tsunami simulation in indonesia`s

TSUNAMI SIMULATION IN INDONESIA’S AREAS
BASED ON SHALLOW WATER EQUATIONS AND
VARIATIONAL BOUSSINESQ MODEL USING
FINITE ELEMENT METHOD
THESIS
Submitted in partial satisfaction of the
requirements for the degree of
Master of Science in the Institut Teknologi Bandung
By
DIDIT ADYTIA
NIM : 20106001
Program Studi Matematika
INSTITUT TEKNOLOGI BANDUNG
2008
ABSTRACT
TSUNAMI SIMULATION IN INDONESIA’S AREAS
BASED ON SHALLOW WATER EQUATIONS AND
VARIATIONAL BOUSSINESQ MODEL USING
FINITE ELEMENT METHOD
By
DIDIT ADYTIA
NIM : 20106001
In many cases, tsunami waveheights and effects show a high variability along
the coast. One way to study this complexity is to simulate the tsunami above
a certain area by using water wave models. Since a tsunami can be considered
as a shallow water wave, we can choose the well-known Shallow Water Equations (SWE) as a non-dispersive water wave model for the tsunami. Dispersive
wave means that harmonic waves of smaller wavelength propagate slower than
waves of larger wavelength. For the dispersive wave model, we used the recently derived Variational Boussinesq Model (VBM). In the SWE model the
vertical variations in the layer of fluid is neglected, different from the VBM
where the vertical variations lead to the effect of dispersion. These models are
derived by using variational formulation. Consistently with the way of their
derivations, these models will be solved numerically by using Finite Element
Method (FEM). In FEM, the solutions are approximated by linear combination
of the basis functions. In this thesis, we used linear basis functions. The radiation boundary condition and hard-wall boundary condition are implemented
for both SWE and VBM. To simulate the tsunami, we use the available data
of the bathymetry of Indonesia which is incorporated into our FEM schemes.
The tsunami simulation in the two areas of Indonesia, which are the area in the
south of Pangandaran and the area in the south of Lampung, will be presented
as a result of FEM’s implementation for the SWE and the VBM.
Keywords.
Tsunami, Simulation, Indonesia, Shallow Water Equations,
Variational Boussinesq Model, Finite Element Method
i
ABSTRAK
SIMULASI TSUNAMI DI DAERAH INDONESIA
BERDASARKAN PERSAMAAN AIR DANGKAL
DAN VARIATIONAL BOUSSINESQ MODEL
MENGGUNAKAN METODE ELEMEN HINGGA
Oleh
DIDIT ADYTIA
NIM : 20106001
Pada banyak kasus tsunami, ketinggian dan akibat dari gelombang ini menunjukkan variasi yang sangat tinggi di sepanjang garis pantai. Salah satu cara untuk mempelajari masalah ini adalah dengan melakukan simulasi tsunami pada
suatu daerah tertentu dengan menggunakan model gelombang air. Karena
tsunami dapat dianggap sebagai gelombang air dangkal, maka dapat digunakan
persamaan gelombang air dangkal atau Shallow Water Equations (SWE) sebagai model non-dispersif. Gelombang yang bersifat dispersif diartikan bahwa
gelombang harmonik yang mempunyai panjang gelombang yang pendek berpropagasi lebih lambat dibanding dengan gelombang dengan panjang gelombang lebih besar. Untuk model gelombang dispersif, akan digunakan Variational Boussinesq Model (VBM). Pada model SWE, variasi pada arah vertikal
diabaikan, berbeda dengan VBM dimana hal tersebut tidak diabaikan sehingga
muncul efek dispersif. Kedua model tersebut diturunkan dengan variational
formulation, konsisten dengan cara penurunannya, model-model tersebut dicari solusi numeriknya dengan metode elemen hingga atau Finite Element
Method (FEM). Pada FEM, solusi dihampiri dengan kombinasi linier dari
fungsi basis. Pada tesis ini digunakan fungsi basis linier. Radiation boundary
condition dan Hard-wall boundary condition akan diimplementasikan baik untuk SWE maupun VBM. Untuk melakukan simulasi tsunami, digunakan data
bathymetry Indonesia. Simulasi tsunami pada dua daerah di Indonesia akan
diperlihatkan sebagai implementasi FEM dari SWE dan VBM.
Kata kunci. Tsunami, Simulasi, Indonesia, Persamaan Air Dangkal, Variational Boussinesq Model, Metode Elemen Hingga
ii
TSUNAMI SIMULATION IN INDONESIA’S AREAS
BASED ON SHALLOW WATER EQUATIONS AND
VARIATIONAL BOUSSINESQ MODEL USING
FINITE ELEMENT METHOD
By
DIDIT ADYTIA
NIM : 20106001
Program Studi Matematika
Institut Teknologi Bandung
Approved
21 June 2008
Supervisor
Dr. Andonowati
GUIDELINES TO USE THE THESIS
This thesis is not published, it is registered and is available in library at the
Institut Teknologi Bandung. This thesis is not open to the public in condition
that the copyright belongs to the author. Permission is granted to quote brief
passages from this thesis provided the customary acknowledgment of the source
is given.
Copying or publishing any material in this thesis is permitted only under
licence from the Director of Program Pascasarjana of the Institut Teknologi
Bandung.
iv
For Husin Mas’ud and Yensi Nio
my best parent ever
ACKNOWLEDGMENTS
In the Name of Allah, I express gratitude to Allah S.W.T for allowing me to
finish this work. Although it is my name who appears in the cover of this
thesis, but there are many people with their help and support who made this
work possible. First of all, sincere thanks to my teacher and my supervisor,
Dr. Andonowati, who has really inspired me about mathematics and natural phenomenon, and also giving me a trust to do this work. This work was
initiated at Labmath-Indonesia in mid 2007. I wishes to thanks to this institution for providing me a financial supports and facilities. This work mostly
has been done in this institution under supervision of Prof. E. (Brenny) van
Groesen and Dr. Ardhasena Sopaheluwakan. I really want say many thanks
to Mr. Brenny for his beautiful Variational Boussinesq Model (VBM) and his
brialliant opinion about mathematics and nature, that is really open my mind
about mathematics. Thank you for choosing me to do this job. Special thanks
to Pak Sena, for introducing me the beauty of Finite Element and the art of
computing. I found it really amazing. In doing this work, I worked with a
young reseacher, L. Oscar Osaputra, who gave a very much contribution to
this work. I really thanks for our 4 months work.
My thanks for Dr. Sri Redjeki for her support, nice sharing and discussion
about mathematics. My thanks are also to all staff and researchers in Labmath
Indonesia for their supports and friendship, I really appreciate that. I express
my thanks to graduate students and all staff in Mathematics Department,
Institute Teknologi Bandung, for all friendship. Finally, to my parent, Husin
and Yensi, my brother and sister, Tinton and Ika, my great friend Fara, and
my families, gratitudes always goes to their deep understanding and supports.
Bandung, Juni 2007
Author
vi
CONTENTS
ABSTRACT
i
ABSTRAK
ii
GUIDELINES TO USE THE THESIS
iv
ACKNOWLEDGMENTS
vi
CONTENTS
vii
LIST OF FIGURES
ix
Chapter I Preliminary
1
I.1
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
I.2
Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . .
1
I.3
Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
I.4
Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Chapter II Basic Theory
4
II.1 Variational Formulation . . . . . . . . . . . . . . . . . . . . . .
4
II.1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
II.1.2 Variational Formulation for Surface Wave
. . . . . . . .
5
II.2 Shallow Water Equations . . . . . . . . . . . . . . . . . . . . . .
6
II.3 Variational Boussinesq Model . . . . . . . . . . . . . . . . . . .
8
II.4 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . 12
II.4.1 Radiation Boundary Condition . . . . . . . . . . . . . . 12
II.4.1 Hard-wall Boundary Condition . . . . . . . . . . . . . . 13
Chapter III Finite Element Method Implementation for SWE
and VBM
14
III.1 Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . 14
vii
III.2 FEM Implementation for SWE . . . . . . . . . . . . . . . . . . 15
III.3 FEM Implementation for VBM . . . . . . . . . . . . . . . . . . 21
III.4 FEM Implementation for Boundary Conditions . . . . . . . . . 25
Chapter IV Tsunami Simulation
28
IV.1 Water Wave Simulation Above Flat Bottom . . . . . . . . . . . 28
IV.2 Indonesia Bathymetry . . . . . . . . . . . . . . . . . . . . . . . 33
IV.3 Tsunami Simulation Using SWE and VBM above Indonesia
Bathymetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Chapter V Conclusions and Recommendations
46
Bibliography
48
viii
LIST OF FIGURES
Figure 3.1 At the left, we show a plot of discretized domain by using
pdetool from MATLAB, and at the right plot show the global
and the local numbering (in bracket). The local numbering is
assumed in counterclockwise direction. . . . . . . . . . . . . . . 15
Figure 3.2 Plot of linear basis function Ti (x) . . . . . . . . . . . . . . 16
Figure 3.3 Linear transformation from Ωe to master triangle and inverse map. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Figure 4.1 Plot of the bipolar initial profile. . . . . . . . . . . . . . . 29
Figure 4.2 Plot of the crosssection of initial wave at y = 0 with 1.7m
amplitude and Λ = 20km. . . . . . . . . . . . . . . . . . . . . . 29
Figure 4.3 Plot of the splitting initial bipolar hump above flat bottom
using SWE model at t = 7 min. . . . . . . . . . . . . . . . . . . 30
Figure 4.4 Plot of SWE simulation above flat bottom when the wave
reached the boundary at t = 12 min. Notice that there is no
reflection from each boundary. . . . . . . . . . . . . . . . . . . . 30
Figure 4.5 Plot of the splitting of initial bipolar hump using VBM,
notice the development of dispersive effect. . . . . . . . . . . . . 31
