Design & Analysis on Manifolds with Irregular Layout

Jorg Peters

IMS-NUS: CdB 17

Support: NSF CCF-0728797 DARPA TRADES ← PhD (postdoc) opening

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---What I would be happy to discuss, but will not present---

SIAC = Smoothness increasing accuracy conservingDG = Discontinuous Galerkin filter → convolution 2

Position-dependent SIAC spline filters for DG

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---What I would be happy to discuss, but will not present---

Design & Analysis on Manifolds with Irregular Layout

Jorg Peters

IMS-NUS 17

Support: NSF CCF-0728797 DARPA TRADES

4

---What I am happy to discuss, and will present---

Design (appearance)

5Straker, styling

Design and Analysis (properties)

6Thin shell elastics of deformation, simulation

Irregular patch layout -- Surface quality

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Farin

Zebra C1 Uniform, parallel → good

A Brief History and Overview of Surface Constructions with Irregular Layout (5 slides)

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Design

Evolution of rendering quality

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Pixel spline

Evolution of surface constructions

10Malcolm Sabin 68 no figures

Evolution of surface quality

Gregory/Zhou Loop et al 11

Irregular patch layoutOverview

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transfinite

Shape, CAD compatibility. degree,...?

Irregular patch layoutTrimmed NURBS

13Automobile styling Autogenerated trim

Analysis & Surface constructionsat irregularities (45 slides)

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Analysis & Surface constructions with Irregular layout

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https://www.cise.ufl.edu/research/SurfLab/pubs.shtml

General Theorem for IGA for Gk

Poisson’s Equation on the disk and other planar domains

New: G1 construction

Analysis & Surface constructions with Irregular layout [NKP13]

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https://www.cise.ufl.edu/research/SurfLab/pubs.shtml

Thin plate

Analysis & Surface constructions with Irregular layout

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Computing with Irregular layout

DDG, triangular macro elements (Powell-Sabin) 18

Computing with Irregular layout

Cirak et al 02, Stam 03, Grinspun&Schroeder 0x, Barendrecht 15, de Goes et al,...Loop subdiv: 19

Catmull-Clark subdivision [CC 78]

20Convex -- saddle `no good’

Analysis & Surface constructions➢ Guided Subdivision (12 slides)

fix the shape (and smoothness) problem

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Guided subdivision [KP07]

Separate shape-finding from mathematical constraints of final output surface

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Guided subdivision [KP07]

Separate shape-finding from mathematical constraints of final output surface

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Tensor border

n=5

Guided subdivision [KP07]

Separate shape-finding from mathematical constraints of final output surface

Guide surface → shape ➢ not exactly fit surrounding surface➢ not nec. smooth everywhere

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Tensor border

n=5

Guided subdivision [KP07]

Separate shape-finding from mathematical constraints of final output surface

Guide surface → shape not exactly fit surrounding surface not nec. smooth

Subdivision rings → flexibility smoothly connect to surface, absorb extra degrees of freedom by closely approximating the guide shape

25Not: gambling with functionals! (similar to surface reconstruction)

Guided subdivision [KP17]

Separate shape finding from mathematical constraints of final output surface

Guide surface Subdivision rings

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Guided subdivision -- what is under the hood

Separate shape finding from mathematical constraints of final output surface

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Corner jet constructor

Guided subdivision -- what is under the hood

Separate shape finding from mathematical constraints of final output surface

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Guided subdivision -- what is under the hood

Separate shape finding from mathematical constraints of final output surface

29Scalable ring

domain subdiv rings

Guided subdivision -- what is under the hood

Separate shape finding from mathematical constraints of final output surface

30scalable

smart sampling !

Range subdiv ring

Smartsampling

Guided subdivision -- Smoothness

Separate shape finding from mathematical constraints of final output surface

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C2 subdivision for irregular layout !!!

Guided subdivision -- implementation via generating functions

Separate shape finding from mathematical constraints of final output surface

n Patches 6 neighbors

Coefficients = control points =nodes of input net

tabulate

tabulate

tabulate

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Stencils

Guided subdivision -- finite caps

Separate shape finding from mathematical constraints of final output surface

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Guided subdivision -- Shape ← obstacle course

Separate shape finding from mathematical constraints of final output surface

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Separate shape finding from mathematical constraints of final output surface

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Guided subdivision -- Shape ← obstacle course

Guided subdivision -- Eigenfunctions (contraction by ½)

Separate shape finding from mathematical constraints of final output surface

36Boundary set to zero

Guided subdivision -- d.o.f. for analysis

Separate shape finding from mathematical constraints of final output surface

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Analysis & Surface constructions➢ Guided Subdivision ➢ Polar constructions (3 slides)

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Computing with Irregular layout

