Transcript Meshless Methods and Nonlinear Optics.

Meshless Methods and Nonlinear Optics
Naomi S. Brown
University of Hawai`i - Mānoa
Junior, Department of Physics
Mentors: Dr. Alvaro Fernández and
Dr. Paul Bennett
ERDC MSRC PET
Summer Internship Presentation
28 Jul 2006
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Nonlinear Optics
• Main study:
– Laser propagation through nonlinear materials
• Apply Maxwell’s 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
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Overview
• Modeling more accurate solutions of convectiondiffusion 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
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Objectives
• Extend current algorithm to model twodimensional convection-dominated flows
• Create new code for integrating the product of
basis functions
…and to learn!
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Getting Started
Information and sources used:
• Project stemmed from Dr. Fernández’s
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
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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
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Application
Convection-Diffusion Equation:
By changing parameters, this equation can represent:
–
–
–
–
Heat conduction
Unsteady diffusion
Linear wave equation
Viscous Burger’s equation
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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
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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
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A disadvantage:
Transformations were more complicated in some cases
y
(-1,1)
(1,1)
x
(-1,-1)
(1,-1)
*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.
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Examples: Different Cases and Shapes
1
2
4
5
3
6
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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:
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The Process
y
x
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Transformations
Using the Jacobian to change variables, the quadrilateral
becomes a perfect square to be integrated
x=
xi Nie
y=
yi Nie
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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
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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?
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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.
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