Figure 4.6 Plot of the VBM simulation when the wave hits the radiation boundary condition, notice the reflected wave from the
left and right boundary. . . . . . . . . . . . . . . . . . . . . . . 31
Figure 4.7 Plot of available bathymetry data with 1′ accuracy. . . . . 33
Figure 4.8 Ilustration of the approximated bathymetry data in triangle domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Figure 4.9 Plot of discretized domain in the south of Java. . . . . . . 34
Figure 4.10 Profile of the bathymetry in the south of Java for Pangandaran case.
. . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Figure 4.11 Plot of the location of earthquake in the south of Java at
July 17, 2006. Courtesy : USGS. . . . . . . . . . . . . . . . . . 35
ix
Figure 4.12 Plot of the splitting of intial bipolar hump using SWE
model at t = 3 minutes. The location of the source is near
9.220 S and 107.320 E. . . . . . . . . . . . . . . . . . . . . . . . . 37
Figure 4.13 Plot of the tsunami simulation on Pangandaran case using
SWE model at t = 9 minutes. . . . . . . . . . . . . . . . . . . . 37
Figure 4.14 Plot of Pangandaran’s tsunami simulation using SWE
model at t = 60 minutes. . . . . . . . . . . . . . . . . . . . . . . 38
Figure 4.15 Plot of the maximum crest-height during 1 hour simulation using SWE model. Crossection near the coast denoted by
white line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Figure 4.16 Plot of Pangandaran’s tsunami simulation using the VBM
at t = 9 min. Notice the appearance of the dispersive tail. . . . . 39
Figure 4.17 Plot of the maximum crest-height near the coast during 1 hour simulation using VBM. The below plot denotes the
crosssection in the white line. . . . . . . . . . . . . . . . . . . . 39
Figure 4.18 Plot of approximated bathymetry for Lampung Case.
Notice the shallow area surrounded by deep area in the south
of Sumatra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Figure 4.19 Plot of the location and the shape of initial wave for
Lampung Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Figure 4.20 Plot of the tsunami simulation for Lampung case by using
SWE model at t = 9 minutes. . . . . . . . . . . . . . . . . . . . 42
Figure 4.21 Plot of tsunami simulation for Lampung case by using
VBM. Notice the appearance of the dispersive tail. . . . . . . . 42
Figure 4.22 The wave after 18 minutes by using VBM. There is delayed and amplified wave above the shallow area. The waveheight reached to more than 10m. . . . . . . . . . . . . . . . . . 43
Figure 4.23 Plot of the maximum waveheight during 1 hour simulation by using SWE model. At the spesific area, the waveheight
reached 9.47m. . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
x
Figure 4.24 Plot of the maximum waveheight during 1 hour simulation by using VBM. At the specific area near the coast in the
south of Sumatra, the wave reached 9.61m. . . . . . . . . . . . . 44
xi
Chapter I
Preliminary
I.1
Motivation
Tsunami is a series of travelling ocean waves of extremely long wavelength
generated by disturbances associated primarily with earthquakes occuring below or near the ocean floor. Volcanic eruptions, landslides, or large meteorid
impact also have the potential to generate a tsunami. Tsunami is shallow water waves, which means that the ratio between waterdepth and wavelength is
very small. The wave moves with speed equal to square root of the product of
acceleration of gravity and water depth. In the deep ocean, the tsunami may
only be a few feet high, but it has a wavelength of hundreds of miles. When
it comes toward the coastline, the amplitude becomes higher and the wavelength become shorter, causing widespread devastation over the land along the
coastline.
Indonesia is a tsunami prone area since it is an active seismic region. The
Krakatoa eruption that generated great tsunami in 1883, the 2004 Indian ocean
tsunami with over one hundred and fifty thousand fatalities in Indonesia or
several other tsunami events in Nias, Pangandaran are examples that showed
the coastal areas of Indonesia are potential to be hit by a tsunami.
I.2
Problem Formulation
In many cases, tsunami waveheights and effects show a high variability along
the coast, as reported on the website of EERI (2005) [9]. The complexity of
tsunami’s behavior near and on the shore is affected by the topography of the
bathymetry underneath the sea.
2
One way to study this complexity is to simulate the tsunami above a certain
area by using water wave models. To do that, we have to choose (or derive) a
good water wave model. Since a tsunami can be considered as a shallow water wave, we choose the Shallow Water Equations (SWE) as a non-dispersive
water wave model. Dispersive effect means that harmonic waves with smaller
wavelength propagate slower than waves with larger wavelength. For the dispersive water wave model, we used the recently derived Variational Boussinesq
Model (VBM) [1]. In the SWE the vertical variations in the layer of fluid is
neglected, different from the VBM where the vertical variations lead to the
effect of dispersion. These models are derived by using variational principle.
Consistently with the way of their derivations, these models will be solved
numerically by using Finite Element Method (FEM). In FEM, the solutions
are approximated by linear combination of the basis functions. This recently
derived VBM is different from other Boussinesq models. It is derived without
introducing higher order derivatives in the dynamic equation so that we can
use linear basis functions. To simulate the tsunami, we use the actual data of
the bathymetry of Indonesia, which is incorporated into our FEM schemes.
We will simulate the tsunami in two areas of Indonesia which are the area in
the south of Pangandaran and the area in the south of Lampung by using both
the SWE model and the VBM. Both of the results of the simulations will be
showed.
I.3
Objectives
Referring to our problem in Problem Formulation, our objectives are to built a
reliable tsunami simulation. Although a tsunami event can not be prevented,
by simulating possible scenarios of tsunami events, we can identify the vulnerable shore lines for early warning purposes and preparedness, also to optimize
future designs of structural tsunami defence. By simulating the tsunami above
3
certain areas, specially in Indonesia, using water wave models, we can identify
(or predict) the waveheight near and on the specific shore that we consider.
I.4
Outline
The organization of this thesis is as follows. Chapter 1 is the introduction part
of this thesis, which describe the general idea of the thesis. This part is devided
into four sections, which are motivation, problem formulation, objectives, and
outline.
In chapter 2, we derived the wave models by using variational principle and
also we choose two kind of boundary conditions which will be implemented in
our water wave models.
Chapter 3 and 4 are the main parts of this thesis. Chapter 3 contains some
explanation about FEM and its implementation for the SWE and the VBM,
and also for the boundary conditions .
In chapter 4, we consider the simple problem of wave propagation above a
flat bottom, after that, we described the bathymetry data of Indonesia and
how to incorporate it into our FEM schemes. In this section, we took two
examples of tsunami simulations in two areas of Indonesia by using both FEM’s
implementation for SWE and VBM.
In the section 5, conclusions and recommendations of this thesis will be given.
Chapter II
Basic Theory
In this chapter, we will describe the basic theory for the derivation of our
water wave models which are the Shallow Water Equations (SWE) and the
Variational Boussinesq Model (VBM) by using variational formulation. The
complete derivation of these models can be read at [1] . We consider a layer of
fluid with free surface and assume water as an idealized fluid, with properties
that are inviscid and incompressible with uniform mass density (ρ = 1). The
flow is assumed to be irrotational. Moreover there is no pressure from the
atmosphere above the fluid layer.
II.1
II.1.1
The Variational Formulation
Notation
We consider three dimensional space, two horizontal directions x = (x, y),
and vertical z-axis. The gravitational acceleration is assumed to be constant
9.81m/s and denoted as g which has direction opposite to the z-axis. The
surface elevation is denoted as η(x, t) and measured from z = 0. The fluid
velocity is denoted as U, and with the assumption that the flow of the fluid
is irrotational we have curl U ≡ ∇3 × U = 0, hence there exist a scalar function Φ, such that U = ∇3 Φ = (∇Φ, ∂z Φ). In the setting of the problem, the
function Φ is called as the velocity potential. The depth given by h(x, t), so
the bathymetry is given by z = −h(x, t), but later in the applications, we will
assume that there will be no bottom motion, so h(x, t) ≈ h(x).
II.1.2
Variational Formulation of Surface Wave
We begin with Luke’s Variational Formulation, which can be stated as
Z
min P(Φ, η)dt
Φ,η
5
where
P(Φ, η) =
Z Z
1
2
∂t Φ + |∇3 Φ| + gz dz dx
2
−h
η
(2.1)
with P is the pressure functional. Minimization of this ”pressure principle”
with respect to (wrt) Φ gives the governing equation for the interior of the fluid
(The Laplace problem), and kinematic boundary condition at the surface and
boundary condition at the bottom. In this case, it is assumed that there is no
friction and no flow though the bottom (impermeable). While minimization
