1. Distribution Statement A: Cleared for Public Release. Distribution is unlimited. 1 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

2. Distribution Statement A: Cleared for Public Release. Distribution is unlimited. 2 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

3. Distribution Statement A: Cleared for Public Release. Distribution is unlimited. 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. Distribution Statement A: Cleared for Public Release. Distribution is unlimited. 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. Distribution Statement A: Cleared for Public Release. Distribution is unlimited. 5 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

6. Distribution Statement A: Cleared for Public Release. Distribution is unlimited. 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. Distribution Statement A: Cleared for Public Release. Distribution is unlimited. 7 Application Convection-Diffusion Equation: By changing parameters, this equation can represent: –Heat conduction –Unsteady diffusion –Linear wave equation –Viscous Burger’s equation

8. Distribution Statement A: Cleared for Public Release. Distribution is unlimited. 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. Distribution Statement A: Cleared for Public Release. Distribution is unlimited. 9 Basis Functions with Circular Support Advantages: Arbitrarily oriented support can be easily rotated Symmetry means less special “cases” to account for y x y x

10. Distribution Statement A: Cleared for Public Release. Distribution is unlimited. 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 x (-1,-1) (-1,1) (1,-1) (1,1)

11. Distribution Statement A: Cleared for Public Release. Distribution is unlimited. 11 Examples: Different Cases and Shapes 132 645

12. Distribution Statement A: Cleared for Public Release. Distribution is unlimited. 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. Distribution Statement A: Cleared for Public Release. Distribution is unlimited. 13 The Process y x

14. Distribution Statement A: Cleared for Public Release. Distribution is unlimited. 14 Transformations Using the Jacobian to change variables, the quadrilateral becomes a perfect square to be integrated x = x i N i e y = y i N i e

15. Distribution Statement A: Cleared for Public Release. Distribution is unlimited. 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. Distribution Statement A: Cleared for Public Release. Distribution is unlimited. 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. Distribution Statement A: Cleared for Public Release. Distribution is unlimited. 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.