method. Dislike the boundary element
method, the fundamental solution is not necessary (Prempramote, 2011). From
Wolf and Song (2005), another unique advantage of SBFEM is that the calculation
of stress intensity factors is in high efficiency and accuracy. Furthermore, for
unbounded domain, the radiation condition at infinite for this SBFEM is satisfied.
At first, a scaling center O needs to be determined. The center should be
located in the problem domain and the whole boundary is visible from O. Then, the
boundary will be discretized into lots of nodes. In two dimensions problem,
curvilinear coordinates , are used to describe the locations of nodes. Symbol is
a dimensionless radial coordinate. It is also a scaling factor to represent the
location of scaling boundaries. For scaling center O, is equal to zero. Symbol
represents circumferential coordinate, which describes the location of nodes along
a specific boundary.
Figure 2.6. 1, Scaled boundary finite element method of spatial discretization of two-dimensional
unbounded domain
In figure 2.6.1, for any scaling boundary in unbounded domain, value of is
larger than one.
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Figure 2.6. 2, Scaled boundary finite element method of three-node line element
For each node along the boundary, the Cartesian coordinates x and y can be
obtained from the shape functions of the element. A typical 3-node line element is
shown in Figure 2.6.2. The coordinates of two end points are known.
For in-plane and out-of-plane motion in semi-infinite soil layer problem, the
major challenge is to find out the displacement at a specific point. At first, the
governing differential equation of displacement, which represents relationship
between nodal displacements and interaction, was required to be found out. Then,
the governing equation should be solved numerically with the method of weighted
residuals. It also needs to be transferred into the local coordinates system. After
simplifying, a new equation called scaled boundary finite element equation in
displacement is obtained. At the same time, the dynamic stiffness [ ] also
needs to be obtained. After processing, dynamic stiffness matrix in frequency is
founded out from the “quadratic equation”. With the help of Fourier
transformation method, scaled boundary finite element equation can be
transferred into time-domain. In my thesis, the critical step is to find out the value
of acceleration unit-impulse response matrix. Finally, those unknowns about nodal
displacement can be calculated out in commercial software-Matlab. The scaled
boundary finite element equations for the out-of-plane motion of a homogeneous
soil layer are derived in detail in Chapter 3. After that, two methods about
calculating the acceleration unit-impulse response matrix will be introduced,
which are called constant scheme and linear scheme. The purpose of this research is
to find out which method is more efficiency.
17
Chapter 3
Theoretical Concepts
After choosing the scaled boundary finite element method, the methodology about
this project needs to be developed. Sub-section 3.1 is about introducing my project
and some basic equations. The governing equation called scaled boundary finite
element equation will also be found, which is in sub-section 3.2 and 3.3. In addition,
there are two schemes about finding the acceleration unit-impulse response
matrix. Those two schemes are constant scheme and linear scheme, which will be
introduced in details in 3.5 and 3.6. Finally, in sub-section 3.6, we will have a look
how to get the in-plane and out-of-plane displacements at a specific point in the
soil layer.
3.1 Introduction of homogeneous, semi-infinite soil layer problem
As mentioned before, the main challenge of this project is to find a suitable way to
solve the dynamic interaction problem in unbounded domain. In most cases of real
conditions, the far field medium we study is usually overlaying on some hard and
fixed soils layers without any nodal displacements, like bedrock. But the whole
layer is unlimited in the horizontal direction. This kind of medium is called
homogeneous semi-infinite soil layer, which is focused on in this thesis. In the first
18
step, a model of homogeneous semi-infinite soil layer needs to be created.
Figure 3.1. 1, Homogeneous, semi-infinite soil layer with bounded and unbounded domain
From figure 3.1.1, there is a structure with dynamic load applied attached to
the soil layer and the layer is unbounded in x direction on both sides. There is a
dynamic interaction between structure and bounded medium. The semi-infinite
soil layer is bounded by the fixed boundary at bottom, which the interaction waves
cannot transmit through and there is no in-plane and out-of-plane displacement
along it. Similar to the example in introduction part, the whole section is divided
into three parts, near field and far field part. For model shown in figure 3.1.1, the
near field part is section A, which includes structure and part of the medium near
the structure. The far field is section B and C. For section A, the finite element
method can be used to model and solve it. In this chapter, only the far field domain
will be discussed. For the soil is homogeneous, we only focus on section B. The far
field region is shown in figure 3.1.2.
Figure 3.1. 2, unbounded domain of the homogeneous, semi-infinite soil layer
19
For simplicity, the depth of thickness of the semi-infinite layer is constant, h.
The upper boundary is stress free boundary, so the stress on each node along the
boundary is zero. For the lower boundary is fixed, the displacement is zero. Along
the left side vertical boundary, where x=0, internal stresses apply on the boundary.
Those stresses are variable along the depth with coupled stresses applied on the
near field part in opposite direction. Thus, the boundary condition is:
1) At ,
2) At ,
3) At ,
Figure 3.1. 3, an infinitesimal element in the homogeneous, semi-infinite soil layer
An infinitesimal element is split out from the medium with dimension
and , shown in figure 3.1.3. Force analysis in z direction of this element will
be discussed and the governing equations will be found out. The only unknown is
the displacement in z direction which depends on the x, y coordinates and time.
Equilibrium of force in z direction is:
3.1. 1
20
Here, is the density of the soil. So, the mass of the infinitesimal element is
Symbol is the second derivative of displacement with respect to time of
this element. It can also be considered as the acceleration. The right hand side of
the equation is the inertia force of this element. After simplify, equation 3.1.1
becomes:
3.1. 2
According to the Hooke’s law:
3.1. 3
3.1. 4
For the strain can be derived as the first derivative of the displacement in each
direction. Thus, the equation 3.1.3 and 3.1.4 can be modified into:
3.1. 5
3.1. 6
In this formula, symbol G [ ] is the shear modulus of the soil. Symbols and
are the strains of elements in x and y directions. After combing equations 3.1.2,
3.1.5 and 3.1.6, the governing differential equation of displacement can be found,
which is:
3.1. 7
21
It can also be considered as the wave equation in far field domain. Soil property
can be derived as symbol c, which is called wave propagation velocity. The unit of c
is m/s and the equation is:
√
3.1. 8
Then, equation 3.1.7 can be transferred into a new equation:
( )
3.1. 9
or
3.1. 10
In equation 3.1.10, the symbol is the gradient operator, which can be derived
as {
}. Thus, equation 3.1.9 and 3.1.10 can be combined as:
{
} {
}
3.1. 11
Then, the governing equation can be obtained later.
3.2 Transforming Cartesian coordinates into local coordinates
In most case, Cartesian coordinates are used to find out the relationship
between nodal displacements and interaction. For this problem, the local
coordinates are more suitable. Before find out the governing equation, the way of
22
translating system from Cartesian coordinates into local coordinates should be
introduced. At first, a 2D far field model in Cartesian coordinates system is created:
Figure 3.2. 1, unbounded domain in Cartesian coordinates
In figure 3.2.1, the boundary in the left edge, which contacts with the near-
filed part, is in curve. This is because, in real condition, the boundaries usually are
not in perfect vertical. The same model in scaled boundary coordinate system is
created in figure 3.2.2.
Figure 3.2. 2, far field model in nodal coordinates system
As the similar physical model created in previous section, there are three
boundaries in this model, and . On boundary , the load { } applied varies
with time and depth. For (y=0) is a fixed boundary, there is no displacement
along it. The boundary is considered as a free surface with a constant depth h.
Thus, the stress on that boundary is zero. In this system, two new coordinates
and are defined. Symbol is called radial coordinate, which is used to represent
the location of boundary . For example, if this boundary is located in its initial
location in left side, is equal to zero. This boundary can shift right horizontally
with an increment , then all the nodes along the boundary shift right with the
same increment without any vertical displacement. As shown in figure 3.2.3:
23
Figure 3.2. 3, shift of the boundary along the horizontal direction
The symbol represents the geometry of the boundary, which is called
circumferential coordinate. The equations of scaled boundary transformation can
be derived as:
3.2. 1
3.2. 2
Here, is the value of Cartesian coordinate at boundary and y is the value
of Cartesian coordinate . These two equations represent the relationship between
the Cartesian coordinates of the nodes with theirs scaled boundary coordinates.
And they are called the scaled boundary transformation. For the boundary ,
there is only one element with two nodes at ends.
Figure 3.2. 4, circumferential coordinate of boundary
For these two end nodes, they both have their known Cartesian coordinates,
and , which can be represented in two vectors:
24
{ } { }
3.2. 3
{ } { }
3.2. 4
According to Finite Element Method for two nodes element, these two end
nodes can represent the coordinates of any other nodes in the boundary. Thus, the
equation can be written as:
[
] { }
3.2. 5
In equation 3.2.5, formulas
and
are the two shape function of
this boundary. They can also be written as and . So, the general
equations of the coordinates for each node along the boundary are:
[ ]{ } 3.2. 6
[ ]{ }
3.2. 7
The major challenge is to represent the equation 3.1.10 by the scaled
boundary coordinates. From equations 3.2.6 and 3.2.7, the relationships between
coordinates x, y and coordinates and are obtained, which are and .
It is also very easy to get the equations of displacements in x and y coordinates u(x,
y). Thus, the displacements of nodes along boundary are represented
as , . Then, we can use the chain rule to deal with that problem.
First of all, the first derivative of displacement with respect to needs to be
calculated out. The equations are represented below:
3.2. 8
25
3.2. 9
We can combine these two equations into a matrix form:
{
} (
) {
}
3.2. 10
In equation 3.2.10, the matrix term (
) is called Jacobian matrix[ ]. Moving
the Jacobian matrix to the left-hand side, the equation becomes:
{
} [ ]
{
}
3.2. 11
According to equations 3.2.1 and 3.2.2, the Jacobian matrix can be simplified into:
[ ] [
]
3.2. 12
From equation 3.2.12, it can be easily find that Jacobian matrix is not depending
on . In that case, this matrix can be symbolized as [J]. According to the method of
inverse function of the two by two matrixes, the inverse of [J] can be easily found,
which looks like:
[ ]
| |[
]
3.2. 13
where, term
| | is the determinant of [J]and[
]
equals to [
].The
gradient operator is defined as:
26
{
} 3.2. 14
Applying equation 3.2.14 to the displacement function gives:
| |{
}
| |{ }
3.2. 15
It can be simplified as:
{ }
{ }
3.2. 16
where:
{ }
| |{
}
3.2. 17
and:
{ }
| |{ }
3.2. 18
3.3 Derivation of the Scaled Boundary Finite Element equation in
frequency domain for out-of-plane motion in a 2D layer
After getting the operator equation in local coordinate, the governing equation
3.1.10 should be solved numerically. The method of weighted residual technique
will be used here. The term ‘residual’ means error in the approximate solution.
