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
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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
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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.
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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
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ABSTRAK
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GUIDELINES TO USE THE THESIS
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ACKNOWLEDGMENTS
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CONTENTS
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LIST OF FIGURES
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Chapter I Preliminary
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I.1
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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I.2
Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . .
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I.3
Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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I.4
Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter II Basic Theory
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II.1 Variational Formulation . . . . . . . . . . . . . . . . . . . . . .
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II.1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . .
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II.1.2 Variational Formulation for Surface Wave
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II.2 Shallow Water Equations . . . . . . . . . . . . . . . . . . . . . .
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II.3 Variational Boussinesq Model . . . . . . . . . . . . . . . . . . .
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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
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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
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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
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Bibliography
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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
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