Lagrangian Surface Propagation

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Lagrangian Surface Propagation

• John C. Hart
• Xiangmin (Jim) Jiao
• Michael Heath
• University of Illinois,Urbana-Champaign
• John Sullivan
• TU Berlin

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Surface Propagation

• Surface/curve x(u,t) evolution
• Evolution direction is normal n(u)
• Evolution controlled by a speed function
  f(u,x,k,t,)
• ?x(u,t)/?t f() n(u)
• Why do this?

3
Applications

• Component of physical simulations
• Solidification
• Extrusion
• Multiphase flow
• Fluid-solid interaction (our favorite)
• Also used for...
• Modeling (blending)
• Animation (morphing, fire, water)
• Vision, IP, Robotics, VLSI
• Books by Sethian and by Fedkiw Osher loaded
  with apps

CSAR
Museth, Breen,Whitaker, Barr
Fedkiw
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Level Sets

• Evolve a voxelized implicit surface
• Automatic Topology
• Isosurface ft-1(0) manifold when 0 regular value
  (and it almost always is), so no self
  intersection
• Entropy-preserving shocks
• No swallowtails (see above)
• Well-studied numerical foundation
• Level set equations belong to well known
  Hamilton-Jacobi equations
• But level sets have drawbacks too

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Memory Consumption

• Level set equations for surface propagation
  solved over (entire) embedding space
• Narrow-band method restricts computation to grid
  points near surface
• Fast marching method evaluates
• t(x) t f(x,t) 0
• Limited to surfaces that only grow (or shrink)
• But were only really interested in the surface

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Fixed Resolution

• Eulerian approach discretized space into a
  uniform rectilinear grid for numerical
  stability, efficient isosurfacing and topological
  simplicity
• Physical apps have smooth interfaces with sparse
  localized fine details
• Eulerian grid must be fine everywhere to resolve
  features, which wastes space
• Recent hierarchical methods (Strain quadtree,
  Fedkiw octree)

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Lagrangian Benefits

• Lagrangian surface propagation would use an
  efficient surface mesh, not a wasteful space mesh
• This mesh could adapt, refined in areas of high
  curvature or computational interest
• Surface mesh would signal when a topology change
  was needed, or needed to be prevented
• Surface mesh would provide more accurately
  estimate volume

!
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Lagrangian Mesh Propagation
Applied to Multiphysics Simulations
Simulation of Space Shuttle solid rocket booster
Full burn of star grain
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Major Ingredients and Progress

• Geometric construction
• Numerical computation
• Mesh quality control
• Mesh optimization and global remeshing
• Local mesh adaptivity
• Implementation
• Software infrastructure
• Parallel algorithms
• Integration with physics codes

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Geometric Construction in 2-D

• Moves nodes by offsetting incident edges
• Approximates Huygens principle
• Treats shocks, rarefactions and saddles
  differently
• Different from marker-particle methods
• Node redistribution by minimizing quadric error

rarefaction
saddle
shock
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Geometric Construction in 3-D

• Decompose motion of edges
• Edge motion within normal plane
• Edge motion along tangential direction
• Node motion by weighted averaging

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Error Analysis

• Translation w/ unit speed
• Level set
• Lagrangian
• Superlinear convergence

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Rotation of Ellipse

• Governing equation
• Superlinear convergence

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Propagation under Constraints

• Part of interface may be constrained
• Enforced automatically by setting speed to zero
  along constrained patch

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Six Types of Corners in 3-D

• All propagated accurately
• Colors indicate magnitude of displacements

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Surface Mesh Smoother

• Optimization-based smoother
• Optimizes algebraic or geometric quality measures
• Projection by minimizes quadric error metric
• Singularity aware
• Preserves ridges and corners
• Avoids pollution errors at rarefactions

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Detection of Ridges and Corners

• Curve-oriented detect curves in decreasing order
  of strongness
• Criteria for strong curves
• Face and turning angles, angle defect
• url-thresholding (Upper bound, signal/noise
  Ratio, and Lower bound)
• Filtration rules for false strong curves short
  or close to stronger

Model courtesy of CAT.
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Propagation in 3-D
Walk-back of ACM Rocket.
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Implementation in 3-D

• Software infrastructure
• Array-based halfedge data structure
• Parallel communication
• Next step
• More accurate handling of rarefactions through
  local mesh adaptivity
• Topological changes
• Coupling with physics codes

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Procedural Level Sets

• Work with Michael Mullan (Illinois student) and
  Ross Whitaker (Utah)
• Level set method based on Eulerian grid of n3
  voxels
• Reconstructed surface can be considered an
  implicit surface with n3 parameters
• Level set motion derived in terms of df/dt the
  change in voxel values
• What about implicit representations other than
  voxel grids?

