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

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!

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

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)

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

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

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!

Examples of Knot Construction

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

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

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

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

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

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!

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

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

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