Physically-based Modelling - Introduction to solid mechanics -

Benjamin GILLES

LIRMM, Equipes ICAR/DEMARCNRS, Université de Montpellier

http://www.lirmm.fr/~gilles/physics.pdf

Videos

Fluidmechanics

Solidmechanics

Interactive simulation

Procedural animation

Computer Graphics

Computational medecine

Mechanical engineering

Human Machine interaction

Images to Models to SimulationsImages to Models to Simulations

Geometric modeling Visualization Diagnosis Comparative anatomy Data fusion

..

Benjamin GILLES Introduction 1/7

measure simulate

Computer animation Mechanical Simulation Surgical planning

...

Model vs. dataModel vs. data

Images Curves Surfaces Volumes Hybrid models

Static

Kinematics

Dynamics

Mechanics

Physiology

Complexity

[delp02]

[weiss05]

[teran05]

[ng-thow-hing00]

[hirota01]

[scheepers97]

[aubel02]

[arnold00]

[badler79]

[maurel98]

Basics

Discrete methods

Continuum based methods

Time integration

Contact handling

OutlineOutline

Mathematical representation

t

Mathematical representations Mathematical representations

First order FEM

Higher order FEM

Mesh-free simulation

• Implicit models– Iso-value of a potential field

in 3D: { p 3 | F(p)=0 }

• Discrete models– Explicit positions in space (vertices)

+ connectivity relationships

• Continuous (or parametric) models– Mapping between material parameters and spatial coordinates

in 3D: t [0,1]p → p = [x(t),y(t),z(t)]T 3

Standard workflow in simulation

- Setup the kinematic degrees of freedom : nodes q- Compute the mass M- Compute forces f

– Internal forces due to deformation– Contact response– Gravity : Mg– User interaction– Damping 

→ space integration

- Solve Newton's equation :

- Update degrees of freedom : → time integration

q = M−1 f

q=∬ q dt

f (q)

f (q)

DEMO

DeformationRestconfiguration Deformed

configuration

q i q i

q (t)

q=∂ q∂ t

q=∂ q∂ t

: coordinates (rest)

: coordinates (deformed)

: velocities

: accelerations

q

For computation, motion is often discretized using a reduced number of independant degrees of freedom = nodes

Nodes can be : points, rigid frames, affine frames, angles, etc...

The internal energy W measures the departure from the rest configuration

Elastic forces minimize the deformation

The stiffness matrix measures local force variations

q=[q0x

q0y

q0z

q1x

q1y

q1z

:

]e.g. points :

f=−W q

K= fq

Multi-model representation

Split computations into independent parts

- Separate problems

- Improve reusability

[SOFA framework]

Kinematic mappingKinematic mapping

Master nodes

Slave nodes

velocities : q

positions: q

forces : f pvelocities : p=J qpositions: p=Φ(q )

J= p q

→forces : f q=JT f p

compute forces

accelerations : q

accelerations : p=J q+dJ q

Time integration

∬q

Example : barycentric mapping

p j=∑ wij qi

q0

q1

q2

p0

J=[w ij ]

q0

q1

q2

q f q= p f pPower conservation :

f p0

f q0

f q1

f q2

Nodes

Deformation

Time integration

Spatial integration

∬q

Visual model

Collision model

Spatial integration

Mass points

Spatial integration

Elastic energyKinetic energy

Multi-model representation

Hybrid models using embeding

Nodes

Time integration

∬q

3D deformation

Spatial integration

Elastic energy

2D deformation

Spatial integration

Elastic energy

1D deformation

Spatial integration

Elastic energy

Discrete representationsDiscrete representations

Particules,points

Spatial dimensions Material dimensions

Material Dimensions =

0 1 2 3

Edges Triangles

Quadrilateral, 2D image

Hexahedra,3D image

Tetrahedra

Discrete representationsDiscrete representations

In general :

Material Discretization Kinematic Discretization ( Material coordinates ) ( spatial coordinates )

Material = fine quadsKinematic model = coarse quads

Material = fine quadsKinematic model = points

Material = Kinematic model= Triangles

Construct volumetric mesh from surface MeshProblem: regular tetrahedra do not tile space

SegmentedSegmented MRI MRI Surface Surface meshmesh Volumetric meshVolumetric mesh

Volumetric mesh generation

Isosurfacing :Marching cubes algorithm

??

