BUCKLING OF BAR BY WAVELET –GALERKIN METHOD A

BUCKLING OF BAR  BY WAVELET –GALERKIN METHOD A
BUCKLING OF BAR BY WAVELET –GALERKIN
METHOD
A
THISIS SUBMITTED IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
MASTER OF TECHNOLOGY
IN
CIVIL ENGINEERING
WITH SPECIALIZATION IN
STRUCTURAL ENGINEERING
BY
SUJI P
ROLL NO: 210CE2027
NATIONAL INSTITUTE OF TECHNOLOGY
ROURKELA – 769008.
2012
BUCKLING OF BAR BY WAVELET –GALERKIN
METHOD
A
THISIS SUBMITTED IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
MASTER OF TECHNOLOGY
IN
CIVIL ENGINEERING
WITH SPECIALIZATION IN
STRUCTURAL ENGINEERING
BY
SUJI P
ROLL NO: 210CE2027
UNDER THE GUIDANCE OF
Dr. M. R BARIK
(Associate professor)
NATIONAL INSTITUTE OF TECHNOLOGY
ROURKELA – 769008.
2012
NATIONAL INSTITUTE OF TECHNOLOGY
ROURKELA-769008, ODISHA
INDIA
CERTIFICATE
This is to certified that the thesis entitled, “BUCKLING OF BAR BY WAVELET
GALERKIN METHOD” submitted by SUJI P bearing Roll No-210CE2027 in partial
fulfillment of the requirement for the award of the degree of Master of Technology in Civil
Engineering with the specialization of “Structural Engineering” of National Institute of
Technology, Rourkela is a record of bonafied work carried by her under my supervision and
guidance in academic year 2011-2012.
This thesis fulfils the requirements relating to the nature and standard work for the award of
Master of Technology in Civil Engineering. To the best of my knowledge the matter embodied in
the thesis has not been submitted to any other Universit/Institute for the award of any degree or
diploma.
Dr. M.R. Barik
(Associate Professor)
Date : 25.05.12
Place: Rourkela
Dept. of Civil Engineering
National Institute of Technology
Rourkela -769008
ACKNOWLEDGEMENT
ACKNOWLEDGEMENT
I wish to express my sincere gratitude to my revered guide Dr. M. R. Barik, Associate
Professor, Department of Civil Engineering, National Institute of Technology, Rourkela for
kindly providing me an opportunity to work under his supervision and guidance.I appreciate
his broad range of expertise and attention to detail, as well as the constant encouragement he
has given me over the years. There is no need to mention that a big part of this thesis is the
result of joint work with him, without which the completion of the work would have been
impossible.
I am grateful to Dr. N. Roy, professor, Head, Department of Civil Engineering for his
valuable suggestions during the synopsis meeting and necessary facilities for the thesis work.I
am also thankful to the Professors and Staff, Department of Civil Engineering, National Institute Of
Technology Rourkela, for their moral support. I am expressing my gratitude to Dr A V Asha,
faculty in Civil Engineering Department.
I should express my special thanks to my friends, Mallikarjun B , Venna Venkateshwara Reddy,
Bijily B, Dhanya V V, Snigdha Mishra and to my classmates of their moral support and advice.
I would like to thanks my parents and other family members for their support, love and
affection. So many people have contributed to my thesis, to my education, and to my life, and
it is with great pleasure to take the opportunity to thank them. I apologize, if I have forgotten
anyone.
Suji P
Roll No:210CE2027
M.Tech. (Structural Engineering)
National Institute of Technology
Rourkela -769008, Orissa, India
i
ABSTRACT
ABSTRACT
Wavelet are used in data compression, signal analysis and image processing and also for
analyzing non stationary time series. Wavelets are the functions which satisfy certain
mathematics requirements and are used in other functions. The use of wavelets in mechanics can
be viewed from two perspectives, first the analysis of mechanical response for extraction of
modal parameters, damage measures, de-noising etc and second the solution of the differential
equations governing the mechanical system. Wavelet theory provides various basis functions and
multi-resolution methods for finite element method. Wavelet-based beam element can be
constructed by using Daubechies scaling functions as an interpolating function. In the present
thesis, the compactly supported Daubechies wavelet based numerical solution of boundary value
problem has been presented for the instability analysis of prismatic members. This problem can
be discretized by the Wavelet-Galerkin method. The evaluation of connection coefficients plays
an important role in applying wavelet galerkin method to solve partial differential equations. The
buckling problem of axially compressed bars by using Wavelet-Galerkin method is explained in
this thesis. The comparisons are made with analytical solutions and with finite element results.
The present investigation indicated that wavelet technique provides a powerful alternative to the
finite element method.
The thesis has been presented in six number of chapters. Chapter 1 deals with the general
introduction to wavelets and different types of wavelet families and their properties. The review
of literature confining to the scope of study has been presented in chapter 2. The Chapter 3
deals with the properties of Daubechies wavelets and determination of scaling function, wavelet
function, filter coefficients and moments of scaling function for different order of Daubechies
ii
ABSTRACT
wavelets. The computation of connection coefficients are described in Chapter 4. Chapter 5
deals with the Wavelet Galerkin method and the buckling problem of prismatic bar by using
wavelet Galerkin method. It also includes the numerical results obtained from the present work,
and comparisons made with analytical solutions and with finite element results. The Chapter6
concludes the present investigation. An account of possible scope of extended study has been
presented to the concluding remarks. At last, some important publications and books referred
during the present investigation have been listed in Reference section.
KEYWORDS: Introduction to wavelet theory, Daubechies wavelets scaling function,
connection coefficients, wavelet galerkin method, buckling of bars.
iii
CONTENTS
Title
Page No.
ACKNOWLEDGEMENTS .......................................................................................... i
ABSTRACT ................................................................................................................. ii
TABLE OF CONTENTS. ........................................................................................... iii
LIST OF TABLES ..................................................................................................... vii
LIST OF FIGURES ................................................................................................. .viii
LIST OF SYMBOLS ....................................................................................................x
CHAPTER 1
INTRODUCTION
1.1.
Overview ...........................................................................................................1
1.2.
Introduction to wavelets ....................................................................................1
1.3.
Wavelet Transform ...........................................................................................2
1.3.1. Discrete Wavelet Transform .............................................................................2
1.3.2. Continuous Wavelet Transform. .......................................................................3
1.3.3. Wavelet Transform Versus Fourier Transform. ................................................4
1.3.4. Similarities between Fourier and Wavelet transform .......................................4
1.3.5. Dissimilarities between Fourier and Wavelet transform. .................................4
1.4.
Types of Wavelets.............................................................................................5
1.4.1. Haar Wavelets. ..................................................................................................6
1.4.2. Daubechies Wavelets. .......................................................................................7
1.4.3. Coiflets wavelets. ............................................................................................10
1.4.4. Symlets Wavelets. ...........................................................................................11
iv
1.4.5. Morlets Wavelets. ...........................................................................................11
1.4.6. Mexican Hat Wavelets. ...................................................................................12
1.4.7. Biorthogonal Wavelets....................................................................................13
1.5.
Introduction to Wavelet Galerkin Method ......................................................14
CHAPTER 2
REVIEW OF LITERATURE
2.1.
Introduction ....................................................................................................15
2.2.
Review of Literature on Wavelet Methods .....................................................15
2.3.
Review of Literature on Wavelet Galerkin Method .......................................18
2.3.1. Review of Literature on connection coefficient ..............................................21
CHAPTER 3
PROPERTIES OF WAVELETS
3.1.
Introduction .....................................................................................................22
3.2.
Daubechies Compactly supported Wavelets ...................................................22
3.3.
Wavelet Function ............................................................................................22
3.4.
Scaling Function .............................................................................................25
3.4.1. Filter Coefficients ...........................................................................................27
3.4.2. Derivatives of Scaling function. .....................................................................29
3.4.3. Integrals of Scaling Function. .........................................................................31
3.4.4. Moments of scaling Function..........................................................................34
3.4.5. Interpolation of Scaling Function. ..................................................................34
v
CHAPTER 4
CONNECTION COEFFICIENTS
4.1.
Introduction .....................................................................................................36
4.2.
Connection coefficients ..................................................................................36
4.3.
Determination of Connection coefficients ......................................................37
4.3.1. Two- Term connection Coefficients. ..............................................................37
4.3.2. Three – Term connection coefficients. ...........................................................42
CHAPTER 5
NUMERICAL EXAMPLE
5.1.
Introduction ....................................................................................................45
5.2.
Critical Load ...................................................................................................45
5.3.
Wavelet Galerkin method. ..............................................................................46
5.4.
Exact Method. .................................................................................................47
5.5.
Numerical example. ........................................................................................50
5.5.1. Clamped supported condition. ........................................................................51
5.5.2. Simply supported condition. ...........................................................................54
5.6.