At first, equation 3.1.10 can be rewritten as:
3.3. 1
27
Where, is in Cartesian coordinates system with respect to time.
Evaluating equation 3.3.1 for an approximate solution and integrating yields:
∫
∫
3.3. 2
In this equation, the term ∫ means the accumulation of the
residual values in Cartesian coordinates with respect to time. To derive a finite-
element approximation, the weighted-residual technique is applied to the
equilibrium equation. The general formula of this method is shown:
∫
3.3. 3
The symbol is the domain of this researched system. According to this general
formula, equation 3.3.1 is transferred into:
∫
3.3. 4
In equation 3.3.4, term represents displacement of soil node in Cartesian
coordinates as a function of time . Assumed in time-harmonic behavior,
the displacements can be derived as:
3.3. 5
The second derivative of displacement (acceleration) is:
3.3. 6
28
Replacing the term in equation 3.3.4 with equation3.3.6 gives:
∫ (
)
3.3. 7
For this equation, it is really hard to integrate the term . So, the method of
integration by parts is employed here. The general formula for 1D integration of
this method looks like:
∫ ∫
3.3. 8
and
∫ [ ] ∫
3.3. 9
In equation 3.3.9, term [ ]
is the boundary term. Employing this method in
equation 3.3.7, integrating by parts for the first term gives:
∫
∫
∫
3.3. 10
Term∫
is the boundary term and .
Figure 3.3. 1, the far-field domain in local coordinates system with boundary condition
29
Substituting equation 3.3.10 into equation 3.3.7 gives:
∫
∫
∫
3.3. 11
The boundary is discretized into a number of elements, and then the
displacements and weighting functions of boundary can be derived as:
[ ]{ }
3.3. 12
[ ]{ } 3.3. 13
The transpose of the weighting function is:
{ } [ ] 3.3. 14
Substituting equation 3.3.12 into equation 3.2.16 gives a new equation for the
gradient operator:
{ }[ ]{ } { }[ ] { } 3.3. 15
The following matrices are introduced as:
[ ] { }[ ] 3.3. 16
[ ] { }[ ] 3.3. 17
Substituting equation 3.3.16 and 3.3.17 in equation 3.3.15 yields:
30
[ ]{ } [ ]{ } 3.3. 18
For the gradient operator of weighting function:
[ ]{ } [ ]{ } 3.3. 19
Gradient operator of the transpose weighting function looks like:
{ } [ ]
{ } [ ] 3.3. 20
Equation 3.3.11 can be rewritten as:
∫
{ } [ ]
{ } [ ] ([ ]{ } [ ]{ })
∫ { } [ ] [ ]
{ } ∫
3.3. 21
where, , which can be transferred into local coordinates system:
| | . 3.3. 22
The term ∫
in equation 3.3.21is the boundary condition corresponds to
applied forces, which can be neglected at the moment. After substituting equation
3.3.22 into 3.3.21 and
, multiply the whole equation by G yields:
∫ { }
∫ [ ]
[ ]| | { } ∫ { }
∫ [ ]
[ ][ ] { }
∫ { } ∫ [ ] [ ]
| |
{ } ∫ { } ∫ [ ] [ ]| |
{ }
∫ { } ∫ [ ] [ ]
| | { }
3.3. 23
31
In equation 3.3.23, the integration of the terms and can be evaluated. The
coefficient matrices are defined as:
∫ [ ] [ ]| |
[ ] 3.3. 24(a)
∫ [ ] [ ]
| | [ ] 3.3. 24(b)
∫ [ ] [ ]| |
[ ] 3.3. 24(c)
∫ [ ] [ ]
| | [ ]
3.3. 24(d)
Matrices [ ] [ ] [ ] are the stiffness matrices in a finite element model.
The matrix term [ ] is the mass matrix.
Substituting all those stiffness matrixes and mass matrix into equation 3.3.23 gives:
∫ { }
([ ]{ } [ ] { }) ∫ { }
([ ]{ } [ ]{ }
[ ]{ })
3.3. 25
Integrating this equation by parts once more yields:
∫ { } [ ]{ } [ ] { } [ ]{ } [ ]
{ }
[ ]{ }
3.3. 26
As said before, this equation represents residual value. In order to get the
appropriate solution, value of residual must be zero. However, the transpose
weighting function { } cannot be zero. So, equation 3.3.26 can only satisfied for
32
arbitrary weighting function if the term in brackets is zero. Then, the scaled
boundary finite element equation in displacement can be obtained:
[ ]{ } [ ] { } [ ]{ } [ ] [ ] { } 3.3. 27
3.4 Derivation of Scaled Boundary Finite Element Equation in time-
domain
After getting the scaled boundary finite element equation in displacement { },
the values of dynamic stiffness [ ] needs to be obtained too. First of all, internal
nodal forces { }should be defined. In John P. Wolf’s book (1994), a method of
equating the virtual work of the internal nodal forces { } to the virtual work of
the surface tractions is discussed, which gives:
{ } [ ]{ } [ ] { } 3.4. 1
For unbounded domain, external nodal force { { }} applied, which is equal to
{ } with opposite direction. So, the relation between these two forces is:
{ { }} { } 3.4. 2
In introduction part, the equation 2.5.1 can be modified into local coordinates,
which is:
{ { }} [ ]{ } { } 3.4. 3
This equation is in frequency domain. In our case, the term { } in equation 3.4.3,
which relates to body loads or surface tractions, can be neglected. Substituting
equation 3.4.2 into 3.4.3 yields:
33
{ } [ ]{ } 3.4. 4
After substituting equation 3.4.1 into 3.4.4, a new equation can be obtained:
[ ]{ } [ ] { } [ ]{ } 3.4. 5
Then, differentiate equation 3.4.5 with respect to yields:
[ ]{ } [ ] { } [ ] { } [ ]{ } 3.4. 6
It can be modified into:
[ ] { } [ ]{ } [ ]{ } [ ] { } 3.4. 7
Substituting scaled boundary finite element equation in displacement 3.3.27 into
equation3.4.7 gives:
[ ] { } [ ]{ } [ ] { } [ ]{ } [ ]{ } 3.4. 8
Equation 3.4.5 can be solved to obtain the value of { } . The equation of { } is
shown below:
{ } [ ] [ ]{ } [ ] [ ] { } 3.4. 9
And then, equation 3.4.8 can be modified as:
[ ] { } [ ][ ] [ ] [ ] { }
[ ] [ ] [ ] [ ] { } [ ]{ } [ ]{ }
3.4. 10
34
For the soil layer has a constant depth with identical material coefficients,
the coefficient term [ ] does not depend on the value of . So, the value of
[ ] is zero. And equation 3.4.10 can be simplified as:
[ ] [ ] { }[ ] [ ] [ ] { }
[ ]{ } [ ]{ }
3.4. 11
All the terms { } in equation 3.4.11 are cancelled, so the equation becomes:
[ ] [ ] [ ] [ ] [ ] [ ] [ ] 3.4. 12
Equation 3.4.12 is called “quadratic equation” for dynamic stiffness matrix[ ].
It is referred to as an algebraic Riccati equation. This equation is also the scaled
boundary finite element equation in frequency domain.
At this stage, the scaled boundary finite element equation in time-domain
can be obtained. At first, the displacement-unit-impulse response [ ] is defined
as:
∫ [ ]{ }
3.4. 13
And the acceleration-unit-impulse response matrix [ ] is defined:
∫ [ ]{ }
3.4. 14
There is a relationship between acceleration dynamic stiffness [ ] and the
dynamic stiffness [ ] (Wolf & Song, 1996), which is shown below:
[ ] [ ]
3.4. 15
Equation 3.4.15 can be modified into:
35
[ ] [ ] 3.4. 16
In this equation, symbol “ ” is imaginary unit. Substituting equation 3.4.16 into
equation 3.4.12 yields:
[ ] [ ] [ ] [ ] [ ]
[ ] [ ]
3.4. 17
For equation 3.4.17, dividing the whole equation by gives:
[ ][ ] [ ]
[ ][ ] [ ] [ ][ ] [ ]
[ ][ ] [ ]
[ ]
[ ]
3.4. 18
In order to transfer this equation into time domain, the Fourier
transformation method needs to be employed. Transforming rules are shown
below:
3.4. 19(a)
3.4. 19(b)
With the help of convolution theorem and integration by parts, scaled boundary
finite element equation in time domain is obtained, which looks like:
∫ [ ]
[ ] [ ]
[ ][ ] ∫ ∫ [ ]
∫ ∫ [ ] [ ] [ ]
3.4. 20
36
[ ] [ ][ ] [ ] [ ]
In equation 3.4.20, the matrix term [ ] is the acceleration unit-impulse
response matrix.
To simplify this equation, Cholesky decomposition of [ ] is utilized.
Equation of [ ] is:
[ ] [ ] [ ] 3.4. 21
Then, inverse matrix of is shown below:
[ ] [ ] [ ]
3.4. 22
Equation 3.4.22 is substituted into equation 3.4.20. At the same time, all terms are
pre-multiplied by [ ] and post-multiplied by [ ] . A new equation is obtained:
For simplify, some multiplying of matrices can be replaced by one term only, for
example:
[ ] [ ][ ] [ ] 3.4. 24(a)
[ ] [ ][ ] [ ]
3.4. 24(b)
[ ] [ ] [ ][ ] [ ] [ ] [ ] 3.4. 24(c)
[ ] ∫ [ ]
[ ] [ ] [ ] [ ]
[ ] [ ][ ] [ ] ∫ ∫ [ ]
[ ]
[ ] ∫ ∫ [ ] [ ] [ ] [ ] [ ]
[ ] [ ][ ]
[ ] [ ][ ] [ ] [ ] [ ] [ ] [ ][ ]
3.4. 23
37
[ ] [ ][ ] [ ] 3.4. 24(d)
Finally, this equation looks like:
∫ [ ][ ] [ ] ∫ ∫ [ ]
∫ ∫ [ ]
[ ]
[ ] [ ]
3.4. 25
where the values of [ ], [ ] and [ ] have been given. The only unknown in this
equation is the term [ ], which is called acceleration unit-impulse response
matrix.
