Refinable C 1 spline elements for irregular quad layout
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1 Refinable C 1 spline elements for irregular quad layout Thien Nguyen Jörg Peters University of Florida NSF CCF , NIH R01-LM T. Nguyen, J. Peters (UF) GMP / 20
2 Outline 1 Refinable, smooth, CAD compatible spline space incl. irregularities 2 Algorithm 3 Applications T. Nguyen, J. Peters (UF) GMP / 20
3 Refinable, smooth, CAD compatible spline space incl. irregularities Outline 1 Refinable, smooth, CAD compatible spline space incl. irregularities 2 Algorithm 3 Applications T. Nguyen, J. Peters (UF) GMP / 20
4 Refinable, smooth, CAD compatible spline space incl. irregularities Challenge: refinable, smooth and CAD compatible multi-sided blends, irregularities T. Nguyen, J. Peters (UF) GMP / 20
5 Refinable, smooth, CAD compatible spline space incl. irregularities Challenge: refinable, smooth and CAD compatible multi-sided blends, irregularities subdivision surface: nested space infinite rings; industrial design infrastructure; integration rules; T. Nguyen, J. Peters (UF) GMP / 20
6 Refinable, smooth, CAD compatible spline space incl. irregularities Challenge: refinable, smooth and CAD compatible multi-sided blends, irregularities subdivision surface: nested space infinite rings; industrial design infrastructure; integration rules; G k spline complex: industrial design infrastructure refinement book keeping (non-local); or: not nested : problem for free-form surfaces! T. Nguyen, J. Peters (UF) GMP / 20
7 Refinable, smooth, CAD compatible spline space incl. irregularities Challenge: refinable, smooth and CAD compatible multi-sided blends, irregularities subdivision surface: nested space infinite rings; industrial design infrastructure; integration rules; G k spline complex: industrial design infrastructure refinement book keeping (non-local); not nested Challenge: combine, for multi-sided configurations, splines with simple nested refinability. T. Nguyen, J. Peters (UF) GMP / 20
8 Refinable, smooth, CAD compatible spline space incl. irregularities Challenge: refinable, smooth and CAD compatible multi-sided blends, irregularities subdivision surface: nested space infinite rings; industrial design infrastructure; integration rules; G k spline complex: industrial design infrastructure refinement book keeping (non-local); not nested singularly parameterized surface nested space, industrial design infrastructure (Peters 91, Neamtu 94) (Reif 97) proves C 1 surface (projection) T. Nguyen, J. Peters (UF) GMP / 20
9 Refinable, smooth, CAD compatible spline space incl. irregularities Challenge: refinable, smooth and CAD compatible multi-sided blends, irregularities subdivision surface: nested space infinite rings; industrial design infrastructure; integration rules; G k spline complex: industrial design infrastructure refinement book keeping (non-local); not nested singularly parameterized surface nested space, industrial design infrastructure (Reif 97) proves C 1 surface (projection) fewer d.o.f. near irregularity than in regular regions T. Nguyen, J. Peters (UF) GMP / 20
10 Refinable, smooth, CAD compatible spline space incl. irregularities Challenge: refinable, smooth and CAD compatible multi-sided blends, irregularities subdivision surface: nested space infinite rings; industrial design infrastructure; integration rules; G k spline complex: industrial design infrastructure refinement book keeping (non-local); not nested singularly parameterized surface nested space, industrial design infrastructure (Reif 97) proves C 1 surface (projection) fewer d.o.f. near irregularity than in regular regions d.o.f. can not be symmetrically distributed as proper control points. T. Nguyen, J. Peters (UF) GMP / 20
11 Refinable, smooth, CAD compatible spline space incl. irregularities Challenge: refinable, smooth and CAD compatible multi-sided blends, irregularities subdivision surface: nested space infinite rings; industrial design infrastructure; integration rules; G k spline complex: industrial design infrastructure refinement book keeping (non-local); not nested singularly parameterized surface nested space, industrial design infrastructure (Reif 97) proves C 1 surface (projection) fewer d.o.f. near irregularity than in regular regions d.o.f. can not be symmetrically distributed as proper control points. Surface shape is poor. T. Nguyen, J. Peters (UF) GMP / 20
12 Refinable, smooth, CAD compatible spline space incl. irregularities 2 2 split construction 2 2 split yields uniform d.o.f.: regardless of vertex valences, each quad has 4 d.o.f.! T. Nguyen, J. Peters (UF) GMP / 20
13 Refinable, smooth, CAD compatible spline space incl. irregularities 2 2 split construction 2 2 split yields uniform d.o.f.: regardless of vertex valences, each quad has 4 d.o.f.! C 1 bi-3 basis functions naturally compatible with bi-cubic PHT refinement T. Nguyen, J. Peters (UF) GMP / 20
14 Algorithm Outline 1 Refinable, smooth, CAD compatible spline space incl. irregularities 2 Algorithm 3 Applications T. Nguyen, J. Peters (UF) GMP / 20