Result and Discussion. ....................................................................................56
CHAPTER 6
CONCLUSION
6.1.
Conclusion. .....................................................................................................59
6.2.
Scope of Future work. .....................................................................................60
REFERENCES. ..........................................................................................................61
vi
LIST OF TABLES
Table 1.1: Values for Nr and Nd for Biorthogonal wavelets ......................................14
Table 3.1: Daubechies wavelet coefficients ................................................................23
Table 3.2: Daubechies wavelet filter coefficients .......................................................27
Table 3.3: Scaling function. .......................................................................................29
Table 3.4: Derivatives and Integrals of scaling function. ...........................................34
Table 4.1: Values of Connection coefficients. ............................................................42
Table 5.1.Values of Buckling load .............................................................................56
Table 5.2: Daubechies filter coefficient for D6. .........................................................57
vii
LIST OF FIGURES
Title
Page No
Figure1.1: Wavelets ......................................................................................................2
Figure1.2:Mallat forward discrete transform ................................................................3
Figure1.3:Windowed Fourier transform .......................................................................5
Figure 1.4:Haar wavelet scaling function .....................................................................6
Figure 1.5: Haar wavelet function ................................................................................7
Figure 1.6:Daubechies wavelets for 6 iteration ............................................................8
Figure 1.7:Different types Daubechies wavelet family ................................................9
Figure 1.8:Coiflets wavelets with order 2 ...................................................................10
Figure 1.9Coiflets wavelets with order 5 ....................................................................11
Figure 1.10:Morlet wavelet function ..........................................................................12
Figure 1.11:Mexican hat wavelet function .................................................................13
Figure 3.1: Wavelet function for D4,D6,D8,D10,D12,D22 .......................................24
Figure 3.2:Scaling function for D4,D6,D8,D10,D12,D22 ..........................................28
Figure 3.3:First derivative of scaling function for D6 ................................................31
Figure 3.4: Second derivative of scaling function ......................................................31
Figure 3.5: Interpolate IN6 scaling function ...............................................................35
Figure 3.6: Interpolate IN8 scaling function ...............................................................35
viii
Figure 5.1: Effective length for different support .......................................................50
Figure 5.2: percentage of error for different level of resolution .................................57
Figure 5.3: Scaling function (db3) ..............................................................................58
Figure 5.4: Derivative of scaling function ..................................................................58
ix
LIST OF SYMBOLS
The principle symbols in this thesis are presented for easy reference. A single symbol is used
different meaning depending on the contest and defined in the text as they occur.
English
ak
Filter coefficients
CWT
Continuous wavelet transform
DWT
Discrete wavelet transform
db
Daubechies wavelet
M
Vanishing moments
N
Order of Wavelets
J
Level of resolution
P
Compressive load
E
Modulus of elasticity
I
Moment of inertia
mk
Moments of scaling function
W
Middle point deflection
L
Length of bar
∆
Two-term connection coefficients
Ω
Three –term connection coefficients
δ
Kronecker delta function
Ψ
Wavelet function
φ
Scaling function
γ
Integral of scaling function
x
CHAPTER 1
INTRODUCTION
INTRODUCTION
1.1.
OVERVIEW
In recent years, the buckling analysis of structures is carried out by using finite difference and
finite element method. Here, we explain buckling analysis of structures by the wavelet Galerkin method. The Wavelet Galerkin method is used for solving partial differential
equations and differential equations. The connection coefficients play an important role in
applying Wavelet-Galerkin method. Daubechies wavelets have been successfully used in as a
base function in wavelet galerkin method, due to their compact support, orthogonality and
multiresolution properties.
1.2.
INTRODUCTION TO WAVELETS
The word wavelet is derived from a French word ondelette means ‘small wave’. Wavelet
analysis was developed in the mathematical literature in 1980. Wavelet can be used in data
compression, signal analysis and image processing. Wavelets are also used for analyzing nonstationary time series. Wavelets are those functions which satisfy certain mathematical
requirements and are used in other functions. The use of wavelets in mechanics can be
viewed from two perspectives; first, the analysis of mechanical response for extraction of
modal parameters, damage measures, de-noising etc, and second, the solution of the
differential equations governing the mechanical system.
The fundamental idea about
wavelets is to analyze according to scale. The wavelet based numerical solution has recently
developed the theory and application of partial differential equations. Wavelet analysis is a
numerical concept which allows representing a function in terms of basic functions, called
wavelets, which are localized both in location and scale. Among the wavelet approximation,
the Wavelet -Galerkin technique is most frequently used scheme now a day. Wavelet theory
provides various basis functions and multi-resolution methods. In finite element method,
1
INTRODUCTION
wavelet-based
based beam element can be constructed by using Daubechies scaling functions as an
interpolating
terpolating function. Since the nodal lateral displacements and rotations are used as element
degrees of freedom, the connection between neighboring elements and boundary conditions
can be processed simply as done for traditional elements.
FIGURE(1.1): Wavelets
1.3.
WAVELET TRANSFORM
According to Daubechies, wavelet transform can be defined as, a tool that cuts up data,
functions or operators in to different frequency
frequency components, and then studied each resolution
component with a resolution matched to its scale. Wavelet transform are transform which
provides a time frequency representation of the signal. The wavelet transform is developed to
overcome the short- coming of the short Fourier transform. Wavelet transform is also used
for multiresolution technique.
2
INTRODUCTION
1) Discrete wavelet transforms (DWT)
2) Continuous wavelet transforms (CWT)
1.3.1. Discrete Wavelet Transforms (DWT)
DWT is a Multiresolution representation of a finite length discretized signal from fine to
coarse scale. It is impossible to analyze a signal using all wavelet coefficients, so one may
wonder if it was sufficient to extract a discrete set of the upper half plane to be enough to
reconstruct a signal from the wavelet coefficients. Discrete wavelet transform has been
performed using a fast algorithm referred to a as Mallat transform[25]. Figure (1.2) shows
Mallat forward transform. The discrete wavelets transform which convert discrete signals to
discrete wavelet representation.
a0
⤓2
F[n]
d1[n]
⤓2
a0
c0
⤓2
⤓2
c0
d2[n]
a0
⤓2
c0
⤓2
d3[n]
Figure(1.2):Mallat Forward Discrete Wavelet Transform
1.3.2. Continuous Wavelet Transform (CWT)
The Continuous Wavelet Transform (CWT) is as given in equation (1.1),
X
,
where,
x(t)
=
1
|s|
x t Ψ
t−τ
dt
s
1.1
= signal
3
INTRODUCTION
Ψ(t) = mother wavelet or wavelet function
= translation parameter
s
= scale parameter
CWT is obtained by using computers sampling the time-scale plane. All the wavelet function
used in the transformation are derived from mother wavelet by translation and dilation.
1.3.3. Wavelet Transforms versus Fourier Transforms
1.3.3.1.Similarities between Fourier and Wavelet transform
The fast Fourier transform (FFT) and the discrete wavelet transform (DWT) are both linear
operations, they generated a data structure that contains ‘log2 n’ segments of several lengths,
usually transforming it into a different data vector of length 2n. The inverse transform for
both the FFT and the DWT matrices are the transpose of the original. As a result, both
transforms can be represented as a rotation function space to a different domain. For the FFT,
contains basis functions that are sines and cosines. For the wavelet transform, contains more
complicated basis functions called wavelets, or analyzing wavelets, or mother wavelets.
1.3.3.2.Dissimilarities between Fourier and Wavelet Transforms
The most interesting dissimilarity between FFT and wavelet transforms are individual
wavelet functions are localized in space. In wavelet transform have number of useful
applications such as data compression, and removing noise from time series. The differences
between the Fourier transform and the wavelet transform in the time-frequency resolution is
to look at the basis function of the time-frequency plane . Figure (1.3) presented a windowed
Fourier transform, where the window was simply a square wave. The square wave window
truncated the sine or cosine function to a window of a particular length,because a single
window is used for all frequencies in the wavelet Fourier transform, the resolution of the
analysis was the same at all locations in the time-frequency plane.
4
INTRODUCTION
Figure (1.3):Windowed Fourier Transform
1.4.
TYPES OF WAVELETS
The followings are the different types of wavelets.
•
Haar wavelet
•
Daubechies wavelet
•
Coiflets wavelet
•
Symlets wavelets
•
Biorthogonal wavelets
•
Morlet wavelets
•
Mexian hat wavelet
1.4.1. Haar Wavelet
The first order Daubechies wavelets are known as Haar wavelet. Haar wavelet is the simplest
and oldest wavelets for all wavelets. It has been used in various mathematical fields. The
Brownian motion can be defined by Haar wavelet. Haar wavelet is the dilation and translation
of wavelet function. Any discussion of wavelet begins with Haar wavelet, they are the first
5
INTRODUCTION
and simplest wavelets. Haar wavelet is discontinuous and used as a step function. It
represents the same wavelet Daubechies db1. Figure (1.4) and figure (1.5) represence the
scaling function and wavelet function for Haar wavelets respectively.
