Animation & Physically-Based Simulation 0368-3236, Spring 2019 - - PowerPoint PPT Presentation

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Animation & Physically-Based Simulation

0368-3236, Spring 2019 Tel-Aviv University Amit Bermano

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Computer Animation

• Describing how 3D objects (& cameras)

move over time

Pixar

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Computer Animation

• Challenge is balancing between …
• Animator control
• Physical realism

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Computer Animation

• Manipulation
• Posing
• Configuration control
• Interpolation
• Animation
• In-betweening

focus.gscept.com https://blenderartists.org/

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Character Animation Methods

• Modeling (manipulation)
• Deformation
• Blendshape rigging
• Skeleton+Envelope rigging
• Interpolation
• Key-framing
• Kinematics
• Motion Capture
• Energy minimization
• Physical simulation
• Procedural

focus.gscept.com https://blenderartists.org/

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Character Animation Methods

• Modeling (manipulation)
• Deformation
• Blendshape rigging
• Skeleton+Envelope rigging
• Interpolation
• Key-framing
• Kinematics
• Motion Capture
• Energy minimization
• Physical simulation
• Procedural

focus.gscept.com https://blenderartists.org/

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Deformation

• How to change a character’s pose?
• Every vertex directly
• Intuitive computation

https://www.youtube.com/watch?v=oxkf_N-QCNI

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Deformation

• A HUGE variety of methods
• Laplacian mesh editing
• ARAP
• CAGE Base
• Barycentric coordinates
• Heat diffusion
• Variational
• …

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Deformation

• A HUGE variety of methods
• Laplacian mesh editing
• ARAP
• CAGE Base
• Barycentric coordinates
• Heat diffusion
• Variational
• …

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Laplacian Mesh Editing

• Local detail representation – enables detail preservation

through various modeling tasks

• Representation with sparse matrices
• Efficient linear surface

reconstruction

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Overall framework

• 1. Compute differential representation
• 2. Pose modeling constraints
• 3. Reconstruct the surface – in least-squares sense

𝜀𝑗 = 𝑀 𝑤𝑗 = 𝑤𝑗 − 1 𝑒𝑗 Σ𝑘∈N 𝑗 𝑤𝑘 𝑤𝑗

′ = 𝑣𝑗,

𝑗 ∈ 𝑫 𝑀 𝑀𝑑 𝑾 = 𝜺 𝑽

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Differential coordinates?

• In matricial form:
• They represent the local detail / local shape description
• The direction approximates the normal
• The size approximates the mean curvature



𝑀𝑗𝑘 = ൞ −𝑥𝑗𝑘 𝑗 ≠ 𝑘 Σ𝑘∈1𝑠𝑗𝑜𝑕𝑗𝑥𝑗𝑘 𝑗 = 𝑘 𝑓𝑚𝑡𝑓

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• In matricial form:

𝑀𝑗𝑘 = ൞ −𝑥𝑗𝑘 𝑗 ≠ 𝑘 Σ𝑘∈1𝑠𝑗𝑜𝑕𝑗𝑥𝑗𝑘 𝑗 = 𝑘 𝑓𝑚𝑡𝑓

P Q

Adding constraints

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Adding constraints

• In matricial form:

𝑀𝑗𝑘 = ൞ −𝑥𝑗𝑘 𝑗 ≠ 𝑘 Σ𝑘∈1𝑠𝑗𝑜𝑕𝑗𝑥𝑗𝑘 𝑗 = 𝑘 𝑓𝑚𝑡𝑓

𝑦1 𝑦2 𝑦3 𝑦4 𝑦5 𝑦6 𝑦7 𝑦8 𝑦9 𝑦10

=

𝜀1 𝜀2 𝜀3 𝜀4 𝜀5 𝜀6 𝜀7 𝜀8 𝜀9 𝜀10 𝑸𝒚 𝑹𝒚

𝑧1 𝑧2 𝑧3 𝑧4 𝑧5 𝑧6 𝑧7 𝑧8 𝑧9 𝑧10

=

𝜀1 𝜀2 𝜀3 𝜀4 𝜀5 𝜀6 𝜀7 𝜀8 𝜀9 𝜀10 𝑸𝒛 𝑹𝒛

P Q

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Demo

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Character Animation Methods

• Modeling (manipulation)
• Deformation
• Blendshape rigging
• Skeleton+Envelope rigging
• Interpolation
• Key-framing
• Kinematics
• Motion Capture
• Energy minimization
• Physical simulation
• Procedural

focus.gscept.com https://blenderartists.org/

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Blendshapes

• Blendshapes are an approximate semantic parameterization
• Linear blend of predefined poses

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Blendshapes

https://www.youtube.com/watch?v=KPDfMpuK2fQ

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Blendshapes

• Usually used for difficult to pose complex deformations
• Such as faces
• Given:
• A mesh 𝑁 = (𝑊, 𝐹) with 𝑛 vertices
• 𝑜 configurations of the same

mesh, 𝑁𝑐 = 𝑊

𝑐, 𝐹 , 𝑐 = 1 … 𝑜

• A new configuration is simply:
• 𝑁′ = (Σ𝑐=1…𝑜wbVb, E)
• Delta formulation:
• 𝑁′ = Σ𝑐=1…𝑜𝑊

0 + wb(Vb − V0 , E)

• A bit more convenient
• 𝑁0 - the rest pose, 𝑥𝑐 blend weights

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Blendshapes

https://www.youtube.com/watch?v=jBOEzXYMugE

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Character Animation Methods

• Modeling (manipulation)
• Deformation
• Blendshape rigging
• Skeleton+Envelope rigging
• Interpolation
• Key-framing
• Kinematics
• Motion Capture
• Energy minimization
• Physical simulation
• Procedural

focus.gscept.com https://blenderartists.org/

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Angel Figures 8.8 & 8.9 Base Arm Hand

Scene Graph

Articulated Figures

• Character poses described by set of rigid bodies

connected by “joints”

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Articulated Figures

• Well-suited for humanoid characters

Rose et al. `96 Root LHip LKnee LAnkle RHip RKnee RAnkle Chest LCollar LShld LElbow LWrist LCollar LShld LElbow LWrist Neck Head

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Articulated Figures

• Animation focuses on joint angles, or general transformations

Watt & Watt

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Forward Kinematics

• Describe motion of articulated character

1 2 X = (x,y)

l2 l1

(0,0) “End-Effector”

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Forward Kinematics

• Animator specifies joint angles: 1 and 2
• Computer finds positions of end-effector: X

1 2 X = (x,y)

l2 l1

(0,0)

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Example: Walk Cycle

• Articulated figure:

Watt & Watt

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Example: Walk Cycle

• Hip joint orientation:

Watt & Watt

Keyframes

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Example: Walk Cycle

• Knee joint orientation:

Watt & Watt

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Example: Walk Cycle

• Ankle joint orientation:

Watt & Watt

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Example: walk cycle

Lague: www.youtube.com/watch?v=DuUWxUitJos

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Character Animation Methods

• Modeling (manipulation)
• Deformation
• Blendshape rigging
• Skeleton+Envelope rigging
• Interpolation
• Key-framing
• Kinematics
• Motion Capture
• Energy minimization
• Physical simulation
• Procedural

focus.gscept.com https://blenderartists.org/

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Beyond Skeletons…

• Skinning

creativecrash.com

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Kinematic Skeletons

• Hierarchy of transformations (“bones”)
• Changes to parent affect

all descendent bones

• So far: bones affect objects in scene
• r parts of a mesh
• Equivalently, each point on a mesh

acted upon by one bone

• Leads to discontinuities when

parts of mesh animated

• Extension: each point on a mesh

acted upon by more than one bone

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Linear Blend Skinning

• Each vertex of skin potentially influenced by all bones
• Normalized weight vector w(v) gives influence of each bone transform
• When bones move, influenced vertices also move
• Computing a transformation Tv for a skinned vertex
• For each bone
• Compute global bone transformation Tb from transformation hierarchy
• For each vertex
• Take a linear combination of bone transforms
• Apply transformation to vertex in original pose
• Equivalently, transformed vertex position is weighted combination of positions

transformed by bones







B b b v b v

T w T

) (

 