3.5 Constant scheme of deriving the Acceleration Unit-Impulse Response Matrix
In the scaled boundary finite element equation in time-domain, the term [ ]
is unknown. The major problem in my project is how to find the values of [ ].
Response matrix [ ] can be solved numerically by splitting the whole time range
into many small time steps. There are two methods of obtaining the values
of [ ]. In this sub-section, the constant scheme is introduced first.
Figure 3.5. 1, constant acceleration unit-impulse response coefficient in each time step
Figure 3.5.1 shows the change of [ ] with respect to time. The constant
time step size equals to . For the original scaled boundary finite element method,
38
term is assumed to be constant in each time interval. The integration of
can be expressed as:
∫ [ ]
∫ [ ]
∫ [ ]
∫ [ ]
3.5. 1
Then, this equation can be simplified into:
∫
∑
3.5. 2
In equation 3.5.2, the values of for each time step are in constant. All other
terms are discretized analogously. First of all, the scaled boundary finite element
equation for the first time interval can be obtained by the initial condition. The
equation is shown below:
[ ] [ ]
[ ] [ ]
[ ]
[ ] [ ]
3.5. 3
For equation 3.4.25, the term t is replaced by
. Dividing the whole equation by
yields:
[ ] [ ]
[ ] [ ]
[ ]
[ ] [ ]
3.5. 4
This quadratic equation for matrices is called Riccati equation. In commercial
software Matlab, the function code called “care.m” can be used to obtain the value
of [ ] .
Now, let’s look at steps that . Equation 3.4.25 is discretized into:
∑ [ ] [ ] [ ] ([ ] [ ]
[ ] )
3.5. 5
39
([ ] [ ]
[ ] ) [ ]
[ ] [ ]
Modifying the equation 3.5.5 yields:
[ ][ ] [ ] [
]
[ ][
]
[ ] [
]
∑ [ ] [ ] ([ ]
[ ] ) [ ]
[ ] [ ]
3.5. 6
This method will be used frequently in numerical examples part. Although
it can give us the results in high accuracy, the calculating process is time
consuming and complicated. Those equations related to constant scheme will be
transferred into Matlab codes for computation. The results obtained will also be
compared with reference solution and the results obtained from Linear Scheme
Method in terms of accuracy and efficiency. The linear scheme is the second
method of getting the acceleration unit-impulse response matrix. Details about
Linear Scheme Method will be introduced in the next sub-section.
3.6 Linear scheme of the obtaining Acceleration Unit-Impulse Response
Matrix
In the method above, scaled boundary finite element method (SBFEM) in time
domain is used for the computation of acceleration unit-impulse response matrix,
which is first developed by Wolf and Song (1996) and extended to non-
homogeneous medium by Bazyar & Song (2006). In order to capture main
characteristics of the wave propagation, the whole time period is discretized into
many time intervals with finer mesh. In addition, the unit-impulse response
coefficient is assumed to be constant within each time step. In that case, the
computational effort is extremely large for this method (Radmanovic & Katz, 2010).
A new integration scheme for the solution of the acceleration unit impulse
response matrix and the evaluation of the soil-structure interaction vector is
developed. There are two characteristics of this method. Firstly, within each time
40
interval, the acceleration unit-impulse response matrix changes linearly.
Furthermore, the extrapolation parameter is used to increase the stability of the
solution, which allows the using of large time step size. Compared to the previous
method, the computational effort decreases significantly.
The transformed scaled boundary finite element equation in acceleration
unit-impulse response for a two-dimensional layered medium is represented as:
∫ [ ][ ]
[ ] ∫ ∫ [ ] ∫ ∫ [ ]
[ ]
[ ]
[ ]
3.6. 1
In equation3.6.1, the value of [ ] [ ] [ ] [ ] are already expressed in
equations 3.4.24. This equation is definitely from the equation (5) in Radmanovic
& Katz’s report (2010). The only difference is the term ∫ [ ]
is missing.
For the first time step m=1, algebraic Riccati matrix equation can be used
for solving[ ] :
[ ] [ ][ ] [ ][
] [ ] [ ] [ ] 3.6. 2
In this equation, terms [ ] and [ ] are represented as:
[ ] [ ] [ ] 3.6. 3
[ ] [ ] [ ][ ] [ ] [
] [ ]
[ ]
3.6. 4
For second time step m=2, the algebraic Riccati equation 3.1.7 is solved for[ ] ,
[ ] [ ][ ] [ ] [ ] [ ] [ ]
[ ] 3.6. 5
with
41
[ ] [ ] [ ] [ ] 3.6. 6
[ ] [ ][ [ ] [ ] ]
[ [ ] [ ] ][ ]
[ [ ] [ ]] [[ ] [ ] [ ] [
] ]
[ ]
3.6. 7
For next few time intervals m 3, the Lyapunov equation 3.6.7 is solved for each
time step,
[ ] [ ] [ ] [ ] [ ] 3.6. 8
with
[ ] [ ] [ ] [ ]
[ ]
3.6. 9
[ ] ∑ [ ] [ [ ] ]
[ ] [ [ ]
[ ] [ ] ] [
[ ]
[ ]
[ ] ] [ ] [[ ] [ ]
[ ] ][ ] [ [ ] [ ]]
[ ] [[ ] [ ] ]
[ ]
3.6. 10
Furthermore, a new and very efficient recursive integration scheme for the
evaluation of the soil-structure interaction vector described by convolution
integral has been developed in Radmanovic & Katz’s report (2010), which is based
on integration by parts.
The soil-structure integration force vector is given by the convolution
integral:
42
∫
3.6. 11
With the help of integration by parts, we can derive efficient integration scheme
for evaluation of the soil-structure interaction vector as follows:
[
] 3.6. 12
The derivation of equation 3.6.12 and definition of the terms
and are described in detail in Radmanovic & Katz’s
reference(2010). The time-domain integration scheme for the layered soils is the
same as for half space. Moreover, integration by parts and linearization allow
truncating convolution integral.
If there is no truncation, the value of acceleration unit-impulse matrix
in equation 3.6.11 grows with time and may cause severe numerical
problem. For the original discretization scheme with n time steps, the matrix-
vector multiplication and vector summations should be carried out at each time
station. If the time step n is huge, large amounts of calculations should be operated,
which leads lots of CPU time consumption. So, it is not efficient. In order to reduce
the computational effort, the truncation time is employed. We may only deal with
the data in the first M time steps, which are only a small portion of the total
number of time steps. After that, the acceleration unit-impulse response matrix is
assumed to change linearly.
Linear scheme with those improvements will be employed in numerical
examples in chapter 4 to compare with the original scheme. In order to show how
the efficiency and accuracy they may achieve, some data analyses will also be
carried out.
43
3.7 Time-domain solution for
After getting the acceleration unit-impulse response matrix , the value of
displacement in each time point can be obtained.
Figure 3.7. 1, the internal reaction forces between near-field and far-field domain
From figure 3.7.1, we can find that the internal reaction forces, which apply on the
boundary of far-field and near-field domain, are in the same value with opposite
direction. The equation about the force-displacement relationship is represented
below:
∫
3.7. 1
In equation 3.7.1, internal reaction forces are given. Term t equals . The
only unknown is the value of acceleration in each time step.
44
Figure 3.7. 2, acceleration unit-impulse response coefficient in each time step
In order to simplify the calculation, the entire period is divided into n time
steps with size equals to . For each time step, the value of acceleration unit-
impulse response coefficient is constant.
During each time step, displacement of any node can be represented by the
displacements of that node at two end points. For example, in the first time step,
displacements of one node at two end points are and . Then, the value in
other time is:
(
)
3.7. 2
According to figure 3.7.2, the internal reaction force can be calculated from the
sum of the integration value in each time step. Then, equation 3.7.1 can be
modified into:
∫ ∫
∫
3.7. 3
with . All the terms t in equation 3.7.3 can be replaced by , then the
equation becomes:
45
∫ ∫
∫
3.7. 4
Equation 3.7.4 can be rewritten as:
∫ ∫
∫
3.7. 5
For each time step, coefficient terms can be split out from the integral sign. A new
equation is shown below:
∫
∫ ∫
3.7. 6
Then, equation 3.7.6 can be simplified as:
∑ ∫
∑
|
3.7. 7
Finally, an equation about the internal reaction forces, with respects to
acceleration unit-impulse response coefficients and the velocities in each time step,
can be obtained.
∑
3.7. 8
In equation 3.7.8, the value of is given or obtained from last time step. The
only unknown is the velocity in the next time step , which need to be calculated
out. In first step, if the term n equals to 1, equation 3.7.8 becomes:
3.7. 9
The initial velocity has been given and equals to zero in this case. Then, the
formula of expressed as:
46
[ ] 3.7. 10
In the second step, the term n equals to 2 and equations 3.7.8 transferred into:
3.7. 11
In equation 3.7.11, term has been obtained from last step and the only unknown
is term . So, the expression of is:
[ ] 3.7. 12
In the case of n=2, terms and the term can be replaced by the coefficients
and respectively. So, equation 3.7.12 can be modified into:
[ ] 3.7. 13
Thus, the general equation of the displacement in the nth time step can be
obtained:
∑
3.7. 14
Now, we can calculate the displacement.
(
)
3.7. 15
Equation 3.7.15 represents the displacement of nodes in each time step. Then, the
velocity in each time step can also be derived as:
(
)
3.7. 16
47
∫
3.7. 17
In equation 3.7.17, term equals to the value of the term . Then, the value of
velocity in each time step can be derived as:
3.7. 18
Substituting equations 3.7.16 and 3.7.18 into equation 3.7.17 yields:
∫ (
(
)
)
3.7. 19
After integral calculation, equation 3.7.19 modifies into:
3.7. 20
In equation 3.7.20, term represents the initial displacement, which is usually
given. In that case, the value of the term can be obtained. The equation of is
shown below:
3.7. 21
Then, substituting equation 3.7.21 into 3.7.15, the displacement for each time step
is obtained.
All those theories and equations will be used in numerical examples in the
next chapter to study about the dynamic interaction problems. Among them, the
two essential steps are the calculation of acceleration unit-impulse response
matrix and in-plane or out-of-plane motions of the soil layer. For simplicity, those
48
equations and theories will be translated into Matlab codes to have a computation.
In next chapter, three different numerical examples will be studied and discussed.
After comparing the results obtained with the reference solutions, there will be a
conclusion about which scheme, constant scheme or linear scheme, is more
suitable for calculating the acceleration unit-impulse response matrix.