wrt η gives the dynamic free surface condition. This leads to the full 3D surface
wave problem.
It is hard to solve the full 3D surface wave problem directly, so we want to
reduce that into two spatial variables only. This dimensional reduction leads us
to introduce a functional (which is the kinetic energy) that should be expressed
in the variables to be introduced. The formulation will involve two physical
quantities, which are η(x, t) and φ (x, t) := Φ (x, z = η (x, t) , t). The second
variable is abtained by prescribing the velocity potential at the free surface.
With these simplifications, the solution of the full 3D surface wave problem can
be determined uniquely. We start with introducing K (φ, η) as kinetic energy
functional of our basic quantities, which is the value function of this following
minimization problem
K(φ, η) = min {K(Φ, η)|Φ = φ at z = η}
Φ
where K(Φ, η) :=
R nR η
1
|∇3 Φ|2 dz
−h 2
(2.1) can be rewritten as
Z Z
(2.2)
o
− Φ∂t h dx. The functional P(Φ, η) at
1
2
∂t Φ + |∇3 Φ| + gz dz dx
P(Φ, η) =
2
−h
Z
Z
Z Z η
1
2
2
=−
φ∂t ηdx−K (φ, η) −
+ ∂t
Φdz dx
g η −h
2
−h
using
Rη
η
∂ Φdz = ∂t
−h t
R
η
−h
Φdz − φ∂t η − ΦB ∂t h , where ΦB is the Φ at the
bottom. Since we assumed that there is no bottom motion, so ∂t h = 0. Now
6
our Luke’s Variational Principle can be rewritten as follow
Z Z
min
φ∂t ηdx − H (φ, η) dt
φ,η
(2.3)
where H (φ, η) is the Hamiltonian Functional (or the total energy) as follow:
H (φ, η) = K (φ, η) +
Z
1
g η 2 − h2 dx
2
(2.4)
where the second term in right hand side (RHS) is the potential energy. The
resulting variational principle in (2.3) is known as a canonical action principle.
The Euler-Lagrange equations which are obtained by taking variations with
respect to φ and η in the action principle are given by
∂t η = δφ H (φ, η)
(2.5)
∂t φ = −δη H (φ, η)
which is known as the Hamilton’s equations. By using (2.4), equations (2.5)
can be rewritten as
∂t η = δφ H (φ, η) = δφ K (φ, η)
∂t φ = − [gη + δη K (φ, η)]
(2.6)
Now our problem is to determine the functional for the kinetic energy. The
Hamiltonian which contains functional K (given by (2.2)), can not be expressed
explicitly in the basic variables ( η and φ ) . This is the essential problem of
surface wave theory. Most of surface wave models deals with the choice of this
kinetic energy, more pricisely, the approximation for the velocity potential Φ.
Examples of such approximation will be given in the following subsections in
the form of the shallow water and Boussinesq type of approximation.
II.2
Shallow Water Equations
The Shallow Water Equations are derived with the assumption that the wavelength of the waves are much larger than the depth of the fluid layer where the
vertical variations are small and would be ignored. This approximation can be
7
applied in the tsunami case since its wavelength reach hundreds of kilometers
above (usually) 4km depth. In this case, there will be no dispersive effect.
Dispersive effect means that harmonic waves of smaller wavelength propagate
slower than waves of larger wavelength, see [2]. This assumption leads to the
idea to approximate the velocity potential at every depth by its value at the
surface, namely Φ (x, z, t) ≈ φ (x, t). With this approximation, the kinetic
energy is approximated with
1
K (φ, η) =
2
Z
(η + h) |∇φ|2 dx
(2.7)
With the above approximation for kinetic energy, our Luke’s Variational Principle in (2.3) turns into
Z Z 1
1
2
2
2
dx dt
−∂t ηφ + |∇φ| (η + h) + g η − h
min
φ,η
2
2
(2.8)
Now, the vanishing of the first variation of (2.8) with respect to variations δφ
in φ results
Z Z
{−∂t ηδφ + (η + h) ∇φ.∇ (δφ)} dx dt = 0,
(2.9)
while the vanishing of the first variation of (2.8) with respect to variations δη
in η results
Z Z 1
2
(∂t φ) δη + |∇φ| δη + (gη) δη dx dt = 0.
2
(2.10)
The resulting Euler-Lagrange equations from the expressions in (2.9) and
(2.10) are the full SWE. In this thesis, we will not use the full SWE, we will
make simplifications. We will linearize the equation and assume that there is
no bottom motion (h (x, t) ≈ h (x)). So the integral expressions in (2.9) and
(2.10) become
Z Z
and
{−∂t ηδφ + h∇φ.∇ (δφ)} dx dt = 0,
Z Z
{(∂t φ) δη + (gη) δη} dx dt = 0.
(2.11)
(2.12)
8
It can be shown that the Euler-Lagrange of (2.9) and (2.10) are the full SWE.
With kinetic energy given by (2.7), the Hamilton equations in (2.6) can be
written as
∂t η = −∇. [(h + η) ∇φ] − ∂t h
∂t φ = − gη + |∇φ|2 /2
(2.13)
Note that (2.13) is the full SWE. Linearization and assuming that there is no
bottom motion (h(x, t) = h(x)), then the equations in (2.13) simplifies to
∂t η = −∇. [h∇φ]
∂t φ = −gη
(2.14)
Also it can be shown that (2.14) are the Euler-Lagrange of (2.11) and (2.12).
Since in this thesis we just deal with the simplified SWE, namely, the linearized
SWE and there is no bottom motion, so in the rest of this thesis we will call
the simplied SWE as the SWE.
II.3
Variational Boussinesq Model
Variational Boussinesq Model (VBM) aimed to make a better approximation
for the kinetic energy that gives rise to the dispersive effect. In shallow water, the velocity potential at every depth is approximated by its value at the
surface ( Φ (x, z, t) ≈ φ (x, t), independent of z ), but in this Boussinesq, the
approximation for Φ will depend on z. Instead of minimizing the kinetic energy
over all possible velocity potentials (Φ), we minimize it only over a subset of
it, namely, we choose a subset
Φ = φ(x) + F (z)ψ(x),
with F (z = η) = 0,
(2.15)
with additional new function ψ on the surface and the vertical profile function
F (will be explained later). The choice such that F (η) = 0 is taken to assure
such that Φ (z = η) = φ. Since the choice for Φ(x, z, t) is given by (2.15), so the
kinetic energy depends also on ψ : K(φ, η, ψ), and besides the two dynamic
equations, we have an additional equation to be solved, namely δψ K = 0.
9
To get the functional for the kinetic energy, observed that |∇3 Φ|2 = (∇Φ)2 +
(∂z Φ)2 = [∇φ + (∇2 ψ)F ]2 +(ψF ′ )2 , where F ′ denotes derivative wrt z. Instead
of (2.7), the kinetic energy for VBM reads
1
K (φ, η) =
2
Z Z
η
−h
2
2
′
(∇φ + F ∇ψ) + (F ψ) dz dx,
by expanding the above equation, we get
1
K (φ, η) =
2
Z Z
η
−h
2
2
2
′
|∇φ| + 2F ∇φ.∇ψ + (F |∇ψ| + (F ψ) dz dx.
Introducing the coefficients (which will depend on x through η and h)
β=
Z
η
F dz,
α=
−h
Z
η
2
F dz,
γ=
−h
Z
η
2
(F ′ ) dz,
(2.16)
−h
we obtain
1
K (φ, η) =
2
Z
|∇φ|2 (η + h) + 2β∇φ.∇ψ + α |∇ψ|2 + γψ 2 dx.
(2.17)
Now we rewritte Luke’s variational principle in (2.3) in a form of a minimization problem with respect to three variables (φ, η, ψ)
min
φ,η,ψ
Z Z
φ∂t ηdx − H (φ, η, ψ) dt.
(2.18)
With the approximation for the kinetic energy given by (2.17), the vanishing
of the first variation of (2.18) with respect to variations δφ of φ , δη of η, and
δψ of ψ resulting three equations below
Z Z
Z
(δφ) ∂t ηdx − [(η + h) ∇φ.∇ (δφ) + β∇ (δφ) .∇ψ] dx dt = 0 (2.19)
Z Z Z
1
2
gη (δη) − |∇φ| (δη) dx dt = 0 (2.20)
− (δη) ∂t φdx −
2
Z Z
{(β∇φ) .∇(δψ) + (α∇ψ) .∇(δψ) + γψ (δψ)} dx dt = 0 (2.21)
The resulting Euler-Lagrange equations from the expressions in (2.19), (2.20),
and (2.21) are the full VBM. As we did in SWE, to simplify the problem, we
linearize the VBM equations in first variations (2.19) and (2.20) , we obtain
10
the linearized VBM equations in integral form
Z Z
Z
(δφ) ∂t ηdx − [h∇φ.∇ (δφ) + β∇ (δφ) .∇ψ] dx dt = 0
Z Z
Z
(δη) ∂t φdx + gη (δη) dx dt = 0
Z Z
{(β∇φ) .∇(δψ) + (α∇ψ) .∇(δψ) + γψ (δψ)} dx dt = 0
(2.22)
(2.23)
Now we will show the VBM in the partial differential equations (PDEs) form,
as follow, the variations of the kinetic energy in (2.17) with respect to φ, η,
and ψ, results
δφ K = −∇. ((η + h) ∇φ) − ∇. (β∇ψ)
δη K =
1
|∇φ|2
2
δψ K = −∇. (β∇φ) − ∇. (α∇ψ) + γψ.
Variations with respect to φ and η of the Luke’s variational formulation in
(2.18) results the dynamic equations for VBM, given by
∂t η = δφ H (φ, η, ψ)
∂t φ = −δη H (φ, η, ψ) ,
which could be rewritten as
∂t η = δφ K (φ, η, ψ)
∂t φ = −gη − δη K (φ, η, ψ) .
Observe that by taking variations with respect to ψ of (2.18) resulting an
additional equation that has to be solved (an elliptic equation):
−∇. (β∇φ) − ∇. (α∇ψ) + γψ = 0,
(2.24)
so the resulting equations to be solved for the VBM would be
∂t η = −∇. ((η + h) ∇φ) − ∇. (β∇ψ)
(2.25)
∂t φ = −gη −
(2.26)
1
(∇φ)2
2
0 = −∇. (β∇φ) − ∇. (α∇ψ) + γψ.
(2.27)
11
if we linearize the three equations above, we obtain
∂t η = −∇. (h∇φ) − ∇. (β∇ψ)
(2.28)
∂t φ = −gη,
(2.29)
the third equation remains the same since it is already linear.
Now observe that the three equations in (2.25), (2.26), and (2.27) are the
Euler-Lagrange equations from the expression (2.19), (2.20), and (2.21). Also
(2.28) and (2.29) are the Euler-Lagrange equations from the expression (2.22)
and (2.23). In this thesis we will just deal with the linearized VBM, so in the
rest of this thesis we will call the linearized VBM as the VBM.
The three coupled PDEs in (2.27), (2.28), and (2.29) have to be solved together.
We will describe the algorithm for solving these equations. First, we need to
have η and φ at t = 0 (initial conditions), then we calculate α, β, and γ (at
t = 0). These are used for solving the elliptic equation (2.27), so we obtained
ψ at t = 0. This ψ will be used for the time stepping in the dynamic equations.
We get η and φ at the next time step and repeat this algorithm again to obtain
the solutions at each time steps.
