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

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

In Introd oduction ion Space-time editing Powerful tool for animation editing § Seeking minimal control forces § Matching the constraints in space-time. § Dynamic or static input Positional and/or keyframe constraints animation

Un Unsolved p problems i in p pract ctice ce ▪ Scalability for complex models ▪ Lack of control due to linearization [Li et al. 2013] [Barbič et al. 2012] ▪ Elastic material significantly affects animation ▪ What is the right material?

Te Technical contributions We propose two new techniques to solve these problems. ▪ Reduced RS (Rotation-Strain) approach Provides tight positional constraints under large deformation. ▪ Material Optimization Provides physically plausible and consistent results.

� Sp Space-Tim Time Edit itin ing For efficiency, we formulate the problem in modal coordinates arg min 𝐹 . 𝑨 + 𝛿𝐹 2 𝑨 , Modal coordinates <=>?@ABC . 𝑨 = 1 Measures control forces ; 𝐹 5 𝑨̈ 7 + 𝐸𝑨̇ 7 + Λ𝑨 7 ; 2 7D; 𝐹 2 𝑨 = 1 F 𝑋, 𝑨 7 − 𝑣 ; Measures error in constraints 7 2 5 𝑣 7 J F ; 7,F ∈M

� Sp Space-Tim Time Edit itin ing arg min 𝐹 . 𝑨 + 𝛿𝐹 2 𝑨 , 𝐹 2 𝑨 = 1 F 𝑋, 𝑨 7 − 𝑣 ; 7 2 5 𝑣 7 J F ; 7,F ∈M Euclidean coordinates reconstruction ▪ Must be robust to large deformation. ▪ Should only require local evaluations for efficiency.

Ro Rotation-St Strain Proposed by [Huang et al. 2011]. RS Euclidian Def. Grad. Coord. Coord. 𝐵 BC 𝑧, 𝑕 𝑧 ▪ Compute for all elements. 𝑕 𝑣 ▪ Solving global linear eq. 𝑓𝑦𝑞 𝑨 R𝑕 𝑧 𝐵𝑣 = 𝐻 P 𝑊 Inefficient for local evaluations

� Re Reduced Rotation-St Strain ▪ Geometric reduction. RS Euclidian Def. Grad. Coord. ▪ Coord. 𝑧 𝑕 𝑣 = 𝐶𝑟 𝑣 𝑓𝑦𝑞 𝑨 𝐶 𝑟 = 5 𝑄 @ 𝑕 @ 𝑟 @∈𝓤 Avoid global linear solve.

Re Reduced Ro Rotation-St Strain ▪ RS Euclidian Def. Grad. ▪ Cubature method. Coord. Coord. 𝑧 𝑕 𝑣 𝑓𝑦𝑞 𝑨 𝐶 𝑕 2 𝑟 𝑧 2 Compute RS for cubature only. Partial shape reconstruction

Co Comparison 𝑨 𝑣 Different method for the mapping from to Rest shape Linear map Reduced RS RS Reduced RS method ▪ Robust to large deformation. ▪ Two orders of magnitude faster than full RS. ▪ Allows local evaluations of 3D coordinates.

Ma Materi rial opti timizati tion How to pick a good elastic material? Introduce material as new DOFs, and Optimize! arg min ,,],^,_ 𝐹 . 𝑨, Λ, 𝐸 + 𝛿𝐹 2 𝑋, 𝑨 Material in modal space: frequency, damping, and modal basis.

Ma Materi rial opti timizati tion arg min ,,],^,_ 𝐹 . 𝑨, Λ, 𝐸 + 𝛿𝐹 2 𝑋, 𝑨 Dimension is too large, so introduce basis sampling, ` 𝑇 𝑋 = 𝑋 𝑋 𝑇 Optimize smaller sampling basis instead of .

Ma Materi rial opti timizati tion Regularization term. Formulation for material optimization arg min 𝐹 . 𝑨, Λ, 𝐸 + 𝛿𝐹 2 𝑇, 𝑨 + 𝜈𝐹 A (𝑇) ,,],^,b subject to 𝜇 g , 𝑒 g ≥ 0 ∀𝑙 ∈ [1, 𝑠] Nonlinear, but all variables are in subspace.

Nu Numeri rical meth thod opt. 𝑨 Optimize the variables one by one opt. Λ, 𝐸 opt. 𝑇 ▪ Fix , optimize 𝑨 Λ, 𝐸, 𝑇 ▪ Fix , optimize Λ, 𝐸 z, 𝑇 ▪ Fix , optimize 𝑇 z, Λ, 𝐸 Guarantees monotone decrease!

Ani Animati tion n edi diti ting ng With material optimization, the resulting animation is more consistent!

Ani Animati tion n edi diti ting ng With material optimization, the resulting animation is more consistent ! Keep Abrupt circular suddenly motion

Co Comparison Our method provides tight positional constraints , even for large edits. Our result [Barbič et al. 2012]

Co Comparison Our method supports large edits without visual artifacts. Our result [Li etal. 2013]

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

Re Recovering material parameters Compare the simulated results with the last 150 frames. Recovered material Uniform material

Re Recovering material parameters Compare the simulated results with the last 150 frames. Recovered material Uniform material

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

Thank you!