B b b v b d transforme

v T w v

) (

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Assigning Weights: “Rigging”

• Painted by hand
• Automatic: function of relative distances to

nearest bones

• Smoothness of skinned surface depends on smoothness of weights!

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Assigning Weights: “Rigging”

• Painted by hand
• Automatic: function of relative distances to

nearest bones

• Smoothness of skinned surface depends on smoothness of weights!
• Other problems with extreme deformations
• Many solutions

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Assigning Weights: “Rigging”

• Painted by hand
• Automatic: function of relative distances to

nearest bones

• Smoothness of skinned surface depends on smoothness of weights!
• Other problems with extreme deformations

https://cgl.ethz.ch /publications/pap ers/paperOzt13.p hp

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Character Animation Methods

• Modeling (manipulation)
• Deformation
• Blendshape rigging
• Skeleton+Envelope rigging
• Interpolation
• Key-framing
• Kinematics
• Motion Capture
• Energy minimization
• Physical simulation
• Procedural

focus.gscept.com https://blenderartists.org/

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Keyframe Animation

• Define character poses at specific time steps

called “keyframes”

Lasseter `87

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Keyframe Animation

• Interpolate variables describing keyframes to determine poses

for character in between

Lasseter `87

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Keyframe Animation

• Inbetweening:
• Linear interpolation - usually not enough continuity

H&B Figure 16.16 Linear interpolation

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Keyframe Animation

• Inbetweening:
• Spline interpolation - maybe good enough

H&B Figure 16.11

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Temporal Enhancement

Start frame End frame

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Time

1 2 3 4 5

Displacement

Temporal Enhancement

x x x x

Database Interpolation

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Results – Hand Animated Rig

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Example: Ball Boy

Fujito, Milliron, Ngan, & Sanocki Princeton University “Ballboy”

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Character Animation Methods

• Modeling (manipulation)
• Deformation
• Blendshape rigging
• Skeleton+Envelope rigging
• Interpolation
• Key-framing
• Kinematics
• Motion Capture
• Energy minimization
• Physical simulation
• Procedural

focus.gscept.com https://blenderartists.org/

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Inverse Kinematics

• What if animator knows position of “end-effector”?

1 2 X = (x,y)

l2 l1

(0,0) “End-Effector”

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Inverse Kinematics

• Animator specifies end-effector positions: X
• Computer finds joint angles: 1 and 2:

1 2 X = (x,y)

l2 l1

(0,0)

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Inverse Kinematics

• End-effector postions can be specified by spline curves

1 2 X = (x,y)

l2 l1

(0,0) y x t

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Inverse Kinematics

• Problem for more complex structures
• System of equations is usually under-constrained
• Multiple solutions

1 2

l2 l1

(0,0) X = (x,y)

l3

3 Three unknowns: 1, 2 , 3 Two equations: x, y

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Inverse Kinematics

• Solution for more complex structures:
• Find best solution (e.g., minimize energy in motion)
• Non-linear optimization

1 2

l2 l1

(0,0) X = (x,y)

l3

3

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Kinematics

• Advantages
• Simple to implement
• Complete animator control
• Disadvantages
• Motions may not follow physical laws
• Tedious for animator

Lasseter `87

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Kinematics

• Advantages
• Simple to implement
• Complete animator control
• Disadvantages
• Motions may not follow physical laws
• Tedious for animator