•Requirements –Element type: Tetrahedron, Hexahedron, etc.

–Element density

•Quality measure–Boundary / input surface matching–Element quality: solid angle, radius ratio, etc.

Input surface meshInput surface mesh

Tetrahedral meshTetrahedral mesh

Tetrahedral meshTetrahedral mesh

Hexahedral meshHexahedral mesh

Different types of degeneracy Different types of degeneracy (slivers, (slivers,

caps, needles and wedges)caps, needles and wedges)

Volumetric mesh generation

Meshing techniques:Rasterization (+ octree recursively Subdivision)

Aliasing, poor quality elements generated near the boundary

Advancing front: cells propagation from boundaries Difficult to compute ideal cell locations (local) Difficult to merge elements when they collide

Owen (1998)Owen (1998)

Owen (1998)Owen (1998)

Volumetric mesh generation

Variational approach Vertex repositioning → avoid degenerate configurations

+ Delaunay triangulation → optimal connectivity

2D Delaunay criterion

(a) Maintained

(b) Not maintained

Volumetric mesh generation

[alliez05]

Basics

Discrete methods

Continuum based methods

Time integration

Contact handling

OutlineOutline

Energy :

Forces :

Momentum conservation :

Problems when changing resolution :

– In parallel : → normalize by S– In serie : → normalize by l

Resolution-independant stiffness :

q0 l q1 q0

lq1

u

f 1 = k ( l−l )u , f 0 = − f 1

pp

Mass-spring models

∑ f i = 0 , ∑ f i∧q i =0

k lS

W = k( l −l )2

2

– Shape matching [mueller04]

• for each cluster: compute closest rigid transform approximating displacements from rest configuration

• Average rigid transforms

As-rigid-as-possible deformation

– Shape matching [mueller04]

• for each cluster: compute closest rigid transform approximating displacements from rest configuration

• Average rigid transforms

As-rigid-as-possible deformation

Basics

Discrete methods

Continuum based methods

Time integration

Contact handling

OutlineOutline

Continuum Mechanics

Continuum Mechanics

Solid

Mechanics

Linear Elasticity (Anisotropic, heterogeneous)

Fluid Mechanics

PlasticityHyper

Elasticity

Continuum Elasticity Basics

Deformation and Force(s)

Undeformed Rest State length L

L LElongation ∆L due to External Force F 1 Newton = 1 N = 1 Kg·m/s2

F

F F F

Discretization (or Cutting Open) reveals Body Forces F (and F = F for Static Equilibrium)

Strains and Stresses

Any deformation (i.e. elongation, compression) is termed the STRAIN (ε) of the spring and generally defined (dimensionless) as ∆L/L

L LF

Any force applied to a cross-sectional surface is termed the STRESS (σ) acting on this surface and is measured as force per unit area (N/m2)

F

σ = F/A

Continuum Elasticity Basics

Soft Tissue Characterization

To characterize a tissue, its stress-strain relationship is studied

Rest configuration

Radius r

Height h

Radius r*

Force F

Height

h*

Stress = F

r2

Strain = h*-h

h

Deformed configuration

Biological TissueBiological Tissue

complex phenomena arise

loading

Unloading

Hysteresis

Linear Domain Plasticity and

rupture

Slope =Young Modulus

Non-Linearity

Anisotropy

V0

V1V2

V3

Visco-elasticity

Elastic potential energy

W (ε0)=∫0

ε0

σ dε

Area under stress/strain curve = Energy density stored in the material

W (ε0) =12

ε0σ =12k ε0

2For linear materials :

Will see later the generalization to 2D and 3D

units : [N.m

m3]=[

Joules

m3]

Computer science

Numerical approximations

Discrete quantities

Discrete geometry

Elastic energy

= displacements*forces

Kinetic energy

=velocity²*mass

Linear systems

Physics

Analytic solutions

Continuous quantities

Differential geometry

Elastic energy

=stress*strain*dv

Kinetic energy

=velocity²*density*dv

Differential equations

Convergence ?

