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

Jul 20, 2023 252 likes โ€ข1k views

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

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SLIDE 1

Towards Real-time Simulation

  • f 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 Oโ€™Brien, Mark Pauly

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SLIDE 2

Towards Real-time Simulation

  • f Deformable Objects

From mass-spring system to general hyper-elastic materials

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SLIDE 3

Towards Real-time Simulation of Deformable Objects

3

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Real-time Physics

4

[Assassin's Creed II, Ubisoft, 2012]

Towards Real-time Simulation of Deformable Objects

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SLIDE 5

Off-line Physics

6 Towards Real-time Simulation of Deformable Objects

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SLIDE 6

Applications with non-negotiable latency and accuracy

E.g. Virtual Surgery

8 Towards Real-time Simulation of Deformable Objects

[VirtaMed]

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SLIDE 7

Goal: Fast simulation of general hyperelastic materials

9 Towards Real-time Simulation of Deformable Objects

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SLIDE 8

Goal: Fast simulation of general hyperelastic materials

Towards Real-time Simulation of Deformable Objects 10

Simple

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SLIDE 9

Related Work: Classic work

Towards Real-time Simulation of Deformable Objects 11

[Baraff and Witkin 1998] [Goldenthal et al. 2007] [Tournier et al. 2015]

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SLIDE 10

Related Work: Position Based Dynamics

12 Quasi-Newton Methods for Real-time Simulation of Hyperelastic Materials

[Mรผller et al. 2007] [Macklin et al. 2016]

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SLIDE 11

Related Work: Projective Dynamics

13 Quasi-Newton Methods for Real-time Simulation of Hyperelastic Materials

[Liu et al. 2013] [Bouaziz et al. 2014] [Narain et al. 2016]

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SLIDE 12

Related Work: Chebyshev Methods

14 Quasi-Newton Methods for Real-time Simulation of Hyperelastic Materials

[Wang 2015] [Wang and Yang 2016]

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SLIDE 13

Related Work: Quasi-Newton Methods in Geometry Processing

15 Quasi-Newton Methods for Real-time Simulation of Hyperelastic Materials

[Kovalsky et al. 2016] [Rabinovich et al. 2017]

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SLIDE 14

๐›ผ๐’‰ ๐’š

๐’š

Quasi-Newton Methods

Towards Real-time Simulation of Deformable Objects 16

๐’‰ ๐’š

๐›ผ2๐’‰ ๐’š โˆ†๐’š = โˆ’

โˆ’1

๐‘ฉ

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SLIDE 15

Spatial Discretization

17 Towards Real-time Simulation of Deformable Objects

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SLIDE 16

Temporal Discretization

18

๐‘ฆ0 โ„Ž = 33๐‘›๐‘ก Time Axis ๐‘ฆ ๐‘ง

Already known

Towards Real-time Simulation of Deformable Objects

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SLIDE 17

Implicit Euler Time Integration

19

๐‘ฆ0 โ„Ž = 33๐‘›๐‘ก Time Axis ๐‘ฆ ๐‘ง min

๐‘ฆ

1 2 ๐‘ฆ โˆ’ ๐‘ง ๐‘ˆ๐‘ต ๐‘ฆ โˆ’ ๐‘ง + โ„Ž2๐น(๐‘ฆ)

Towards Real-time Simulation of Deformable Objects

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SLIDE 18

๐นโ€ฆelastic potential (energy)

Variational Implicit Euler

20

min

๐‘ฆ

1 2 ๐‘ฆ โˆ’ ๐‘ง ๐‘ˆ๐‘ต ๐‘ฆ โˆ’ ๐‘ง + โ„Ž2๐น(๐‘ฆ) ๐‘งโ€ฆpure inertial motion (Newtonโ€™s 1st Law) ๐‘ง = ๐‘ฆ๐‘œ + โ„Ž๐‘ค๐‘œ

inertial potential Elastic potential

Towards Real-time Simulation of Deformable Objects

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SLIDE 19

Variational Implicit Euler

21

min

๐‘ฆ

1 2 ๐‘ฆ โˆ’ ๐‘ง ๐‘ˆ๐‘ต ๐‘ฆ โˆ’ ๐‘ง + โ„Ž2๐น(๐‘ฆ) Implicit Euler:

Compromise between inertia and elasticity

inertial potential Elastic potential

Towards Real-time Simulation of Deformable Objects

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SLIDE 20

Mass-spring System: Basis

22 Towards Real-time Simulation of Deformable Objects

Hookeโ€™s Law: ๐น ๐’’๐Ÿ, ๐’’๐Ÿ‘ = 1 2 ๐‘™ ๐’’๐Ÿ โˆ’ ๐’’๐Ÿ‘ โˆ’ ๐‘  2

๐’’1 ๐’’2 ๐‘ 

Non-quadratic Non-convex

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SLIDE 21

Non-convex Potential

Towards Real-time Simulation of Deformable Objects 23

๐น( 1 โˆ’ ๐‘ข ๐’ƒ + ๐‘ข๐’„) ๐’„ ๐’ƒ rest length ๐Ÿ. ๐Ÿ”๐’ƒ + ๐Ÿ. ๐Ÿ”๐’„

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SLIDE 22

Standard Solution: Newtonโ€™s Method

Towards Real-time Simulation of Deformable Objects 24

min

๐‘ฆ

1 2 ๐‘ฆ โˆ’ ๐‘ง ๐‘ˆ๐‘ ๐‘ฆ โˆ’ ๐‘ง + โ„Ž2๐น(๐‘ฆ)

๏Š ๏Œ ๏Œ ๏ต Slow

๏ต ๐›ผ2๐น depends on ๐‘ฆ

๏ต Non-convex

๏ต The Hessian ๐‘ + โ„Ž2๐›ผ2๐น can be indefinite

๏Œ

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SLIDE 23

Standard Solution: Newtonโ€™s Method

Towards Real-time Simulation of Deformable Objects 25

๐น( 1 โˆ’ ๐‘ข ๐’ƒ + ๐‘ข๐’„) ๐Ÿ. ๐Ÿ”๐’ƒ + ๐Ÿ. ๐Ÿ”๐’„ ๐’„ ๐’ƒ

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SLIDE 24

Ideal Problem Reformulation

Towards Real-time Simulation of Deformable Objects 26

Large Convex Quadratic Problem (Ideally with Constant System Matrix) Many Small Non-convex Problems (Ideally Independent)

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SLIDE 25

Hookeโ€™s Law with auxiliary variables

Towards Real-time Simulation of Deformable Objects 27

๏ตFor the i-th spring: ๏ต๐น๐‘— ๐’š =

1 2 ๐‘™๐‘—

๐’’๐‘—1 โˆ’ ๐’’๐‘—2 โˆ’ ๐‘ 

๐‘— 2

๏ตIntroduce auxiliary variable ๐’†๐‘— where ๐’†๐‘—

= ๐‘ 

๐‘—

๐’’๐‘—1 ๐’’๐‘—2 ๐’†๐‘—

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SLIDE 26

Hookeโ€™s Law with auxiliary variables

Towards Real-time Simulation of Deformable Objects 28

๐’’๐‘—1 ๐’’๐‘—2 ๐’†๐‘—

๏ต min

๐’†๐‘— =๐‘ ๐‘— 1 2 ๐‘™๐‘—

๐’’๐‘—1 โˆ’ ๐’’๐‘—2 โˆ’ ๐’†๐‘—

2

=

1 2 ๐‘™๐‘—

๐’’๐‘—1 โˆ’ ๐’’๐‘—2 โˆ’ ๐‘ 

๐‘— 2

๏ตWhen ๐‘’๐‘— = ๐‘ 

๐‘— ๐‘ž๐‘—1โˆ’๐‘ž๐‘—2 ๐‘ž๐‘—1โˆ’๐‘ž๐‘—2

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SLIDE 27

Hookeโ€™s Law with auxiliary variables

Towards Real-time Simulation of Deformable Objects 29

๐’’๐‘—1 ๐’’๐‘—2

๐น ๐’š =

๐‘—

min

๐’†๐‘— =๐‘ ๐‘—

1 2 ๐‘™๐‘— ๐’’๐‘—1 โˆ’ ๐’’๐‘—2 โˆ’ ๐’†๐‘—

2

๐น ๐’š = min

๐’†โˆˆโ„ณ ๐‘—

1 2 ๐‘™๐‘— ๐’’๐‘—1 โˆ’ ๐’’๐‘—2 โˆ’ ๐’†๐‘—

2

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SLIDE 28

Variational Time Integration with Auxiliary Variable

Towards Real-time Simulation of Deformable Objects 30

min

๐‘ฆ

1 2 ๐‘ฆ โˆ’ ๐‘ง ๐‘ˆ๐‘ต ๐‘ฆ โˆ’ ๐‘ง + โ„Ž2๐น(๐‘ฆ) min

๐‘ฆ,๐‘’