49
Chapter 4
Numerical Examples
There are three numerical examples in my thesis. In the first example, there is a
study about the out-of-plane motions, which are caused by dynamic load, in a
homogeneous-soil layer. For the second example, we will focus on the in-plane
motion. In those two examples, constant scheme and linear scheme will be tested
and find out which one is more efficiency. In the last example, a trench is added in
the soil layer near the structure. Then, a study about how the trench influences the
displacements in soil layer will be carried out.
4.1 Out-of-plane motion of homogeneous-soil layer
At present, a new model of homogeneous, semi-infinite soil layer is created. It will
be analyzed and solved by Matlab. Then, the results of acceleration unit-impulse
response matrix will be compared with the standard solution, which is
represented in Wolf and Song’s book (1996).
In this model, the thickness of the layer is 2 meters. The vertical boundary
is discretized into two elements with two nodes each. For the soil properties, the
density is assumed to be 2000kg/m^3, Young’s modulus is 1.25e8 N/m^2 and the
Passion’s ratio is 0.25. The soil is considered as isotropic. The dimensionless end
time (tend) calculates up to 4 and the dimensionless time step size is 0.01.
The value of the stiffness and mass matrices in this example are already given
below:
[ ] [
]
4.1. 1
50
[ ] [
] 4.1. 2
[ ] [
]
4.1. 3
[ ] [
]
4.1. 4
The relationship between the dimensionless time and the real time t in [ ] is as
follows:
4.1. 5
In equation 4.1.5, term represents the time in units, means the dimensionless
time, h is the thickness of the layer and term denotes the shear wave velocity in
soil.
First of all, an upper triangular matrix [ ] is obtained through the Cholesky
decomposition of [ ], which has been explained in equations 3.4.21 and 3.4.22. In
Matlab, the code of Cholesky factorization is:
[ ] 4.1. 6
The value of matrix u is:
[ ] [
]
4.1. 7
According to equations 3.4.24, 3.4.25, 3.4.26 and 3.4.27, those simplified stiffness
and mass matrices, like [ ] [ ] and [ ] can be obtained:
51
[ ] [
]
4.1. 8
[ ] [
] 4.1. 9
[ ] [
]
4.1. 10
With the help of a function called “care”, equations 3.5.4 and 3.5.5 can be
solved in Matlab. Then, the values of for each time step can be obtained. All
these values need to be transferred into acceleration unit-impulse response
matrices[ ]. The equation of transformation is represented as:
[ ] [ ] [ ][ ] 4.1. 11
In most case, discretization of structure-medium interface can be much
finer, like dozens or even hundreds of nodes in each element. In order to simplify
the calculation, a method called impose quadratic variation will be used to transfer
the enormous original acceleration unit-impulse response matrices into a very
simple two by two matrix. For example, the structure medium interface consists of
2 elements, with 5 nodes in each. The graph is shown below:
Figure 4.1.1, structure medium interface with 9 nodes
52
In figure 4.1.1, within those nine nodes, only the displacements of node 5
and 9 are known, which can represent the displacements of other seven nodes. For
example, for the displacement of node 2, the equation looks like:
[
(
)
] {
}
4.1. 12
From the equation 4.1.12, the term [
(
)
] is the shape
function of the displacement of node 2. The equation of the displacements of these
nine nodes can be represented as:
{
}
[ ] {
}
4.1. 13
The term [ ] denotes the shape functions of nine nodes’ displacements, which is a
nine by two matrix. Then, acceleration unit-impulse response matrices [ ]
can be transferred into a 2 by 2 matrix from multiplying [ ] forward and [ ]
backward respectively. The equation looks like:
[ ] [ ] [ ] [ ] 4.1. 14
with
[ ] [
]
4.1. 15
After getting the acceleration unit-impulse response matrices, for the different
unit-impulse response coefficients ,
and , three figures are created to
53
show the change of each coefficient’s value with respect to dimensionless time.
Before plotting, real time in units should be changed back into dimensionless time
again.
54
Figure 4.1.2, acceleration unit-impulse response matrix of out-of-plane motion of semi-infinite layer of
constant depth discretized with quadratic finite element
In figure 4.1.2, the acceleration unit-impulse response coefficients in y-axes
also need to be changed into dimensionless. The tendency of each curve is the
same as the reference solutions. This references solution of acceleration unit-
impulse response coefficients are from figure A-9 in Wolf & Song’s reference
(1996)(see in Appendix A). But there are still some errors here, for example the
value of some key points. With the purpose of diminishing the errors, the
structure-medium interface will be discretized into a finer mesh. In this case, there
are five nodes in each element, but the number of elements along the boundary
keeps the same.
0
1
2
3
4
5
6
0 1 2 3 4 5
2 nodes
5 nodes
U
nit
-Im
pu
lse
Res
po
nse
Co
effi
cien
t
Dimensionless Time
55
Figure 4.1.3, curves of diagonal coefficients for 2 nodes mesh VS 5 nodes mesh
Figure 4.1.3 show the curves of diagonal coefficients for 2 and 5 nodes
meshes. As can be seen, for the curve of the coefficient , the difference between
two cases is quite small. But there is a big distinction for the curves of
coefficients
. The curves for the case of 5 nodes per element are much
closer to the reference solution. In next step, we will try the finer the mesh with 10
nodes per element for constant scheme. The comparison of the acceleration unit-
impulse response coefficients within three different meshes are shown in the
figures below:
-1.6
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0 1 2 3 4 5
2 nodes
5 nodes
U
nit
-Im
pu
lse
Res
po
nse
Co
effi
cien
t
Dimensionless Time
0
0.5
1
1.5
2
2.5
3
0 1 2 3 4 5
2 nodes
5 nodes
U
nit
-Im
pu
lse
Res
po
nse
Co
effi
cien
t
Dimensionless Time
56
Figure 4.1.4, curves of diagonal coefficients with respect to time in 2, 5 and 10 nodes meshes
0
1
2
3
4
5
6
0 1 2 3 4 5
2 nodes
5 nodes
10 nodes
U
nit
-Im
pu
lse
Res
po
nse
Co
effi
cien
t
Dimensionless Time
-1.6
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0 1 2 3 4 5
2 nodes
5 nodes
10 nodes
U
nit
-Im
pu
lse
Res
po
nse
Co
effi
cien
t
Dimensionless Time
0
0.5
1
1.5
2
2.5
3
0 1 2 3 4 5
2 nodes
5 nodes
10 nodes
U
nit
-Im
pu
lse
Res
po
nse
Co
effi
cien
t
Dimensionless Time
57
From these three figures, it can be easily found that the curves of the 5
nodes and 10 nodes converge. As the mesh finer, the results are much closer to
reference solutions. In addition, 5 nodes per element are fine enough.
At present, a model with 5 nodes per element is created in Matlab to
compare the stability of the linear and constant scheme. All related equations are
transferred into Matlab code to compute the acceleration unit-impulse response
matrix. In Matlab code, the linear scheme is named as intTyp=1, while the constant
scheme is defined as intTyp=0.
Figure 4.1 5, structure medium interface with 5 nodes per element
The code of constant scheme is expected to run with two different
dimensionless time steps 0.1 and 0.13. The linear scheme will also run in two
different conditions, time step is 0.1, theta is 1.0 and time step is 0.5, theta is 1.7.
Then, those four curves, which show the change of coefficient M11 respect to time
in four different cases, can be displayed in the same graph to make a comparison in
terms of stability. The figure is shown below:
58
Figure 4.1.6, comparison of the stability of the linear and constant scheme for determination of
acceleration unit-impulse response matrix of 5 nodes element model
As can be seen from figure 4.1.6, for the constant scheme, the result is stable
in 0.1 dimensionless time step size. But the curve is unstable when the size
increases to 0.13. If we use the linear scheme with the step size equals to 0.1, the
result is unstable quite early and becomes even worse when theta equals to 1.0. If
the theta increases to 1.7, the result will be stable again even for larger time step
size, like 0.5. In conclusion, the linear scheme can be used efficiently for a larger
time step, if we just increase the value of theta.
Figure 4.1.7, effect of the extrapolation parameter on the stability of the linear integration scheme
For linear scheme only, the program runs four times with the same time
step 0.5, but in different value of theta. From figure 13, we can find that the result
is unstable for 1.0 and 1.4, but it becomes stable when the theta increases to 1.7
0
5
10
15
20
25
30
35
40
45
0 10 20 30 40
const, timestep=0.1
const, timestep=0.13
theta=1.0,time step=0.1
theta=1.7,time step=0.5
Un
it I
mp
uls
e R
esp
on
se C
oef
fici
ent
Dimensionless TIme
0
5
10
15
20
25
30
35
40
45
50
0 10 20 30 40
theta=1.0
theta=1.4
theta=1.7
theta=1.8
U
nit
- Im
pu
lse
Res
po
nse
Co
effi
cien
t
Dimensionless Time
59
and 1.8. In addition, compare to the curve with 1.7 theta value, the curve with theta
equals to 1.0 becomes unstable quite early. Thus, as the value of theta increase, the
results are more stable.
Now, the same research is carried out for a finer mesh. All the coefficient
values keep the same. For this new model, each element consists of 9 nodes in the
structure medium interface, which is represented in the graph below:
Figure 4.1.8, structure medium interface model with 9 nodes per element
Figure 4.1.9, comparison of the stability of the linear and constant scheme for determination of
acceleration unit-impulse response matrix for finer mesh
0
5
10
15
20
25
30
35
40
45
0 10 20 30 40
const, timestep=0.02
theta=1.0, timestep=0.02
const, timestep=0.0375"
theta=1.7, timestep=0.5
Dimensionless Time t
Un
it-I
mp
uls
e R
eso
np
on
se C
oef
fici
ent
60
Figure 4.1.10, zoom in of the curves of coefficient for two stable cases in the initial steps.
In this figure, the curve for the larger time step (0.5) is quite rough. While
for smaller time step size 0.02, the curve looks smoother. Thus, the results will be
more accurate, if the time step size decreases.