15 Algorithm Algorithm Input Input: B-spline-like control points cij l Recall: regular double-knot bi-3 B-spline coefficients are co-located with inner Bézier coefficients: c b 00 2b 01 42b 10 4b 11 T. Nguyen, J. Peters (UF) GMP / 20
16 Algorithm Algorithm Output Output: Bézier points b k,11 αβ obtained by projection P T. Nguyen, J. Peters (UF) GMP / 20
17 Algorithm Algorithm: PSc Conversion to BB form: copy a k ij := c k ij for i, j {1, 2} make C 1 except a k 00 := n k=1 ck 11 /n. T. Nguyen, J. Peters (UF) GMP / 20
18 Algorithm Algorithm: PSc Conversion to BB form: copy a k ij := c k ij for i, j {1, 2} make C 1 except a k 00 := n k=1 ck 11 /n. Subdivide h k,ij := Sa k, i, j {1, 2}. When i + j > 2 then b k,ij := h k,ij. T. Nguyen, J. Peters (UF) GMP / 20
19 Algorithm Algorithm: PSc Conversion to BB form: copy a k ij := c k ij for i, j {1, 2} make C 1 except a k 00 := n k=1 ck 11 /n. Subdivide h k,ij := Sa k, i, j {1, 2}. When i + j > 2 then b k,ij := h k,ij. Subpatches h αβ := h k,11 αβ : b 11 h 11 b 21 := P h 21 b 12 h 12 For all edges make C 1 : b k 10 := (bk 11 + bk+1 11 /2). T. Nguyen, J. Peters (UF) GMP / 20
20 Properties of the splines Algorithm singular parameterization at irregularities. T. Nguyen, J. Peters (UF) GMP / 20
21 Properties of the splines Algorithm singular parameterization at irregularities. C 1 continuity (Reif 97: invertible Jacobian exists regular reparameterization at 0) T. Nguyen, J. Peters (UF) GMP / 20
22 Properties of the splines Algorithm singular parameterization at irregularities. C 1 continuity (Reif 97: invertible Jacobian exists regular reparameterization at 0) For 1 < p < 4, the L p norms of the main curvatures are finite T. Nguyen, J. Peters (UF) GMP / 20
23 Properties of the splines Algorithm singular parameterization at irregularities. C 1 continuity (Reif 97: invertible Jacobian exists regular reparameterization at 0) For 1 < p < 4, the L p norms of the main curvatures are finite Refinable (nested spaces) PSc P S SPSc P (1) PSc S SPSc (PSPS = SPS since S does not change the projected function) T. Nguyen, J. Peters (UF) GMP / 20
24 Properties of the splines Algorithm singular parameterization at irregularities. C 1 continuity (Reif 97: invertible Jacobian exists regular reparameterization at 0) For 1 < p < 4, the L p norms of the main curvatures are finite Refinable (nested spaces) PSc P S SPSc P (1) PSc S SPSc (PSPS = SPS since S does not change the projected function) Linear independence of f k ij associated c k ij (proof via functionals) T. Nguyen, J. Peters (UF) GMP / 20
25 Applications Outline 1 Refinable, smooth, CAD compatible spline space incl. irregularities 2 Algorithm 3 Applications T. Nguyen, J. Peters (UF) GMP / 20
26 Applications Applications: free-form surfaces T. Nguyen, J. Peters (UF) GMP / 20
27 Applications Applications: free-form surfaces T. Nguyen, J. Peters (UF) GMP / 20
28 Applications Applications: free-form surfaces T. Nguyen, J. Peters (UF) GMP / 20
29 Applications Applications: Poisson local refinement Poisson s equation on the square [0, 6]2. Error -0.08:0.08, T. Nguyen, J. Peters (UF) -0.02:0.02, GMP : / 20
30 Applications Applications: Poisson Error and approximate convergence rate (a.c.r.): close to 2 4 l u u h L 2 a.c.r.. L 2 u u h L a.c.r.. L u u h H 1 a.c.r e e e e e e-04 6 T. Nguyen, J. Peters (UF) GMP / 20
31 Applications: Thin shell Applications Scordelis-Lo roof T. Nguyen, J. Peters (UF) GMP / 20
32 Applications Summary Irregularities in a C 1 bi-3 spline surface T. Nguyen, J. Peters (UF) GMP / 20
33 Applications Summary Irregularities in a C 1 bi-3 spline surface Refinable (nested) T. Nguyen, J. Peters (UF) GMP / 20
34 Applications Summary Irregularities in a C 1 bi-3 spline surface Refinable (nested) Degrees of freedom = as for PHT splines (C 1 bi-3 spline) T. Nguyen, J. Peters (UF) GMP / 20
35 Applications Summary Irregularities in a C 1 bi-3 spline surface Refinable (nested) Degrees of freedom = as for PHT splines (C 1 bi-3 spline) isogeometric analysis T. Nguyen, J. Peters (UF) GMP / 20
36 Applications Summary Irregularities in a C 1 bi-3 spline surface Refinable (nested) Degrees of freedom = as for PHT splines (C 1 bi-3 spline) isogeometric analysis shape T. Nguyen, J. Peters (UF) GMP / 20
37 Applications Summary Irregularities in a C 1 bi-3 spline surface Refinable (nested) Degrees of freedom = as for PHT splines (C 1 bi-3 spline) isogeometric analysis shape Thank You & Questions? T. Nguyen, J. Peters (UF) GMP / 20
38 Applications Summary Irregularities in a C 1 bi-3 spline surface Refinable (nested) Degrees of freedom = as for PHT splines (C 1 bi-3 spline) isogeometric analysis shape Thank You & Questions? T. Nguyen, J. Peters (UF) GMP / 20
39 Applications T. Nguyen, J. Peters (UF) GMP / 20
40 Applications linear independence Nonzero BB coefficients of the basis function f21 l. The coefficient marked additionally with an is nonzero only for f21 l. It is zero for f 12 k or f 11 k, k = 0,..., n 1 and for f21 k, k l. T. Nguyen, J. Peters (UF) GMP / 20
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