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Polar surface constructions

Surfaces with Polar Structure Karciauskas Peters Dagstuhl 2005

A C² Polar Jet Subdivision Karciauskas, Peters, Eurographics/SGP 2006

Bicubic Polar Subdivision Karciauskas,Peters ACM Transactions on Graphics 2007

Extending Catmull-Clark Subdivision and PCCM with Polar Structures Myles Karciauskas Peters

Pacific Graphics 2007 40

Polar surface constructions

Pairs of Bi-Cubic Surface Constructions Supporting Polar Connectivity Feb. 2008

An introduction to guided and polar surfacing June 2008

Bi-3 C2 Polar Subdivision SIGGRAPH 2009

Finite curvature continuous polar patchworks Karciauskas Peters IMA Surfaces, 2009

C2 Splines Covering Polar Configurations Myles Peters Computer Aided Design , 2011

Multi-degree smooth polar splines: A framework for geometric modeling and isogeometric analysis Toshniwal Speleers Hughes CMAME 2017 41

Analysis & Surface constructions➢ Guided Subdivision ➢ Polar constructions➢ G-splines (4 slides)

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Computing with Irregular layout

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← as many as spline families

G-splines

General Theorem for geometrically continuous constructions

➢ in any number of variables, ➢ for any smoothness, ➢ any manifold (including, of course, planar ones),...

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https://www.cise.ufl.edu/research/SurfLab/pubs.shtml

gIGA for irregularities on manifolds

Finite Element Obstacle course: meshing-less analysis

use spline for geometry & displacement function → g(eneralized) IGA

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gIGA: Layered trivariate manifolds

https://www.cise.ufl.edu/research/SurfLab/pubs.shtml

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https://www.cise.ufl.edu/research/SurfLab/pubs.shtml

https://www.cise.ufl.edu/research/SurfLab/pubs.shtml

gIGA: basis functions

https://www.cise.ufl.edu/research/SurfLab/pubs.shtml

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https://www.cise.ufl.edu/research/SurfLab/pubs.shtml

Analysis & Surface constructions➢ Guided Subdivision ➢ Polar constructions➢ G-splines➢ Corner Collapsed Control Nets (10 slides)

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https://www.cise.ufl.edu/research/SurfLab/pubs.shtml

C1 singular

Computing with Irregular layout: singular parameterization

Refinable $C^1$ spline elements for irregular quad layout Thien Nguyen and Jörg Peters CAGD, 2016

C1 singular

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Refinable C1 spline elements for irregular quad layout Thien Nguyen and Jörg Peters CAGD, 2016

Linearly independent

2⨉2=4 degrees of freedom

per quadrilateral

Computing with Irregular layoutComputing with Irregular layout: singular parameterization

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T-junctions (trivial) B-spline-level mesh

Refinable C1 spline elements for irregular quad layout Thien Nguyen and Jörg Peters CAGD, 2016

➢ Linearly independent➢ 2x2 dof/quad ➢ C^1 proof: Reif 95 ! ➢ poor shape➢ robust

Computing with Irregular layout: singular parameterization

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B-spline-like

Bezier, collapsed

Distribution Statement 53

heat dissipation on thin shell 14K bi-3 pieces, 27.5K spline-dof, 100 time steps =3.6 secs on NVidia 1080 GPU

Spline manifold

Quad mesh

Design and Analysis

Refinable C1 spline elements for irregular quad layout Thien Nguyen and Jörg Peters CAGD, 2016

Computing with Irregular layout: singular parameterization

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➢ Linearly independent➢ 2x2 dof/quad ➢ C1

➢ poor shape ➢ robust➢ refinable ➢ Naturally integrates

with THP splines B-spline-like

Bezier

B-spline-like

Improved shape: SPM 2017

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Refinable C1 spline elements for irregular quad layout Thien Nguyen and Jörg Peters CAGD, 2016

Computing with Irregular layoutComputing with Irregular layout: singular parameterization

Parametrizing singularly to enclose vertices by a smooth parametric surface 91

Degenerate polynomial patches of degree 4 and 5 Neamtu,Pfluger 93

A note on degenerate triangular Bézier patches, Reif 95

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Refinable C1 spline elements for irregular quad layout Thien Nguyen and Jörg Peters CAGD, 2016

➢ Generalizes to m dimensions (m variables)➢ Challenge (recently solved): multi-dimensional irregularities ➢ Projection & proof of C1

Computing with Irregular layoutComputing with Irregular layout: singular parameterization

57Irregular edges, vertices,...

Analysis & Surface constructions on manifolds with irregularities

➢ Guided Subdivision ➢ Polar constructions➢ G-splines➢ Collapsed Control Nets

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C1 singular

THANK YOU

Support: NSF CCF-0728797 DARPA TRADES

Design & Analysis on Manifolds with Irregular Layout

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Evolution ...

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200 × 193 - en.wikipedia.org