1.4
1.2
scaling function
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
grid points[0, 2M-1]
Figure(1.4): Haar Wavelet Scaling Function
6
1
1.2
1.4
INTRODUCTION
1.5
1
psi
0.5
0
-0.5
-1
-1.5
0
0.2
0.4
0.6
0.8
grid points [0,2M-1]
1
1.2
1.4
Figure(1.5): Haar Wavelet Function
1.4.2. Daubechies Wavelet
The family of this type of wavelet constructed by Daubechies includes members from
highly localized to highly smooth. The orthonormal basis of compactly supported wavelet
is generated by dilation and translation of wavelet function. The regularity order increases
linearly with the support condition. All the examples of compactly supported wavelet
bases are conspicuously non symmetric in contrast to infinitely support wavelet bases.
Compactly supported wavelets have been successfully applied in numerical simulation.
Daubechies proposed orthogonal compactly supported wavelets are known as Daubechies
wavelet. The first order Daubechies wavelet becomes the well known Haar wavelet.
Daubechies orthogonal wavelets db1 to db20 are commonly used. Figure (1.7) shows
different types of Daubechies wavelet family. The higher order Daubechies function are
7
INTRODUCTION
not easy to describe with an analytical expression. The order of Daubechies functions are
denoted by the vanishing moments. The larger the number of vanishing moments, better
the frequency localization of the decomposition.. Daubechies wavelets are commonly
used in solving a broad range of problems the order of the wavelet function can be
compared to the order of a linear filter. Figure (1.6) shows Daubechies wavelet function
for 6 iteration.
2
1.5
1
0.5
0
-0.5
-1
-1.5
0
1
2
3
4
Figure(1.6): Daubechies wavelets for 6 iteration
8
5
6
7
2
2
1.5
1.5
1
1
wavelet function
wavelet function
INTRODUCTION
0.5
0
-0.5
-1
1
2
grid points[0 2M-1]
(a)
1
2
3
grid points[0 2M-1]
(b)
4
5
1.5
1
wavelet function
1
wavelet function
-0.5
-1.5
0
3
1.5
0.5
0
-0.5
0.5
0
-0.5
-1
0
2
4
grid points[0 2M-1]
(c)
-1.5
6
0
2
4
6
grid poits [0 2M-1]
(d)
8
1
1.5
1
0.5
wavelet function
wavelet function
0
-1
-1.5
0
-1
0.5
0.5
0
-0.5
0
-0.5
-1
-1.5
-1
0
2
4
6
8
grid points[0 2M-1]
(e)
10
12
0
5
10
15
20
grid points[0 2M-1]
(f)
Figure(1.7): Different Types of Daubechies Wavelets Family
9
25
INTRODUCTION
1.4.3. Coiflets Wavelet
Coiflets are discrete wavelets designed by Daubechies. Coiflet wavelets are near symmetric,
their wavelet function
3, vanishing moments and scaling function
scaling function and wavelet function can be normalized by a factor 1
3 − 1. Both the
. Coiflet wavelets
√2
are orthogonal compactly supported wavelets with the highest number of vanishing moments
for wavelet function for given support width. The Coiflet wavelets are more symmetric and
have more vanishing moments than Daubechies wavelets. The Coiflet wavelet function with
order 2 and 5 shows respectively figure(1.8) and figure (1.9).
coif 2 wavelet
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
0
2
4
6
Figure(1.8): Coiflet wavelet function with order 2
10
8
10
12
INTRODUCTION
coif 5 wavelet
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
0
5
10
15
20
25
30
Figure(1.9): Coiflet wavelet function with order 5
1.4.4. Symlets wavelets
The properties of symlets wavelets are near symmetric, orthogonal and biorthogonal. Symlets
wavelet families are Symlets 2 to Symlets 20. Symlets wavelets’ are commonly used for
removal of Gaussian additive noise from speech signal. Symlets wavelets are proposed by
Daubechies as modification to the Daubechies family. The properties of the two wavelet
families are similar.
1.4.5. Morlet wavelets
The morlet wavelet is defined as:
#
= ! " $%& 5
1.2
11
INTRODUCTION
The constant C was used for normalization in view of reconstruction. The support width of
this wavelets is infinite. The effective width of Morlet wavelets is [-4,4]. The morlet wavelet
functions are shown in figure (1.10)
Morlet wavelet
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-4
-3
-2
-1
0
1
2
3
4
Figure(1.10): Morlet Wavelet Function
1.4.6. Mexican hat wavelet.
Mexican hat wavelet is proportional to the second derivative function of the Gaussian
probability density functions. The wavelet is a special case of a larger family of derivative of
Gaussian wavelets. There is no scaling function associated with this type of wavelet. The
support width of these wavelets is infinite. The effective width of Mexican hat wavelets is [5,5]. The Mexican hat wavelet shown in figure (1.11). The Mexican hat wavelet can be
defined as:
12
INTRODUCTION
=
2
√3
(
)*
+
1−
,
!
)" #
1.3
,
mexican hat wavelet
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-8
-6
-4
-2
0
2
4
6
8
.
Figure(1.11): Mexican Hat Wavelet
1.4.7. Biorthogonal wavelets
Biorthogonal family of wavelets exhibits the property of linear phase, which is used for
signal and image reconstruction. Two types of wavelets are used; one for decomposition (on
the left side) and the other for reconstruction (on the right side) instead of the same single
one, interesting properties are derived. He order of the these wavelets are Nr and Nd. Nr for
reconstruction and Nd for decomposition. The possible values for Nr and Nd are shown in
table (1.1).
13
INTRODUCTION
Table (1.1): Values of Nr And Nd for Biorthogonal Wavelets
1.5.
Nr
Nd
1
1 , 3 or 5
2
2,4,6,or 8
3
1,3,5,7,or 9
4
4
5
5
6
8
INTRODUCTION TO WAVELET GALERKIN METHOD
The Wavelet Galerkin method is a powerful tool for solving partial differential
equations.Wavelet Galerkin technique is the most frequently used in these days. Daubechies
wavelet as bases in a Galerkin method to solve differential equations requires a
computational domain of simple shape. The wavelet method has been shown to be a powerful
numerical tool for the fast and accurate solution of differential equations.The connection
coefficients plays and important role in applying wavelet galerkin method. Daubechies
wavelets have been successfully used as a base function in Wavelet Galerkin method, due to
their compact support, orthogonality and Multiresolution properties. The connection
coefficients are employed in the calculation of stiffness, mass and geometry matrices. The
one of the most possible applications of wavelet theory is numerical solution of differential
equations. The Wavelet Galerkin method is applied to solve differential equations for
structures like beam, column, plates etc. wavelet galerkin method require different types of
connection coefficients for different differential or integral equations.
14
CHAPTER 2
REVIEW OF LITERATURE
REVIEW OF LITERATURE
2.1.
INTRODUCTION
This chapter reviews the most relevant literatures on the wavelet Galerkin finite element
method. Over the last two decades wavelets are being effectively used for signal processing
and solution of differential equations. Wavelets have several properties which are
encouraging their use for numerical solutions of partial differential equations (PDEs). The
Wavelet-Galerkin method is a powerful tool for solving partial differential equations. The
orthogonal, compactly supported wavelet basis of Daubechies exactly approximates
polynomial of increasingly higher order. These wavelet bases can provide an accurate and
stable representation of differential operations even in region of strong gradients or
oscillations. In addition, the orthogonal wavelet bases have the inherent advantage of multi
resolution analysis over the traditional methods. Some of the pertinent studies done recently
are reviewed elaborately and critically discussed to identify the lacunae in the existing
literature. The studies in this chapter are grouped into two major parts as follows
2.2.
•
Review of literature on wavelets method
•
Review of literature on wavelet-Galerkin method
REVIEW OF LITERATURE ON WAVELETS METHOD
Wavelets are mathematical function that cut data into different frequency component.
Wavelets methods are more advantageous than Fourier methods. Wavelets are developed
independently in the field of mathematics, quantum physics etc.
Chui [9] presented the history of wavelets beginning with Fourier, compare wavelet
transform with Fourier transform, state properties and other special aspects of wavelets and
finishing with some interesting applications. This book gives some similarities and difference
between wavelet transform and fourier transform
15
REVIEW OF LITERATURE
Daubechies [10] studied the construction of orthonormal bases of compactly supported
wavelets with arbitrarily very high regularity. The order of regularity increases linearly with
the support width. They start by reviewing the concept of Multiresolution analysis as well as
several algorithms in vision decomposition and reconstruction. Then the construction follows
from a synthesis of different approaches.
