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
p=3
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
Sederberg’s 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

9. T-Spline Surfaces Evaluating Points on Surface T-NURBS:
Point (s,t)
Relevant control point
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

10. 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

11. Local Refinement with T-Splines
Figures from Doerfel, Buettler, & Simeon, Adaptive refinement with T-Splines

12. 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

14. The Biggest Problem With T-Splines?
Refinement Isn’t 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

16. Adaptive Refinement Blow-Up
Hughes et al: Stayed Local
Doerfel et al: Cascade Triggered
FAIL
Good

17. 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