The choice for function F (z) is explained thoroughly in [1], we choose a normalized function F (devided by the depth : h), given by
2z z 2
+ 2,
F =
h
h
(2.30)
this choice is motivated by the fact that the solution of Φ for linear wave
approximation (small variation in surface elevation and mildly sloping bottom)
is a cosh (z) function. This leads to the idea to choose the function F (z) as
a parabolic profile. The choice for function F can be a function that depend
on surface elevation η, but for the tsunami case we assumed that the depth
is much larger than the surface elevation, so we just choose function F as in
(2.30).
12
II.4
Boundary Conditions
To implement either SWE or VBM as wave models for tsunami simulation,
we need boundary conditions. In this thesis, we use two types of boundary
conditions (BCs). They are radiation boundary condition (RBC) and hardwall
boundary condition (HBC). The RBC will be used to terminate the computational domain in the ocean. Meanwhile, the HBC will be used for boundary
between land and sea. RBC means that when waves hit the boundary, it
will propage through the boundary without reflection. On the contrary, when
waves hit the land, it will be fully reflected or no wave propagate through the
boundary.
II.4.1
Radiation Boundary Condition
First we will derive the RBC for the SWE. We start with the derivation of
RBC for 1D case, then we will extend it to 2D case. Suppose our domain is
an interval (for instance [0, L] ) in the x-axis. We want to place the RBC at
the left side of our domain, namely at x = 0. The idea for deriving the RBC is
that we have to know the general solution in the exterior of our computational
domain (in this case, the solutions at left side of x = 0). For the 1D linear
SWE
∂t η = −∂x [h∂x φ]
∂t φ = −gη
we assume that on the left side of the boundary (at x = 0), the bathymetry
extends at a constant depth h0 towards −∞. Hence the general solution for
outgoing waves at the left exterior can be written down as
p
gh0 t)
p
φ(x, t) = G(x + gh0 t)
η(x, t) = F (x +
with F and G are arbitrary differentiable functions. By way of ilustration, we
will only derive the boundary condition for φ. The derivation for η is the same.
13
Differentiation of G wrt t and x at the neighborhood of this point results
∂x φ = G ′
p
∂t φ = ghG′
Elemination of G′ resultin the RBC as follow
∂t φ −
p
gh∂x φ = 0
(2.31)
for the 2D case, the RBC would be
∂t φ −
p
gh∇φ.n = 0
(2.32)
where n is outward normal direction at the boundary. Condition (2.32) is the
radiation BC for SWE. For the VBM, the derivation of the RBC is rather
difficult. In this thesis, we do not have the RBC for the VBM yet, so we will
use the RBC for the SWE for implementing radiation BC for VBM. In the
next chapter, it will be shown that the RBC for the SWE is not appropriate to
be implemented in the VBM, it will gives some reflections on this boundary.
However, since the RBC is used to terminate the computational domain in
the open ocean, the reflection that apprear from this boundary is rather small
compared to the outgoing wave, so this reflected wave will not much contribute
to the waveheight near coastline.
II.4.2
Hard-wall Boundary Condition
In the HBC, the normal flow through the boundary is assumed to be zero
or U.n = 0, where U is fluid velocity. For SWE, note that we approximate
the velocity potential at each depth by its value at the surface : Φ (x, z, t) ≈
φ (x, t), so the condition for hardwall can be rewritten as ∇φ.n = 0 (since
U = ∇3 Φ and Φ ≈ φ). Same problem appear as in the RBC, we do not have
the HBC for the VBM yet, so we will use the HBC of SWE for implementing
the HBC for the VBM.
Chapter III
Finite Element Method Implementation for
SWE and VBM
III.1
Finite Element Method
The finite element method (FEM) is computational technique for obtainning
approximate solutions for partial differential equations (PDEs). Rather than
approximating the PDEs point-wise as finite difference methods, the FEM use
a variational problem that involves a functional of the differential equation over
the domain of problem. This domain is devided into a number of subdomain
called elements. The solution of PDE is approximated by polynomial basis
function on every elements. These polynomials have to be pieced together so
that the approximate solution has an appropriate degree of smoothness over
the entire domain, after that, the variational integral is evaluated as a sum of
contributions over each element (see [3]).
There are several ways to so solve PDEs or variational problem by using FEM,
the two common methods are through the variational problem and through a
weak formulation. The first method is to substitute the approximate solutions
into the functional of the problem then minimize it with respect to certain
quantities. In the weak formulation, we substitute the approximate solutions
into the PDEs then multiply it with an arbitrary test function and integrate
it over the problem domain. The second method is known as Ritz-Galerkin
method. In this thesis we will use both of these methods. The first method
is used to implement the interior problem and the second method is used to
implement the boundary problem.
15
III.2
FEM Implementation for SWE
To implement FEM for the SWE, we start with describing the essential part
of FEM which is to provide FEM basis functions. After that we devide the
domain into finite number of elements and define polynomial basis functions
on each elements.
First, we discretize the domain into n elements. In this case, the computational
domain is discretized by using a mesh generator to make a triangulation from
the domain. There are many kind of mesh generator (commercial or noncommercial), but in this thesis, we use the mesh generator from MATLAB
and FEMLAB. FEMLAB’s mesh generator will be used for discretizing the
domain for Indonesia’s bathymetry. An illustration of a discretization of a
domain is given by a the left plot of Figure 3.1.
Figure 3.1: At the left, we show a plot of a discretized domain by using
pdetool from MATLAB, and at right plot show the global and the local
numbering (in bracket). The local numbering is assumed in counterclockwise
direction.
The mesh generator gives the information of the numbering and the coordinate of every nodes that is generated in the domain. There are two types of
numbering, global and local numbering. Local numbering means the number
of nodes in each element (triangle), which are 1, 2 and 3 (in counterclockwise
direction). This local numbering is important in the calculation in FEM which
will be explained later on. Meanwhile, the global numbering is the numbering
16
of points in the whole domain. These numbering are ilustrated in the right
plot of Figure 3.1.
As we mentioned before, in this thesis, we will use linear basis function for
finite element implementation, denoted as Ti (x). Ilustration for this linear
basis function is given by Figure 3.2. Note that this basis function has a strict
local character, with has a value 1 at one point, namely at point i and zero at
other points.
Figure 3.2: Plot of linear basis function Ti (x).
In order to construct a linear polynom which is defined on each triangle we
need 3 parameters. A natural choice is to use the function values in the three
vertices of the triangle. The linear polynom is defined by
L (x) =
ai + b i x + c i y
2∆
(3.1)
where ∆ is the area of the triangle in which the vertices are given by the three
points (x1 , y1 ), (x2 , y2 ), (x3 , y3 ), its value is given by
1 x1 y1 1
1 ∆ = 1 x2 y2 = [(x2 − x1 ) (y3 − y2 ) − (y2 − y1 ) (x3 − x2 )]
2
2
1 x3 y3 (3.2)
There are three unknowns (ai , bi , ci ) in (3.1), so we need three points (x1 , y1 ),
(x2 , y2 ), (x3 , y3 ) to get the expression for L (x). Assuming that L (x1 , y1 ) = 1,
17
and zero at the other vertices, we have
1 = ai + b i x 1 + c i y 1
0 = ai + b i x 2 + c i y 2
0 = ai + b i x 3 + c i y 3
where the values of the coefficients ai , bi ,and ci are given by
ai = x 2 y 3 − x 3 y 2
bi = y2 − y3
(3.3)
c i = x3 − x2 .
so now we have an explisit expression for L (x) per element, and note that the
indexing in the above equation refers to the local numbering.
In FEM, the solutions are approximated by linear combination of the basis
functions. We proceed with the approximation for solutions of SWE which are
η and φ with
η (x, t) =
φ (x, t) =
n
X
k=1
n
X
k=1
ηbk (t) Tk (x)
φbk (t) Tk (x)
(3.4)
where Tk (x) is ilustrated by Figure 3.2. In the FEM implementation of SWE
we use variational principle, which means, we substitute the approximation
(3.4) into the SWE in the integral form which is given by (2.11) and (2.12).
We get the following functionals
!
!
)
Z (
n
n
X
X
− ∂t
ηbk (t) Tk (x) δφ + h (x) ∇
φbk (t) Tk (x) .∇ (δφ) dx = 0
k=1
Z (
∂t
n
X
k=1
!
k=1
φbk (t) Tk (x) δη + g
n
X
k=1
!
ηbk (t) Tk (x) δη
)
dx = 0
(3.5)
Note that δφ and δη are arbitrary admissible variations with respect to φ and η.
We can also approximate these variations using the same basis function as δφ =
18
Pn
i=1
v1i Ti (x) and δη =
Pn
i=1
v2i Ti (x). If we substitute these approximations
into (3.5) then say that the integrals vanish for arbitrary nonzero v1 and v2,
we conclude that
Z (
−∂t
n
X
k=1
!
ηbk (t) Tk (x) Ti (x) + h (x) ∇
Z (
∂t
n
X
k=1
n
X
k=1
!
φbk (t) Tk (x) Ti (x) + g
!
)
φbk (t) Tk (x) .∇Ti (x) dx = 0
n
X
k=1
!
(3.6)
)
ηbk (t) Tk (x) Ti (x) dx = 0
(3.7)
which hold for every i. We can rewrite the approximation in (3.6) and (3.7)
in a matrix equations as follow :
−
→
→
M∂t −
η = Gφ
−
→
→
η
M∂t φ = −gM−
or