Hypothesis ?

Definitions

p

pp

δ v

Restconfiguration Deformed

configuration

p

u( p)=p− p

δ v

: material coordinates of a point (rest configuration)

: spatial coordinates of a point (deformed configuration)

: deformation function

: displacement function

: infinitesimal volume

: density

: traction (or stress vector)

ρ

tδ s

t=δ fδ s

Ω

σ .n=t N on ∂ΩN

u=uD on ∂ΩD

Boundary conditions :- Neumann conditions : - Dirichlet conditions :

Φ( p)=p

Solution of elasto-static problem

∇ .σ+ρ f=0 p∈ΩEquilibrium equation (strong form):

subject to boundary conditions

Galerkin approach → approximate solution based on a discretization

Integration by part → Variational formulation:Find u such that is minimal

Body forces (ie. gravity)

Can be solved using finites differences.. but they are very little used in solid mechanics (due to complex geometries and boundary condition enforcement)

∫Ω(∇ .σ+ρ f ). δw δ v=0 δw∈CPrinciple of virtual work (weak form):

Displacement test function

∫ΩW (u)δ v−∫

Ωρ f.u δ v−∫

∂ΩN t

N .u δ s

div

Discretization

3 types of discretizations :

- Deformation → kinematic degrees of freedom

- Spatial integration → material points → mass points

→ contact points

- Temporal integration → time steps

Kinematic discretization

pp

δ v

Restconfiguration Deformed

configuration

q i q i

q

q=∂ q∂ t

q=∂ q∂ t

: generalized coordinates (rest)

: generalized coordinates (deformed)

: generalized velocities

: generalized accelerations

q

For computation, motion is often discretized using a reduced number of independant degrees of freedom = nodes

Nodes can be : points, rigid frames, affine frames, angles, etc...

p=J q

p=Φ( p , q , q)

J= p q

Nodes and material points are related through a mapping :with :

q=[q0x

q0y

q0z

q1x

q1y

q1z

:

]e.g. points :

Lagrangian mechanics

W = ∫ΩW ( p)δ vTotal elastic potential energy in the material:

T =12∫

ΩpT pρδ vTotal kinetic energy in the material:

Lagrangian: L=T-W

ddt

(∂ L∂ q

) =∂ L∂q

Euler-Lagrange equation (generalized coordinates):

(∫Ω

∂ p∂ q

T∂ p∂ q

ρδ v) q = −∂W∂q

→ Newton law of motion:

D =12∫

ΩΨ( pT p)δ v

Can add dissipation/ damping/ viscous forces:

with the dissipation energy :

M q = −∂W∂ q

−∂D∂ q

M fgeneralized mass and forces :

Modeling pipeline

Define control nodes

= kinematic Degrees Of Freedom

Generate a smooth deformation function

based on nodes equiped with shape functions

Then, follow the classic pipeline :

Initial

Deformed

p

p

Φ( p)

Node influence= shape function

Nodes

Material

positions ← forces ↓ ↑

strain → stress

Modular vision of solid mechanics

Nodes

Integration points

Time integration

Spatial integration

velocities : q

positions: q

forces : f p=− E p

∫E

∬q

Independent degrees of freedome.g. : Points, rigid frames, affine frames, angles...

Interpolation method using shape functione.g. : barycentric, moving least squares, skinning..

Solve equations of motione.g. : static, explicit, implicit..

p

velocities : p=J qpositions: p=Θ(q)

J= p q

forces : f q=JT f p

Mapped quantitiese.g. : strain, displacement..