1 2 ๐’š๐‘ˆ๐‘ฉ๐’š + ๐’š๐‘ˆ ๐‘ช๐’† + ๐’… , ๐‘ก. ๐‘ข. ๐’† โˆˆ ๐“ ๐’† โˆˆ ๐“ ๐’†๐’‹ = ๐‘ 

๐‘—

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SLIDE 29

Optimization

Towards Real-time Simulation of Deformable Objects 31

min

๐‘ฆ,๐‘’

1 2 ๐’š๐‘ˆ๐‘ฉ๐’š + ๐’š๐‘ˆ ๐‘ช๐’† + ๐’… , ๐‘ก. ๐‘ข. ๐’† โˆˆ โ„ณ

๏ต ๐‘ฉ, ๐‘ช, ๐’… does not depend on ๐’š or ๐’† ๏ต If we fix ๐’š -> easy to solve for ๐’† ๏ต If we fix ๐’† -> easy to solve for ๐’š ๏ต Invites alternate solver (local/global)

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SLIDE 30

Local Step

Towards Real-time Simulation of Deformable Objects 32

๐’’๐‘—1 ๐’’๐‘—2 ๐’†๐‘—

๏ตFor each spring, project to unit length using the

current ๐’š to find ๐’†๐’‹

๏ตTrivially Parallelizable

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SLIDE 31

Global Step

Towards Real-time Simulation of Deformable Objects 33

min

๐‘ฆ,๐‘’

1 2 ๐’š๐‘ˆ๐‘ฉ๐’š + ๐’š๐‘ˆ ๐‘ช๐’† + ๐’… , ๐‘ก. ๐‘ข. ๐’† โˆˆ โ„ณ

๏ตMatrix ๐‘ฉ is: ๏ตIndependent of ๐’š and ๐’† (Constant) ๏ตPositive Definite ๏ตThus can be pre-factorized (using e.g. Cholesky)

๐บ๐‘—๐‘ฆ ๐’†: ๐’šโˆ— = โˆ’๐‘ฉโˆ’1(๐‘ช๐’† + ๐’…)

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SLIDE 32

Alternating Solver

Towards Real-time Simulation of Deformable Objects 34

Large Convex Quadratic Problem (with Constant System Matrix) Many Small Non-convex Problems

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SLIDE 33

Performance

Towards Real-time Simulation of Deformable Objects 35

Our Method Newtonโ€™s Method

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SLIDE 34

Performance

Towards Real-time Simulation of Deformable Objects 36

Our Method Newtonโ€™s Method

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SLIDE 35

Remark: Fast Mass-spring Systems

Towards Real-time Simulation of Deformable Objects 37

min

๐‘ฆ

1 2 ๐‘ฆ โˆ’ ๐‘ง ๐‘ˆ๐‘ต ๐‘ฆ โˆ’ ๐‘ง + โ„Ž2๐น(๐‘ฆ) min

๐‘ฆ,๐‘’

1 2 ๐’š๐‘ˆ๐‘ฉ๐’š + ๐’š๐‘ˆ ๐‘ช๐’† + ๐’… , ๐‘ก. ๐‘ข. ๐’† โˆˆ ๐“ min

๐’†๐‘— =๐‘ ๐‘—

1 2 ๐‘™๐‘— ๐’’๐‘—1 โˆ’ ๐’’๐‘—2 โˆ’ ๐’†๐‘—

2

= 1 2 ๐‘™๐‘— ๐’’๐‘—1 โˆ’ ๐’’๐‘—2 โˆ’ ๐‘ 

๐‘— 2

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SLIDE 36

Beyond Mass-spring Systems

Towards Real-time Simulation of Deformable Objects 38

๐’’๐‘—1 ๐’’๐‘—2 ๐’†๐‘— ๐’’๐’‹๐Ÿ โˆ’ ๐’’๐’‹๐Ÿ‘ ๐’†๐‘—

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SLIDE 37

Distance from Constraint Manifold

Towards Real-time Simulation of Deformable Objects 39

๐’’๐’‹๐Ÿ โˆ’ ๐’’๐’‹๐Ÿ‘: naรฏve differential operator ๐’’๐’‹๐Ÿ โˆ’ ๐’’๐’‹๐Ÿ‘ = ๐‘ฏ๐’‹๐’š โ‹ฎ 1 โ‹ฎ โˆ’1 โ‹ฎ ๐‘ฏ๐’‹ ๐’‹๐Ÿ ๐’‹๐Ÿ‘

