Example-based Plastic Deformation of Rigid...

Example-based PlasticDeformation of Rigid Bodies

Tae-Woo Lee

2019.04.04

Computer Graphics @ Korea University

Ben Jones (Denver Univ.) et al.SIGGRAPH 2016

Tae-Woo Lee | 2019-04-22| # 2Computer Graphics @ Korea University

Index

• Abstract

• Introduction

• Related Work

• Method

• Authoring Simulation Assets

• Results and Discussion

Abstract


Abstract

An example-based plasticity model based on linear blend skinning

Deformation

Artists author simulation objects using familiar tools

Introduction


Introduction

• Large-scale destruction scene is used in physics-based animation

• The materials in these scenes are often man-made,carefully engineered and designed to be nearly rigid

• During failure, man-made materials such as steel and concreteexhibit plastic deformation and fracture

• Most of the existing approaches are based on continuum theory


Non-continuum Method Example(1/2)

• “Transformer 2 – breaking buildings”[Criswell (Industrial Light&Magic.) et al. / SIGGRAPH Conference 2010]

In advance, the object is broken to pieces of geometryand then integrate it seamlessly


Non-continuum Method Example(1/2)

• “Real-time deformation and fracture in a game environment”[Parker and O’Brien (Pixelux Entertainment) / SCA 2009]

Replaces the underlying geometry when destructive events occur,employ finite state machines and to hide temporal discontinuities


Contributions

• A similarly practical approach to animating plastic deformationin destructive scenes

• An example-based deformation model based on linear blend skinning with a spatially varying blend of examples

• A method for mapping from discrete rigid body impulses to deformations

• A method for incorporating energy dissipation due to plastic deformation in system dynamics by modulating the coefficient of restitution

Related Work


Related Work

Methods Based on Continuum Mechanics(1/2)

• Spring-mass system

“A peridynamic perspective on spring-mass fracture”[Levine (Clemson Univ.) et al. / SCA 2014]


Related Work

Methods Based on Continuum Mechanics(2/2)

• Lagrangian finite element method

“Simulating liquids and solid-liquid interactions withlagrangian meshes”[Clausen (California, Berkeley Univ.) et al. / SIGGRAPH 2013]


Related Work

Fracture Effect

• “Fracturing rigid materials”[Bao (Stanford Univ.) et al. / IEEE 2007]


Related Work

Denting and Bending

• “Efficient denting and bending of rigid bodies”[Patkar (Stanford Univ.) et al. / SCA 2014]


Related Work

Linear-Blend Skinning

• “Frame-based elastic models”[Gilles (British Columbia Univ.) et al. / SIGGRAPH 2011]


Related Work

Exampled-Based Simulation(1/2)

• “Drivenshape – a data-driven approach for shape deformation”[Kim and Vendrovsky (Rhythm and Hues Studios) / SCA 2008]

• “Sensitivity-optimized rigging for example-basedreal-time clothing synthesis”[Wang (Hangzhou normal Univ.) et al. / SIGGRAPH 2014]


Related Work

Exampled-Based Simulation(2/2)

• “Real-time example-based elastic deformation”[Koyama (Tokyo Univ.) et al. / SCA 2012]

Method


Method Overview(1/2)


Method Overview(2/2)


Initial Setting(1/3)

• Artists begin by rigging their model with bones.

• Then, they use this rig to deform their input meshinto a set of characteristic example poses.

• Each object stores standard rigid body properties:

density, 𝜌, and mass, 𝓂 (both constant during simulation)

inertia tensor, 𝚰

linear position, x𝑐𝑜𝑚, and momentum p

orientation, 𝛀, and angular momentum, 𝐋

coefficient of restitution, 𝐶𝑟



• Geometry of object is modeled usinga tetrahedral mesh with 𝑁 vertices.

• The skinned vertex positions are determinedby the transformations of the 𝐵 bones.

• The user provides a set of 𝐸 example poses for the object,where each example contains a rotation and translationfor each bone.



• To track skinning properties over time, each object stores

undeformed mesh vertex positions, 𝐮 ∈ ℝ𝑁×3

skinned mesh vertex positions, x ∈ ℝ𝑁×3

skinning weights, 𝐖 ∈ ℝ𝑁×𝐵

example weights, 𝐄 ∈ ℝ𝑁×𝐸

deformation accumulator, ∆𝐄 ∈ ℝ𝑁×𝐸

example transformation, 𝐓 ∈ ℝ7𝐵×𝐸

• 𝐮, 𝐖 and 𝐓 are constantwhile the remaining properties change during simulation


Skinning

• What is skinning?