Figure 4.1.11, effect of the extrapolation parameter on the stability of the linear integration scheme for
finer mesh
From figure 4.1.9 and 4.1.11, for the finer mesh, we can find that the
conclusion will be the same as the previous study. In addition, for both constant
and linear scheme, the coefficient values will be stable with smaller critical time
step sizes in a finer mesh. For the new scheme only, the critical time step sizes of
the stabilization of coefficient values does not depend on the theta value.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 0.5 1 1.5
const,timestep=0.02
theta=1.7,timestep=0.5
Dimensionless Time
Un
it I
mp
uls
e R
esp
on
se C
oef
fici
ent
-20
-10
0
10
20
30
40
50
60
70
0 10 20 30 40
theta=1.0
theta=1.4
theta=1.6
theta=1.7
U
nit
-Im
pu
lse
Res
po
nse
Co
effi
cien
t
DImensionless Time
61
The accuracy and efficiency of the new procedure will be tested in a similar
numerical example. In order to compare the accuracy, the curves of out-of-plane
displacements with respect to time in each case need to be obtained at first. For
the same homogeneous, semi-infinite soil layer, there is a uniformly distributed
load applied on the boundary. The figure of the model looks like:
Figure 4.1.12, time-dependent uniformly distributed load applied on the boundary of the
homogeneous, semi-infinite soil layer.
The load is a time dependent cosine function.
4.1. 16
with
4.1. 17
In equation 4.1.17, term is the first natural frequency of homogeneous layer.
After applying the dynamic load, the out-of-plane displacements for each soil
element are generated. The equation below represents the getting of the out-of-
plane displacements, which is from the honor thesis of Truong (2012).
∑ [ (
)]
√
√
4.1. 18
62
with
(Where j=0, 1, 2…. etc.)
4.1. 19
where is the eigenvalue.
In this example, a homogeneous, semi-infinite soil layer model with 2
meters thickness is still employed. But the vertical boundary is discretized into
two elements with 5 nodes each. The material properties keep the same and
dimensionless time step size for time-integration is 0.01. The dimensionless
finishing time is 30. At first, the curve of displacements for constant scheme is
created, which compared with the reference solution. The figure is shown below:
Figure 4.1.13, out-of-plane motion of numerical result VS analytical result
From figure 4.1.13, we can find that the numerical result from constant
scheme is quite close to the exact solutions, except for some small errors. There
are two kinds of errors here, the amplitude errors and phase errors. For the case
that creates the exact solution, there is only force vibration response. But for the
constant scheme, free vibration response also needs to be considered, which leads
to the errors. But the free vibration response can be diminished after at certain of
time.
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
0 10 20 30 40
exactsolutions
const
Dimensionless Time
Dis
pla
cem
ent
𝑣
63
Figure 4.1.14, numerical result VS analytical result over longer time
As can be seen from this figure, two curves are slightly different on the
amplitude values at the beginning. Then, they converge together. In the next few
studies, the numerical results from constant scheme can be used directly as the
reference solutions.
For the new scheme, the value of theta keeps constant at 1.7 in different
cases. Time step size for the computation of the displacements is taken as
0.01, which is the same as original scheme. The time step size for the
derivation of is , which is N time larger than the time step size of
displacements. For the linear scheme only, the Matlab program runs several time
with different value of N, but there is no truncation time. Through observing the
results, we can find that if the value of N is small, the displacements curve is
usually unstable. It becomes stable until N increases to around 50. But the results
are still inaccurate when compared with the reference solution. If the truncation
time is employed, the curve looks stable even for a smaller N.
At first, there is no truncation and a big value of is chosen to investigate
the error. In this example, the value of N is 50, so equals 0.5. Other parameters
keep the same.
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
0 20 40 60 80 100 120
const
exactsolution
Dimensionless Time 𝑡 /h
Dis
pla
cem
ent 𝑣
64
Figure 4.1.15, curve of new scheme case with N=50 and no truncation employed.
From this figure, there is almost no difference between the reference
solution and the new scheme, no truncation results in the first half period. As time
progress, the new scheme curve looks increasingly inaccurate. Those errors
include both the amplitude errors and the shift errors, which are mainly from the
choosing of the large time step size.
Now, truncation time is employed and we would like to know how it
influences the results. In this case, the truncation time is varied, while the value
of N keeps constant.
Figure 4.1.16, influence of the truncation time on results
-12
-10
-8
-6
-4
-2
0
2
4
6
8
0 10 20 30 40
const
N=50, notruncation
Dimensionless Time𝑡
Dis
pla
cem
ent 𝑣
-6
-4
-2
0
2
4
6
0 5 10 15 20 25 30 35
const
N=30,tc=3
N=30,tc=2
N-30,tc=1
Dimensionless Time𝑡
Dis
pla
cem
ent 𝑣
65
In figure 4.1.16, for the new scheme cases with later truncation time, like
or , there are some amplitude and shift errors. So, with the increasing of
truncation time , the results become more accurate. Compared with the no
truncation, larger time step size case in figure 4.1.15, the results of case N=30 and
will be more accurate. In the next part of study, the truncation time keeps
constant and we would like to find out how the value of N affects the results.
Figure 4.1.17, influence of the value of N on results
From this figure, large value of N means large time step size of derivation of
coefficients, which leads to inaccurate results.
At present, we would like to look at CPU time consumption of some new
scheme cases, with different values of N and truncation time.
-6
-4
-2
0
2
4
6
0 5 10 15 20 25 30 35
const
N=30, tc=1
N=40, tc=1
N=50, tc=1
Dimensionless Time 𝑡
Dis
pla
cem
ent 𝑣
66
Figure 4.1.18, vertical displacements vs. time in original and new schemes
Figure 4.1.19, CPU time required for the computation
As can be seen from the figure 4.1.18 and 4.1.19, although there are some
errors for linear scheme, the computational CPU time is quite small. From the bar
chart in figure 4.1.19, no matter how much the N value and truncation time are,
time spending of new scheme cases are all less than 5% of that of the constant
scheme. Moreover, there is no big difference within each other. Thus, in order to
find out which method is more efficiency, we will look at the accuracy of the result
more, while not taking its CPU time consumption into a serious consideration. In
-6
-4
-2
0
2
4
6
0 5 10 15 20 25 30 35
const
N=30,tc=1.0
N=40,tc=3.0
N=30,tc=3.0
Dimensionless Time 𝑡
Dis
pla
cem
ent 𝑣
0
0.2
0.4
0.6
0.8
1
1.2
const
N=30, tc=1
N=30, tc=3.
N=40, tc=3
% C
PU
tim
e o
f co
nst
ant
and
new
sch
eme
67
this example, the case with N=30 and will be banned, for its lower accuracy.
Although the CUP time consumption of N=30, is slightly longer than that of
N=40, , the former one is chosen as the best scheme since its results are more
accurate.
In conclusion, for first example, linear scheme is more suitable. The errors
can be diminished if we choose the appropriate values of parameters, like N,
truncation time, and theta value. In the next sub-section, the in-plane-motion
responses to dynamic load in a homogeneous semi-infinite soil layer will be
studied and discussed. And then, find out which scheme, linear scheme or linear
scheme, are more suitable.
4.2 In-plane motion of homogeneous-soil layer
In this example, a model of in-plane motion of a homogeneous-soil layer is created,
which is quite similar to the soil layer model in the last sub-section. But some
parameters and dimensions have been changed. The thickness of the layer is
shorted into 1 meter. In horizontal direction, the layer is still extended to infinite.
However, it is impossible to get the analytical solutions. In that case, the reference
solutions are needed. The model for reference solution will be created in ANSYS
with 50 meters in length, which will be discussed later. The soil properties, like
young’s modulus, density and Passion’s ratio, keep the same as those in example 1.
Figure 4.2. 1, model of a homogeneous soil layer
From this graph, the soil layer is overlaying on hard rock layer, which
means there is no displacement in both horizontal and vertical directions along the
bottom surface. On the boundary in the left end, a time-dependent surface traction
68
applied and it is described as a triangular function. The plotting of this
function load is shown in figure 4.2.2.
Figure 4.2. 2, time history of the triangular function surface traction
In figure 4.2.2, the time in x-axis is in dimensionless. The amplitude of the
triangular force is 10,000 N.
First of all, this kind of model is created in commercial software-ANSYS (see
Appendix B). The solutions, which generated from ANSYS model, can be used as
reference solution in later studies. The homogeneous layer is in the dimension of
1meter height and 50 meters length. The whole layer is discretized into 8 elements
in y-direction and 400 elements in x-direction with 4 nodes in each element. In this
example, we will only look at the horizontal and vertical displacements responses
at the upper-left corner point of this layer. In addition, the soil layer in ANSYS is
recreated into finer meshes, which are 16 elements in y- direction and 800
elements in x-direction. The displacement responses of this point to the horizontal
surface traction under two different meshes are represented in the figures below:
69
(a)
(b)
Figure 4.2. 3, Displacement responses to horizontal surface traction at the upper-left corner point for a
finer mesh model and a coarse mesh model computed using ANSYS: (a) horizontal displacement and (b)
vertical displacement
As can be seen, the curves for finer and coarser meshes converge and match
perfectly in both horizontal and vertical displacements. In that case, the results,
which obtained from coarse mesh model, are accurate enough as the reference
solutions.
In Matlab, the same model is also created. Similar to the example 1 in last
sub-section, two methods will be tested here, constant scheme and linear scheme,
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 10 20 30 40
Finermesh
Coarsemesh
Horizontal Displacement
Dimensionless Time
Dis
pla
cem
ent
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 10 20 30 40
Finermesh
Coarsemesh
Verical Displacement
Dimension less Time
D
isp
lace
men
t
70
which has been explained in details in chapter 3. In order to get the stable results,
the biggest dimensionless time step size for constant scheme is 0.07, which is
obtained from running the program many times with different values of . In next
step, I am trying to find out which scheme is more suitable for this example. First
of all, the constant scheme is used with dimensionless time step size equals to 0.07
and finished at 30.Then, the results will be compared with the reference solutions
from ANSYS.
(a)
(b)
Figure 4.2. 4, Displacement responses at the upper-left corner point in Matlab model with constant
scheme VS reference solutions: (a) horizontal displacement and (b) vertical displacement
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 5 10 15 20 25 30 35
ReferenceSolution
Dimensionless Time
Dis
pla
cem
ent
Horizontal Displacement
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 10 20 30 40
ReferenceSolution
ConstantScheme,timestep=0.07
Dimensionless Time
D
isp
lace
men
t
Vertical Displacement
71
From figure 4.2.4, it can be seen that the curves of constant scheme perfectly
match the curves of reference solution. The high accuracy of the results is caused
by the small time step size 0.07.
At this stage, the linear scheme with no truncation is employed here. After
running the Matlab program with different time step size, we can find that the
critical dimensionless time step size for stable and accurate displacement curves is
0.008. When enlarge the time step size, the curves are unstable at first. It becomes
relatively stable again until time step equals to 0.5 (Nint=50). But the results are
very inaccurate.