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SLIDE 38

Distance from Constraint Manifold

Towards Real-time Simulation of Deformable Objects 40

๐’’๐’‹๐Ÿ โˆ’ ๐’’๐’‹๐Ÿ‘ = ๐‘ฏ๐’‹๐’š ๐น ๐’š = min

๐’†โˆˆโ„ณ ๐‘—

1 2 ๐‘™๐‘— ๐’’๐‘—1 โˆ’ ๐’’๐‘—2 โˆ’ ๐’†๐‘—

2

๐น ๐’š = min

๐’†โˆˆโ„ณ ๐‘—

๐‘ฅ๐‘— ๐‘ฏ๐’‹๐’š โˆ’ ๐’†๐‘—

2

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SLIDE 39

Deformation Gradient

Towards Real-time Simulation of Deformable Objects 41

Rest pose ๐’€ Current pose ๐’š ๐‘ฏ๐’š

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SLIDE 40

๐น ๐’š = min

๐’†โˆˆโ„ณ ๐‘—

๐‘ฅ๐‘— ๐‘ฏ๐’‹๐’š โˆ’ ๐’†๐‘—

2

Distance from Constraint Manifold

Towards Real-time Simulation of Deformable Objects 42

๐‘ฏ๐’‹๐’š ๐’†๐’‹

๐‘ฏ๐’‹๐’š ๐’†๐‘—

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SLIDE 41

Intuitive Projection Manifold: SO(3)

43

๏ตSO(3) โ€ฆ Best Fit Rotation Matrix ๏ตโ€œAs Rigid As Possibleโ€ [Chao et al. 2010]

Towards Real-time Simulation of Deformable Objects

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SLIDE 42

Intuitive Projection Manifold: SL(3)

44

๏ตSL(3) โ€ฆ Group of Matrices with det = 1 ๏ตVolume Preservation

Towards Real-time Simulation of Deformable Objects

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SLIDE 43

Other Constraint Manifolds (Example Based)

Towards Real-time Simulation of Deformable Objects 45

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SLIDE 44

Other Constraint Manifolds (Laplace-Beltrami operator)

Towards Real-time Simulation of Deformable Objects 46

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SLIDE 45

Remark: Projective Dynamics

Towards Real-time Simulation of Deformable Objects 47

๐น ๐’š, ๐’† = min

๐’š,๐’† ๐‘—

๐‘ฅ๐‘— ๐‘ฏ๐’‹๐’š โˆ’ ๐’†๐‘—

2

๐‘ก. ๐‘ข. ๐’† โˆˆ โ„ณ min

๐’š,๐’† ๐‘• ๐’š, ๐’† = min ๐’š,๐’†

1 2 ๐’š๐‘ˆ๐‘ฉ๐’š + ๐’š๐‘ˆ(๐‘ช๐’† + ๐’…) ๐‘ก. ๐‘ข. ๐’† โˆˆ โ„ณ

๏ต Like before, ๐‘ฉ, ๐‘ช, ๐’… does not depend on ๐’š and ๐’† ๏ต If we fix ๐’š -> easy to solve for ๐’†: Projection ๏ต If we fix ๐’† -> easy to solve for ๐’š: ๐’š โˆ—= โˆ’๐‘ฉโˆ’1(๐‘ช๐’† + ๐’…)

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SLIDE 46

Problems: Projective Dynamics

Towards Real-time Simulation of Deformable Objects 48

๐น ๐’š, ๐’† = min

๐’š,๐’† ๐‘—

๐‘ฅ๐‘— ๐‘ฏ๐’‹๐’š โˆ’ ๐’†๐‘—

2

๐‘ก. ๐‘ข. ๐’† โˆˆ โ„ณ min

๐’š,๐’† ๐‘• ๐’š, ๐’† = min ๐’š,๐’†

1 2 ๐’š๐‘ˆ๐‘ฉ๐’š + ๐’š๐‘ˆ(๐‘ช๐’† + ๐’…) ๐‘ก. ๐‘ข. ๐’† โˆˆ โ„ณ

๏ต Like before, ๐‘ฉ, ๐‘ช, ๐’… does not depend on ๐’š and ๐’† ๏ต If we fix ๐’š -> easy to solve for ๐’†: Projection ๏ต If we fix ๐’† -> easy to solve for ๐’š: ๐’š โˆ—= โˆ’๐‘ฉโˆ’1(๐‘ช๐’† + ๐’…)