Lasseter `87

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Character Animation Methods

• Modeling (manipulation)
• Deformation
• Blendshape rigging
• Skeleton+Envelope rigging
• Interpolation
• Key-framing
• Kinematics
• Motion Capture
• Energy minimization
• Physical simulation
• Procedural

focus.gscept.com https://blenderartists.org/

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Motion Capture

• Measure motion of real characters and then

simply “play it back” with kinematics

Captured Motion

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Motion Capture

• Measure motion of

real characters and then simply “play it back” with kinematics

https://www.youtube.com/watch?v=MVvDw15-3e8

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Motion Capture

• Could be applied on different parameters
• Skeleton Transformations
• Direct mesh deformation
• Advantage:
• Physical realism
• Challenge:
• Animator control

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Character Animation Methods

• Modeling (manipulation)
• Deformation
• Blendshape rigging
• Skeleton+Envelope rigging
• Interpolation
• Key-framing
• Kinematics
• Motion Capture
• Energy minimization
• Physical simulation

focus.gscept.com https://blenderartists.org/

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Animation Techniques

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Physically Based Simulation

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Physically-based simulation

• Computational Sciences
• Reproduction of physical phenomena
• Predictive capability (accuracy!)
• Substitute for expensive experiments
• Computer Graphics
• Imitation of physical phenomena
• Visually plausible behavior
• Speed, stability, art-directability

PBS and Graphics

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Simulation in Graphics

• Art-directability

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Simulation in Graphics

• Speed

https://www.youtube.com/watch?v=-x9B_4qBAkk

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Simulation in Graphics

• Stability

https://www.youtube.com/watch?v=tT81VPk_ukU

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Applications in Graphics

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Applications in Graphics

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Mass – Spring Systems

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Spatial Discretization

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Forces

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Forces

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Forces

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Example: Rope

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Particle System Forces

• Spring-mass mesh

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Example: Cloth

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Demo

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Equations of Motion

• Newton’s Law for a point mass
• f = ma
• Computing particle motion requires solving

second-order differential equation

• Add variable v to form coupled

first-order differential equations: “state-space form”

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Solving the Equations of Motion

• Initial value problem
• Know x(0), v(0)
• Can compute force (and therefore acceleration)

for any position / velocity / time

• Compute x(t) by forward integration

f

x(0) x(t)

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Solving the Equations of Motion

• Forward (explicit) Euler integration

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Solving the Equations of Motion

• Forward (explicit) Euler integration
• x(t+Δt)  x(t) + Δt v(t)
• v(t+Δt)  v(t) + Δt f(x(t), v(t), t) / m

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Solving the Equations of Motion

• Forward (explicit) Euler integration
• x(t+Δt)  x(t) + Δt v(t)
• v(t+Δt)  v(t) + Δt f(x(t), v(t), t) / m
• Problem:
• Accuracy decreases as Δt gets bigger

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Single Particle Demo

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Mass Spring Systems

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Alternative

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Continuum Mechanics

https://www.youtube.com/watch?v=BOabEZXm9IE

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1D Continuous Elasticity

undeformed state

1D elastic solid

←

Given 𝑢, how to determine deformed configuration?

Principle of minimum potential energy A mechanical system in static equilibrium will assume a state of minimum potential energy.

t

deformed state

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1D Continuous Elasticity

• Strain:

𝜁 =

Δ𝑚 𝑀

(relative stretch)

• Stress:

𝜏 =

𝑔𝑗𝑜𝑢 𝐵

(internal force density)

• Hooke’s law:

𝜏 = 𝑙𝜁 (𝑙 material constant)

• Strain energy density: Ψ =

1 2 𝑙𝜁2

(postulate via 𝜏 =

𝜖Ψ 𝜖𝜁 )

L Δl t A

1D elastic solid

←

𝑔

𝑗𝑜𝑢

undeformed state deformed state

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1D Continuous Elasticity

• Discretize domain into elements
• Element strain:

𝜁𝑗 =

𝑦𝑗+1

′

−𝑦𝑗

′ −𝑀𝑗

𝑀𝑗

with 𝑀𝑗 = 𝑦𝑗+1 − 𝑦𝑗

• Element strain energy:

𝑋

𝑗 = Ψ𝑗 ⋅ 𝑀𝑗 = 1 2 𝑙𝜁𝑗 2 ⋅ 𝑀𝑗

• Total strain energy:

𝑋 = ∑𝑋

𝑗

t

… … 𝑦1 𝑦2 𝑦3 𝑦𝑜 𝑦1

′

𝑦2

′

𝑦3

′

𝑦𝑜

′

undeformed state deformed state

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1D Continuous Elasticity

Minimum energy principle: at equilibrium

• system assumes a state of minimum total energy
• total forces vanish for all nodes
• 𝑋

𝑗 = 1 2 𝑙𝜁𝑗 2 ⋅ 𝑀𝑗 and 𝜁𝑗 = 𝑦𝑗+1

′

−𝑦𝑗

′−𝑀𝑗

𝑀𝑗

→

𝜖𝑋𝑗 𝜖𝑦𝑗

′ =

𝜖𝑋𝑗 𝜖𝜁𝑗 𝜖𝜁𝑗 𝜖𝑦𝑗

′ = −𝑙𝜁𝑗

• 𝑔

𝑗 = − 𝜖𝑋 𝜖𝑦𝑗

′ = −

𝜖𝑋𝑗−1 𝜖𝑦𝑗

′ −

𝜖𝑋𝑗 𝜖𝑦𝑗

′ = −𝑙(𝜁𝑗−1 − 𝜁𝑗) for 𝑗 = 2 … 𝑜 − 1

• 𝑔

1 = 𝑙𝜁1 and 𝑔 𝑜 = −𝑙𝜁𝑜−1

t

… … 𝑦1 𝑦2 𝑦3 𝑦𝑜 𝑦1

′

𝑦2

′

𝑦3

′

𝑦𝑜

′

undeformed state deformed state

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1D Continuous Elasticity

Equilibrium conditions

𝑔

𝑗 =

∀𝑗 ∈ 2 … 𝑜 − 1 −𝑢 𝑗 = 𝑜

→ n-2 linear equations for n-2 unknowns 𝑦𝑗

′

→ solve linear system of equations to obtain deformed configuration.

t

𝑦1 … … 𝑦2 𝑦3 𝑦𝑜 𝑦1

′

𝑦2

′

𝑦3

′

𝑦𝑜

′

𝑢 𝑗 = 1

undeformed state deformed state

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Stress Strain Curve

• What do these represent?

A: Paper B: Fabric / unfilled plastic C: Rubber

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Material Models – Linear

𝛀 = 𝟐 𝟑 𝝁𝐮𝐬(𝜻)𝟑+𝝂𝒖𝒔(𝜻𝟑)

99

Material Models – Linear

100

Material Models – Linear

𝜈 = − Δ𝑧 Δ𝑦 = 0.5

101

Negative Poisson’s Ratio

https://www.youtube.com/watch?v=5wpRszZZhYQ

103

Nonlinear Elasticity

• Idea: replace Cauchy strain with Green strain

→ St. Venant-Kirchhoff material (StVK)

• Energy Ψ

𝑇𝑢𝑊𝐿 = 1 2 𝜇tr(𝐅)2+𝜈tr(𝐅2)

• Component 𝑚 of force on node 𝑙 is 𝐠𝑙𝑚

𝑓 = − 𝜖𝑋𝑓 𝜖𝐲𝑙 = − ∑𝑗𝑘 𝜖𝑋𝑓 𝜖𝐆𝑗𝑘

𝑓

𝜖𝐆𝑗𝑘

𝑓

𝜖𝐲𝑙𝑚

• Note:
• Energy is quartic in 𝐲, forces are cubic
• Solve system of nonlinear equations

104

Nonlinear Elasticity

105

Limits

• Real-world materials are not perfectly (hyper)elastic
• Viscosity (stress relaxation, creep)
• Plasticity (irreversible deformation)
• Mullins effect (stiffness depends on strain history)
• Fatigue, damage, …

106

Finite Elements

What is a finite element?

