Computing IGA-suitable Planar Parameterizations by PolySquare-enhanced

Domain Partition

Shiwei Xiao1, Hongmei Kang2, Xiao-Ming Fu1 , Falai Chen 1

1 University of Science and Technology of China

2 Soochow University , China

Background

• Isogeometry analysis (IGA)

Structural vibrations[Cottrell et al. 2006]

Phase transition phenomena[Gómez et al. 2008]

Shape optimization[Qian et al. 2010]

Shell analysis [Benson et al. 2010]

Background

• Parameterization

Find a mapping 𝐵 𝑢, 𝑣 with the given boundaries 𝑦

𝑢 𝑥

𝑣

𝐵

𝐵−1

Ω0 Ω𝐵 𝑢, 𝑣 : Ω0 = 𝑎, 𝑏 × 𝑐, 𝑑 ↦ Ω

Good parameterization

• Complex and high genus domains

• Injective

• Orthogonal

• Few control points

Good parameterization

• Complex and high genus domains

• Injective

• Orthogonal

• Few control points

Good parameterization

• Complex and high genus domains

• Injective

• Orthogonal

• Few control points

Good parameterization

• Complex and high genus domains

• Injective

• Orthogonal

• Few control points #Con: 5150 #Con: 735

Related work

• IGA-suitable parameterization• Single-patch

[Xu et al. 2013] [Nian et al. 2016]

Related work

• IGA-suitable parameterization• Partition-driven

[Xu et al. 2018][Xu et al. 2015] [Buchegger et al. 2017]

Related work

• Automatic generation of coarse quad patches

• Drawback• Narrow rectangles

• Dependence on the cross field

[Bommes et al. 2011] [Bommes et al. 2013]

[Razafindrazaka et al. 2015]

[Pietroni et al. 2016]

Our contribution

• Efficient and robust method

• IGA-suitable planar parameterizations

• High genus and complex domains

• No extra inputs

Our approachPolySquare-enhanced Domain Partition

Central to our method: using PolySquare structure !

Deformation Pixelation Quad meshing

Subdivision SimplificationParameterization

Deformation

min𝐯𝐸iso + 𝜆 𝐸align

Isometric AMIPS energy[Fu et al. 2015]

Normal alignment energy

Pixelation

Embed the normal-aligned mesh into a planar grid

Pick the pixels which are inside or partially inside the mesh

Obtain the initial PolySquare

Morphological optimization

• Morphological operations: opening & closing [Yu et al. 2014]

𝐸mor ≔ 𝐸simp + 𝛼 𝐸dev

Simplicity metric

Deviation metric

Quad meshing

• Initial back projection

• Foldover elimination

• Quad optimization

• Re-projection and optimization

Quad meshing

• Initial back projection

• Foldover elimination

• Quad optimization

• Re-projection and optimizationProject Obtain

Quad meshing

• Initial back projection

• Foldover elimination

• Quad optimization

• Re-projection and optimization

𝐸fold =

𝑘=1

𝑁𝑓𝑞

𝐽𝑘 𝐹2

det𝐽𝑘 + det𝐽𝑘2 + 휀

Foldover-penalized energy [Escobar et al. 2003]

Quad meshing

• Initial back projection

• Foldover elimination

• Quad optimization

• Re-projection and optimization

Conformal AMIPS energy [Fu et al. 2015]

𝐸conf =

+∞, ∃𝑓𝑘 , det𝐽𝑘 ≤ 0;

1

𝑁𝑓𝑞

𝑘=1

𝑁𝑓𝑞

exp 𝛿𝑘conf , otherwise.

Quad meshing

• Initial back projection

• Foldover elimination

• Quad optimization

• Re-projection and optimization

Boundary-preserving decimation

Boundary-preserving metric

Polychordcollapse operation

𝜌 𝒞 = 𝛽 1 − 𝑒−𝜌𝑞 𝒞 + 1 − 𝛽 1 − 𝑒−𝜌𝑑 𝒞

Geometric loss Area loss

Boundary-preserving decimation20 collapses 70 collapses50 collapses

94 collapses 80 collapses

Layout subdivision and optimization

(a)

Layout subdivision and optimization3 subdivisions2 subdivisions1 subdivision

4 subdivisions5 subdivisions

Parameterization Computation

• Each patch is fitted by a B-spline surface 𝐵 𝑢, 𝑣 ：

• Adjacent patches are stitched with 𝐶0 continuity.

min𝐵 𝑢,𝑣

𝑖=1

𝑁

𝐵 𝑠𝑖 , 𝑡𝑖 − 𝑃𝑖2

Our optimization solver: PAPG

• Problem formulation:

min𝐱𝐸 𝐱

s. t. 𝐴𝐱 = 𝐛

Deformation Quad meshing Subdivision

Our optimization solver: PAPG

• Preconditioned Accelerated Proximal Gradient method (PAPG)

PAPG = APG + preconditioned technique

Find the descent direction 𝐩𝑘 by the quadratic proxy method [Kovalsky et al. 2016]

𝐻 𝐴𝑇

𝐴 0

𝐩𝑘𝛌=−𝛻𝐸 𝐱𝑘𝟎

KKT condition

𝐻: preconditioner,we choose it as mesh Laplacian

[Li et.al 2015]

Superiority of PAPG

time(s)

log 𝐸fold

#iter

(b) APG(a) L-BFGS (c) PAPGlog 𝐸fold

(a) (b)(c)

Experiments

Quality metric of parametrization

𝐽 = 𝐵𝑢 𝐵𝑣 =

𝜕𝑥(𝑢, 𝑣)

𝜕𝑢

𝜕𝑥(𝑢, 𝑣)

𝜕𝑣𝜕𝑦(𝑢, 𝑣)

𝜕𝑢

𝜕𝑦(𝑢, 𝑣)

𝜕𝑣

𝐵 𝑢, 𝑣 = 𝑥 𝑢, 𝑣 , 𝑦 𝑢, 𝑣𝑇∈ 𝑅2

• Scaled Jacobians of 𝐵:

det𝐽 𝑢, 𝑣

𝐵𝑢 𝐵𝑣

• Condition number of 𝐽:

𝐽 𝐹 𝐽−1𝐹

Comparison with [Nian et al. 2016]

[Nian et al. 2016] Ours[Nian et al. 2016]Ours

18 min 30 min29.09 s 8.71 min

Comparison with [Xu et al. 2018]

[Xu et al. 2018] [Xu et al. 2018]

Ours Ours

50.14 s

18.01 s

251.08 s

7.87 s

Conclusion

• An efficient and practical method

• Works for complex and high genus domains

• Higher parameterization quality

• Fewer patches

Discussion

(a) Input (b) SA (c) Ours

Comparison with the SA [Fu et al. 2016] on foldover elimination

Running timeSA: 3.47sOurs: 1.54s

More Discussion

• Minimum scaled Jacobian

More Discussion

• Minimum scaled Jacobian

• Integrate B-splines into domain partition

More Discussion

• Minimum scaled Jacobian

• Integrate B-splines into domain partition

• IGA-suitable volumetric parameterizations

[Xu et al. 2013]

More Discussion

• Minimum scaled Jacobian

• Integrate B-splines into domain partition

• IGA-suitable volumetric parameterizations

• Limitation