Meshless Methods and Nonlinear Optics

1
Meshless Methods and Nonlinear Optics
Naomi S. Brown University of Hawaii -
Manoa Junior, Department of Physics Mentors Dr.
Alvaro Fernández and Dr. Paul Bennett
ERDC MSRC PET Summer Internship Presentation 28
Jul 2006
2
Nonlinear Optics

• Main study
• Laser propagation through nonlinear materials
• Apply Maxwells equations to existing code to
  ensure energy is conserved for various scenarios
• Compile and run the code in parallel
• Motivation
• Simulations may seem correct, but using
  established physics laws provides proof that code
  is robust
• Computational testing vs. physical experiments
• Understanding nature of light

3
Overview

• Modeling more accurate solutions of
  convection-diffusion problems
• My focus find a way to represent integral inner
  product of two basis functions
• Trial and error, and researching different
  methods to solve special cases
• Result
• Approximate solutions were within 0.014 of exact
  solution for particular shapes of overlapping
  bases
• More tests needed, possibly better transformations

4
Objectives

• Extend current algorithm to model two-dimensional
  convection-dominated flows
• Create new code for integrating the product of
  basis functions
• and to learn!

5
Getting Started

• Information and sources used
• Project stemmed from Dr. Fernándezs dissertation
  used as a guideline for understanding concepts
  and mathematics
• Books, journal articles, and mini-tutorial
  sessions
• Online sources, especially in finding examples to
  test the code

6
Motivation

• The advantage of meshless methods
• Discontinuities are handled better, which allows
  more accuracy for fewer basis functions
• Making the code two-dimensional
• Real-life problems are seldom 1-D, but
  transforming them to 2-D can be difficult

7
Application

• Convection-Diffusion Equation
• By changing parameters, this equation can
  represent
• Heat conduction
• Unsteady diffusion
• Linear wave equation
• Viscous Burgers equation

8
Methods

• Optimizing the approximation place basis
  functions where Galerkin residual is greatest
• Galerkin implies
• Integral inner products (of basis functions)
• The shape of the support these functions can vary
  (and so do the difficulties with each!)
• Triangular, quadrilateral, circular
• Transformations make integration more accurate
• Account for the shape of intersection
• Different shapes use different transformations
• What we chose functions with circular support

9
Basis Functions with Circular Support
y
y
x
x

• Advantages
• Arbitrarily oriented support can be easily
  rotated
• Symmetry means less special cases to account
  for

10

• A disadvantage
• Transformations were more complicated in some
  cases
• Most people who have worked with a similar
  algorithm have not transformed the region to use
  quadrature points. Differently-shaped regions are
  treated the same, producing less accurate results.

y
(-1,1)
(1,1)
x
(-1,-1)
(1,-1)
11
Examples Different Cases and Shapes
1
3
2
6
4
5
12
Transforming the Region

• Transformations use
• Determinant of the Jacobian
• 2-D Gauss quadrature rules (giving us points and
  weights within 2 x 2 square)
• For lens-shaped overlap
• Isoparametric Quadrilaterals make region easier
  to work with
• The following slide shows a typical scenario

13
The Process
y
x
14
Transformations

• Using the Jacobian to change variables, the
  quadrilateral becomes a perfect square to be
  integrated
• x xi Nie
• y yi Nie

15
Results

• For integration over an entire circle several
  functions tested with known answers
• Approximations were within 0.014 of exact
• Accuracy depends on number of quadrature points
• Code must be integrated into practical
  application, using appropriate basis functions
• Quartic splines
• Many cases accounted for, but others may exist

16
Conclusion

• All objectives completed, with more work to be
  done on each project
• 2-D development of code will more accurately
  model flows, especially convection-dominated
• Further work to be done
• Parallelize the code and make it 3-D
• Deal with lost space quadratic triangles?

17
Mahalo!

• My thanks to both Dr. Fernández and Dr. Bennett
  for their patience and willingness to share
  knowledge, and to the PET program coordinators
  for all of their hard work.