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SLIDE 47

Spline-Based Materials [Xu et al. 2015]

49

Soft ARAP Stiff ARAP

Polynomial Material [Xu et al. 2015]

Towards Real-time Simulation of Deformable Objects

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SLIDE 48

How to generalize Projective Dynamics?

50 Towards Real-time Simulation of Deformable Objects

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SLIDE 49

Reformulation of Projective Dynamics

51

min

๐’š,๐’†

1 2 ๐’š๐‘ˆ๐‘ฉ๐’š + ๐’š๐‘ˆ(๐‘ช๐’† + ๐’…) ๐‘ก. ๐‘ข. ๐’† โˆˆ โ„ณ min

๐’š

1 2 ๐’š๐‘ˆ๐‘ฉ๐’š + ๐’š๐‘ˆ(๐‘ช๐’†(๐’š) + ๐’…) ๐’‰(๐’š)

Towards Real-time Simulation of Deformable Objects

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SLIDE 50

Reformulation of Projective Dynamics

52

min

๐’š

1 2 ๐’š๐‘ˆ๐‘ฉ๐’š + ๐’š๐‘ˆ(๐‘ช๐’†(๐’š) + ๐’…) ๐’‰(๐’š) ๐›ผ๐’‰ ๐’š = ๐‘ฉ๐’š + ๐‘ช๐’† ๐’š + ๐’… + ๐‘ช๐›ผ๐’†(๐’š) ๐‘ˆ๐’š

Towards Real-time Simulation of Deformable Objects

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SLIDE 51

Projection Differential

53

๐œ€ ๐‘ฏ๐’š โˆ’ ๐’† ๐’š

2 = ๐‘ฏ๐’š โˆ’ ๐’† ๐’š ๐‘ˆ๐ป๐œ€๐’š

โˆ’๐œ€๐’† ๐’š ๐‘ˆ ๐‘ฏ๐’š โˆ’ ๐’† ๐’š ๐‘ฏ๐’š ๐’†(๐’š) ๐œ€๐’† ๐’š

Towards Real-time Simulation of Deformable Objects

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SLIDE 52

Reformulation of Projective Dynamics

54

min

๐’š

1 2 ๐’š๐‘ˆ๐‘ฉ๐’š + ๐’š๐‘ˆ(๐‘ช๐’†(๐’š) + ๐’…) ๐’‰(๐’š) ๐›ผ๐’‰ ๐’š = ๐‘ฉ๐’š + ๐‘ช๐’† ๐’š + ๐’… + ๐‘ช๐›ผ๐’†(๐’š) ๐‘ˆ๐’š ๐โˆ’1๐›ผ๐’‰ ๐’š = ๐’š + ๐‘ฉโˆ’1 ๐‘ช๐’† ๐’š + ๐’… โˆ’๐’šโˆ— ๐’šโˆ— = ๐’š โˆ’ ๐‘ฉโˆ’1๐›ผ๐’‰ ๐’š

Towards Real-time Simulation of Deformable Objects

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SLIDE 53

55

๐’šโˆ— = ๐’š โˆ’ ๐‘ฉโˆ’1๐›ผ๐’‰ ๐’š

๐’šโˆ— = ๐’š โˆ’ ๐œท ๐›ผ2๐’‰(๐’š)

โˆ’1๐›ผ๐’‰ ๐’š

Compare to one Newton step:

Reformulation of Projective Dynamics

๏ต ๐›ฝ: Step size, usually decided by linesearch, typical value is 1.