캐릭터의 뼈대에 물체(피부)를 연결하는 작업

• Skinning example

출처 : Vertex Skinning (in C++ & OpenGL)https://www.youtube.com/watch?v=0K2W7I9c4ms


Skinning Equation(1/2)

• Traditional linear blend skinning

x𝑖 = σ𝑏∈𝐵 𝐖𝑖𝑏𝐓𝑏𝐮𝑖

• skinned mesh vertex positions, x ∈ ℝ𝑁×3

• skinning weights, 𝐖 ∈ ℝ𝑁×𝐵

• example transformation, 𝐓 ∈ ℝ7𝐵×𝐸

• undeformed mesh vertex positions, 𝐮 ∈ ℝ𝑁×3

• bones – rig’s degrees of freedom

• 𝐓𝑏 : current transformation of bone, 𝑏

• To incorporate artist examples, if transformations blended linearly,

x𝑖 = σ𝑏∈𝐵 𝐖𝑖𝑏 σ𝑒=1𝐸 𝐄𝑖𝑒𝐓𝑏𝑒 u𝑖

• 𝐄 : example weights


Skinning Equation(2/2)

• Linearly blending the translations and interpolating rotations with,

x𝑖 = σ𝑏∈𝐵 𝐖𝑖𝑏QLERP 𝐄𝑖 , 𝐓𝑏 u𝑖

“Spherical blend skinning: a real-time deformation of articulated models”[Kavan and Zara (Czech Technical Univ.) / I3D 2005]

• To avoid extrapolation outside of the example space

each entry of 𝐄 is between 0 and 1 → ∀𝑖∀𝑒, 0 ≤ 𝐄𝑖𝑒 ≤ 1

each row of 𝐄 sums to 1 → σ𝑒 e𝑖𝑒 = 1

each row of 𝐄 stores barycentric coordinates thatdescribe how to combine the example transformations for vertex 𝑖


Impulse-Based Deformation

• Traditional deformable body simulation

Each vertex has own positional degrees of freedom

Elastoplastic response is computed by analyzing stresses

⟶ This approach is unsuitable

• An impulse 𝒋𝑖 at vertex 𝑖 is mapped to a change

in the barycentric coordinates in row 𝑖 of the matrix, 𝐄

• Using Barycentric coordinates

somewhat localized plastic deformations that we expect fromcolliding, otherwise rigid, bodies

• To ensure smooth deformation, use a smoothing kernel



Projection(1/2)

• First, Map the impulse to a desired change in position by

Δ𝐱𝑖 =Δ𝑡

𝑚max( 𝐣𝑖 − 𝛽, 0)

𝐣𝑖

𝐣𝑖

• Δ𝑡 : time step

• 𝑚 : mass of the object

• 𝛽 : plastic yield threshold

• Jacobian matrix with respect to change of example weights

𝐉𝑖 =𝜕𝐱𝑖

𝜕𝐄𝑖𝑒, 𝐉𝑖 ∈ ℝ3×𝐸

• “Fadbad++, flexible automatic differentiation using templatesand operator overloading in C++”[Ole Stauning and Claus Bendten / 2007]



Projection(2/2)

• Applying the Jacobian transpose

Δ𝐞𝑖 = 𝐉𝑖𝑇Δ𝐱𝑖

Δ𝐞𝑖 is the change in example weights at vertex 𝑖 anda non-physical quantity

“Virtual model control: an intuitive approach for bipedal locomotion”[Pratt (IHMC.) et al. / INT J ROBOT RES 2001]

• Alternative and appealing approach is

Δ𝐞𝑖 = α𝐉+Δ𝐱𝑖

𝛼 is user-defined scaling parameter

• Consequently, Our method uses

Δ𝐞𝑖 = α𝐉𝑖𝑇Δ𝐱𝑖

very fast to compute



Propagation and Application(1/2)

• Propagate the change to nearby vertices using a smoothing kernel

Rules out Euclidean distances⟶ Because our skinning weights are smooth by design

Using a few passes of Dijkstra’s algorithm

• Update each row of the matrix Δ𝐄 by

Δ𝐄𝑗 += 𝜙𝑥𝑗−𝑥𝑖

𝛾Δei

𝛾 > 0 is the radius of the kernel, the deformation radius

𝜙 represents a user tunable parameter

• Use the cubic kernel

𝜙 𝑥 = ൜2𝑥3 − 3𝑥2 + 1 ∶ 𝑥 < 10 ∶ 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒



Propagation and Application(2/2)