Figure 4.2. 5, horizontal displacement curve for the linear scheme case with critical dimensionless
time step 0.008, no truncation VS reference solution
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 5 10 15 20 25 30 35
Linearscheme,0.008, notruncation
ReferenceSolution
Horizontal Displacement
Dimensionless Time
D
isp
lace
men
t
72
Figure 4.2. 6, horizontal displacement curve for linear scheme case with time step size equals to 0.5, no
truncation VS reference solution
Figure 4.2. 7, zoom in the first two periods of figure 4.2 7
From the figure 4.2.6 and 4.2.7, the curve is only stable up to 5 in the first
period in a low accuracy. The main reason of that is the time step size is too big.
But for the displacement curve with 0.008-dimensionless time step size, it is more
stable and accurate. However, it still becomes unstable after dimensionless time 20.
As discussed before, the values of acceleration unit-impulse response
matrix may also influence the values of displacement. For the constant scheme, the
-3
-2
-1
0
1
2
3
4
0 10 20 30 40
ReferenceSolution
Linear Scheme,dt=0.5, notruncation
Dimensionless Time
Dis
pla
cem
ent
Horizontal Displacement
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0 2 4 6 8 10 12
LinearScheme,0.5, notruncationReference solution
Dimensionless Time
D
isp
lace
men
t
Horziontal Displacement
73
dimensionless values of coefficients , and at the dimensionless time
point 1.0 are 2.9846, -0.20701and 3.0505 respectively. But those values for the
linear scheme with 1.0 time step size equal to 2.9056, -0.20739 and 2.9953
respectively.
Figure 4.2. 8, curves of acceleration unit-impulse response coefficient in constant scheme case and
linear scheme case with 0.5 dimensionless time step size
From this figure, we can find that those two curves are not converging.
Compared to the linear scheme, the curve of constant scheme is smoother since the
smaller time step size. That is why the displacement curves of constant scheme are
much closer to the reference solutions.
At present, the truncation will be employed and see how it affects the
results. Firstly, a linear scheme case with a large time step 0.5 is tested. If no
truncation, the horizontal and vertical curves are relatively stable, which are
represented below:
0
0.5
1
1.5
2
2.5
3
3.5
0 0.2 0.4 0.6 0.8 1 1.2
ConstantScheme,dt=0.07
LinearScheme,dt=0.5
DImensionless TIme
U
nit
-Im
pu
lse
Res
po
nse
Co
effi
cien
t
𝑀
∞𝑡′
density
𝑠
74
(a)
(b)
Figure 4.2. 9, Displacement response of upper-left corner point to horizontal surface traction in linear
scheme with , no truncation VS reference solution: (a) horizontal displacement and (b) vertical
displacement
As can be seen, if no truncation, the displacement curves are only stable in
the first two periods in very low accuracy. The results obtained contain lots of
errors and we can consider that this method is unsuitable. Now, the model is
-3
-2
-1
0
1
2
3
4
0 10 20 30 40
ReferenceSolution
LinearScheme,dt=0.5, notruncation
Dimensionless Time
Dis
pla
cem
ent
Horizontal Displacement
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
0 10 20 30 40
ReferenceSolution
LinearScheme,dt=0.5, notruncation
Dimensionless Time
D
isp
lace
men
t
Vertical Displacement
75
truncated at dimensionless time 5. The horizontal and vertical displacements
curves look like:
(a)
(b)
Figure 4.2. 10, Displacement responses of upper-left corner point to horizontal surface traction in
linear scheme with , truncate at 5 VS reference solution: (a) horizontal displacement and (b)
vertical displacement
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
0 5 10 15 20 25 30 35
Referencesolution
Lienarscheme,dt=0.5,truncate at 5
Dimensionless Time
Dis
pla
cem
ent
Horizontal Displacement
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
0 5 10 15 20 25 30 35
"Reference solution
Linearscheme,dt=0.5,truncateat 5
Dimensionless Time
D
isp
lace
men
t
Vertical Displacement
76
From figure 4.2 10, with the help of truncation, the curves can keep stable
over longer time. But results still have certain amount of errors. In that case, since
the low accurate results, the linear schemes with large time step size are not
workable whenever truncation or not.
Then, we will look at the linear scheme case with a smaller time step size
0.008 and discuss how the truncation influences the results.
(a)
(b)
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 10 20 30 40
Linearscheme,0.008, notruncation
Referencesolution
Dimensionless Time
D
isp
lace
men
t
Horizontal Displacement
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0 10 20 30 40
Linearscheme,0.008, notruncationReferencesolution
Dimensionless Time
D
isp
lace
men
t
Vertical Displacement
77
Figure 4.2. 11, Displacement responses of upper-left corner point to horizontal traction in linear
scheme with and no truncation VS reference solution: (a) horizontal displacement and (b)
vertical displacement
As can be seen, because of the smaller time step size, the curves are more
accurate and quite close to the reference solution. They become unstable only after
dimensionless time point 25. Moreover, amplitude values of two curves are also
identical in the first five periods. Then, we still try to truncate it at dimensionless
time 5. The displacements curves are represented below:
(a)
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 10 20 30 40
Lienarscheme,0.008,truncateat 5Referencesolution
Horziontal Displacement
Dimensionless Time
D
isp
lace
men
t
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 10 20 30 40
Linearscheme,0.008,truncateat 5Referencesolution
Dimensionless Time
D
Isp
lace
men
t
Vertical Displacement
78
(b)
Figure 4.2. 12, Displacement responses of upper-left corner point to horizontal surface traction in
linear scheme with and truncate at 5 VS reference solution: (a) horizontal displacement
and (b) vertical displacement
From figure 4.2 12, we can find that the curves of linear scheme case can be
stable for a long time. But, for both the horizontal and vertical curves, the accuracy
of amplitude values incline after a certain time. It seems that there is a slight over
damping in this system. In conclusion, for a smaller time step size, the truncation
causes some error after a certain time. There are mainly two methods to improve
the accuracy: making time step smaller or truncating at a later time. Now, we will
have a look at how these two methods influence the results.
At first, for the same truncation time 5, different time step size will be
tested. Those chosen dimensionless time step sizes are 0.008, 0.006 and 0.002,
whose results will be compared with the reference solution. The curves are
represented below:
Figure 4.2. 13, horizontal displacement responses of upper-left corner point to horizontal surface
traction in linear scheme with different time step size, truncate at 5 VS reference solution
From figure 4.2.13, those curves for different linear schemes are identical
and the errors are not diminished. So, for the linear scheme under truncation,
decreasing of time step size does not improve the accuracy. For the second method,
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 10 20 30 40
ReferenceSolution
t=0.008
t=0.006
t=0.002
Dimensionless Time
Dis
pla
cem
ent
Horizontal Displacement
79
the dimensionless time step size keeps constant as 0.008, while the truncation
time increases to 10 and 15. The curves look like:
Figure 4.2. 14, horizontal displacement responses of upper-left corner point to horizontal surface
traction in linear scheme with , truncate at 5, 10 and 15 VS reference solution
From figure 4.2.14, as the truncation time increased to 10, the horizontal
displacement curve converges to the reference solution’s curve. But it gets worse
when the truncation time increases to 15. There is some instability of the curves.
Then, it is hard to say that increasing the truncation time can always improve the
accuracy of results. But the optimal results can be obtained by employing a critical
value of truncation time.
In summary, the idea of using a big time step ( ) is not suitable here,
because of the loss of accuracy in both truncation and non-truncation conditions.
But for small time step size condition, which is smaller than or equal to 0.008, the
results is much more accurate, no matter truncate it or not. The accuracy can be
optimized if the appropriate truncation time is chosen.
Finally, we will look at the CPU time consuming of four different cases.
Those cases are constant scheme with time step size equals to 0.07, linear scheme
with 0.008 time step size and truncate at 5, 10 and 15 respectively.
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0 10 20 30 40
Reference Solution
t=0.008, truncate at 5
t=0.008, truncate at 10
t=0.008, truncate at 15
Dimensionless Time
Dis
pla
cem
ent
Horizontal Displacement
80
Figure 4.2. 15, CPU time required for the computation
As can be seen from figure 4.2.15, the time consumptions for linear schemes
are much larger than that of constant scheme with 0.07 dimensionless time step
size. With the increasing of truncation time, the costs become larger. In this case,
we can make a conclusion that the constant scheme is the best choice since its high
accuracy and less time consumption. In the future, truncation may be employed in
the constant scheme. That may lead to a higher efficient method.
4.3 Homogeneous semi-infinite soil layer with trench
In this example, there will be a study about a new homogeneous soil layer, with a
building foundation laying on the top and a trench nearby. The main purpose of
this example is to find out how the trench influences the dynamic wave
propagation and the displacement responses to dynamic load at some specific
points. The layer is constant in depth and extends to infinite on both sides in
horizontal direction. It is overlaying on a hard concrete layer and no displacement
along the lower boundary. The simple sketching of the layer looks like:
0
10
20
30
40
50
60
70
Constant Scheme,t=0.07
Linear Scheme,t=0.008, truncateat 5Linear Scheme,t=0.008, truncateat 10Linear Scheme,t=0.008, truncateat 15
CP
U t
ime
of
con
stan
t an
d n
ew s
chem
e
81
Figure 4.3. 1, sketch of the homogeneous soil layer with a building foundation and a trench
For the construction in site, there is always some dynamic load applied on
the building foundation, which may damage the structures nearby. So, the function
of the trench here is to weak the propagation of the dynamic loading waves.
The whole layer will be divided into three parts, a bounded domain and two
unbounded domains. The bounded domain includes the foundation, the trench and
surrounding soils. Those three parts independently looks like:
Figure 4.3. 2, independent bounded and unbounded parts
At present, we will only study the bounded domain. In numerical study, the
trench can be considered as a crack in the layer. For traditional finite element
method or scaled boundary finite element method, the mesh around the crack is
much finer than other locations, which indeed increases the computer resource
and human effort. In order to overcome that problem, the super-elements method
will be employed. Super-elements method is usually suitable for some huge models
with different material properties or loading conditions in different parts. The big
model can be divided into several small sections and each section will be meshed
according to specific method, like finite element method and scaled boundary
82
finite element method. The figure of super-elements of the bounded domain is
represented below:
Figure 4.3. 3, super-elements of bounded domain
In our case, the dynamic stress intensity factors can be determined based
on scaled boundary finite-elements formulation in high efficiency with the help of
super-elements. According to Song, although the advantages of the scaled
boundary finite element method in modeling stress singularities are retained, the
size of the super-elements is limited by the highest frequency of interest. In order
to simulate the response at high frequencies, the problem domain should be
divided into smaller elements, which increases complexity and effort (Song C. ,
2008).
