T-Splines and T-NURCCs

1
T-Splines and T-NURCCs

• Toby Mitchell
• Main References
• Sederberg et al, T-Splines and T-NURCCs
• Bazilevs et al, Isogeometric Analysis Using
  T-Splines

2
What T-Splines Do

• T-Juntions in NURBS Mesh

• Reduces number of unnecessary control points
• Allows local refinement
• NURBS models
• Subdivision surfaces
• Can merge NURBS patches
• Unlike subdivision surfaces, compatible with
  NURBS (superset of NURBS)
• Much more acceptable to engineering industry!

3
Overview The Main Idea

• Break-Down and Reassemble

• Basis Functions Review

• Start with B-spline or NURBS mesh
• Write in basis function form
• Break apart mesh structure Point-Based (PB)
  splines
• Reassemble a more flexible mesh structure
  T-splines
• Done in terms of basis functions

p3
2/3
Linear interpolation by de Casteljau (Bezier) or
Cox-de Boor (B-spline) can be expanded in terms
of polynomial sum basis functions
4
Start With B-Spline Surfaces
kt4

• B-spline surfaces are tensor products of curves
• Need local knot vectors
• Rewrite basis function

D
Nj
bi,j
kt3
Ni
(s,t)
kt2
j,t
kt1
i,s
ks1
ks2
ks3
ks4
b(s,t)
5
Break into Point-Based (PB) Splines

• Each control point and basis function has own
  knot vectors
• No mesh, points completely self-contained
• bi,j becomes ba a loops over all PB-splines in a
  given set
• Domains should overlap
• One PB-spline point
• Two line
• Need at least 3 for surface
• Domain of surface a subset of the union of all
  domains
• No obvious best choice

ba
t
s
6
Build T-splines from PB-Spline Basis

• Sederbergs Key Insight

• Building Blocks of T-splines

• Can construct mesh-free basis functions that
  still satisfy partition of unity
• Normalized over domain
• Rational, but not NURBS

• Need to impose structure(?)
• Once done, have T-splines
• Evaluate by PB-spline basis
• Same as B-splines, except
• One sum over all control points in domain instead
  of two in each direction

Can represent any polynomial up to the order of
the basis
7
T-Meshes Structure of the Domain

• Define a T-mesh

• Grid of airtight but possibly non-regular
  rectangles Rule 1
• Each edge has a knot value
• Control points at junctions
• Basis functions centered on anchors
• Knot values for basis functions collected along
  rays
• Intersection of ray with edge add knot to local
  vector
• Rule 2 designed to avoid ambiguity in knot
  collection

Not discussed in paper!
8
Examples of Knot Construction
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T-Spline Surfaces

• Evaluating Points on Surface

• Query for all domains that enclose point (s,t)
• Gives all basis functions and points that must be
  summed
• Price for flexibility more complex data
  structure
• T-NURBS
• Group weight with control point
• Replace B-spline basis function with NURBS basis
  functions
• Microwave 3 minutes and serve

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Merging NURBS with T-Splines

• Insert new knots to align knots between patches
• Create T-junctions to stitch patches together
• Average boundary control points across patches
• One row C0 merge
• Three rows C2 merge
• Resulting merge very smooth

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Local Refinement with T-Splines
Figures from Doerfel, Buettler, Simeon,
Adaptive refinement with T-Splines
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Local Refinement for Subdivision Surfaces

• T-NURCCs Catmull-Clark subdivision surfaces with
  non-uniform knots T-junctions
• Do a few global refinements
• Subsequent steps can be purely local (to smooth
  out extraordinary points)
• Shape control available through parameter in
  subdivision rule
• Rather complex rules required

13
Finite Elements with T-Splines

• Local Refinement

• Have a system to model
• Want a solution at a given accuracy level
• Local refinement is tricky in standard methods
  get excess DOFs, expense
• T-splines allow local refinement around features
  of interest
• Big savings?

Regular
T-Spline
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The Biggest Problem With T-Splines?

• Refinement Isnt THAT Local

• Need to keep T-mesh structure with refinement
• Must add new knots besides the ones you actually
  want to add
• Sederberg et al. improved this a bit in a later
  paper
• Better local refinement algorithm, but with
• No termination condition!

15
Local Refinement Test Problem

• Advection-Diffusion Problem

• Pool with steady flow along 45-degree angle
• Pollutant flows in one side and flows out the
  other
• No diffusion line between polluted and
  unpolluted water should stay perfectly sharp
• Requires high refinement, but only along boundary
  layer
• Perfect test for T-splines

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Adaptive Refinement Blow-Up

• Hughes et al Stayed Local

• Doerfel et al Cascade Triggered

FAIL
Good
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Conclusion

• T-splines introduce T-junctions into NURBS
• Reduce complexity by orders of magnitude
• Allow smooth merges of NURBS patches
• Pretty clever, careful formulation props
• BUT local refinement requires more work,
  especially for adaptive refinement