 
−
→
0
G
M 0
η

=

 ∂t 
−
→
−gM 0
φ
0 M




−
→
η

−
→
φ
(3.8)
−
→
→
where −
η and φ denote the vectors solution for η and φ, the entry of a matrix
R
M is given by mij = Ω Ti (x) Tj (x) dΩ, and the entry of a matrix G is given by
R
gij = Ω (h (x)) ∇Ti (x) .∇Tj (x) dΩ. To evaluate the entries of these matrices
directly, is rather difficult. Note that we have an unstructured form of triangles
in our domain (see Figure 3.1). This cause the difficulty in calculation when
we calculate integrals of the basis function directly. One way to solve this
problem is to transform any triangles into ’natural’ coordinate where we can
calcute the integrals within these triangles easily. We will call this transformed
triangle in natural coordinate as master triangle, see Figure 3.3.
19
Figure 3.3: Linear transformation from Ωe to master triangle and inverse
map.
There are three non-zero basis functions over this master triangle, they are
T n1 (ξ, η) = 1 − ξ − η,
T n2 (ξ, η) = ξ,
T n3 (ξ, η) = η
(3.9)
The linear transformation from a triangle (x1 , y1 ) , (x2 , y2 ) , and (x3 , y3 ), arranged in the counter clockwise direction, to the master triangle is
x = Σ3j=1 xj T nj (ξ, η) ,
y = Σ3j=1 yj T nj (ξ, η)
or
1
[(y3 − y1 ) (x − x1 ) − (x3 − x1 ) (y − y1 )]
2∆
1
η=
[− (y2 − y1 ) (x − x1 ) + (x2 − x1 ) (y − y1 )]
2∆
ξ=
where ∆ is the area of the triangle which is given by (3.2). Now with the
linear transformation of the coordinate, we can calculate the integral of the
basis function by using
Z
Z
∂ (x, y)
|dξdη
f (x, y) Ti (x) Tj (x) dx = f (ξ, η) T ni (ξ, η) T nj (ξ,η) |
(∂ξ, η)
Ω
Ωe
(3.10)
for some function f (x, y), where ∂(x,y)
is known as determinant of the Jaco(∂ξ,η) bian matrix.
20
In assembling M and G matrices, we have to work element-wise, since the
global numbering also has an unstructured numbering as ilustrated in the
right plot of Figure 3.1. For assembling M, for each element, we have a 3 × 3
elementary matrix
mij =
Z
Ti (x) Tj (x) dΩe
,with i, j = 1, 2, 3
Ωe
where i and j are the local numbering. By using the linear transformation of
the coordinate in (3.10), we get
mij =
(1 + δij )
∆,
12
with i, j = 1, 2, 3
(3.11)
where δij is Kronecker’s delta and ∆ is given by (3.2). We proceed to assemble
P
hs Ts (x) (we choose to apmatrix G. First, we approximate h by h (x) = ns b
proximate h (x) in this way since we assumed that there is no bottom motion).
The same procedure is done for calculating the matrix G, we also have 3 × 3
elementary matrix for G
gij =
Z
Ωe
=
Z
Ωe
(h (x)) ∇Ti (x) .∇Tj (x) dΩe
!
n
X
b
hs Ts (x) ∇Ti (x) .∇Tj (x) dΩe
,with i, j = 1, 2, 3
s=1
where i and j are the local numbering. The value of gij is given by
bi bj + c i c j
gij =
4∆2
"
1 Xb
hs as +
2 s
#
P
X
X
x
y
s
s
s
b
b
hs bs + s
hs cs
6
6
s
s
P
(3.12)
where s = 1, 2, 3, and a, b, c are the coefficients of the basis function given by
(3.3). Note that system in (3.8) is system of ordinary differential equations
(ODEs). This system can be easily calculated by using Runge-Kutta method
or using ODE solver in MATLAB to obtain the solutions in each time steps.
Note that until now, we already get complete matrix sistem in (3.8) except
for boundary conditions. We will implement the RBC and HBC in our FEM
scheme, but we will explained it later. In the following, we will proceed with
FEM implementation for VBM.
21
III.3
FEM Implementation for VBM
Similar with FEM implementation for SWE, for the VBM, the variables (φ,η,ψ,α,β,γ,h)
are approximated by linear combinations of basis function Ti (x). Approximation for φ and η are given by (3.4), and the approximations for ψ, h, α ,β and
γ are
ψ (x) =
α (x) =
n
X
s=1
n
X
s=1
ψbs Ts (x) , h (x) =
α
bs Ts (x) , β (x) =
n
X
s=1
n
X
s=1
b
hs Ts (x)
βbs Ts (x) , γ (x) =
n
X
s=1
γ
bs Ts (x)
(3.13)
We will explain the reason why we choose (3.13) the approximation for α, β
and γ. Note that we have defined α,β,γ in (2.16). Originally, α, β and γ are
functions of x and t (through η and h), but in the tsunami case, we assume
that the depth h is much larger than the elevation η. So we approximate α,β,γ
R0
R0
R0
by β = −h F dz, α = −h F 2 dz, γ = −h (F ′ )2 dz. For the parabolic profile
function (F (z)) is given by (2.30), the values of α,β,γ are given by
α(x) =
8
h(x),
15
2
4
β(x) = − h (x) , γ (x) =
,
3
3h (x)
and now we can rewrite the approximation for α,β,γ in (3.13) as
n
α (x) =
n
n
8 Xb
2 Xb
4X 1
Ts (x)
hs Ts (x) , β (x) = −
hs Ts (x) , γ (x) =
15 s=1
3 s=1
3 s=1 b
hs
As we did in the FEM implementation for SWE, we use variational principle for
FEM implementation, so we substitute the approximation for the variables φ,η
and ψ (we keep variables α,β,γ,h in α (x),β (x),γ (x), and h (x) first) into the
linear VBM in integral form in (2.21), (2.22), and (2.23), we get the following
22
integrals