• Update the barycentric matrix by

𝐄 += (1 − λ)Δ𝐄

• Then, apply Δ𝐄 ∗= 𝜆, 𝑤ℎ𝑒𝑟𝑒 𝜆 ∈ [0,1)

⟶ 𝜆 is a user-tunable parameter that should vary with the inverseof the time-step. Namely, 𝜆 is the plasticity rate

• This approach has several advantages,

1 − 𝜆 σ𝑖=0∞ 𝜆𝑖 = 1 →

Total accumulated deformation is independent of 𝜆 ∈ [0,1)

Need to store only a single accumulation buffer Δ𝐄

Expose a single, fairly intuitive parameter; for values of 𝜆close to 0 the deformation happens very quickly



Comparison of Kernel Radius and Plasticity Rate



Dynamics

• Object dynamics are computed usingan unmodified rigid body simulator(Bullet-Physics)

• Perform triangle based collision detection

• The effective coefficient of restitution 𝐶𝑟

𝐶𝑟 ≔ min(𝐶𝑟∗ , 𝐶𝑟 + 𝜇Δ𝑡 , exp −𝑣 𝚫𝐄 𝑓 𝐶𝑟

∗)

• 𝐶𝑟∗ : minimum of the default value

• 𝜇, 𝑣 : the parameters that control the coefficient of restitution



Comparison of Restitution Modification

Authoring Simulation Assets


Authoring Algorithm


Simulation Analysis

• ”Bounded biharmonic weights for real-time deformation”[Jacobson (New York Univ.) et al. / SIGGRAPH 2011]⟶ Use skinning weight

• “Automatic construction of coarse, high-quality tetrahedraliza-tions that enclose and approximate surfaces for animation”[Stuart (Utah Univ.) et al. / MIG 2013]⟶ Pipeline makes use of the approximating tetrahedralization

• Other auxiliary structures⟶ Regular grid (Wall, Bar, Ground)


How to Make Bones


Artist Can Set Parameters

• Once the mesh is rigged,⟶ Artist manipulates the control handle

• Characteristic deformations are created⟶ Artist can perform a rigid body simulation

• Before the simulation, Artist sets six new parameters

𝛼 : the scaling constant for mapping from displacementsto changes in barycentric example-coordinates

𝛽 : the threshold for plastic deformation

𝛾 : the deformation radius

𝜆 : the plasticity rate

𝜇, 𝜈 : the parameters that control the coefficient of restitution

Results and Discussion


Pachinko Object Vertex Bone Example

Barrel 1790 12 3

Scene ∆𝒕 RB Solver Deformation Average FPS

seconds 4e-3 64.7 47.8 10.6

• Show the range of deformations that can be achieved frombarrels with identical material parameters and example poses

• The graph shows the per-frame computational cost of rigid body simulation and deformation computations for the scene


Hailstorm Object Vertex Bone Example

Car 4695 32, 48 6


seconds 4e-3 0.6 2.4 197.1

• How the colors vary over the surface and change over time


Car Crashes Object Vertex Bone Example

Car 4695 32 6


seconds 5e-3 17.3 8.8 13.79

• How the example weights vary over the objects


Spaceships crash Object Vertex Bone Example

Big Spaceship 4178 64 6

Little Spaceship 1188 12 3


seconds 4e-3 64.7 47.8 10.6

• A small fleet of spaceships crashes into an enemy vessel


• Most of our examples required less than one minute to run

• The short runtimes of the method allowed artist to literatequickly to tune parameters and refine the simulation.

• He said “for non-physical deformation, it really looks like physics.”

Artist Experience Object Vertex Bone Example

Crane 1939 14 3


seconds 2e-3 381 160 1.1


Performance

• Depending on the scene,out method adds 42% to 59% overhead per timestep on average(excluding the scene of spaceships crash)

• Several of our examples run in real-time, while the rest are fastenough to allow artists to interactively tune parameters

• The examples in this paper were run on a 2013 Macbook Prowith an integrated GPU

• The projection and propagation were parallelized on the CPUwhile skinning was performed on the GPU


Summary

• Our method targets extreme destructive scenes

• The underlying examples have a complex relationshipwith the final animation

• A technique for animating the failure of near-rigidman-made materials

• Fast, artist friendly, easily implemented,integrates smoothly into existing pipeline!


Limitations and Future Work

• Automatically generated rig was only able to approximatethe shapes in the simulation output.⟶ Improved automatic rig generation

• Examples deliberately use a small number of example deformation

⟶ Reducing the burden on the artist

• Smoothness of the example weights or favoring using a single example over an average of all examples

⟶ Computing the example weights

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