Siraj –ul – Islam Laboratory for Multiphase Processes University of Nova Gorica, Slovenia.
-
Upload
scot-greer -
Category
Documents
-
view
216 -
download
2
Transcript of Siraj –ul – Islam Laboratory for Multiphase Processes University of Nova Gorica, Slovenia.
Some Applications of Wavelets
Siraj –ul – Islam
Laboratory for Multiphase ProcessesUniversity of Nova Gorica, Slovenia
Some Applications of Wavelets
Siraj –ul – Islam
Laboratory for Multiphase ProcessesUniversity of Nova Gorica, Slovenia
Some Applications of Wavelets
Siraj –ul – Islam
Laboratory for Multiphase ProcessesUniversity of Nova Gorica, Slovenia
Khyber Pass"Khyber is a Hebrew word meaning a fort"
• Alexander the Great and his army marched through the Khyber to reach the plains of India ( around 326 BC)
• In the A.D. 900s, Persian, Mongol, and Tartar armies forced their way through the Khyber
• January 1842, in which about 16,000 British and Indian troops were killed
• Mahmud of Ghaznawi, marched through with his army as many as seventeen times between 1001-1030 AD• Shahabuddin Muhammad Ghaur, a renowned ruler of Ghauri dynasty, crossed the Khyber Pass in 1175 AD to consolidate the gains of the Muslims in India
• In 1398 AD Amir Timur, the firebrand from Central Asia, invaded India through the Khyber Pass and his descendant Zahiruddin Babur made use of this pass first in 1505 and then in 1526 to establish a mighty Mughal empire
Khyber Pass"Khyber is a Hebrew word meaning a fort"
Some Applications of Wavelets
Siraj –ul – Islam
Laboratory for Multiphase ProcessesUniversity of Nova Gorica, Slovenia
Some Applications of Wavelets
Siraj –ul – Islam
Laboratory for Multiphase ProcessesUniversity of Nova Gorica, Slovenia
Some Applications of Wavelets
Siraj –ul – Islam
Laboratory for Multiphase ProcessesUniversity of Nova Gorica, Slovenia
What are Wavelets?
10
A wavelet is a function which
• maps from the real line to the real line• has an average value of zero• has values zero except over a bounded domain
What are Wavelets?
11
• A small wave• Extends to finite interval
The word wavelet refers to the function h(t) that generates a basis for the orthogonal complement of V0 in V1
Wavelets analysis is a procedurethrough which we can decompose a given function into a set of elementarywaveforms called wavelets
The Haar Scaling Functions and Haar Wavelets
13
a) Haar scaling function (Father function) b) Haar Wavelet function (Mother wavelets)
The Haar Wavelets and its Integrals
16
with the collocation points
The repeated integral of Haar wavelet is given by
Some applications of wavelets
• Numerical Analysis• Ordinary and Partial Differential
Equations• Signal Analysis• Image processing and Video Compression
(FBI adopting a wavelet-based algorithm as
a the national standard for digitized finger
prints)• Control Systems• Seismology
Multi-Resolution Analysis
22
2
20 0 1 2
2
0 0 00
The space L (R) can be decomposed as an infinite orthogonal direct sum
L (R) V W W W .
In particular, each L (R) can be written uniquely as
where v belongs to V and belongs j jj
f
f v w w
jto W
Multi-Resolution Analysis
24
Scaling function (Father wavelet) basis in V 0h
Wavelet function (Mother wavelets) basis in Wih
Problems with Gaussian Quadrature
28
• Solution 2n by 2n system• Search for better nodal values• Finding optimized values for the unknown weights
Numerical double integration with variable limits
35
To extend the present idea to numerical integration with variable limits and make it more efficient, we use an iterative approach instead of using two and three dimensional wavelets
Numerical Solution of Ordinary Diff. Eqs.
44
Existing Methods
• Runge-Kutta family of Methods (Need shooting like to convert BVP into IVP, Stability limits)• Finite difference Methods (Low accuracy and large matrix inversion)• Asymptotic Methods (Series solution convergence problem)
Shooting method
• Idea: transform the BVP in an initial value problem (IVP), by guessing some of the initial conditions and using the B.C. to refine the guess, until convergence is reached
Target
Too high: reduce the initial velocity!
Too low: increase the initial velocity!
Use the same algorithms used for IVPConvergence can be problematic
Shooting Method for Boundary Value Problem ODEsShooting Method for Boundary Value Problem ODEs
Definition: a time stepping algorithm along with a root finding method for choosing the appropriate initial conditions which solve the boundary value problem.
