Towards Real-time Simulation of Deformable Objects From mass-spring - PowerPoint PPT Presentation

Towards Real-time Simulation of Deformable Objects From mass-spring system to general hyper-elastic materials GAMES Webinar Presentation Tiantian Liu Joint Work with: Adam Bargteil, Sofien Bouaziz, Ladislav Kavan, Sebastian Martin, James O’Brien, Mark Pauly

Towards Real-time Simulation of Deformable Objects From mass-spring system to general hyper-elastic materials

Towards Real-time Simulation of Deformable Objects 3

[Assassin's Creed II, Ubisoft, 2012] Real-time Physics Towards Real-time Simulation of Deformable Objects 4

Off-line Physics Towards Real-time Simulation of Deformable Objects 6

[VirtaMed] Applications with non-negotiable latency and accuracy E.g. Virtual Surgery Towards Real-time Simulation of Deformable Objects 8

Goal: Fast simulation of general hyperelastic materials Towards Real-time Simulation of Deformable Objects 9

Goal: Fast simulation of general hyperelastic materials Simple Towards Real-time Simulation of Deformable Objects 10

Related Work: Classic work [Goldenthal et al. 2007] [Tournier et al. 2015] [Baraff and Witkin 1998] Towards Real-time Simulation of Deformable Objects 11

Related Work: Position Based Dynamics [Müller et al. 2007] [Macklin et al. 2016] Quasi-Newton Methods for Real-time Simulation of Hyperelastic Materials 12

Related Work: Projective Dynamics [Liu et al. 2013] [Bouaziz et al. 2014] [Narain et al. 2016] Quasi-Newton Methods for Real-time Simulation of Hyperelastic Materials 13

Related Work: Chebyshev Methods [Wang 2015] [Wang and Yang 2016] Quasi-Newton Methods for Real-time Simulation of Hyperelastic Materials 14

Related Work: Quasi-Newton Methods in Geometry Processing [Kovalsky et al. 2016] [Rabinovich et al. 2017] Quasi-Newton Methods for Real-time Simulation of Hyperelastic Materials 15

Quasi-Newton Methods 𝛼 2 𝒉 𝒚 −1 ∆𝒚 = − 𝛼𝒉 𝒚 𝑩 𝒉 𝒚 𝒚 Towards Real-time Simulation of Deformable Objects 16

Spatial Discretization Towards Real-time Simulation of Deformable Objects 17

Temporal Discretization Already known 𝑧 𝑦 0 𝑦 ℎ = 33𝑛𝑡 Time Axis Towards Real-time Simulation of Deformable Objects 18

Implicit Euler Time Integration 𝑧 𝑦 0 𝑦 ℎ = 33𝑛𝑡 Time Axis 1 2 𝑦 − 𝑧 𝑈 𝑵 𝑦 − 𝑧 + ℎ 2 𝐹(𝑦) min 𝑦 Towards Real-time Simulation of Deformable Objects 19

Variational Implicit Euler inertial potential Elastic potential 𝑧 …pure inertial motion (Newton’s 1 st Law) 𝑧 = 𝑦 𝑜 + ℎ𝑤 𝑜 1 2 𝑦 − 𝑧 𝑈 𝑵 𝑦 − 𝑧 + ℎ 2 𝐹(𝑦) min 𝐹 …elastic potential (energy) 𝑦 Towards Real-time Simulation of Deformable Objects 20

Variational Implicit Euler 1 2 𝑦 − 𝑧 𝑈 𝑵 𝑦 − 𝑧 + ℎ 2 𝐹(𝑦) min 𝑦 inertial potential Elastic potential Implicit Euler: Compromise between inertia and elasticity Towards Real-time Simulation of Deformable Objects 21

Mass-spring System: Basis Hooke’s Law: 𝐹 𝒒 𝟐 , 𝒒 𝟑 = 1 𝒒 𝟐 − 𝒒 𝟑 − 𝑠 2 2 𝑙 Non-quadratic Non-convex 𝒒 2 𝑠 𝒒 1 Towards Real-time Simulation of Deformable Objects 22

Non-convex Potential rest length 𝐹( 1 − 𝑢 𝒃 + 𝑢𝒄) 𝒃 𝟏. 𝟔𝒃 + 𝟏. 𝟔𝒄 𝒄 Towards Real-time Simulation of Deformable Objects 23