The exact and explicit representation of differential operators, in orthonormal bases of
compactly support wavelets, is explained in Beylkin [5]. The method of computing this
representation is directly applicable to multidimensional convolution operators. This
describes a brief idea about wavelets, Hilbert transform, pseudo differential, shift operators,
fractional derivatives and also describes numerical algorithms for finding connection
coefficients.
Amaratunga and Williams [2] developed the use of wavelet basis function in solving partial
differential equations. Wavelet theory provides various basis functions and multi-resolution
methods for finite element method. In this paper, a wavelet-based beam element is
constructed by using Daubechies scaling functions as an interpolating function. Since the
nodal lateral displacements and rotations are used as element degrees of freedom, the
connection between neighboring elements and boundary conditions can be processed simply
as done for traditional elements.
Goedecker [35] presented the wavelets and their applications of partial differential equation
in various field. This article is intended to make the theory of wavelets understandable to the
audience. In addition to a self-contained and intuitive presentation of the theory of wavelets,
extensive tables with the basic filter coefficients of differential operators in several wavelet
families can be found in this article.
16
REVIEW OF LITERATURE
Kozbial [36] presented a new wavelet-based approach
for solving two-dimensional
boundary-value mechanical problems on the example of plate bending. The deflection
equation of a bending plate is approximated by two-dimensional Daubechies wavelets using a
least-squares Galerkin method. Due to the order of the differential equation in mechanics of
plate structures is four, a way to perform the calculations of high order connection
coefficients (that is, integrals of products of basis functions with their high order derivatives)
is suggested. The implementation of two-dimensional Daubechies scaling functions
approximation to plate bending is exhibited numerically in some examples. The results show
that this method has good precision and reliability.
Amaratunga and Williams [3], have described how wavelets may be used for the temporal
discretization of ordinary differential equation and partial differential equations. The first
problem associated with the use of wavelets in is that initial conditions are difficult to
impose. They are developed a method for solving initial value problem using wavelet
extrapolation and they are demstrated the usage of the method for a linear mode problem.
Wave propagation deal with loadings that have very high frequency content. Hence the
element size has to be comparable to wavelengths, which are very small at high frequencies.
Fourier methods for frequency domain used for solving this type of problem, which can in
principle achieve high accuracy in numerical differentiation. One such method is spectral
finite element method (SFEM) developed by Doyle [15].
Mira and Gopalakrishnan [27] studied the spectral finite element modeling. Exact shape
functions are derived and finite element procedure is followed in the transformed frequency
domain. Here spectral finite element formulation is done using Daubechies scaling function
bases for temporal approximation. In comparison to the conventional Fourier transform based
spectral finite element method, the use of localized bases functions in the Daubechies scaling
17
REVIEW OF LITERATURE
function based spectral finite element method allows accurate wave propagation analysis of
finite length structures. First, numerical experiments are performed with narrow banded
modulated pulse loading to obtain the location of damage from wave arrival time. Next, a
broad banded impulse load is considered and effects of parameters like damage width, depth,
and location on the responses are studied in time and frequency domains. WSFE is
formulated for a notched Euler–Bernoulli beam. The effect of the notch is introduced in the
formulation using perturbation technique. It should be mentioned here that as the analysis is
restricted to damages with much smaller dimension, compared to that of the beam, the mode
conversion resulting in axial and flexural coupling is not considered in the formulation. The
present method is developed for pure flexural wave propagation and the effect of axial
coupling due to the presence of the notch is neglected.
2.3.
REVIEW OF LITERATURE ON WAVELET-GALERKIN METHOD
Wavelet Galerkin method is a powerful tool for solving partial differential equation. Wavelet
galerkin method is the one of the best method for finding numerical solution of partial
differential equation. The wavelet-Galerkin method providing an improvement over other
methods by using compactly support wavelets.
Deka and Choudary [13] studied some special type of integrals of Daubechies wavelets,
which are used as galerkin basis function for numerical solution of partial differential
equation. Numerical and theoretical results are obtained for elliptical problem for second
order with different types of boundary conditions. Instead of scaling functions, wavelet
functions can also be used for this problem. Finally compare this result with finite difference
method shows that the wavelet method is right competator of the classical methods.
Dahmen and Micchelli [12] developed the Wavelet Galerkin method used for solving partial
differential equations leads to problem of computing integrals of products of derivatives of
18
REVIEW OF LITERATURE
wavelets. In this paper studies the problem from point of view of stationary subdivision
scheme. One of the main results is to identify these integrals as components of the unique
solution of a certain eigenvector moment problem associated with the coefficient of refined
equation.
Amaratunga and Williams [1] studied review of application of wavelet into the solution of
partial differential equation. The theory of wavelets is described here using the language and
mathematics of signal processing. This paper shows a method of adapting wavelets to an
interval using an extrapolation technique called wavelet extrapolation. Wavelet extrapolation
can be regarded as a solution to the problem of wavelets on an interval. In wavelet
extrapolation technique a polynomial of p-1, p generally taken as the vanishing moments are
assumed to be extrapolate the values at the boundaries.
Chen and Hawang [7] have described an exact evaluation of various finite integrals whose
integrands involves product of Daubechies compactly support wavelets and their derivatives
and integrals. These finite integral plays an important role in the wavelet-Galerkin
approximation of differential or integral equations on bounded interval.
Besora [21] developed Galerkin wavelet method for global waves in 1D. Galerkin wavelet
method for solving partial differential equations has been tested for two particular cases. The
results of the first case, a harmonic wave equation. The results of the second case, a
biharmonic wave equation, show that a more accurate method for the solution of wavelet
overlap integrals, involving more than two basis functions are required.
Jin and chang [20] presented wavelet function apply to the numerical solution of differential
equation. Antiderivatives of wavelets are used for numerical solution for differential
equation. The orthogonal property of the wavelet is used to construct efficient interactive
methods for the solution of the resultant linear algebraic system. And also give some
19
REVIEW OF LITERATURE
numerical examples. Optimal error estimates are got in the application of two point boundary
value problem of second order differential equation.
Jin and Ye [16] presented the compactly supported wavelet- based numerical solution of
boundary value problem This problem can be discretized by the wavelet-Galerkin method.
The evaluation of connection coefficients plays an important role in applying wavelet
galerkin method to solve partial differential equations.
Santos and Burgos [33] presented the use of compactly support wavelet functions has become
increasingly popular in the development of numerical solution of partial differential
equations, especially for problems with local high gradient. He presented the formulation and
validation of the Wavelet Galerkin method using Deslauriers-Dubuc interpolates. It was also
shown that the wavelets have the ability of capturing the discontinuities without the need to
place nodes where they occur.
Vinod and Sabina [37] presented the wavelet techniques and apply Galerkin procedure to
analyze one dimensional harmonic wave equation as a test problem using fictitious boundary
approach. Solutions of differential equation obtained using the Daubechies 6, 8 and 12
coefficients wavelets have been compared with the exact solution.
2.2.1. Review of Literature on Connection Coefficients
The connection coefficient is an integral of products of wavelets basis function their
derivative and translation. Latto, Resnikoff and Tenenbaum [23], have described an exact
20
REVIEW OF LITERATURE
method for evaluating connection coefficients. It is essential for the application of wavelets to
the numerical solution of partial differential equations.
Romie and Peyton [32], have shown proper connection coefficients for compactly support
wavelets. Proper connection coefficients are important for the solution of non periodic partial
differential equation. They have demonstrated a technique for deriving a linear system whose
solution in the set of proper connection coefficients needed to compute the natural inner
product on bounded interval. They exhibit a simple one dimensional test problem that
illustrated the use of proper connection coefficients for partial differential equations on
bounded domains with dirchilet boundary condition.
Popovici [28], dealt with solutions to partial differential equations or ordinary differential
equations using the wavelet galerkin method. For finding the connection coefficients several
algorithms have been worked out and MATLAB programs. In those papers they described the
MATLAB evaluation of connection coefficient.
Latto [23] described a connection coefficient is an integral of products of wavelet basis
function, their derivatives and translates and also give the exact method for calculating the
connection coefficients. This is essential for the application of wavelet to the numerical
solution of partial differential equations. This gives the solutions of three term and two term
connection coefficients. A Fortran code written by Restrepo [31] exists to compute 2-term
and 3-term connection coefficients using periodized wavelets.
21
CHAPTER 3
PROPETIES OF WAVELETS
PROPERTIES OF WAVELETS
3.1
INTRODUCTION
The discrete wavelet transforms are developed as a special family of wavelet
functions.This type of wavelets are compactly supported, orthogonal or biorthogonal
and are characterized by low pass and high pass analysisand synthesis filters.. The
orthonormal basis of compactly supported wavelet is generated by dilation and
translation of wavelet function. The complete basis of wavelet can be formed through
dilation and translation of mother wavelet scaling function. The wavelet consists of
two functions, the scaling function and wavelet function. The scaling function
describes the low pass filter for the wavelet transform and the wavelet function
describes the band pass filter for the wavelet transform.