Energy, constitutive lawe.g. : kinetic, elastic, external..

Quadrature methode.g. : midpoint, Gauss, elastons..

accelerations : q

accelerations : p=J q+dJ q

Modular vision of solid mechanics

Independent degrees of freedom Points

Interpolation method : Linear

Shape function : barycentric

First order finite elements

p

Mapped degrees of freedom : Green strain

Constitutive law : Hooke's law

Quadrature method : midpoint

St Venant Kirchoff material

Nodes

Integration points

Time integration

Spatial integration

velocities : q

positions: q

forces : f p=− E p

∫E

∬q

velocities : p=J qpositions: p=Θ(q)

J= p q

forces : f q=JT f p

accelerations : q

accelerations : p=J q+dJ q

Modular vision of solid mechanics

Independent degrees of freedom Points

Interpolation method : moving least squares

Shape function : radial

Meshless simulation

Mapped degrees of freedom : Green strain

Constitutive law : Hooke's law

Quadrature method : midpoint

St Venant Kirchoff material

Nodes

Integration points

Time integration

Spatial integration

velocities : q

positions: q

forces : f p=− E p

∫E

∬q

velocities : p=J qpositions: p=Θ(q)

J= p q

forces : f q=JT f p

accelerations : q

accelerations : p=J q+dJ q

Modular vision of solid mechanics

Independent degrees of freedom : Frames

Interpolation method : Linear Dual-quaternion blending

Shape function : voronoi-based

Frame-based simulation

Mapped degrees of freedom : Green strain

Constitutive law : Hooke's law

Quadrature method : midpoint

St Venant Kirchoff material

Nodes

Integration points

Time integration

Spatial integration

velocities : q

positions: q

forces : f p=− E p

∫E

∬q

velocities : p=J qpositions: p=Θ(q)

J= p q

forces : f q=JT f p

accelerations : q

accelerations : p=J q+dJ q

Kinematics- Deformation gradient

- Strain

- Discretization

- Interpolation

Time integration

Spatial integration

∫E

∬q

Nodes

Strain

Deformation grad.

Deformation gradient

p

pp

Restconfiguration Deformed

configuration

p

u( p)=p− p

Ω

: material coordinates of a point (rest configuration)

: spatial coordinates of a point (deformed configuration)

: deformation function

: displacement function

pF=∇ Φ=

d pd p

Deformation gradient F : Captures the local affine deformation

l l

F =d pd p

=ll

pp

In 1D :

F=[∂ px∂ px

∂ px∂ p y

∂ px∂ p z

∂ p y∂ px

∂ p y∂ p y

∂ p y∂ p z

∂ p z∂ px

∂ pz∂ p y

∂ p z∂ p z

]In 3D :

Φ( p)=p

Distance between point may not be preserved

Distance between deformed points

Right Cauchy-green Deformation tensorMeasures the change of metric in the deformed body

C=FT F

Right Cauchy deformation tensor

p p+d p

Restconfiguration Deformed

configuration

Ω

p=Φ( p)Φ( p+d p)

dl2=Φ( p+d p)−Φ( p)2≈d pT (FT F )d p

Strain tensor

Example : Rigid Body motion entails no deformation

the strain tensor captures the amount of deformationIt is defined as the “distance between C and the Identity matrix”

p=R p+t F=d pd p

=R C=RT R=I

E=12(FT F−I )=

12(C−I )

Green-Lagrangian strain

l lE = (

ll)2

−1pp

In 1D :

dE=12(FT dF+dFT F)

Diagonal Terms : iiCapture the length variation along the 3 axis

Off-Diagonal Terms :ijCapture the shear effect along the 3 axis

Strain tensor

E=[εxx εxy εxzεxy ε yy ε yzεxz ε yz εzz ]

Linearized Strain Tensor

Use displacement rather than deformation

Assume small displacements

ε =12

(∂u∂ p

+∂u∂ p

T

) =12

( FT + F ) − I

E =12

(dud p

+dud p

T

+dud p

T d ud p

)