Standard Solution: Newton’s Method 1 2 𝑦 − 𝑧 𝑈 𝑁 𝑦 − 𝑧 + ℎ 2 𝐹(𝑦) min 𝑦     Slow  𝛼 2 𝐹 depends on 𝑦   Non-convex  The Hessian 𝑁 + ℎ 2 𝛼 2 𝐹 can be indefinite Towards Real-time Simulation of Deformable Objects 24

Standard Solution: Newton’s Method 𝐹( 1 − 𝑢 𝒃 + 𝑢𝒄) 𝒃 𝟏. 𝟔𝒃 + 𝟏. 𝟔𝒄 𝒄 Towards Real-time Simulation of Deformable Objects 25

Ideal Problem Reformulation Large Convex Quadratic Problem (Ideally with Constant System Matrix) Many Small Non-convex Problems (Ideally Independent) Towards Real-time Simulation of Deformable Objects 26

Hooke’s Law with auxiliary variables  For the i-th spring: 1 𝑗 2  𝐹 𝑗 𝒚 = 2 𝑙 𝑗 𝒒 𝑗1 − 𝒒 𝑗2 − 𝑠  Introduce auxiliary variable 𝒆 𝑗 where 𝒆 𝑗 = 𝑠 𝑗 𝒆 𝑗 𝒒 𝑗2 𝒒 𝑗1 Towards Real-time Simulation of Deformable Objects 27

Hooke’s Law with auxiliary variables 1 1 2 𝑗 2  min 2 𝑙 𝑗 𝒒 𝑗1 − 𝒒 𝑗2 − 𝒆 𝑗 = 2 𝑙 𝑗 𝒒 𝑗1 − 𝒒 𝑗2 − 𝑠 𝒆 𝑗 =𝑠 𝑗 𝑞 𝑗1 −𝑞 𝑗2  When 𝑒 𝑗 = 𝑠 𝑗 𝑞 𝑗1 −𝑞 𝑗2 𝒆 𝑗 𝒒 𝑗2 𝒒 𝑗1 Towards Real-time Simulation of Deformable Objects 28

Hooke’s Law with auxiliary variables 1 2 𝐹 𝒚 = min 2 𝑙 𝑗 𝒒 𝑗1 − 𝒒 𝑗2 − 𝒆 𝑗 𝒆 𝑗 =𝑠 𝑗 𝑗 1 2 𝐹 𝒚 = min 2 𝑙 𝑗 𝒒 𝑗1 − 𝒒 𝑗2 − 𝒆 𝑗 𝒆∈ℳ 𝑗 𝒒 𝑗2 𝒒 𝑗1 Towards Real-time Simulation of Deformable Objects 29

Variational Time Integration with Auxiliary Variable 1 2 𝑦 − 𝑧 𝑈 𝑵 𝑦 − 𝑧 + ℎ 2 𝐹(𝑦) min 𝑦 1 2 𝒚 𝑈 𝑩𝒚 + 𝒚 𝑈 𝑪𝒆 + 𝒅 , 𝑡. 𝑢. 𝒆 ∈ 𝓝 min 𝑦,𝑒 𝒆 ∈ 𝓝 𝒆 𝒋 = 𝑠 𝑗 Towards Real-time Simulation of Deformable Objects 30

Optimization 1 2 𝒚 𝑈 𝑩𝒚 + 𝒚 𝑈 𝑪𝒆 + 𝒅 , 𝑡. 𝑢. 𝒆 ∈ ℳ min 𝑦,𝑒  𝑩, 𝑪, 𝒅 does not depend on 𝒚 or 𝒆  If we fix 𝒚 -> easy to solve for 𝒆  If we fix 𝒆 -> easy to solve for 𝒚  Invites alternate solver (local/global) Towards Real-time Simulation of Deformable Objects 31

Local Step  For each spring, project to unit length using the current 𝒚 to find 𝒆 𝒋  Trivially Parallelizable 𝒆 𝑗 𝒒 𝑗2 𝒒 𝑗1 Towards Real-time Simulation of Deformable Objects 32

Global Step 1 2 𝒚 𝑈 𝑩𝒚 + 𝒚 𝑈 𝑪𝒆 + 𝒅 , 𝑡. 𝑢. 𝒆 ∈ ℳ min 𝑦,𝑒 𝐺𝑗𝑦 𝒆: 𝒚 ∗ = −𝑩 −1 (𝑪𝒆 + 𝒅)  Matrix 𝑩 is:  Independent of 𝒚 and 𝒆 (Constant)  Positive Definite  Thus can be pre-factorized (using e.g. Cholesky) Towards Real-time Simulation of Deformable Objects 33