Z 






Pn
− (∂t k=1 ηbk (t) Tk (x)) δφ
P
n
bk (t) Tk (x) .∇ (δφ) dx = 0
φ
+h (x) ∇
k=1







 +β(x)∇ (δφ) .∇ Pn ψb T (x)
k
k
k=1
! )
!
Z (
n
n
X
X
ηbk (t) Tk (x) δη dx = 0
φbk (t) Tk (x) δη + g
∂t
k=1
k=1
P

n
b

β
(x)
∇
φ
(t)
T
(x)
.∇(δψ)
k
k=1 k

Z 

P
n
b
+α (x) ∇
k=1 ψk Tk (x) .∇(δψ)

P


n

ψbk Tk (x) (δψ)
+γ (x)
k=1









dx = 0
(3.14)
Note that δφ, δη and δψ are arbitrary admissible variations with respect to
φ, η and ψ. We can also approximate these variations using the same basis
P
P
P
function as δφ = ni=1 v1i Ti (x), δη = ni=1 v2i Ti (x) and δψ = ni=1 v3i Ti (x).
If we substitute these approximations into (3.14) then let that the integrals
vanish for arbitrary nonzero v1, v2 and v3, we conclude that



Z 

Z (




∂t
n
X
k=1
P
− (∂t nk=1 ηbk (t) Tk (x)) Ti (x)
P
n
bk (t) Tk (x) .∇Ti (x)
+h (x) ∇
φ
k=1
P
n
bk Tk (x)
+β(x)∇Ti (x) .∇
ψ
k=1
!
φbk (t) Tk (x) Ti (x) + g
n
X
k=1
!




)
dx = 0
(3.15)
ηbk (t) Tk (x) Ti (x) dx = 0
P

n
b

β
(x)
∇
φ
(t)
T
(x)
.∇Ti (x)
k
k=1 k

Z 

P
n
b
+α (x) ∇
k=1 ψk Tk (x) .∇Ti (x)


Pn b


+γ (x)
ψk Tk (x) Ti (x)
k=1





(3.16)









dx = 0
(3.17)
23
which hold for every i. We can rewrite the approximations for (3.15), (3.16)
and (3.17) in the system of matrix equations as follow
−
→
−
→
→
M∂t −
η = Gφ + Rψ
−
→
→
η
M∂t φ = −gM−
−
→
−
→
Lψ = Kφ.
(3.18)
(3.19)
(3.20)
Note that the system of (3.18) and (3.19) is similar with the system of matrix
equations for SWE(3.8) except for additional term in the first equation (Rψ).
The M and G matrices are given by (3.11) and (3.12), while for assembling
matrix R, for each element, we have a 3 × 3 elementary matrix
rij =
Z
Ωe
=
Z
Ωe
β (x) ∇Ti (x) .∇Tj (x) dΩe
Σns=1 βbs Ts
(x) ∇Ti (x) .∇Tj (x) dΩe
,with i, j = 1, 2, 3
where i and j are the local numbering. By using linear transformation of the
coordinate in (3.9), we get:
#
"
P
P
X
X
x
y
bi bj + c i c j 1 X b
s
s
βs as + s
βbs bs + s
βbs cs
rij =
4∆2
2 s
6
6
s
s
where s = 1, 2, 3, and a, b, c are the coefficients of the basis function given by
hs . Now for the third equation in
(3.3). The value for βbs is given by βbs = − 23 b
VBM (3.17) which can be rewritten as (3.20) where matrix L is assembled for
each element, from a 3 × 3 elementary matrix
lij =
Z
Ωe
=−
Z
[−α (x) ∇Ti (x) .∇Tj (x) − γ (x) Ti (x) Tj (x)] dΩe
Ωe
Σ3s=1 α
bs Ts (x) ∇Ti (x) .∇Tj (x) + Σ3s=1 γ
bs Ts (x) Ti (x) Tj (x) dΩe
with i, j = 1, 2, 3 are the local numbering. lij is given by
#
"
P
P
X
X
x
y
bi bj + c i c j 1 X
s
s
lij = −
α
b s as + s
α
b s bs + s
α
b s cs
4∆2
2 s
6
6
s
s
Z
Z
Z
− γ
b1
T1 Ti Tj dΩe + γ
b2
T2 Ti Tj dΩe + γ
b3
T3 Ti Tj dΩe
Ωe
Ωe
Ωe
24
where
Z
∆
,
10
∆
= ,
30
∆
= ,
60
Ti Tj Tk dΩe =
Ωe
if i = j = k,
if i = j 6= k,
if i 6= j 6= k
with s = 1, 2, 3, and a, b, c are the coefficients of the basis function given by
(3.3). The value for γ
bs and α
bs are γ
bs =
4
3b
hs
and α
bs =
8b
h.
15 s
The matrix K matrix is assembled for each element, from a 3 × 3 elementary
matrix where the entries are given by
Z
kij =
β (x) ∇Ti (x) .∇Tj (x) dΩe
Ωe
Z Σ3s=1 βbs Ts (x) ∇Ti (x) .∇Tj (x) dΩe
=
,with i, j = 1, 2, 3
Ωe
or
bi bj + c i c j
kij =
4∆2
"
1Xb
βs as +
2 s
#
P
X
X
x
y
s
s
s
βbs bs + s
βbs cs
6
6
s
s
P
where s = 1, 2, 3, and a, b, c are the coefficients of the basis function given by
(3.3). Note that matrix K is exactly the same as matrix R.
Now we already have a complete system of matrix equation in (3.18), (3.19),
and (3.20). The dynamic equations (3.18) and (3.19) can be rewritten as a
system of ordinary differential equations

 

 
−
→
0
G
M 0
η

=

 ∂t 
−
→
−gM 0
φ
0 M

 
−
→
−
→
Rψ
η

+
−
→
0
φ
(3.21)
and the third equation (3.20) can be expressed as
−
→
−
→
ψ = L−1 K φ
(3.22)
The system in (3.21) and (3.22) should be solved together. As we already
described before, we use the initial conditions η and φ (at t = 0) to calculate
−
→
α, β, and γ (at t = 0) . These are used to get a ψ (at t = 0) by using (3.22).
−
→
Then this ψ will be used for the time stepping in the dynamic equations
25
in (3.21). So we obtain η and φ at the next time step. By repeating this
algorithm we will obtain the solutions at each time steps the system in (3.21)
and (3.22). Same as in SWE, the system in (3.21) can be easily calculated by
using Runge-Kutta method or using ODE solver in MATLAB.
III.4
FEM Implementation for Boundary Conditions
The derivation of the governing equations from the first variation (2.11) and
(2.12) does not give the corresponding boundary condition because the functional that we minimized accounts only for the interior (from Pressure principle
in (2.1)). We need to modify the weak formulation in order to incorporate the
boundary conditions. Therefore, we use the corresponding governing equations
from the weak formulation of (2.14) only for implementing the BC. To have
the weak formulation of (2.14), we multiply the first equation of (2.14) by a
test function and integrate it over a domain, then we do partial integration for
the right hand side term, so we obtain the following equation
Z
v∂t ηdx =
Ω
Z
Ω
h (x) ∇φ.∇vdx −
Z
∂Ω
(vh (x) ∇φ.n) d∂Ω
(3.23)
where v is arbitrary test function. Note that this is essentially the same with
(2.11) without the boundary term (second term at right hand side). Including
the RBC in (2.32) into the last term on the RHS of (3.23) gives
!
Z
Z
Z
vh (x) ∂t φ
p
d∂Ω
v∂t ηdx =
h (x) ∇φ.∇vdx −
gh(x)
Ω
Ω
∂Ω
(3.24)
Note that from (2.12), we have the relation ∂t φ = −gη, so (3.24) is equivalent
with
Z
Ω
v∂t ηdx =
Z
Ω
h (x) ∇φ.∇vdx +
Z
∂Ω
vh (x) gη
p
gh(x)
!
d∂Ω
(3.25)
For FEM implementation for the SWE with RBC, we will use this last equation
instead of (2.11), by setting that test function v = δφ. Now we will only
treat the boundary term, because the rest is the same with (3.8). Note that
the boundaries in a 2D domain are basicallly a curve (or a path), so the
26
boundary integration in (3.25) is implemented as if in 1D FEM. In principal,
the procedure for deriving the FEM scheme in 1D case is the same with FEM
in 2D. The different is only for the basis functions. In 1D case, we use a linear
basis function which is given by
 x−xk−1



 xk −xk−1 x−xk+1
T 1k (x) =
xk −xk+1



 0
,
x ∈ [xk−1 , xk ]
,
x ∈ [xk , xk+1 ]
,
elsewhere
(3.26)
For every boundary, we will not discretize the boundary again since the boundary is already discretized by the mesh generator when we generate the finite
element mesh in 2D. We will use the nodes at the boundary as discretized by
the mesh generator. The same like other FEM implementation, we approximate v, η and h in boundary term of (3.25) by linear combination of basis
functions in 1D (3.26), then substitute these approximations into the boundary term. This will gives a matrix called B which is assembled from a 2 × 2
elementary matrix,in which the entries are given by
Z p
√
h(x)T 1i (x)T 1j (x) d∂Ω,
with i, j = 1, 2
bij = − g
∂Ω
then approximation
p
h(x) =
Σni=1
q
b
hi T 1i (x), gives
 √
√
Bek = − g∆k 
√ h1
h2
+
4
12
√
√ h1 + h2
12
√
√ h1 + h 2
√ 12 √ h1
+ 4h2
12


where ek denotes the k-th element in the boundary, ∆k is the length between
two points of the element in the boundary, and h1 and h2 are the depth in
the points of element at the boundary (the indices correspond to the local
numbering, note that the local numbering for 1D case are 1 and 2). With the
presence of the boundary term, our system of matrix equation in (3.8) would
be
−
→
→
→
M∂t −
η = G φ + B−
η
−
→
→
η
M∂t φ = −gM−
27
or