Second-order Boundary-Value Problem
),',,(''1 yyxfy y(a)=A and y(b)=B
Computational Algorithm Computational Algorithm Based on Haar Wavelets
1. Contrary to the existing methods, the new method based on wavelets can be used directly for the numerical solution of both boundary and initial value problems
2. Stability in time integration is overcome.
3. Variety of boundary condition can be implemented with equal ease
4. Simple applicability along with guaranteed convergence.
Computer Math. Model. 2010
Haar Wavelets for Boundary Value Problem in ODEsHaar Wavelets for Boundary Value Problem in ODEs
Consider the following coupled nonlinear ODEs
Along with boundary conditions
Haar Wavelets for Boundary Value Problem in ODEsHaar Wavelets for Boundary Value Problem in ODEs
Wavelets approximation for and can be given by,
f
Adaptivity Through Non-uniform Haar Wavelets Adaptivity Through Non-uniform Haar Wavelets
Inter. J. Comput. Method Eng Science & Mechanics (2010)
Nodes Generations Through Cubic SplineNodes Generations Through Cubic Spline
1 1.5 2 2.5 3 3.5 4-4
-2
0
2
4x 10
-15 residuals1 1.5 2 2.5 3 3.5 4
-1.5
-1
-0.5
0
0.5
1
Cubic spline interpolant
data 1
spline
1 1.5 2 2.5 3 3.5 4-2
-1
0
1
2residuals
1 1.5 2 2.5 3 3.5 4-1
-0.5
0
0.5
1
data 1 linear
quadratic
cubic
4th degree 5th degree
6th degree
7th degree 8th degree
Nodes Generations Through Cubic SplineNodes Generations Through Cubic Spline
-5 -4 -3 -2 -1 0 1 2 3 4 50
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5Cubic spline interpolant
Nodes Generations Through Cubic SplineNodes Generations Through Cubic Spline
0 2 4 6 8 10 12 14 16-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
data
spline
Nodes GenerationsNodes Generations
initial temperature
calculate deformation of the slice
at the new position
solve temperature of the slice
at the new position
initial shape
initial velocityIn rolling direction
final velocityin rolling direction
0v
v
initial shape
final shape
initial nodes
final nodes renoding
Nodes GenerationsNodes Generations
Nodal points are generated through the following procedures:
Transfinite Interpolation
Elliptic Grid Generation
Nodes GenerationsNodes Generations
TRANSFINITE INTERPOLATION
Through this technique we can generate initial grid which is confirming to the geometry we encounter in different stages of plate and shape rolling.We suppose that there exists a transformation
which maps the unit square, in the computational domain onto the interior of the region ABCD in the physical domain such that the edges
map to the boundaries AB, CD and the edges are mapped to the boundaries AC, BD.
The transformation is defined as
Where represents the values at the bottom, top, left and right edges respectively
( , ) [ ( , ), ( , )]tx y r
0 1, 0 1
0, 1 0,1
0 0 1l r b t b t b t, r ( ) = (1- )r ( ) + r ( ) + (1- )r ( ) + r ( ) - (1 - )(1 - )r ( ) - (1 - ) r ( ) - (1 - ) r ( ) - r
, , ,b t l rr r r r
Nodes GenerationsNodes Generations
An example of transformation from computational domain to physical domain.
( )tr
( )br
( )lr ( )rr
Nodes GenerationsNodes Generations
ELLIPTIC GRID GENERATION
The mapping procedure defined above form the physical domain to the computational domain is described by are continuously differentiable maps of all order.
The grid generated through transfinite interpolation can be made more conformal to the geometry by using the following elliptic grid generators
where is the Jacobean of the transformation.
2 2 2
22 12 112 2
2 2 2
22 12 112 2
2 0
2 0
x x xg g g
y y yg g g
2 2
22 122 2
2 2
11 2
1 1, ,
1
x x x x y yg g
J J
x xg
J
J
( , ), ( , )x y x y
Nodes GenerationsNodes Generations
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8title here
x
y
0 0.5 1 1.5 2 2.50
0.5
1
1.5title here
x
y
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.5
0
0.5
1
1.5
2title here
x
y
0 0.5 1 1.5 2 2.5 3 3.5 4-4
-3
-2
-1
0
1
2title here
x
y
Application of Meshless Method to Hyperbolic PDEsApplication of Meshless Method to Hyperbolic PDEs
Submitted to journal
Comparison of Local and Global Meshless MethodsComparison of Local and Global Meshless Methods
CMES. 2010