Highorder Surface Representation

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High-order Surface Representation
Matthew M. Pohlman, California Institute of
Technology
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Motivation

• A fast, high-order surface scattering method
• Solves integral equation formulation of Helmholtz
  (acoustic) or Maxwell equations (electromagnetic)
  using high-order integrators and analytic
  resolution of singularities
• Computational complexity is O(N6/5log(N)) to
  O(N4/3log(N))
• Several surface derivatives needed

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Accuracy of the basic non-accelerated solver
Scattering by a sphere of radius 2.7 l
Doubling the discretization density improves the
accuracy by 200 to 300 times!
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Large spheres (comparison w/ O(N log(N)) FISC)
Bruno and Kunyansky, JCP 2001
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High-order Integration and the Trapezoidal Rule
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Partitions of Unity...
localize integration problem
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Surface Representation

• Goals
• Fast
• High-order
• Flexible
• Standard Methods
• Piecewise interpolation methods are fast, but
  only marginally smooth for surfaces
  (discontinuous first or second derivatives)
• Trigonometric interpolation methods are
  high-order and give smooth (all derivatives
  continuous) surfaces, but efficient
  implementation (FFT) requires evenly spaced data

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New Approach

• Use Fourier Series for smooth representation of
  surface patches with necessary parameterization
• Take advantage of an Unevenly Spaced Fast Fourier
  Transform (USFFT) to obtain/evaluate these
  Fourier Series with irregularly spaced data with
  accuracy ? in O(N log N N log 1/?) time Dutt
  Rokhlin,1993
• Continue discrete data to smooth periodic
  functions in order to preserve spectral
  convergence of Fourier methods (Continuation
  Method)

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Unevenly Spaced Fast Fourier Transform

• From unevenly spaced data f, sample the
  convolution fg, at evenly spaced locations in
  O(MN) time
• the convolution filter g is known analytically
• for error tolerance ?, the necessary sample rate
  of g for discrete convolution step is MO(log
  1/?) ! M¼32 when ?¼1e-16
• Use standard FFT to move from spatial domain to
  frequency domain, O(N log N) time
• Divide by the Fourier coefficients of filter g to
  obtain Fourier coefficients of f, O(N) time

Patches and Partitions of Unity

• Use partitions of unity (POU) to divide surface
  into patches, just like in surface scattering
  Bruno, et. al.
• unknowns become smooth periodic functions on each
  patch
• overlapping patches take care of regions where
  POU is small

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1-Dimensional USFFT Example

• The periodic function exp(sin(2?x)) is
  interpolated using Fourier modes obtained from
  the USFFT algorithm
• Data points are from randomly spaced values on
  curve
• Approximation is accurate to machine precision,
  yet costs still scale with FFT of same size

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Interpolation using POU
FFT and division by POU (regions where POU 1
are discarded)
partition of unity
data POU
original 1D and 2D unevenly spaced data
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Difficulties

• Use this technique on all the patches for any
  smooth object
• POU forces these patches to overlap
• What about objects with edges or corners?

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Interpolation of Non-periodic Data

• patches cant overlap an edge or corner smoothly,
  so POU cant solve all issues for a complicated
  geometry
• need a spectrally accurate interpolation method
  for smooth non-periodic data as well
• IDEA
• find smooth periodic function (over a larger
  period) that coincides with original data
• finding such a periodic function is easy using a
  least-squares fit of truncated Fourier series
• need to also ensure that the result is smooth
• Fourier coefficients of smooth periodic functions
  decay exponentially, so make sure the
  coefficients of the continuation decay
  exponentially too!
• this can be accomplished during the least-squares
  step by multiplying Fourier coefficients by
  appropriate factors and setting equal to zero

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Continuation of Non-periodic Data
Direct FFT Gibbs phenomenon
Given data
Continuation function is a Fourier series, whose
coefficients are obtained by solving a linear
system
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Convergence of Continuation Method
Generalizes to any number of dimensions!
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Surface Parameterization

• Parameterization of each patch is done with the
  Intrinsic Parameterization initially developed in
  the CG community to minimize texture map
  deformation Desbrun, Meyer and Alliez, 2002
• Human intervention is currently needed to rescale
  singularities in the parameterization (which
  occur where the geometry has large curvature)

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Surface Interpolation of Wing Patch
Refined mesh shown here generated via surface
interpolation
Wing represented by eleven overlapping patches,
each patch given explicitly by three coordinate
functions (which are Fourier Series!)
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Wing Edges
A change of variables in parameter space gives an
unevenly sampled surface for accurate resolution
of edge-scattering
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Wing Normals
Fine array of surface normals plotted on
interpolated wing surface
It is easy to compute surface normals and
curvatures by differentiation of Fourier Series
representation!
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A Few Other Patches
Canopy
Nose
Top
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Conclusions

• interpolation using USFFT together with POU or
  Periodic Continuation is spectrally accurate
• USFFT is O(N log N) for fixed accuracy (?¼1e-16)
• Continuation is O(N3) due to SVD but necessary
  only for small patches of a geometry
• evaluation of Fourier Series is O(N log N) in
  either case
• can accurately evaluate the surface derivatives
  needed by scattering codes for what began as a
  triangle mesh

Acknowledgements

• Oscar Bruno, Randy Paffenroth
• AFOSR

References

• Dutt and Rokhlin, Fast Fourier Transforms for
  Nonequispaced Data, 1993
• Duijndam and Schonewille, Nonuniform fast Fourier
  transform, 1999
• Desbrun, Meyer and Alliez, Intrinsic
  Parameterizations of Surface Meshes, 2002
• Bruno and Kunyansky, A Fast High-Order Algorithm
  for the Solution of Surface Scattering Problems,
  2001