3.2
DAUBECHIES COMPACTLY SUPPORTED WAVELET
The family of this type of wavelet constructed by Daubechies includes members from
highly localized to highly smooth.The orthonormal basis of compactly supported
wavelet is generated by dilation and translation of wavelet function.The regularity
order increases linearly with the support condition. All the examples of compactly
supported wavelet bases are conspicuously non symmetric in contrast to infinitely
support wavelet bases. Compactly supported wavelets have been successfully applied
in numerical simulation.The higher order Daubechies function are not easy to describe
with an analytical expression. The order of Daubechies functions are denoted by the
vanishing moments.
3.3
WAVELET FUNCTION
Thewavelet function describes the band pass filter for the wavelet transform. The
wavelet function is also called mother wavelets. The wavelet function can be defined
as
22
PROPERTIES OF WAVELETS
=
−1
2 −
3.1
The fundamental support of the scaling function
is in the interval [0. N-1]
Figure (3.1) shows different families of Daubechies wavelets. The wavelet
coefficients are calculated by reversing the order of the filter coefficients or
coefficients of scaling function and reversing the sign of every second one.
Mathematically, wavelet coefficients are expressed as:
= −1
3.2
−1−
The Daubechies wavelet coefficients are presented in table (3.1)
Table 3.1 :Daubechies Wavelet coefficients
for N= 4, 6,8,10
K
N=4
N=6
N=8
N=10
0
-0.18301
0.0498
-0.0150
0.0047
1
-0.31698
0.1208
-0.0465
0.0178
2
1.18301
-0.1909
0.0436
-0.0088
3
-0.683012
-0.6504
0.2645
-0.1097
4
1.1411
-0.0396
-0.0456
5
-0.4705
-0.8922
0.3427
6
1.0109
0.1958
7
-0.3258
-1.0243
8
0.8539
9
-0.2264
23
2
2
1.5
1.5
1
1
wavelet function
wavelet function
PROPERTIES OF WAVELETS
0.5
0
-0.5
-1
1
2
grid points[0 2M-1]
(a)
1
2
3
grid points[0 2M-1]
(b)
4
5
1.5
1
wavelet function
1
wavelet function
-0.5
-1.5
0
3
1.5
0.5
0
-0.5
0.5
0
-0.5
-1
0
2
4
grid points[0 2M-1]
(c)
-1.5
6
0
2
4
6
grid poits [0 2M-1]
(d)
8
1
1.5
1
0.5
wavelet function
wavelet function
0
-1
-1.5
0
-1
0.5
0.5
0
-0.5
0
-0.5
-1
-1.5
-1
0
2
4
6
8
grid points[0 2M-1]
(e)
10
12
0
5
10
15
20
grid points[0 2M-1]
(f)
Figure 3.1: Wavelet function for different families of Daubechies wavelet:
24
25
PROPERTIES OF WAVELETS
3.4
SCALING FUNCTION
The scaling function describes the low pass filter for the wavelet transform. In the
following expression, known as the two-scale relation, ak are the filter coefficient of
the wavelet scale function
=
and N is the wavelet order.
2 −
3.3
The fundamental support of the scaling function
coefficients
is in the interval [0. N-1] .the
in the above relation (3.3) is called wavelet filter coefficients.
The scaling functions have the following properties
=1
3.4
−
−
ψ
=
3.5
,"
=0
3.6
The scaling functions are obtained by solving recursively the dilation equation(3.3)
which can be expanded as
=
0 =
1 =
2
0
+
2 +
2 − 1 + ⋯+
1 +
2 −
0
+1
3.7
3.8
3.9
25
PROPERTIES OF WAVELETS
2 =
4 +
−2 =
3 +
2 +
−1 +
,
−1 =
,
1 +
-
0
−2 +
3.10
−3
3.11)
−1
3.12
This can also be written as a matrix form
0
0
/
/ /⋯
/0
/0
.0
,
⋯
0
0
0
0
⋯
0
0
0
⋯
⋯
⋯
⋯
⋯
⋯
⋯
0
0
0
⋯
0
,
0
0
0
⋯
0
-
0
0
0
⋯
40
3/
3/
3 //
13
/
,3 /
2.
0
1
2
⋯
−3
−2
−1
4 0
3 /
3 /
3=/
3 /
3 /
3 /
2 .
0
1
2
⋯
−3
−2
−1
4
3
3
3
3
3
3
2
3.13
i.e,
5 =
3.14
Above equation indicates that the unknown vector ϕ is the eigenvector of the matrix
[A]. As in all eigenvalue problems the solution to the system is not unique, and so a
normalizing condition is required in order to determine a unique eigenvector.
The values of
ϕ(x) are known at the integer values of x between 0 to N-1 , the values at the
points in between the integers can be obtained from the equation(3.3) modified as
6 7=
2
−
3.15
Figure (3.2) shows Daubechies scaling function at for different orders. Scaling
function corresponding to N=6 presented in table (3.3).
26
PROPERTIES OF WAVELETS
3.4.1
Filter coefficients
The filter coefficients can be calculated by using MATLAB wavelet tool box function
dbwavf. The filter coefficients of order of Daubechies wavelet for 4, 6,8,10 are given
in table(3.2). The filter coefficients satisfy the following conditions:
=2
"
−1
3.16
=2
.
−1
8
3.17
,"
=0
9 = 0,1,2, … … ;2 − 1
"
=0
3.18
9 = 0,1,2, … … ;2 − 1
Table 3.2 :Daubechies Wavelet filter coefficients
3.19
for N= 4, 6,8,10
K
N=4
N=6
N=8
N=10
0
-0.18301
0.0498
-0.0150
0.0047
1
-0.31698
0.1208
-0.0465
0.0178
2
1.18301
-0.1909
0.0436
-0.0088
3
-0.683012
-0.6504
0.2645
-0.1097
4
1.1411
-0.0396
-0.0456
5
-0.4705
-0.8922
0.3427
6
1.0109
0.1958
7
-0.3258
-1.0243
8
0.8539
9
-0.2264
27
PROPERTIES OF WAVELETS
1
scaling function
scaling function
1
0.5
0
0
0
1
2
gridpoints[0 2M-1]
(a)
3
0
0.5
0
2
3
grid points[0 2M-1]
(b)
4
5
0.5
0
0
2
4
6
grid points [0 2M-1]
(d)
8
0
1
0.5
0.5
scalinf function
1
0
-0.5
1
1
scaling function
scaling function
1
scaling function
0.5
0
5
10
15
grid poits[0 2M-1]
(e)
0
5
10
15
grid poits[0 2M-1]
(f)
Figure3.2: scaling function for different order of Daubechies wavelets
28
6
0
-0.5
20
2
4
grid points[2M-1]
(c)
20
25
PROPERTIES OF WAVELETS
Table (3.3): Scaling Functions
x
3.4.2
0
0
0.5
0.6505178
1
1.2863351
1.5
0.4411248
2
-0.385836
2.5
-0.014970
3
0.0952675
3.5
-0.031541
4
0.004234
4.5
0.002109
5
0
Derivatives of Scaling Function
As there is no analytical expression for wavelets, derivatives are obtained in dyadic
grid points and the refinement of the solution depends on the level of resolution
needed. The scale relation can be differentiated n times, generating the following
expression:
<
<
=
=
3.20
<
Therefore,
=
3.21
29
PROPERTIES OF WAVELETS
Let
be the nth derivative of the scaling function, its denoted by:
= 2<
<
<
2 −
3.22
Where n=0, 1, 2,…..N/2-1
The above relation is the two scale relation for
<
find the values of
<
. This two scale relation can be used to
for all dyatic points. To find out the values of
<
at integer the
values of x are substituted from 1 to N-2 in equation (3.22. The homogeneous linear system
of equationsobtained as
2
<
Where
==
5==
=5
<
3.23
1
?
>
<
2
<
3 ………
−2 >
<
3.24
3.25
@?, @
In the equation(3.23) the unknown vector
corresponding to the eigenvalue2
<
is the eigenvector of the matrix [A], it
. For all eigenvalue problems, the solution to the
system was not unique. The values of
can be determined uniquely by first finding
the eigenvector of the matrix [A] corresponding to the eigenvalue 2
<
, and then
normalizing with the condition
−1
<
<
= A!
3.26
For n=0, equation (3.22) is similar to the two scale relation (3.3) for
. Table (3.4)
give the values of first, second and third order of scaling function corresponding to
N=6. Figure (3.3) and figure (3.4) shows the first and second derivatives of scaling
functions for N=6.