 
−
→
B
G
M 0
η

=

 ∂t 
−
→
−gM 0
φ
0 M




−
→
η

−
→
φ
Now we discuss about HBC’s FEM implementation for SWE. Note that we
already show in subsection boundary condition (II. 4) that in order to implement the HBC, the condition ∇φ.n = 0 has to be satisfied in the boundary
term of (3.23). This condition makes the boundary term in (3.23) vanishes.
With this condition, we will have the system of matrix equation exactly the
same with our system in (3.8). So the HBC’s FEM implementation for SWE
gives the system of matrix equation that we derive based on the variational
principle (2.11), where the that term is not present.
Until now, we do not have a precise radiation boundary condition for VBM.
So for the VBM’s FEM implementation on the boundary conditions, we will
implement it in the same way as we did for the case of the SWE.
Chapter IV
Tsunami Simulation
In this section we will show several simulations of our code that we derived at
the previous chapters. We start with the wave propagation above a flat bottom,
by using SWE and VBM, in order to give an intuition about the evolution of
these waves. After that, we will deal with our main purpose which is the
tsunami simulation above the real bathymetry of Indonesia. We will choose
two areas in Indonesia which have the potential to be hit by a tsunami. They
are the area in the south of Pangandaran coast and the area in the south of
Krakatau volcano.
IV.1
Water Wave Simulation Above Flat Bottom
In this part we will show the water wave simulation above a flat bottom by using both SWE and VBM. Suppose we have a rectangle domain, and discretized
using mesh generator form MATLAB (pdetoolbox), with size 110km × 110km.
In the four sides of the domain, the RBC is implemented both for the SWE
and VBM. In order to capture the propagation history, we use a tool called
Maximum Temporal Amplitude (MTA) that is introduced in [5]. In this thesis we will use the variants of MTA. They are maximal temporal crest-height
(MTC) and minimal temporal through-depth (MTT), also called the maximal
positive and negative amplitude respectively. These are a curve that represent
the maximal and minimal surface elevation at position x as a function of time
M T C (x) = max η (x, t) ,
t
M T T (x) = min η (x, t) .
t
We consider an initial wave (a bipolar hump with initial amplitude 1.7m) with
a wavelength Λ = 20km released without a speed above a horizontal bottom
at the depth 1000m. Plot of the initial condition is shown at Figure 4.1, and
the plot of the crosssection at y = 0 is shown at Figure 4.2.
29
Figure 4.1: Plot of the bipolar initial profile.
Crosssection at y=0 of the initial wave profile
2
1.5
1
η [m]
0.5
0
−0.5
−1
−1.5
−2
−40
−30
−25
−20
−10
0
10
20
30
40
45
x [km]
Figure 4.2: Plot of the crosssection of initial wave at y = 0 with 1.7m
amplitude and Λ = 20km.
30
Figure 4.3: Plot of the splitting initial bipolar hump above flat bottom using
SWE model at t = 7 min.
Figure 4.4: Plot of SWE simulation above flat bottom when the wave
reached the boundary at t = 12 min. Notice that there is no reflection from
each boundary.
31
Figure 4.5: Plot of the splitting of initial bipolar hump using VBM, notice
the development of dispersive effect.
Figure 4.6: Plot of the VBM simulation when the wave hits the radiation
boundary condition, notice the reflected wave from the left and right
boundary.
32
The resulting evolution for SWE is shown at Figure 4.3. The plot below it
denote the crosssection at y = 0. It illustrates the splitting of the bipolar hump
into every direction. It also can be seen that when the wave hit the boundary,
Figure 4.4, the reflection is rather small. For SWE, this RBC is good enough
for our purposes (tsunami simulations).
The evolution for the VBM is shown at Figure 4.5. In this evolution, it can
be seen clearly the effect of dispersion. Observe that slower wave with smaller
amplitude is propagating behind the wave with the largest amplitude. This
”dispersive tail” appear in our VBM, because we incorporated vertical variations in the approximation of the velocity potential Φ. In the SWE the waves
run with the same velocity which is square root of gravitational accelleration
times the depth, but in the VBM, the velocity of the waves depend on its
wavelength. The waves with larger wavelength will propagate faster than the
waves with smaller wavelength. Observe again the different between the SWE
and VBM simulations at Figure 4.3 and 4.5, with the presence of this dispersive tail in the VBM simulation, it produces a different wave elevation than in
the SWE simulation.
We can see at the VBM’s crossection, the wave that propagate to the right has
the positive wave larger than the SWE. This is caused by ”the exchanging of
amplitude” between the first negative and the successive second positive wave,
the positive wave compensate the decreased negative wave. But for the wave
that propagate to the left the amplitude are smaller than the SWE’s result, this
is expectable, because dispersion will decrease the amplitude. In general, the
presence of dispersive effect will decrease the amplitude, the fact that happen
in the wave that propagate to the right is accidentally because the limitation
of the domain, for the longer domain, the amplitude will decrease. Since we
use the radiation BC of the SWE for the VBM, the result is not so good, it
can be seen in Figure 4.6 that there are reflections from the boundaries.
33
IV.2
Indonesia Bathymetry
In order to simulate the tsunami, we need to incorporate the real bathymetry
of Indonesia. There are several sources on internet which provide the world’s
bathymetry, some of them are free and the others are not. Indonesia’s bathymetry
that we used is taken from The General Bathymetric Chart of the Oceans
(GEBCO). This a free bathymetry which provides the bathymetry data in
global grid with one arc-minute spacing. For information, one degree is equal
to 60 minutes (or written as 10 ≈ 60′ ) and 1′ ≈ 1830m. The Ilustration for
this bathymetry data is given by Figure 4.7.
Figure 4.7: Plot of available bathymetry data with 1′ accuracy.
A problem appears when we want to incorporate this square grid data into our
domain (a triangular grid, see Figure 3.1). In this case, we approximate the
n
X
b
function of depth h (x) by h(x) =
hi Ti (x), hence we need to know the value
i=1
for b
hi which is given by the value of h at the point xi : b
hi = h(xi ). In this
thesis, we choose to approximate b
hi by the average of the four nearest nodes
from the original data, as ilustrated in Figure 4.8.
34
Figure 4.8: Ilustration of the approximated bathymetry data in triangle
domain.
An example for a triangular mesh is given by Figure 4.9 which is generated
by using FEMLAB, while the plot of approximated bathymetry of Indonesia
is shown at Figure 4.10. This area is in the south of Java, near Pangandaran.
Discretized Domain of Pangandaran Case
longitude[deg]
−7.3
−8.3
−9.3
106.2
107.2
108.2
109.2
latitude[deg]
Figure 4.9: Plot of discretized domain in the south of Java.
Figure 4.10: Profile of the bathymetry in the south of Java for Pangandaran
case.
35
IV.3
Tsunami Simulation Using SWE & VBM Above Indonesia
Bathymetry
In this subsection we will describe the main results of this thesis which is the
tsunami simulation in two areas of Indonesia. They are the areas in the south
of Pangandaran and in the south of Lampung (near Krakatau).
We start with Pangandaran’s tsunami. This is a historical tsunami in Indonesia which happened in July 17, 2006. As reported by U.S. Geological
Survey (USGS), the location of the earthquake was 9.220 S and 107.320 E at
the depth of 34km. The earthquake with magnitude 7.7 occured as a result of
thrust-faulting on the boundary between the Australian plate and the Sunda
plate. On this part of their mutual boundary, the Australian plate moves
north-northeast with respect to the Sunda plate at about 59mm/year. The
Australian plate thrusts beneath the Sunda plate of the Java trench, south of
Java, and is subducted to progressively greater depth beneath Java and north
of Java. This subduction produced the bipolar initial condition with the negative hump into the coast direction and the positive hump into the oceanic
direction. These informations are very important for simulating the tsunami.
Figure 4.11: Plot of the location of earthquake in the south of Java at July
17, 2006. Courtesy : USGS.
36
The location of the souce of the tsunami ilustrated in Figure 4.11 is taken from
USGS. For Pangandaran’s tsunami simulation, we start by using our SWE’s
FEM code, then later we will compare it with the result of VBM’s FEM code.
We consider an initial wave (a bipolar hump with amplitude 2m) with a wavelength Λ = 40km is generated by a fast bottom excitation of that form. We
assumed that this initial wave profile is released without a speed at the same
location in the historical Pangandaran tsunami. As we described previously,
we will implement the RBC for the ocean and the HBC for the islands. Also
we will use the MTC and MTT curves at the crossection area to analyze the
propagation of the tsunami. The resulting simulations are shown at Figure
4.12 and 4.13. The wave reached the nearest coast line after 18 minutes and
reached the Pangandaran’s coast line after 30 minutes. The profile of the wave
