We only consider geometric nonlinearity, with linear strain-stress - - PowerPoint PPT Presentation

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We only consider geometric nonlinearity, with linear strain-stress - - PowerPoint PPT Presentation

Jan 21, 2024 177 likes β€’886 views

We only consider geometric nonlinearity, with linear strain-stress relationship = : In Introd oduction ion Space-time editing Powerful tool for animation editing Seeking minimal control forces Matching the constraints in

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We only consider geometric nonlinearity, with linear strain-stress relationship 𝜏 = 𝐷: ϡ

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Positional and/or keyframe constraints

In Introd

  • duction

ion

Space-time editing Β§ Powerful tool for animation editing Β§ Seeking minimal control forces Β§ Matching the constraints in space-time.

Dynamic or static input animation

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Un Unsolved p problems i in p pract ctice ce

β–ͺ Lack of control due to linearization

[Barbič et al. 2012] [Li et al. 2013]

β–ͺ Scalability for complex models β–ͺ Elastic material significantly affects animation β–ͺ What is the right material?

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Te Technical contributions

We propose two new techniques to solve these problems.

β–ͺ Reduced RS (Rotation-Strain) approach

Provides tight positional constraints under large deformation.

β–ͺ Material Optimization

Provides physically plausible and consistent results.

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For efficiency, we formulate the problem in modal coordinates

Sp Space-Tim Time Edit itin ing

arg min

,

𝐹

. 𝑨 + 𝛿𝐹2 𝑨 𝐹

. 𝑨 = 1

2 5 π‘¨Μˆ7 + 𝐸𝑨̇7 + Λ𝑨7

; ; <=>?@ABC 7D;

𝐹2 𝑨 = 1 2 5 𝑣7

F 𝑋, 𝑨7 βˆ’ 𝑣

JF

7 ; ;

  • 7,F ∈M

Measures control forces Measures error in constraints Modal coordinates

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Sp Space-Tim Time Edit itin ing

Euclidean coordinates reconstruction

β–ͺ Must be robust to large deformation. β–ͺ Should only require local evaluations for efficiency.

arg min

,

𝐹

. 𝑨 + 𝛿𝐹2 𝑨 𝐹2 𝑨 = 1 2 5 𝑣7

F 𝑋, 𝑨7 βˆ’ 𝑣

JF

7 ; ;

  • 7,F ∈M
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Ro Rotation-St Strain

𝐡𝑣 = 𝐻Pπ‘Š R𝑕 𝑧

β–ͺ Compute for all elements. β–ͺ Solving global linear eq.

𝑧, 𝑕

Proposed by [Huang et al. 2011].

  • Def. Grad.

Euclidian Coord. RS Coord.

𝑨 𝑣 𝑧 𝑕 π‘“π‘¦π‘ž 𝐡BC

Inefficient for local evaluations

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Re Reduced Rotation-St Strain

β–ͺ Geometric reduction. β–ͺ

𝑣 = πΆπ‘Ÿ π‘Ÿ = 5 𝑄

@𝑕@

  • @βˆˆπ“€

Avoid global linear solve.

  • Def. Grad.

Euclidian Coord. RS Coord.

𝑨 𝑣 𝑧 𝑕 π‘Ÿ 𝐢 π‘“π‘¦π‘ž

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Re Reduced Ro Rotation-St Strain

β–ͺ β–ͺ Cubature method.

Compute RS for cubature only. Partial shape reconstruction

  • Def. Grad.

Euclidian Coord. RS Coord.

𝑨 𝑣 𝑧 𝑧2 𝑕2 𝑕 π‘Ÿ 𝐢 π‘“π‘¦π‘ž

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Co Comparison

Different method for the mapping from to

Reduced RS RS Linear map Rest shape

Reduced RS method

β–ͺ Robust to large deformation. β–ͺ Two orders of magnitude faster than full RS. β–ͺ Allows local evaluations of 3D coordinates.

𝑨 𝑣

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arg min

,,],^,_ 𝐹 . 𝑨, Ξ›, 𝐸 + 𝛿𝐹2 𝑋, 𝑨

Ma Materi rial opti timizati tion

How to pick a good elastic material? Introduce material as new DOFs, and Optimize!

Material in modal space: frequency, damping, and modal basis.

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Ma Materi rial opti timizati tion

arg min

,,],^,_ 𝐹 . 𝑨, Ξ›, 𝐸 + 𝛿𝐹2 𝑋, 𝑨

Dimension is too large, so introduce basis sampling,

𝑋 = 𝑋 ` 𝑇

Optimize smaller sampling basis instead of .

𝑋 𝑇

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Ma Materi rial opti timizati tion arg min

,,],^,b

𝐹

. 𝑨, Ξ›, 𝐸 + 𝛿𝐹2 𝑇, 𝑨 + 𝜈𝐹A(𝑇)

subject to πœ‡g, 𝑒g β‰₯ 0 βˆ€π‘™ ∈ [1, 𝑠]

Formulation for material optimization

Regularization term.

Nonlinear, but all variables are in subspace.

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Nu Numeri rical meth thod

Optimize the variables one by one

  • pt.
  • pt.
  • pt.

𝑨 𝑇 Ξ›, 𝐸

Guarantees monotone decrease!

β–ͺ Fix , optimize β–ͺ Fix , optimize β–ͺ Fix , optimize

Ξ›, 𝐸, 𝑇 z, 𝑇 z, Ξ›, 𝐸 𝑨 Ξ›, 𝐸 𝑇

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Ani Animati tion n edi diti ting ng

With material optimization, the resulting animation is more consistent!

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Ani Animati tion n edi diti ting ng

With material optimization, the resulting animation is more consistent!

Keep circular motion Abrupt suddenly

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Co Comparison

Our method provides tight positional constraints, even for large edits.

[Barbič et al. 2012] Our result

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Co Comparison

Our method supports large edits without visual artifacts.

[Li etal. 2013] Our result

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Re Recovering material parameters

Non-uniform material for the experiments. Input animation, with first 150 frames as constraints, last 150 frames for comparison. We use input animation as keyframes in the space-time editing.

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Re Recovering material parameters

Recovered material Uniform material

Compare the simulated results with the last 150 frames.

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Re Recovering material parameters

Recovered material Uniform material

Compare the simulated results with the last 150 frames.

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Future work

  • Intrinsic representation of elasticity
  • Redundant DOFs: 9|T| v.s 3|N|
  • Pure strain representation
  • Embeddable condition (integratable condition)
  • Physically accurate warping
  • Change rotation extrapolation function, e.g. Cayley mapping
  • Introduce material-aware metric for Poisson construction
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Thank you!