Accurate Implementation of the Schwarz-Christoffel Tranformation Evan Warner.
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Transcript of Accurate Implementation of the Schwarz-Christoffel Tranformation Evan Warner.
![Page 1: Accurate Implementation of the Schwarz-Christoffel Tranformation Evan Warner.](https://reader036.fdocuments.us/reader036/viewer/2022082611/56649e225503460f94b0f39d/html5/thumbnails/1.jpg)
Accurate Implementation of the Schwarz-Christoffel Tranformation
Evan Warner
![Page 2: Accurate Implementation of the Schwarz-Christoffel Tranformation Evan Warner.](https://reader036.fdocuments.us/reader036/viewer/2022082611/56649e225503460f94b0f39d/html5/thumbnails/2.jpg)
What is it?
A conformal mapping (preserves angles and infinitesimal shapes) that maps polygons onto a simpler domain in the complex plane
Amazing Riemann Mapping theorem: A conformal (analytic and bijective) map always
exists for a simply connected domain to the unit circle, but it doesn't say how to find it
Schwarz-Christoffel formula is a way to take a certain subset of simply connected domains (polygons) to find the necessary mapping
![Page 3: Accurate Implementation of the Schwarz-Christoffel Tranformation Evan Warner.](https://reader036.fdocuments.us/reader036/viewer/2022082611/56649e225503460f94b0f39d/html5/thumbnails/3.jpg)
Why does anyone care?
Physical problems: Laplace's equation, Poisson's equation, the heat equation, fluid flow and others on polygonal domains
To solve such a problem: State problem in original domain Find Schwarz-Christoffel mapping to simpler
domain Transform differential equation under mapping Solve Map back to original domain using inverse
transformation (relatively easy to find)
![Page 4: Accurate Implementation of the Schwarz-Christoffel Tranformation Evan Warner.](https://reader036.fdocuments.us/reader036/viewer/2022082611/56649e225503460f94b0f39d/html5/thumbnails/4.jpg)
Who has already done this?
Numerical methods, mostly in FORTRAN, have existed for a few decades
Various programs use various starting domains, optimizations for various polygon shapes
Long, skinny polygons notoriously difficult, large condition numbers in parameter problem
Continuous Schwarz-Christoffel problem, involving integral equation instead of discrete points, has not been successfully implemented
![Page 5: Accurate Implementation of the Schwarz-Christoffel Tranformation Evan Warner.](https://reader036.fdocuments.us/reader036/viewer/2022082611/56649e225503460f94b0f39d/html5/thumbnails/5.jpg)
How to find a transformation...
State the domain, find the angles of the polygon, and come up with the function given by the formula:
http://math.fullerton.edu/mathews/c2003/SchwarzChristoffelMod.html
B and A are constants determined by the solution to the parameter problem, the x's are the points of the original domains, the alphas are the angles
![Page 6: Accurate Implementation of the Schwarz-Christoffel Tranformation Evan Warner.](https://reader036.fdocuments.us/reader036/viewer/2022082611/56649e225503460f94b0f39d/html5/thumbnails/6.jpg)
How to find a transformation...
Need a really fast, accurate method of computing that integral (need numerical methods) many many times.
Gauss-Jacobi quadrature provides the answer: quadrature routine optimized for the necessary weighting function.
Necessary to derive formulae for transferring the idea to the complex domain.
![Page 7: Accurate Implementation of the Schwarz-Christoffel Tranformation Evan Warner.](https://reader036.fdocuments.us/reader036/viewer/2022082611/56649e225503460f94b0f39d/html5/thumbnails/7.jpg)
How to find a transformation...
The parameter problem must be solved – either of two forms, constrained linear equations or unconstrained nonlinear equations (due to Trefethen)
Solve for prevertices - points along simple domain that map to verticies
Once prevertices are found, transformation is found
![Page 8: Accurate Implementation of the Schwarz-Christoffel Tranformation Evan Warner.](https://reader036.fdocuments.us/reader036/viewer/2022082611/56649e225503460f94b0f39d/html5/thumbnails/8.jpg)
Examples
Upper half-plane to semi-infinite strip; lines are Re(z)=constant and Im(z)=constant
![Page 9: Accurate Implementation of the Schwarz-Christoffel Tranformation Evan Warner.](https://reader036.fdocuments.us/reader036/viewer/2022082611/56649e225503460f94b0f39d/html5/thumbnails/9.jpg)
Examples
Mapping from upper half-plane to unit square; lines are constant for the opposite image
![Page 10: Accurate Implementation of the Schwarz-Christoffel Tranformation Evan Warner.](https://reader036.fdocuments.us/reader036/viewer/2022082611/56649e225503460f94b0f39d/html5/thumbnails/10.jpg)
What have I done so far?
Implementation of complex numbers in java ComplexFunction class Implementation of Gauss-Jacobi quadrature Basic graphical user interface with capability to
calculate Gauss-Jacobi integrals Testing done mostly in MATLAB (quad routine)
![Page 11: Accurate Implementation of the Schwarz-Christoffel Tranformation Evan Warner.](https://reader036.fdocuments.us/reader036/viewer/2022082611/56649e225503460f94b0f39d/html5/thumbnails/11.jpg)
![Page 12: Accurate Implementation of the Schwarz-Christoffel Tranformation Evan Warner.](https://reader036.fdocuments.us/reader036/viewer/2022082611/56649e225503460f94b0f39d/html5/thumbnails/12.jpg)
What's next?
Research into solving the nonlinear system parameter problem – compare numerical methods
Independent testing program for a variety of domains, keeping track of mathematically computed maximum error bounds
User-friendly GUI for aids in solving physical problems and equations