30
PROPERTIES OF WAVELETS
1.4
1.2
1
fig(a)
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
0
1
2
3
4
5
6
Figure3.3: first derivative of scaling functions for N=6
Figure3.4: second derivative of scaling functions for N=6
3.4.3
Evaluation of Integral of
o Scaling Function
as C
Define the integaral of scaling function,
denoted by C<
C<
. The n-tuple
tuple integrals of
, i.e,
E
C<
D
D
3.27
31
PROPERTIES OF WAVELETS
C<
=2
C
C< 2 −
<
=
C
D
=
D+
C D
3.28
E
D
E
D+
D=1
C D
E
= C F−1 +
= F−1 +
C,
=
C D
= C,
−
D+
E
C<
=
M
3.30
+1
=C
3.31
−1 + D−A+1 > D
−1 +C
x−N+1 J
γL
i!
The value of C
D
1 D
−1
From the above equation, we can C<
L
3.29
M
−
for
+1 +
≥
1
C
2!
3.32
−1
−
+1
− 1 as follows:
N−1
− 1 = 1, butthe values of γL
be still determined. For determine C
M
N − 1 for i= 0,1,2,….n-2 are to
− 1 for n=2,3,…, we go back to equation
(3.27)
C<
−1 =2
<
From equation(3.31), C<
C< 2 − 2 −
3.34
− 1 can be expressed as
32
PROPERTIES OF WAVELETS
C<
−1 =2
=2
<
L
<
M
P C<
−1
N−1−k J
γL
i!
+
M
N−1
<
3.35
−1−
Q!
?
?
C<
?
−1 R
From the relation(3.15) , we then have the recursive formula
C<
−1 =
1
2<
2
The values of C<
<
?
S
−1−
!
T C<
?
−1
3.36
for x =1,2,….N-2 and can be determined from the following
linear system of equations:
U−2
Where,
5==
<
5 C< = V
?
>
?
=
……
?
W
, ,…
3.38
@?, @
C< = =C< 1 C< 2
V==
3.37
… … C<
>
−2 >
3.39
3.40
C< 2Q −
3.41
The values of integral of scaling function for n=1 is given in table (3.4).
33
PROPERTIES OF WAVELETS
Table 3.4: Derivatives and integrals of scaling function for N=6
ϕ ( x)
x
3.4.4
r ( x)
ϕ ( x)
1
2
0
0
0
0
1
1.6385
0.9042
0.6007
2
-2.2328
-1.7127
1.0967
3
0.5502
0.7127
0.9854
4
0.0441
0.0958
0.9996
5
0
0
1
Moments of Scaling Function
The another important property of wavelet is calculating the moments of scaling
8
function. The moments of scaling function
function
8
denotes the kth moment of scaling
8
, with the initial condition
=
= 1 and expressed as
3.43
Applying the scaling relationship in the above equation, we get a recursive relation,
=
8
3.4.5
2
1
+1
−1
− 2 Q=0
=1
X Y
QQ
9
−
3.44
Interpolation of scaling function
The common characteristics of interpolating wavelets require that the scaling function
satisfies the following conditions:
=
.
1
,Z [
0
=0
3.45
Figure(3.5) and figure(3.6) shows the interpolate IN6 and IN8 respectively. All
expression used for calculation of scaling function, derivatives of scaling function,
34
PROPERTIES OF WAVELETS
connection coefficients and moments of Daubechies wavelet can be used to
interpolate. Due to correlation, the support [0 N-1] in the expression for Daubechies
becomes [1-N N-1] for interpolation.
1.200
1.000
fig(a)
0.800
0.600
0.400
Series1
0.200
0.000
-0.200
-3
-2
-1
0
1
2
3
Figure (3.5): Interpolet IN6 Scaling Function
1.200
fig(b)
1.000
0.800
0.600
0.400
Series1
0.200
0.000
-0.200
-5 -4 -3 -2 -1
0
1
2
3
4
5
Figure (3.6): Interpolet IN8 Scaling Function
35
CHAPTER 4
CONNECTION COEFFICIENTS
CONNECTION COEFFICIENTS
4.1
INTRODUCTION
A connection coefficient is an integral of products of wavelet basis function, their derivatives
and translates. Connection coefficients play an important role in wavelet galerkin method..
Connection coefficients are essential for application of wavelets to the numerical solution of
partial differential equations, since numerical approximations of connection coefficients are in
general unstable due to oscillatory nature of integrands. Here, we explain two term and three
term connection coefficients.
4.2
CONNECTION COEFFICIENTS
The Wavelet-Galerkin scheme involves the evaluation of connection coefficients, to approximate
derivatives as well as non-linear terms. The connection coefficients are the integrals with the
integrands being product of wavelet bases and their derivatives. Owing to derivatives of
compactly supported wavelet being highly oscillatory, it is difficult to compute the connection
coefficients by the numerical evaluation of integrals. In order to overcome this problem,
dedicated algorithms have been devised for the exact evaluation of connection coefficients. The
connection coefficients and associated computation algorithms are essentially based on
unbounded domain. Wavelet Galerkin method are limited to cases where the problem domain is
unbounded or the boundary condition is periodic.
36
CONNECTION COEFFICIENTS
4.3
DETERMINATION OF CONNECTION COEFFICIENTS
4.3.1
Determination of two-term connection coefficients
The connection coefficients can be defined as the integral product of the scaling function
and its nth derivative
∆
=
−
−
as
4.1
The determination of above connection coefficient plays an important role in applying the
wavelet-Galerkin method to solve differential equations. Substituting the equation (3.5) and
(3.25) to equation (4.1), we have
∆
=
2
2 −2 −
=2
2 −! "
2 −2 −
=2
2 −!
2 −2 −
%
=2
2 −!
#−2 − +!
=2
4.2
∆%
&
2 −!
#
#
2 − ! 4.3
Followings are the important properties of the two term connection coefficients:
(ℎ*+ | | ≥ - − 1,
(ℎ*+
≤ 0 23
(ℎ*+
≥0
≥-−1
≤
∆
∆
∆
= ∆ - − 1 4.4
=0
- − 1 = −1 ∆ - − 1 4.6
37
4.5
CONNECTION COEFFICIENTS
(ℎ*+
−
≥-−1
∆
- − 1 = −1 ∆ - − 1
4.7
Evaluation of ∆87 9 − :
To compute the values of ∆ - − 1 for k=0,1,2,…..,N-2.
From equation (3.48), we have,
∆ -−1
=2
=2
∆%
∆%
&
&
-−1
2- − 2 − !
4.8
For n>0, the equation substituting k= 0,1,2…..N-2 into equation(3.49), we obtain following
homogeneous linear system of equation.
2
∆ - − 1 = <∆ - − 1
where,
∆ - − 1 = >∆ - − 1 ∆ - − 1
< = >BC.D @
EC.DE
BC,D = 2
H
G
%
E, E
F%C&
D
+ −1
… ….
E, E
%C&
4.9
∆
D
%
- − 1 @A
K
J
I
4.10
4.11
4.12
In the formulation of equation (4.8) the property (4.4) to (4.7) has been used. From equation
(4.9) the unknown vector ∆ - − 1 is the eigen vector of the matrix [B] that corresponds to the
unity eigenvalue. However, the homogeneous linear system of equation does not admit the
38
CONNECTION COEFFICIENTS
unknown vector ∆ - − 1 to be determined uniquely. Therefore an additional normalizing
condition is required.
We have,
L
−
L
= +!
4.13
Multiplying both side of equation (3.58) by
L
L
L
L
L
= +!
−
and integrating from -∞ to +∞.
4.14
L
where,
L
L
L
+!
L
L
L
−
=
L
L
∆ -−1
4.15
= +!
4.16
We obtain the normalizing condition as,
%
∆ -−1 =
+!
2
4.17
The above equation (4.17) is desired normalizing condition for the eigenvector of the matrix [B]
associated with the unity eigenvalue for n>0
Evaluation of ∆87 N for x=1,2,3,…, N-2
Using the properties (3.49) to (3.52) that there are only (N-2)2 independent members.
39
CONNECTION COEFFICIENTS
Substituting k= x-N+2,…..x-1 and x=1,,3,….N-2 into equation (3.48), we obtain following linear
system of equation:
2
O−P ∆ =
where,
∆ = >∆ 1 ∆ 2 …
= >∆
∆
U
U
&% ∆
U,
T U%,
S ⋮
P=S ⋮
S
S ⋮
RU %,
U
= \U
]
,C
,C,D,
=>
=
1
,C,D,
%
C
U ,%
U%,%
⋮
⋮
⋮
%,%
ED, E
4.18
∆ - − 2 @A
&Q
⋯ ⋯ ⋯
U,
⋯ ⋯ ⋯
U%,
⋱ ⋱ ⋱
⋮
⋱ ⋱ ⋱
⋮
⋱ ⋱ ⋱
⋮
⋯ ⋯ ⋯ U %,
@A
%
%
4.20
[
Z
Z
Z
Z
%Y
4.21
4.22
%
% &D
2 …….
