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