Alternating Solver Large Convex Quadratic Problem (with Constant System Matrix) Many Small Non-convex Problems Towards Real-time Simulation of Deformable Objects 34

Performance Our Method Newton’s Method Towards Real-time Simulation of Deformable Objects 35

Performance Newton’s Method Our Method Towards Real-time Simulation of Deformable Objects 36

Remark: Fast Mass-spring Systems 1 2 𝑦 − 𝑧 𝑈 𝑵 𝑦 − 𝑧 + ℎ 2 𝐹(𝑦) min 𝑦 1 = 1 2 𝑗 2 min 2 𝑙 𝑗 𝒒 𝑗1 − 𝒒 𝑗2 − 𝒆 𝑗 2 𝑙 𝑗 𝒒 𝑗1 − 𝒒 𝑗2 − 𝑠 𝒆 𝑗 =𝑠 𝑗 1 2 𝒚 𝑈 𝑩𝒚 + 𝒚 𝑈 𝑪𝒆 + 𝒅 , 𝑡. 𝑢. 𝒆 ∈ 𝓝 min 𝑦,𝑒 Towards Real-time Simulation of Deformable Objects 37

Beyond Mass-spring Systems 𝒒 𝒋𝟐 − 𝒒 𝒋𝟑 𝒆 𝑗 𝒒 𝑗2 𝒒 𝑗1 𝒆 𝑗 Towards Real-time Simulation of Deformable Objects 38

Distance from Constraint Manifold 𝒒 𝒋𝟐 − 𝒒 𝒋𝟑 : naïve differential operator 0 0 ⋮ 1 𝒋𝟐 ⋮ 𝒒 𝒋𝟐 − 𝒒 𝒋𝟑 = 𝑯 𝒋 𝒚 −1 𝒋𝟑 ⋮ 0 0 𝑯 𝒋 Towards Real-time Simulation of Deformable Objects 39

Distance from Constraint Manifold 𝒒 𝒋𝟐 − 𝒒 𝒋𝟑 = 𝑯 𝒋 𝒚 1 2 𝐹 𝒚 = min 2 𝑙 𝑗 𝒒 𝑗1 − 𝒒 𝑗2 − 𝒆 𝑗 𝒆∈ℳ 𝑗 2 𝐹 𝒚 = min 𝑥 𝑗 𝑯 𝒋 𝒚 − 𝒆 𝑗 𝒆∈ℳ 𝑗 Towards Real-time Simulation of Deformable Objects 40

Deformation Gradient 𝑯𝒚 Rest pose 𝒀 Current pose 𝒚 Towards Real-time Simulation of Deformable Objects 41

Distance from Constraint Manifold 2 𝐹 𝒚 = min 𝑥 𝑗 𝑯 𝒋 𝒚 − 𝒆 𝑗 𝒆∈ℳ 𝑗 𝑯 𝒋 𝒚 𝑯 𝒋 𝒚 𝒆 𝒋 𝒆 𝑗 Towards Real-time Simulation of Deformable Objects 42

Intuitive Projection Manifold: SO(3)  SO(3) … Best Fit Rotation Matrix  “As Rigid As Possible” [Chao et al. 2010] Towards Real-time Simulation of Deformable Objects 43

Intuitive Projection Manifold: SL(3)  SL(3) … Group of Matrices with det = 1  Volume Preservation Towards Real-time Simulation of Deformable Objects 44

Other Constraint Manifolds (Example Based) Towards Real-time Simulation of Deformable Objects 45

Other Constraint Manifolds (Laplace-Beltrami operator) Towards Real-time Simulation of Deformable Objects 46

Remark: Projective Dynamics 2 𝐹 𝒚, 𝒆 = min 𝑥 𝑗 𝑯 𝒋 𝒚 − 𝒆 𝑗 𝑡. 𝑢. 𝒆 ∈ ℳ 𝒚,𝒆 𝑗 1 2 𝒚 𝑈 𝑩𝒚 + 𝒚 𝑈 (𝑪𝒆 + 𝒅) min 𝒚,𝒆 𝑕 𝒚, 𝒆 = min 𝑡. 𝑢. 𝒆 ∈ ℳ 𝒚,𝒆  Like before, 𝑩, 𝑪, 𝒅 does not depend on 𝒚 and 𝒆  If we fix 𝒚 -> easy to solve for 𝒆 : Projection  If we fix 𝒆 -> easy to solve for 𝒚 : 𝒚 ∗= −𝑩 −1 (𝑪𝒆 + 𝒅) Towards Real-time Simulation of Deformable Objects 47