Recently, a continued fraction solution of the scaled boundary finite
element equation in the dynamic stiffness of bounded domain is employed, which
can greatly improve the efficiency. This method is developed from the derivation of
dynamic stiffness matrix of unbounded domain. As Song said, with applying this
continued fraction solution, the force-displacement relationship on the boundary
is formulated as an equation of motion expressed by symmetric, spare, high-order
static stiffness and mass matrices (2008). The derivation of continued fraction
solution is introduced here briefly.
First of all, the scaled boundary finite element equation in dynamic stiffness
is derived and the details are shown in Song’s thesis (2008). The equation looks
like:
[ ] [ ] [ ] [ ] [ ] [ ] [ ]
[ ] [ ]
4.3. 1
83
In this equation, the terms [ ], [ ], [ ]and [ ]are coefficient matrices,
which obtained by assembling the element coefficient matrices. They are derived
from the equation 3.3.24. Term [ ] is the dynamic stiffness matrix of a bounded
domain. In Song’s thesis (2008), based on the equation 4.3.1, the derivation of the
continued fraction solution is illustrated step-by-step by using a simple example.
Then, the equation of the continued fraction is expressed as:
4.3. 2
with the coefficients K, M,
and
( =1,2… ). Term is the order of the
continued fraction. Term equals to (
𝑠)
. According to Song, the accuracy of the
continued fraction solution at high frequency can be improved, as the order of
continued fraction increases (Song C. , 2008). In addition, a continued fraction
solution for the dynamic stiffness matrix of a bounded domain of arbitrary
geometry can be obtained. For simplicity in notation, the scaled boundary finite
element equation in dynamic stiffness 4.3.1 can be rewritten as:
[ ] [ ] [ ] [ ] [ ] [ ] [ ]
[ ] [ ]
4.3. 3
where
4.3. 4
Equation 4.3.2 can also be rewritten as the matrix equation, which looks like:
84
[ ] [ ] [ ] ([
] [
] ([
] [
]
([
( )] [ ( )])
)
)
4.3. 5
From Song’s thesis, after getting the continued fraction for a given order , the
dynamic stiffness matrix at specific frequency can be obtained by substituting
equation 4.3.4 into equation 4.3.5 (Song C. , 2008).
From continued fraction solution for the dynamic stiffness matrix, the force
displacement relationship of a bounded domain can be expressed as:
[ ] [ ] { } { } 4.3. 6
with coefficient matrices
[ ] ([ ] [
] [
] [
]) 4.3. 7(a)
[ ]
[ [ ] [ ]
[ ] [ ] [ ]
[ ]
[ ]
[
]]
4.3. 7(b)
{ }
{
{ }
{ }
{ }
{ }}
, { }
{
{ } }
4.3.7(c)
85
Equation 4.3.6 can be transferred into time domain, which looks like:
[ ]{ } [ ]{ } { } 4.3. 8
According to Song, the coefficient matrices of continued fraction solution
are determined for each super-element. Through assembling the equations of
motion of individual super-element, the global equation of motion is obtained,
which is represented below:
[ ]{ } [ ]{ } { } 4.3. 9
In equation 4.3.9, matrices terms [ ] and [ ] are the global static stiffness and
mass matrices respectively. Term { } is the displacement vector, includes all the
auxiliary variables. Term { } is the external fore vector (Song C. , 2008).
Same as the last two examples, commercial software- Matlab will be used
here to establish the model and then find out the results. First of all, a simple
homogeneous layer, without trench and the building foundation, is created, which
is same as last example. It is still overlaying on a hard rock with a time dependent
surface traction applying on the left boundary surface. In addition, it extends to
infinite in horizontal direction and has a constant depth. All the other properties
will be the same as the last example, like dimensions, loading conditions and soil
properties. The way of meshing will be different.
Figure 4.3. 4, a homogeneous layer divided into two sub-domains
86
From figure 4.3.4, the whole layer is divided into two sub-domains, a
bounded domain and an unbounded domain. For bounded domain, the scaled
boundary finite element method is employed here with a scaling center . The
boundary of bounded domain is discretized into 8 elements with 5 nodes in each.
For unbounded domain, only the left boundary is discretized into two elements.
The values of horizontal and vertical displacements response at the upper-left
corner point of the bounded domain will be calculated out and then compared with
the reference solution obtained in last sub-section.
(a)
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 5 10 15 20 25 30 35
Continuedfractionsolution,dt=0.01
Constantscheme,dt=0.01
Dimensionless Time
Dis
pla
cem
ent
Horizontal Displacement
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 5 10 15 20 25 30 35
Constantscheme,dt=0.01
Continuted fractionsolution,dt=0.01
Dimensionless Time Dis
pla
cem
ent
Vertical Displacement
87
(b)
Figure 4.3. 5, Displacement responses to horizontal surface traction at the upper-left corner point of
the continued fraction solution VS constant scheme: (a) horizontal displacement and (b) vertical
displacement
From figure 4.3.5, we can find that curves of constant scheme and continued
fraction solution are completely identical for both horizontal and vertical
displacements. The new results are in high accuracy. Thus, the new discretizing
method is practicable and can be used in this example.
The particular homogeneous soil layer for example 3 and its dimensions are
represented below:
Figure 4.3. 6, sketch of the homogeneous soil layer with a building foundation and a trench for example
3
In order to find out the effects of the trench, the in-plane motion responses
will be monitored in point A and B, which are shown in figure 4.3.6. Point A locates
in front of the trench with distance of 0.5 meter, while point B is 0.8 meter from
another side of the trench. With the employment of super-elements, the whole
section is divided into several parts according to different loading and structure
conditions. After subdividing, the whole layer looks like:
88
Figure 4.3. 7, the super-elements mesh plot of the homogeneous soil layer with a trench
As can be seen from 4.3.7, the bounded domain is divided into eight sub-
domains. The sub-domain 9 is the unbounded domain. For each sub-domain, the
scaled boundary finite element method is used and the scaling center is located at
the center point. Each element contains 5 nodes. The similar mesh plot is also
created by Matlab (see in Appendix C). Before finding the displacement values, we
need to check whether this mesh is fine enough.
A time-dependent triangular function load, which is the same as the load in
example 2, applies on the building foundation in vertical direction. In
Prempramote’s PHD thesis, the same load is also used in his second example,
which is called semi-infinite layer subjected to horizontal surface traction. The
figures of time history and Fourier transform is shown in figure 7.7.9
(Prempramote, 2011)(see in Appendix D). From the Fourier transform figure, the
maximum value of dimensionless frequency is 5. Then, the minimum
requirement value of the wavelength can be obtained for this example. The
equation of the dimensionless frequency looks like:
4.3. 10
Where, term c is the wave velocity, term h is the depth of the layer. The equation of
traveling wave velocity is also represented below:
89
4.3. 11
In equation 4.3.11, term is the wave length and term f is the frequency of the
wave. Combination of equations 4.3.10 and 4.3.11 gives us the minimum value of
wavelength:
4.3. 12
After substituting in the values of and h into equation 4.3.12, the value
of is around 2.51m. In that case, the mesh of the subdomains is fine enough. In
addition, the longest radial distance contained in each subdomain also meets the
requirement. Thus, the meshing of the domain is fine enough to get the accurate
results.
With running the Matlab program, the in-plane motion at a specific point in
this soil layer can be obtained. In order to find out the effects of the trench, the
data of different two points, which locate in front and at the back of the trench
respectively, will be collected. In addition, when the trench has been removed, the
data will also be collected at the same pointsand then compared with the previous
data.
At present, we will look at how the trench influences the propagation of
dynamic waves. In the first case, the trench is kept and the values of horizontal and
vertical displacements of point A and B are collected. The curves look like:
90
(a)
(b)
Figure 4.3. 8, displacement responses to the dynamic load point A and B when the trench exists: (a)
horizontal displacement and (b) vertical displacement
From the figure 4.3.8, for both horizontal and vertical displacements, the
amplitude values of the curves of the point B are much smaller than those of point
A. So, the trench indeed weakens the propagation of the dynamic waves in the soil
layer. Moreover, reduction in amplitude values for horizontal displacement curves
is more pronounced.
In the second case, the trench is removed. The mesh plot is shown below:
-150
-100
-50
0
50
100
150
200
250
0 5 10 15 20 25 30 35
Horizontal Displacements
Point A
Point B
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Vertical Displacements
Point A
Point B
91
Figure 4.3. 9, the super-elements mesh plot of the homogeneous soil layer without trench
Since the trench does not exist, there are only five subdomains in this soil layer,
four bounded subdomains and one unbounded subdomain. This kind of mesh is
also fine enough to give us the accurate results. In this mesh plot, node points 9
and 10 have the same locations as the point A and B respectively. The horizontal
and vertical displacement curves of these two points are represented in the figure
4.3.10 below:
(a)
-40
-20
0
20
40
60
80
100
120
0 5 10 15 20 25 30 35
Horizontal Displacements
Point A
Point B
92
(b)
Figure 4.3. 10, displacement responses to the dynamic load at point A and B when the trench is
removed: (a) horizontal displacement and (b) vertical displacement
As can be seen, for the horizontal and vertical displacements curves, the
differences in the amplitude values are quite small. Two curves are not identical at
the beginning. But they converge as time goes. In that case, if there is no trench, the
wave can propagate freely. The dynamic waves can affect the structures or
buildings, which is far away from the loading point.
Furthermore, a study about how trench influences the displacement
responses at two points individually is also carried out:
-40
-20
0
20
40
60
80
100
120
0 5 10 15 20 25 30 35
Vertical Displacements
Point A
Point B
93
(a)
(b)
Figure 4.3. 11, displacement responses to the dynamic load at the point A with and without trench: (a)
horizontal displacement and (b) vertical displacement
From figure 4.3.11, for both the horizontal and vertical displacements at point A,
there are some differences of the amplitude values. The using of the trench leads to
larger displacement responses of soil layer, which is caused by the wave reflection
from the boundary and trench. For there is no material damping, the dynamic
waves can be trapped.
