Engineering Computation and Simulation

1
Engineering Computation and Simulation
EE317

• Conor Brennan
• Dublin City University brennanc_at_eeng.dcu.ie

2
FRACTALS
GROUP 4 - Manpreet Singh Nanda Varun
Raina Kanika Poply
3
What is a Fractal?

• Irregular shape or curve which when reduced in
  size or zoomed into gives a similar copy of the
  original.
• Main Characteristic
• Self Similarity

4
Uses of Fractals in Engineering

• Fractals not only make pretty pictures but
  they have many diverse applications as well.
• Fractal Antennas
• - Uses Fractals or self-similar design to
  increase receiving or transmitting area of the
  Antenna
• - Next generation cell phones/ communication
  devices
• - Smaller pieces resemble the whole

5
Uses of Fractals in Engineering(2)

• Frequency Selective Surfaces
• - Filters that allow only certain frequencies
  to pass
• - Deflected frequencies used to hide objects
• - Finds use in Military and Satellite
  Communications
• Fluid Scaling using Fractals
• - Narrow down broad distribution of fluid
  properties
• - Hence a solution to problem of Fluid
  Scaling

6
Newton Raphson Method

• Method to find the best approximation of roots of
  a real function
• Initial Guess -x0
• Better Approximation-x1
• Principle
• - Let initial guess of a root of f (x) 0 be
  at xi
• - Draw a tangent to the curve at f (xi)
• - The point xi1 where tangent crosses the
  x-axis is an improved estimate.

7
Newton Raphson Method in Engineering

• Analysis of Electric Networks Nodal voltages are
  calculated through iterative calculations via
  newton raphson method.
• Complex Functions Finding the roots of complex
  functions like x5-1.
• Non-Linear Equations Finding the zeros of
  continuously differentiable functions.

8
Our First Fractal

• Number of iterations
• Checking condition
• fetching just 2 roots.

9
Changes Made in Code to Improve the Fractal

• Introducing functions .
• Use of Newton fractal loop.
• Matrix to store values for iterations required,
  roots reached and combined fractals.

10
Number of Iterations
LEVEL 1 ZOOMING ALONG THE WINGS
real_lower -0.1 real_upper 0.0 imag_lower
0.0 imag_upper 0.1
11
Red Points along the wings
POINTS MERGING AT LATEST STAGE
12
Combining Roots Reachedand Iterations
13
Zoom In Effects
SHOWS CONTIINOUS FRACTALS SYSTEM
14
An Improved Version (Root Evaluation)

• Q. What do the different colours represent?
• Values towards which the roots converge.

LEVEL 1 real_lower -2.0 real_upper
2.0 imag_lower -2.0 imag_upper 2.0
15
First Zoom

• Changing the Co-Ordinates
• real_lower -1.0
• real_upper 1.0
• imag_lower -1.0
• imag_upper 1.0

16
Second Zoom

• Changing the Co-Ordinates
• real_lower 0.3
• real_upper 1.0
• imag_lower -0.8
• imag_upper 0.0

17
Third Zoom

• Further zooming on petal of fractal gets the same
  shape.

18
Fourth Zoom

• Observation
• Same Continuous
• Fractal

19
(No Transcript)
20
Thank You !! QUESTIONS ???