Transcript Continuous Schwarz

Accurate Implementation of the
Schwarz-Christoffel Tranformation
Evan Warner
What is it?
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A conformal mapping (preserves angles and
infinitesimal shapes) that maps polygons onto a
simpler domain in the complex plane
Amazing Riemann Mapping theorem:
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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
Why does anyone care?
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Physical problems: Laplace's equation,
Poisson's equation, the heat equation, fluid flow
and others on polygonal domains
To solve such a problem:
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State problem in original domain
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Find Schwarz-Christoffel mapping to simpler
domain
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Transform differential equation under mapping
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Solve
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Map back to original domain using inverse
transformation (relatively easy to find)
Who has already done this?
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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
How to find a transformation...
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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
How to find a transformation...
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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.
How to find a transformation...
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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
Examples
Upper half-plane to semi-infinite strip; lines are Re(z)=constant and Im(z)=constant
Examples
Mapping from upper half-plane to unit square; lines are constant for the opposite image
What have I done so far?
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Implementation of complex numbers in java
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ComplexFunction class
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Implementation of Gauss-Jacobi quadrature
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Basic graphical user interface with capability to
calculate Gauss-Jacobi integrals
Testing done mostly in MATLAB (quad routine)
What's next?
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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