-150
-100
-50
0
50
100
150
200
250
0 5 10 15 20 25 30 35
With trench
Withouttrench
Horizontal displacement of point A
-80
-60
-40
-20
0
20
40
60
80
100
120
0 5 10 15 20 25 30 35
With trench
Withouttrench
Vertical displacement of point A
94
(a)
(b)
Figure 4.3. 12, displacement responses to the dynamic load at point B with and without trench: (a)
horizontal displacement and (b) vertical displacement
For the point at the back of trench, there is no big difference in the
amplitude values of displacements in both horizontal and vertical directions. For
horizontal displacement only, there are some shifts of the curves. So, if trench is
-30
-20
-10
0
10
20
30
40
50
0 5 10 15 20 25 30 35
Horizontal displacement of point B
Withtrench
Withouttrench
-20
-15
-10
-5
0
5
10
15
20
0 5 10 15 20 25 30 35
Vertical displacement of point B
Withtrench
Withouttrench
95
removed, the horizontal displacement responses may happen earlier at point B. So,
the existing of trench can prevent the propagation of waves and delay the
responses of soil in horizontal direction. But for vertical displacements, there is
almost no influence and two displacements curves are almost identical.
In conclusion, the using of trench can lead to a reduction of the
displacement amplitudes. Compared to the vertical displacements, the horizontal
displacement responses of the soil are more easily to be influenced at any place in
the soil layer.
96
Chapter 5
Conclusion and Future Work
5.1 Summary
The main objective of this research is to develop a reliable and efficient method to
obtain the acceleration unit-impulse response matrix in the dynamic soil-structure
interaction problems in homogeneous, semi-infinite soil layer. According to the
literature review part, the scaled boundary finite element method is chosen as the
theoretical framework of this study. The major advantage of using SBFEM is that it
can be used to model the unbounded domain soil layer. In addition, the
fundamental solution is not required. Only the boundary of the domain needs to
be discretized. So, the spatial dimension reduced by one, which significantly
decreases the computational effort. Other advantages include the using in a soil
layer with anisotropic materials and cracks.
This research focuses on the homogeneous, semi-infinite soil layer. Based
on the scaled boundary finite element method, the scaled boundary finite element
equation in frequency-domain of this system is derived. There is a challenge to get
the scaled boundary finite element equation in time-domain. In my thesis, the key
step is to find out the acceleration unit-impulse response matrix. According to the
previous studies (Radmanovic & Katz, 2010), two different technologies have been
developed, constant scheme and linear scheme. These two schemes are tested in a
homogeneous, semi-infinite soil layer for both in-plane and out-of-plane motion
responses to dynamic load. This research is also extended to a soil layer with a
trench near a foundation, whose purpose is to find out how the trench influences
of the displacements of the soil layer. A summary of conclusion for each chapter is
represented here.
In Chapter 1, the introduction of this thesis was presented. The background
and statement of problem are also introduced. Two different kinds of methods to
97
analyze the unbounded domain are also talked about, which are direct method and
substructure method.
In Chapter 2, a detailed literature of existing approach for soil-structure
interaction problems in unbounded domains was represented. Those approaches
are divided into two groups, global procedures and local procedures. Global
procedures include boundary element method and scaled boundary finite element
method. Local procedures include most absorbing boundary methods and finite
elements method. After discussing the advantages and disadvantages of each
method, scaled boundary finite element is chosen as the theoretical framework of
this research.
In Chapter 3, a detailed about derivation of the theoretical concepts of this
research was represented, which includes some essential equations and important
schemes. First of all, properties of homogeneous, semi-infinite soil layer were
introduced and the basic equation called operator equation was obtained. Then, a
method of transforming Cartesian coordinates into local coordinates was
described, which was used in obtaining the operator equation in local coordinates.
With the help of weighted residuals technique and method of integration by parts,
the scaled boundary finite element equation of displacement in frequency domain
was obtained. By using the Fourier transformation method, the scaled boundary
finite element equation in time domain was derived. Within this equation, the only
unknown is the acceleration unit-impulse response matrix [ ]. There were
two methods introduced to get the values of [ ], which are constant scheme
and linear scheme. For constant scheme, the coefficient value is assumed to be
constant. But for linear scheme, the coefficient value changes linearly within each
time interval. The main purpose of this research was to find out which scheme is
more efficiency and reliable, which had been done in chapter 4. Finally, the way of
calculating the in-plane and out-of-plane motions of soil layer under dynamic loads
were also represented.
In Chapter 4, three numerical examples were studied. First two examples
were about the out-of-plane and in-plane motion of homogeneous semi-infinite
soil layer problem. After comparing the results and the CPU time consumption of
running program, there is a conclusion that the constant scheme can always give
98
us the high accurate results. But it was usually time consuming. For linear scheme,
it might safe some time with using truncation, but the results contained lots of
errors. In order to get the results in high accuracy for linear scheme, very small
time step size can also be chosen. But it may cost more. So, linear scheme may still
not that efficient for certain problems. In addition, a trench was added in the same
soil layer nearby the structure, which was the third example. In-plane motion at
two separate points, which were in front and at the back of the trench respectively,
were collected and studied. Then, it could be easily found that the trench indeed
affects the propagation of dynamic waves in soil layer and the values of
displacement amplitudes were reduced significantly. Moreover, horizontal
displacements of soil were more easily to be influenced.
5.2 Recommendations for future research
In Chapter 3, the Fourier transformation method was employed to transfer
the scaled boundary finite element method from frequency domain into
time domain, which increased the computational efforts a lot. In future
work, a more reliable and efficiency method can be used to replace the
Fourier transformation method, for example approximating the exact DtN
map by a frequency-dependent rational function.
For the second numerical example in chapter 4, the linear scheme was not
that efficiency. The using of truncation always leads to some errors. Future
work of this research might involve the employment of truncation time for
constant scheme. In that case, the cost of using constant scheme may be
reduced. At the same time, the high level of accuracy can still be kept.
For the third example in chapter 4, a reference solution of that problem can
be created by commercial software. With the help of reference solutions, it
will be much easier to find out how the trench influences the displacements
of soil layer.
99
Appendix A
Reference solution of references solution of acceleration unit-impulse
response coefficients (Wolf & Song, 1996)
100
Appendix B
B.1 Mesh plot of the model of numerical example 2 from ANSYS
B.2 Zoom in of the mesh plot of the model of numerical example 2
101
Appendix C
Mesh plot of Homogeneous semi-infinite soil layer with trench
102
Appendix D
The figures of time history and Fourier transform (Prempramote, 2011)
103
Bibliography Bazyar, M. H., & Song, C. (2006). Time-harmonic response of non-homogeneous elastic unbounded domains using the scaled boundary finite-element method. Sydney: University of New South Wales. Bettess, P. (1977). Infinite elements. International Journal for Numerical Methods in Engineering (Vol. 11). Bikri, I., Guenther, R. B., & Thomann, E. A. (2010). The Dirichlet to Neumann Map-An Application to the Stokes Problem In Half Space. Oregon State University, Department of Mathematics. Corvallis. Bouchon, M. (1981). A siimple method to calculate Green's functions for elastic layeredmedia. . Bulletin of the Seismological Society of America. Fan, S., Li, S., & Yu, G. (2005). Dynamic fluid-structure interaction analysis using boundary finite element method-finite element method. Journal of Applied Mechanics (72), 591-598. Green, G. (1828). An essay on the application of mathematical analysis to the theories of the electricity and magnetism. Nottingham: Gonville-and Cains-Colleges at Cambridge. James, J. (2011). A Student's Guide to Fourier Transforms with Application in Physics and Engineering (Thrid Edition ed.). New York, United States of America: Cambridge University Press. Lehmann , L., Langer, S., & Clasen, D. (2006). Scaled boundary finite element method for acoustics. Journal of Computational Acoustics (14), 489-506. Luco, J., & Apsel, R. (1983). On the Green's functions for a layered half-space. Part 1. . Bulletin of the Seimological Society of America. Lysmer, J., & Kuhlemeyer, R. L. (1969). Finite dynamic model for infinite media. Journal of Engineering Mechanics , 95 (28), 859-877. Lysmer, J., & Waas, G. (1975). Shear waves in plane infinite structures. Journal of the Engineering Mechanics Division (101), 771-785. Nikishkov, G. (2001). Introduction to the Finite Element Method. Lecture Notes, University of California, Los Angles. Prempramote, S. (2011). Development of high-order doubly asymptotic open boundaries for wave propogation in unbounded domains by extending the scaled boundary finite element method. PhD Thesis, Unversity of New South Wales.
104
Radmanovic , B., & Katz, K. (2010). A high-performance scaled boundary finite element method, (Vol. 10). IOP Conf. Series: Materials Science and Engineering. Radmanovic, B., & Katz, C. (2010). A High Performance Scaled Boundary Finite Element Method. SOFiSTiK AG, Bruckmannring 38, 85764 Oberschleissheim. Song, C. (2005). Evaluation of power-logarithmic singularities, T-stress and higher order terms of in-plane singular stress fields at cracks and multi-material corners. Engineering Fracture Mechanics (72), 1498-1530. Song, C. (2008). The scaled boundary finite element method in structural dynamics. University of New South Wales, Depart of Civil and Environmental Engineering. Sydney: Wiley InterScience. Song, C., & Wolf, J. (1997). The scaled boundary finite-element method-allias consistent infinitesimal finite-element cell method-for elastodynamics. Computer Methods in Applied Mechanics and Engineering (147), 329-355. Song, C., & Wolf, J. (1999). The scaled boundary finite-element method-alias consistent infinitesimal finite-element cell method-for diffuson. International Journal for Numerical Methods in Engineering (45), 1403-1431. Song, C., & Wolf, J. (2002). Semi-analytical representation of stress singularity as occuring in cracks in anisotropic multi-materials with the scaled boundary finite-eement method. Computers and Structures (80), 183-197. Trinks, C. (2005). Consistent absorbing boundaries for time-domain interaction analysis using the fractional calculus. PhD Thesis, Technische Universitaet Dresden. Truong, W. (2012). Numerical Modelling of Scalar Wave Propagation in Unbounded Domains. Sydney: The University of New South Wales. Wolf, J. (1994). Foundation Vibration Analysis using Simple Physical Models. Englewood Cliffs, NJ, U.S.A.: Prentice-Hall. Wolf, J., & Meek, J. (1994). Cone models for embedded foundaton. Journal of Geotechnical Engineering (120), 60-80. Wolf, J., & Song, C. (1996). Finite-element modeling of unbounded media. John Wiley & Sons. Wolf, J., & Song, C. (1996). Finite-Element Modelling of Unbounded Media. Lausanne, Switzerland: Elsevler Science Ltd. Zienkiewicz, O., Taylor, R., & Zhu, J. (2005). The Finite Element Method: Its Basic and Fundamentals (Sixth ed.). Butterworth-Heinemann.