A finite element is a triplet consisting of

• a closed subset Ω𝑓 ⊂ 𝑺𝑒 (in 𝑒 dimensions)
• 𝑜 vectors of nodal variables ഥ

𝒚𝑗 ∈ 𝑺𝑒 describing the reference geometry

• 𝑜 nodal basis functions, 𝑂𝑗: Ω𝑓 → 𝑺

→ 𝑜 vectors of degrees of freedom (e.g., deformed positions 𝒚𝑗)

𝑦 𝑣 𝑦𝑗 𝑣𝑗 = 𝑣𝑗+1 ⋅

𝑦−𝑦𝑗 𝑦𝑗+1−𝑦𝑗 + 𝑣𝑗 ⋅ 𝑦𝑗+1−𝑦 𝑦𝑗+1−𝑦𝑗

𝑦 ∈ 𝑦𝑗, 𝑦𝑗+1 ,𝑗 = 0 … 𝑜 − 1 = ෍

𝑗

𝑣𝑗 ⋅ 𝑂𝑗 𝑦

107

FEM – 1D – Basis Functions

108

Linear Simplicial Elements

1D: line segment 2D: triangle 3D: tetrahedron

• Simplicial elements admit linear basis functions
• Basis functions are uniquely defined through
• reference geometry ഥ

𝒚𝑗 and

• interpolation requirement 𝑂𝑗 ഥ

𝒚𝑘 = 𝜀𝑗𝑘

ഥ 𝒚𝑗 = ҧ 𝑦𝑗 in 1D ഥ 𝒚𝑗 = ( ҧ 𝑦𝑗, ത 𝑧𝑗) in 2D ഥ 𝒚𝑗 = ( ҧ 𝑦𝑗, ത 𝑧𝑗, ҧ 𝑨𝑗) in 3D ഥ 𝒚1 ഥ 𝒚2 ഥ 𝒚3 ഥ 𝒚1 ഥ 𝒚2 ഥ 𝒚4 ഥ 𝒚3 ഥ 𝒚1 ഥ 𝒚2

109

Computing Basis Functions – 2D

                               

i i i i i i

c b a y x y x y x

3 2 1 3 3 2 2 1 1

1 1 1                                   

 i i i i i i

y x y x y x c b a

3 2 1 1 3 3 2 2 1 1

1 1 1   

• Due to

, we have

ij j i

N   ) (x

Example: 3-node elements with linear basis functions

e



1

N

2

N

3

N

• Basis functions are linear:

c y b x a y x N

i i i

   ) , (

110

Computing Basis Functions – 3D

110 𝑂𝑗 ҧ 𝑦, ത 𝑧, ҧ 𝑨 = 𝑏𝑗 ҧ 𝑦 + 𝑐𝑗 ത 𝑧 + 𝑑𝑗 ҧ 𝑨 + 𝑒𝑗 Ω𝑓 ത 𝐲1 ത 𝐲2 ത 𝐲3 ത 𝐲4 ഥ V𝑓 4-node tetrahedron with 4 linear basis functions … … … …

• Basis functions are linear, 𝑂𝑗

ҧ 𝑦, ത 𝑧, ҧ 𝑨 = 𝑏𝑗 ҧ 𝑦 + 𝑐𝑗 ത 𝑧 + 𝑑𝑗 ҧ 𝑨 + 𝑒𝑗

• From 𝑂𝑗 ഥ

𝒚𝑘 = 𝜀𝑗𝑘 we obtain

112

Example

https://www.youtube.com/watch?v=I46ly-ubzYQ

113

Example

https://vimeo.com/245424174