+- − 2+ + ^ =
…… ∆
4.19
,C∈a D, ,
The b ^, +, - is given by:
- − 2 % @A
C ∆%D& _C
b ^, +, - = c , d : - − 1 ≤ 2+ − 2^ −
4.23
-−1
23 - − 1 ≤ 2+ − d , 0 ≤ , d ≤ - − 1 f
4.24
4.25
4.26
For n=0, the values of ∆ can be obtained by solving the equation (4.18) directly. When n>0,
equation (4.18) is a linear dependent system and the corresponding rank deficiency is n. In order
to eliminate the singularity of equation (4.19) additional relations are required for the members
∆ .
40
CONNECTION COEFFICIENTS
L
−
L
= +!
4.27
Multiplying both side by
L
L
−
L
L
and integrating from y=0 to y=x:
∆
= +!
4.28
= +! g
4.29
When x =m the above equation can be written as :
D
D
&%
∆ ^ = +! g
−∇
4.30
Where,
∇= −1
%
E ED
&
∇
4.31
-−1
Table (4.1) presents the values of connection coefficients for N=6.
41
CONNECTION COEFFICIENTS
Table(4.1): The Values of Connection Coefficients for n=1 and N=6
X
1
2
3
4.3.2
k
Λ ( x)
x
k
Λ ( x)
-4
-3
-2
-1
0
0.00034
-0.00960
0.24682
-1.06420
0.82732
4
-4
-3
-2
-1
0
1
0.000342
0.014611
0.14376
-0.77113
0.074435
0.56789
-4
-3
-2
-1
0
1
2
3
0.00034
0.014611
-0.14539
0.745205
0.000008
0.745285
-0.14539
0.015053
-4
-3
-2
-1
0
1
2
0.00034
0.014611
-0.14539
-0.74488
0.00453
0.73437
-0.12428
-4
-3
-2
-1
0
1
2
3
4
-0.00034
-0.014611
0.14539
-0.745205
0
0.00034
0.014611
-0.14539
0.745205
n
k
5
n
k
Three-term connection coefficients
The integral is defined as:
i D,
,
=
D
−!
−
4.32
The three -term connection coefficients play an important role in wavelet galerkin method.
i D,
,
i D,
,
have the following properties:
=0
j23 |!|, | |, 23 |! − | ≥ - − 1
42
4.33
CONNECTION COEFFICIENTS
i D,
,
=0
i D,
,
j23
− !, −
= i D,
-−1
,
23
j23
≤0
− !, −
4. 34
23
≥-−1
The properties (4.33) and (4.34) comes from the fact that supports of
4.35
for i= 0,1,2,3,…. are
all in the interval [0, N-1]. The two scale relations are,
=
2 −
=2
4.36
2 −
4.37
Substituting two scale relation (4.36) and (4.37) into equation (4.32).
i D,
,
= 2D&
k
m
l
k
m l
i%D,&m
k,% &l k
2 −n
4.38
The values of i D,
- − 1 for integers j,k,m,n can be computed. For this purpose from the
,
properties (4.33) to (4.35), for fixed integers m and n, there are 3- % − 9- + 7 unknown
Ωr,s
p,q N − 1 . The equation (4.38) associated with these unknown coefficients to form a linear
homogeneous equation as follows:
u=2
D
vu
4.39
Where,
43
CONNECTION COEFFICIENTS
u = >u%
uQ
……….
A
%@
4.40
… … … Ωr,s
p,x N − 1 ]
r,s
u \Ωr,s
p,w N − 1 Ωp,w& N − 1
y = max ! + 2 − -, 2 − -
u
+ b = min ! + - − 2, - − 2
are sum of product of the form
k m l.
4.41
and the elements of matrix [S]
The matrix [S has the eigenvalue 2
, k=0,1,2,…N-2
with their multiplicity order k+1. In equation (4.39) the vector R is the eigenvector of matrix [S]
corresponding to the eigenvalue of 2
D
. It is not sufficiently to determine the vector R
uniquely from (4.39). For solving the unknown vector R, we need extra equations. The required
extra equation is derived from the moment equation (4.27). First multiplying both sides of
equation (4.27) by
D
−!
+
−
, respectively, and taking integration from y=0 to
y=x, we have,
i D,
,
! D i D,
,
= +!∆D
4.42
= ^!∆
4.43
For fixed integers m and n , we can form an independent system of equations by eliminating the
corresponding rows of the unknowns i D,
, , ! = 2 − -, 3 − -, … . . ,2 − - + +, and of the
unknowns i D,, ,
= 1,2,3, … . . , ^. It is noted that such a replacement is rather difficult. In
practice, however, the rank of the resultant system of equation can be checked by a numerical
computation, and obtained the values of connection coefficients and verified by using the
equation (4.42) and (4.43).
44
CHAPTER 5
NUMERICAL EXAMPLES
NUMERICAL EXAMPLE
5.1.
INTRODUCTION
The wavelet based numerical solution is recently used in the application of partial differential
equation. The wavelet method is an efficient alternative of numerical solutions. In this
chapter, the problems are discretized by wavelet galerkin method. The coefficient matrix of
the wavelet methods are more difficult to calculate in practical application. The Daubechies
wavelet has gained much interest in the solution of partial differential equations. The
connection coefficients play an important role in applying wavelet galerkin method. The
evaluation of connection coefficients are explained in chapter 3.
5.2.
CRITICAL LOAD
Critical load was the only load for which the structure would be in equilibrium in the just
disturbed position. If the axial load is very less than critical load the effect of the moment in
the spring dominated and the structure returns to the vertical position. If the axial load of the
member is larger than the critical load the effect of the axial force predominates and the
structure buckles to unstable condition.
The boundary position between stability and instability is called the neutral equilibrium
condition. At which the deflections of the member become very large, that critical point is
called Bifurcation point of the system
45
NUMERICAL EXAMPLE
5.3.
WAVELET GALERKIN METHOD
The wavelet galerkin method is a powerful method for solving partial differential equations.
Wavelet Galerkin technique is the most frequently used scheme these days. Daubechies
wavelet as bases in a Galerkin method to solve differential equations requires a
computational domain of simple shape. The wavelet method has been shown to be a powerful
numerical tool for the fast and accurate solution of differential equations. The connection
coefficients plays an important role in applying wavelet galerkin method. Daubechies
wavelets have been successfully used as a base function in Wavelet Galerkin method, due to
their compact support, orthogonality and Multiresolution properties. The connection
coefficients are employed in the calculation of stiffness, mass and geometry matrices. The
one of the most possible applications of wavelet theory is numerical solution of differential
46
NUMERICAL EXAMPLE
equations. The wavelet galerkin method can be applied to solve differential equations for
structures like beam, column, plates etc. wavelet galerkin method require different types of
connection coefficients for different differential or integral equation.
5.4.
EXACT METHOD
Considering an ideal prismatic bar made of a linearly elastic material and subjected to an
axial compressive load.
The governing equation is given by
+
+
=0
(5.1)
=0
(5.2)
This is a second order homogeneous ordinary differential equation with constant coefficients
that has a solution of the form
=
+
( 5.3)
Boundary conditions for simply supported ends are
=0;
=
;
=0
=0
=0
=0
Applying above boundary conditions to equation (5.3) we have;
B=0;
=0
,
= 0 , ! , 2! … … … ….
=
47
!
NUMERICAL EXAMPLE
Buckling load must satisfy the relation
=
The critical load is taken as
#$
=
!
(5.4)
for n=1
#$
=
!
&#$ =
(5.5)
=
!2
(5.6)
2
The effective length of column is defined as the length of the column between point of
inflection or point of zero moments . The effective length expressed in terms of total length
by:
(=)
(5.7)
where K is the "effective length factor" of the member depends on the support conditions.
Effective length factors for different conditions are given in Figure (5.1)
48
NUMERICAL EXAMPLE
K = 1.0
Pinned Pinned
Pinned -
K = 0.707
Fixed
Fixed – Fixed
K = 0.5
49
NUMERICAL EXAMPLE
Cantilever
K=2
Figure (5.1): Effective Length for Different Support Condition
5.5.
PROBLEM FORMULATION
In this chapter solving the buckling problem of pressurized bar by using wavelet galerkin
method. And demonstrate the effectiveness of the ‘connection coefficient method. The
connection coefficient method explained in third chapter.
Consider an ideal prismatic bar of a linearly elastic material and subjected to an axial
compressive load.
The bending differential equation of pressurized bar for buckling analysis is
+( )
+ +( ) = 0
(5.8)
+ -- ( ) + + ( ) = 0
(5.9)
Where,
P
= axial pressure on the bar
W
= Difflection of the bar
L
=length of the bar
E
= modulus of elasticity of bar
I
= moment of inertia of the bar
50
NUMERICAL EXAMPLE
The equation ( 5.9) can be converted into its non dimensional form. The non dimensional
form of equation is given by
/ -- (01) + +
/ (01) = 0
+
(5.10)
The non dimensional parameters in the above equations are given by
/ =
+
01 =
+
(5.11)
0
(5.12)
1=
(5.13)
.