Towards Real-time Simulation of Deformable Objects

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SLIDE 54

Quasi-Newton Formulation

56

๐’š โˆ’ ๐›ฝ๐‘ฉโˆ’1๐›ผ๐’‰ ๐’š

๐‘• ๐’š = 1 2 ๐’š โˆ’ ๐’› ๐‘ˆ๐‘ต(๐’š โˆ’ ๐’›) + โ„Ž2

๐‘—,๐‘’โˆˆโ„ณ

๐‘ฅ๐‘— ๐‘ฏ๐’‹๐’š โˆ’ ๐’’๐‘—

2

๐‘ฉ = ๐‘ต + โ„Ž2

๐‘—

๐‘ฅ๐‘—๐‘ฏ๐’‹

๐‘ผ๐‘ฏ๐’‹ = ๐‘ต + โ„Ž2๐‘ด

๐›ฝ = 1

Towards Real-time Simulation of Deformable Objects

Projective Dynamics:

A Quasi Newton method applied on a special type of energy

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SLIDE 55

Supporting More General Materials

57

๐’š โˆ’ ๐›ฝ๐‘ฉโˆ’1๐›ผ๐’‰ ๐’š This quasi-Newton formulation can be used for any hyperelastic material, but:

  • We need to do line-search
  • ๐›ฝ = 1 only works for Projective Dynamics
  • We need to define the proper weights ๐‘ฅ๐‘—
  • ๐‘ฉ = ๐‘ต + โ„Ž2 ๐‘— ๐‘ฅ๐‘—๐‘ฏ๐’‹

๐‘ผ๐‘ฏ๐’‹ = ๐‘ต + โ„Ž2๐‘ด

Towards Real-time Simulation of Deformable Objects

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SLIDE 56

๐‘ฅ๐‘—

Strain-Stress Curve for PD

Towards Real-time Simulation of Deformable Objects 58

  • ๐‘ฉ = ๐‘ต + โ„Ž2 ๐‘— ๐‘ฅ๐‘—๐‘ฏ๐’‹

๐‘ผ๐‘ฏ๐’‹ = ๐‘ต + โ„Ž2๐‘ด

Strain Stress

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SLIDE 57

Supporting More General Materials

59

  • ๐‘ฉ = ๐‘ต + โ„Ž2 ๐‘— ๐‘ฅ๐‘—๐‘ฏ๐’‹

๐‘ผ๐‘ฏ๐’‹ = ๐‘ต + โ„Ž2๐‘ด

Towards Real-time Simulation of Deformable Objects

Strain Stress

๐‘ฅ๐‘—

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SLIDE 58

Supporting More General Materials

60 Towards Real-time Simulation of Deformable Objects

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SLIDE 59

We can do more

61 Towards Real-time Simulation of Deformable Objects

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SLIDE 60

L-BFGS Acceleration

62

Projective Dynamics Our Method Exact Solution

Towards Real-time Simulation of Deformable Objects

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SLIDE 61

L-BFGS Acceleration

63

Our Method Projective Dynamics

Towards Real-time Simulation of Deformable Objects

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SLIDE 62

L-BFGS with rest-pose Hessian

Towards Real-time Simulation of Deformable Objects 65

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SLIDE 63

L-BFGS with Scaled Identity

Towards Real-time Simulation of Deformable Objects 66

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SLIDE 64

L-BFGS with updating Hessian

Towards Real-time Simulation of Deformable Objects 67

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SLIDE 65

Performance of L-BFGS family

68 Towards Real-time Simulation of Deformable Objects

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SLIDE 66

Iterative Solvers (CG)

69 Towards Real-time Simulation of Deformable Objects

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Results: Accuracy

70 Towards Real-time Simulation of Deformable Objects

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SLIDE 68

Results: Robustness

71 Towards Real-time Simulation of Deformable Objects

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SLIDE 69

Results: Collision

72 Towards Real-time Simulation of Deformable Objects

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SLIDE 70

Results: Anisotropy

73 Towards Real-time Simulation of Deformable Objects

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SLIDE 71

Results: Spline-Based Materials

Towards Real-time Simulation of Deformable Objects 74

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SLIDE 72

Remark

๏ต Our method is:

๏ต General: supports a variety types of hyperelastic materials ๏ต Fast: >10x faster compared to Newtonโ€™s method to achieve similar accuracy level ๏ต Simple: avoids Hessian computation, avoids definiteness fix

75 Towards Real-time Simulation of Deformable Objects

Simple

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SLIDE 73

Future Work

๏ต More Generalization? Relaxation, creep, hysteresis ๏ต Performance? Collision and topology change on the fly. ๏ต Other Integrators? How to make symplectic integrators fast and stable? ๏ต More Applications? Virtual surgery, real-time physics in VR, fast prototyping

for fabrication.

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SLIDE 74

Thank You

http://www.seas.upenn.edu/~liutiant/