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Parameterized Implicits
x1
f10
r
(a,b)
x2
f30
f20
x3

• Let fi f(xiq)
• q (qj) is a vector of shape parameters
• Shape corresponding to parameters q represented
  by the isosurface
• x f(xq) 0
• Assume f(x) lt 0 ? x inside isosurf.
• Surface normal in direction of gradient
• N(x) ?f / ?f

Circle example f(xq) (x-a)2 (y-b)2 r2 x
(x,y), q (a,b,r)
r
b
a
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Isosurface Particles
f1(0) 0 f1(1)
x1(0)
x1(1)
4
3
x2(1)

• Assume fi(t) f(xi(t)q(t)) 0 ?? xi
• Q What functions xi(t), q(t) keep it so?
• A Ones that satisfy, for all i,
• fi?(t) df(xi(t)q(t))/dt 0
• where
• df(xiq)/dt dfi/dx ? dxi/dt dfi/dq ? dq/dt
• ?fi ? vi dfi/dq ? q?

x2(0)
(4,0)
(0,0)
x3(1)
x3(0)
q(0) (0,0,4)
q(1) (4,0,3)
t
x
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Particles on Surface
x1
?f1
x2
?f2

• Q How do particles move to adhere to a evolving
  surf.?
• A Adjust vi to accommodate q?
• ?fi ? vi dfi/dq ? q? 0
• Any of part. vels. vi when dotted with ?fi
  cancel dfi/dq ? q?
• Find vi that move particles least
• Move particles in ?fi direction to get greatest
  change in value with least change in position
• Let vi li ?fi
• Then ?fi ? li ?fi dfi/dq ? q? 0
• So Thus

?f3
x3
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Surface Thru Particles
x1
x2(1)
v2
x2(0)
?f2

• Q How to evolve surf. to pass thru particles?
• A Adjust q? to accomodate vi
• ?fi ? vi dfi/dq ? q? 0
• Need max change in f with min change in q
• So move in param. gradient dfi/dq dir.
• But fi changes differently at different parts.
• Answer is weighted sum of q-gradients at each
  particle position xj
• q? S lj dfj/dq
• Which yields for each particle xi
• df(xi)/dx ? vi df(xi)/dq ? Sjlj df(xj)/dq 0
• Leading to the Axb system
• Sj df(xi)/dq ? df(xj)/dq lj -df(xi)/dx ? vi

x3
q?
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Level Sets
ft1-1(0)
x(1)
ft2-1(0)
f1 gt 0 f2 lt 0

• Time varying implicit surface
• ft-1(0) x f(x,t) 0
• Let x(t) be a surface point at time t
• f(x(t),t) 0
• Time derivative leads to field motion
• df(x,t)/dx dx/dt df(x,t)/dt 0
• df(x,t)/dt -?ft(x) ? x(t)
• Change in function value changes by its gradient
  magnitude scaled by speed function f(x)
• df(x,t)/dt -f(x) ?ft(x)

x(2)
f1(x)
x(1)
?ft(x)
f2(x)
x(2)
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Derivation
xi(1)
xi(0)
f(x,0)0

• Level set propagation of a single point on the
  evolving surface
• Witkin-Heckbert 94
• f(x,q) 0, q parameter vector
• Constrain motion of floater particle xi to
  adhere to surface
• How can we connect the speed function to the
  parameterization velocity?

f(x,1)0
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Juxtaposition

• From level sets
• From Witkin-Heckbert
• Stan Jamie, meet Andy Paul
• Now bring in the speed function
• Yielding

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Solution

• Solve via least squares optimization
• Matrix ?qfij(t) is rank deficient
• n floaters interrogating surface
• m parameters (m ltlt n)
• Compute SVD to find m floater particles that
  combine to yield change in parameterization under
  speed func.
• These floater subsets change during evolution

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Algebraic Surface Fitting

• Scattered data fitting
• d(x) vector to nearest data point
• f(x) ?f(x)?? d(x)
• Algebraics
• q algebraic coefficents
• Sphere to teapot
• quartic fit of teapot bowl

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Radial Basis Functions

• Minimal variation function whose implicit surface
  passes through target points
• Use dipoles for normal control
• pairs of target points with target values ?e
  around desired surface
• Requires derivation of dipole propagation under
  sphere function
• Adaptively add dipoles in areas of high curvature
  (as detected by floater particles) and remove
  redundant dipoles

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