5.5.1. Clamped Support
The boundary condition for non dimensional clamped support is as follows
For x=0;
/ (0) = 0
+
(5.14)
/ - (0) = 0
+
(5.15)
For x=1;
/ (1) = 0
+
(5.16)
/ - (1) = 0
+
(5.17)
/ (01) can be approximated by the jth–level wavelet series is given by:
The +
; <=
56
/ (01) = 2 34 2
+
4> <?
7825 01 − :
(5.18)
51
NUMERICAL EXAMPLE
The integer j is used to control the smoothness of the solution. The integer j value is larger
more accurate result will be obtained.The equation (5.17) is a function may be represented in
terms of the Daubechies scaling function at scale J, 34 is the expansion coefficients give the
magnitude of each scaling function component of the solution. Substituting the equation
(5.17) in to (5.10).obtain wavelet galerkin discretization as follows:
=
@6
2 @ A2
D
@6
7 -- ([email protected] ̅ − )2
=
@6
+ 1A2
D
7 (2 @ ̅ − C )
@6
7 (2 @ ̅ − )2
̅ . 34
7 (2 @ ̅ − C )
̅ 34
(5.18)
Where,
m= 2-N, 3-N,………,2J-1
where.
=
A2
D
@6
@6
7 -- ([email protected] ̅ − )2
7 (2 @ ̅ − C )
̅.
=
= A 7 -- ([email protected] ̅ − )7([email protected] ̅ − C) ([email protected] ̅ − C).
D
F <G
= A 7 -- (E −
+ C )7 (E ) E
<G
= ∆4<G ([email protected] − C) − ∆4<G (−C)
Similarly
(5.19)
=
@6
A2
D
@6
7 (2 @ ̅ − )2
52
7 (2 @ ̅ − C )
̅
NUMERICAL EXAMPLE
=
= A 7 (2 @ ̅ − )7 (2 @ ̅ − C ) (2 @ ̅ − C ).
D
F <G
= A 7 (E −
<G
+ C )7 (E ) E
= ∆D4<G ([email protected] − C) − ∆D4<G (−C)
(5.20)
Substituting equation (5.19) and (5.20) in equation (5.18), we have,
2 @ I∆4<G ([email protected] − C) − ∆4<G (−C)J34 + 1 I∆D4<G ([email protected] − C) − ∆D4<G (−C)J34 = 0 (5.21)
The values for calculating the connection coefficients explain in the chapter 3.
From the equation (5.14) to (5.17), W(x) satisfy the boundary condition as follows,
For x = 0;
K<?
/ (0) = 2 34 7(− ) = 0
+
(5.22)
4> <?
/ -(
K<?
+ 0) = 2 34 7 - (− ) = 0
(5.23)
4> <?
For x = 1
/ (1) =
+
/ - (1) =
+
F <=
4>
2
F <?L
F <=
4>
2
F <?L
34 7([email protected] − ) = 0
(5.24)
34 7 - ([email protected] − ) = 0
(5.25)
The buckling load corresponding to the least eigenvalue problem. The equation (5.21)
together with the equation (5.22) to (5.25) to form a over determined equation of 34 .
53
NUMERICAL EXAMPLE
5.5.2. Simply Supported
The boundary condition for non dimensional clamped support is as follows
For x=0;
/ (0) = 0
+
(5.26)
/ -- (0) = 0
+
(5.27)
For x=1;
/ (1) = 0
+
(5.28)
/ -- (1) = 0
+
(5.29)
the derivation are similar to clamped support.W(x) satisfy the boundary condition as
follows,
For x = 0;
K<?
/ (0) = 2 34 7(− ) = 0
+
(5.30)
4> <?
K<?
/ -- (0) = 2 34 7 - (− ) = 0
+
(5.31)
4> <?
For x = 1
/ (1) =
+
F <=
4>
2
F <?L
34 7([email protected] − ) = 0
(5.32)
54
NUMERICAL EXAMPLE
/ -- (1) =
+
F <=
2
4> F <?L
34 7 - ([email protected] − ) = 0
(5.33)
The equation (5.18) can be writtern in another form as follows:
F <=
F <=
5
5
2 M4,G
34 + 2 N4,G 34 = 0 C = 2 − O, 3 − O … … … , [email protected] − 1
4> <?
4> <?
(5.34)
Where,
=
5
@6
M4,G = A 2
D
5
N4,G
=
@6
= A2
D
@6
7 (2 @ ̅ − C )
̅ . 34
(5.35)
@6
7 (2 @ ̅ − C )
̅ . 34
(5.36)
7 -- ([email protected] ̅ − )2
7 (2 @ ̅ − )2
The equation (5.34) can be expressed in matrix vector form as follows:
+3 = 0
(5.37)
+ = 2 5M + / N
(5.38)
Where,
5
M = PM4,G Q
5
N = PN4,G Q
And 3 = I3
<?R4,GR F <=
(5.39)
<?R4,GR F <=
(5.40)
<? 3S<?
… … .. 3
F <=
JT
(5.41)
Now we have a linear system of [email protected] + O − 2 equation of the [email protected] + O − 2 unknown
coefficients. We can obtained the coefficient of the approximate solution by solving this
55
NUMERICAL EXAMPLE
linear system of equation. The solution of 3 gives the expansion coefficients of Wavelet
Galerkin approximation.
5.6.
RESULT AND DISCUSSION
The buckling load
#$
of structure is corresponding to the least eigenvalue of the problem.
The equation (5.21) together with the boundary condition to form a over determined equation
of 34 . In practical computation, QR decomposition is a powerful tool for solving least square
fitting problem, to change the discretized equation to be an upper triangular matrix, let the
determinate of matrix is zero, the least root of the eigenvalue problem is required as
#$ .
The
results are listed in table (5.1). Figure (5.2) shows the percentage of error for different levels
of resolution.
Table (5.1): values of
Boundary
condition
Simply
supported
#$
for N=6, J=0,1,3
Wavelet galerkin method
J=0
J=1
FEM
Exact method
J=3
9.58
9.65
9.685
9.8
37.21
37.78
38.21
40
9.8696
Clamped
supported
56
39.478
NUMERICAL EXAMPLE
Table (5.2): Daubechies filter coefficients for N=6
4
k
0
0.0498
1
0.1208
2
-0.1909
3
-0.6504
4
1.1411
5
-0.4705
7
6
5
4
3
2
1
0
0
0.5
1
1.5
2
2.5
3
Figure (5.2): Percentage of error for different levels of resolution.
57
3.5
NUMERICAL EXAMPLE
1.4
1.2
1
0.8
0.6
0.4
scaling function
0.2
-1E-15
-0.2
-0.4
-0.6
0
1
2
3
4
5
6
Figure (5.3): Scaling Function for N=6
2
1.5
1
0.5
first oeder scaling
function
0
-0.5
2nd order scaling
function
-1
-1.5
-2
-2.5
0
1
2
3
4
5
6
Figure (5.4): Derivatives of Scaling Function for N=6
58
CHAPTER 6
CONCLUSION
CONCLUSION
6.1.
CONCLUSION
The Wavelet Galerkin method is a good alternative for solving partial differential equations. In
present project work, the Daubechies family of wavelet have been consider because they posses
several useful properties, such as orthogonality, compact support and ability to represent function
at different levels of resolution. Wavelet galerkin Method has been shown to be a powerful
numerical tool for accurate solution of partial differential equation. Wavelet Galerkin methods
have more advantages than standard Galerkin methods by using compactly supported orthogonal
functional basis. In wavelet Galerkin method, it uses Daubechies coefficients and the scaling
functions. Daubechies wavelets are more useful in the numerical solution of ordinary differential
equation and partial differential equation.
The wavelet Galerkin approximation relies on evaluation of connection coefficients. In present
project work, described the algorithms for the exact evaluations of connection coefficients of
Daubechies compactly supported wavelets. The buckling load of pressurized bar obtained using
Daubechies 6 or D6 coefficients wavelets have been compared with the exact solution. This
leads to considerable savings of the time and improves numerical results through the reduction of
round of errors.
59
CONCLUSION
6.2.
SCOPE OF FUTURE WORK
Through in this project work, some studies attempted for buckling of bar but still many areas are
left which require further investigation. The possible extensions to the present study as below:
•
In present project work, described the algorithms for the exact evaluations of connection
coefficients of Daubechies compactly supported wavelets, These algorithm also used for
finite domain problem.
•
The present study deals with buckling of bar. This nay extended for bending of beams
and plates.
•
In the place of scaling functions, wavelet functions can also be used for this type of
problems.
.
60
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