after 1 hour is shown at Figure 4.14, the waveheight reaches to more than 5m in
certain coastline. Notice that the appearance of reflected wave is caused by the
complexity of the bathymetry and also by the HBC at the island. Meanwhile
the RBC at the ocean looks very appropriate (Figure 4.13).
37
Figure 4.12: Plot of the splitting of intial bipolar hump using SWE model at
t = 3 minutes. The location of the source is near 9.220 S and 107.320 E.
Figure 4.13: Plot of the tsunami simulation on Pangandaran case using SWE
model at t = 9 minutes.
38
Figure 4.14: Plot of Pangandaran’s tsunami simulation using SWE model at
t = 60 minutes.
Figure 4.15: Plot of the maximum crest-height during 1 hour simulation
using SWE model. Crossection near the coast denoted by white line.
39
Figure 4.16: Plot of Pangandaran’s tsunami simulation using the VBM at
t = 9 min. Notice the appearance of the dispersive tail.
Figure 4.17: Plot of the maximum crest-height near the coast during 1 hour
simulation using VBM. The below plot denotes the crosssection in the white
line.
By taking the same initial condition, the resulting Pangandaran’s tsunami
simulation using the VBM’s FEM code is shown at Figure 4.16 and 4.17.
Through the crossection plot in Figure 4.16, we can see the dispersive tail in
40
the VBM result. Again we can see that our RBC does not perform well with
the VBM code. But the reflections at the boundaries, acceptable since it is
rather small so will not have much contribution for the amplitude amplification
near the coast.
The important results of these simulations is to give a public awareness about
how fast the propagation of this tsunami since it is generated above the earthquake source and which areas will be hit by waves with bigger amplitude than
others. At the specific area, compared to the SWE’s result, the VBM produces larger surface elevation because the presence ot the dispersive tail in
VBM gives different wave elevation. It can be seen at Figure 4.15 and 4.17, at
the same location in the coastline, the SWE produces 8.9m waveheight, but
the VBM produces 10.0m waveheight.
For the second case of tsunami simulation, we choose the area in the south
of Lampung (between Sumatra and Java), we took a new area instead of the
historical tsunami from the Krakatau’s eruption in 1883. The same with the
Pangandaran case, this area also has a potential to generate an earthquake
which results in a tsunami. The location of Lampung’s case is also in the
region of the plate boundary between the Australia plate and Sunda plate
which is seismically active region.
Although there is no historical tsunami generated from this specific area that
we will choose, still, there were many historical tsunami from the surrounded
area near it. As the information from USGS, there were tsunami on June 2,
1994 with magnitude 7.8, killed over 200 people, and on August 20, 1977, a
magnitude 8.3 normal-fault earthquake occured within the Australian plate
about 1200km east-southeast of the earthquake that happen in Pangandaran
in 2006. Also in 2006 Yogyakarta’s earthquake with magnitude 6.3, occured
at shallow depth within overriding Sunda plate. So it not imposible at all that
tsunami can be generated from this Lampung area.
41
Figure 4.18: Plot of approximated bathymetry for Lampung Case. Notice the
shallow area surrounded by deep area in the south of Sumatra.
We start Lampung’s case by using SWE’s FEM code. We took the initial wave
profile the same like in the Pangandaran’s case, but we adjusted its location
so that it is on the top of the mutual boundary of Australian plate and Sunda
plate, see Figure 4.19. As we did before, in this case it is assumed that the
tsunami is caused by a fast bottom excitation so the initial wave profile is
released without a speed. The resulting simulation by using SWE’s FEM is
shown at Figure 4.20 and 4.23.
Figure 4.19: Plot of the location and the shape of initial wave for Lampung
Case.
42
Figure 4.20: Plot of the tsunami simulation for Lampung case by using SWE
model at t = 9 minutes.
Figure 4.21: Plot of tsunami simulation for Lampung case by using VBM.
Notice the appearance of the dispersive tail.
43
Figure 4.22: The wave after 18 minutes by using VBM. There is delayed and
amplified wave above the shallow area. The waveheight reached to more than
10m.
Figure 4.23: Plot of the maximum waveheight during 1 hour simulation by
using SWE model. At the spesific area, the waveheight reached 9.47m.
44
Figure 4.24: Plot of the maximum waveheight during 1 hour simulation by
using VBM. At the specific area near the coast in the south of Sumatra, the
wave reached 9.61m.
Meanwhile, the resulting simulation for the VBM is shown at Figure 4.21, 4.22
and 4.24. As already mentioned previously, the difference between the VBM’s
simulation and the SWE’s simulation are the appearance of the dispersive tail
that results in different wave amplitudes and also the reflection of the RBC in
VBM’s case. It can be seen that since the initial profile is released, the wave
reached the nearest coast (at Panaitan island, small island in the west side of
Java and Ujung Kulon in Java) after 11 minutes (see Figure 4.20 and 4.16)
and reached the south of Sumatra after 27 minutes for both wave models. For
the SWE model, the maximum waveheight reached 9.47m at the south coast
of Sumatra, and 9.6m for the VBM, see Figure 4.23 and 4.24. Figure 4.23
and 4.24 show the maximum wave amplitudes during 1 hour simulation along
the coast at Sumatra, Java, and other small islands between them.
There is something ”special” in this Lampung’s case. It is the topography
of the bathymetry at southest area of Sumatra. There is a very shallow area
between two deep areas (a ridge) as shown at Figure 4.18. According to the
√
linear wave theory, the wave speed c is related to the depth h by c = gh.
45
In the shallower area, the wave will propagate with slower speed, whereas in
contrary, in the deeper area, the wave will have faster speed. Based on this fact,
the topography like in Figure 4.18 supports a phenomena called near-coast
tsunami waveguiding, as introduced in [3,4]. At the top of the waveguide
(above the ridge) where the speed of the wave slower that the surrounding
area, the wave will have an amplification of the waveheight, but outside of
the waveguide (above the two deeper area surrounding the waveguide) the
distorted wave has to adjust with the delayed wave above the waveguide, see
Figure 4.22. Observe that the amplified waveheight reached 13.3m for the
SWE’s simulation and 10.8m for the VBM’s simulation above the waveguide
as shown at the Figure 4.23 and 4.24. Futher studies about this phenomenon
will be investigated in the future.
Chapter V
Conclusions and Recommendations
V.1
Conclusions
In the derivation of the SWE, the vertical variations of velocity potential Φ
are neglected, different then the derivation of the VBM where the vertical variations of the velocity potential lead to the effect of dispersion. This dispersive
effect is very crusial thing in tsunami simulation, with the presence of this
dispersive effect in the VBM, it produces a different wave elevation than in
the SWE simulation. At a certain area in the coast line in the Pangandaran’s
tsunami simulation, the wave amplitude reached 8.9m for the SWE’s simulation and 10m for the VBM (at the same position). Meanwhile for the tsunami
simulation in the south of Lampung, there is a ’focusing’ of the wave above a
shallow area that surrounded by deeper area in the south of Lampung. This
kind of phenomenon has been investigated in [3,4], and it is known as nearcoast tsunami waveguiding. Above this waveguide, the waveheight reached
13.3m for the SWE’s simulation and 10.8m for the VBM’s simulation.
Since we do not have the boundary conditions (either radiation or hard-wall
boundary conditions) for the VBM yet, when we used the SWE’s boundary
conditions for the VBM, it gives a problem. For radiation boundary condition
(RBC) for the VBM, it gives some reflection in the boundary. However, since
the RBC is used to terminate the computational domain in the open ocean,
the reflection that apprear from this boundary is rather small compared to
the outgoing wave, so this reflected wave will not much contribute to the
waveheight near coastline.
47
V.2
Recommendations
From the simulation results, it can be seen that the RBC that we incorporated
does not perform well for the VBM, so it needs further study to improve this
type of boundary condition. In order to get more realistic tsunami simulation,
these codes can be improved by adding an appropiate boundary condition
between land and sea. At this thesis the nonlinearity of the SWE and VBM
are not incorporated, but it will be a part of future research.
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Physical Sciences. SIAM, Mathematical Modelling and Computation,
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[3] Van Groesen, E. , Adytia, D. , Andonowati and Klopman, G. 2007. Nearcoast tsunami waveguiding: simulations for various wave models. Technical Report of LabMath–Indonesia, LMI-GeoMath-07/02; ISBN 90-3652352-4; http://www.labmath-indonesia.or.id/Reports/Reports.php.
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