F =d pd p

= I +dud p

Cauchy strain

l lε =

ll

− 1ppIn 1D :

d ε =12

(dFT+dF )

Shortcomings of linear elasticityShortcomings of linear elasticity

Non valid for « large rotations and displacements »

Corotational strain

Polar decomposition of deformation gradient :

F = RS Where R=rotation and S=small affine excursion from rotation

FR

ε =12

( ST + S ) − I =12

( FT R + RT F ) − I

Remove rigid component from the deformation gradient

d ε ≈12

(dFT R+RT dF )

Classification of interpolation methods :

Point-based methods [Nealen05]:- Mesh-based :

– Barycentric (edges, triangles, tetrahedra)– Bilinear, trilinear (quads, hexahedra)– Higher order interpolations (2d order triangles, etc.)

- Mesh-free :– Shepard method (SPH)– Radial basis functions– Moving least squares– Natural element method (Sibson, Laplace)

…

Frame-based methods : - Skinning ...

Interpolation

Standard point-based methods are based on a weighted combination :

p=Θ( p , q , q)

p− p=∑iwi [qi−qi ]

Mapping from kinematic model to material :

Desired properties :– Smooth and locally supported weights → sparse matrices

– Linear completeness (can reproduce linear displacements)

– Interpolating

– Can reproduce rotations

: weight (or shape) function of node i = influence of node iw i( p)

u=∑iwiu → ∑i

w i=1

w i(qi)=1 , wi(q j)=0 i≠ j

R p=∑iwiR qi → p=∑i

wi qi , p=∑iwiqi

Barycentricinterpolation

Partition of unity

First order edge elements

q0

lq1 q0 l

q1

u

pp

1 w1w0

w0=1−p−q0

l

w1=1+p−q1

l

∇w 0=∂w0

∂ p=−

1l

∇ w1=∂w1

∂ p=+

1l

Linear shape function :

u = p− p = ∑ wi(qi−q i)Linear mapping of material points :

= barycentric coordinates

, dp = ∑ widqi

Mapping of deformation gradient :

F =∂ p∂ p

= ∑ (q i−q i)∇ wi+ I =llu , dF = ∑ dq i∇ wi

constant JacobianUniform in element

constant Jacobian

First order triangle elements

p=∑ w i q i

p=∑ w iq i

q0

q1

q2

p

q0

q1

q2

p

u = p− p = ∑ wi(qi−q i)

w i=1−( p−qi). ni

h i

n2

n1

n0h2

∇w i=∂wi

∂ p=(−

nihi

)T

Linear shape function :

Mapping of deformation gradient : F =∂ p∂ p

= ∑ (q i−q i)∇ wi+ I , dF = ∑ dq i∇ wi

, dp = ∑ widqi

FF

= barycentric coordinates

Linear mapping of material points :constant Jacobian

constant JacobianUniform in element

First order quad elements

p=∑ w i q i p=∑ w iq i

q0 q1

q2

pq0

q1

q2

p

q3 q3

Lx

lx

with w x=lxLx

, w y=l yL y

[w0

w1

w2

w3]=[

(1−w x)(1−w y)

w x (1−w y)

(1−w x)w y

w xw y]Bilinear shape function :

u = p− p = ∑ wi(qi−q i)Linear mapping of material points :

Mapping of deformation gradient : F =∂ p∂ p

= ∑ (q i−q i)∇ wi+ I , dF = ∑ dq i∇ wi

, dp = ∑ widqi

→ not uniform inside an element ! → need several evaluations for energy integration

[∇ w0

∇ w1

∇ w2

∇ w3]=[

(w x−1)∇ w y+(w y−1)∇ wx

w x∇ w y+(w y−1)∇w x

(w x−1)∇w y+w y∇w x

wx∇ w y+w y∇ w x]

constant Jacobian

constant Jacobian

Second order edge elements

q0

l

q1 q0

q1p

p

1

w0=1−3 (p−q0

l)+2(

p−q0

l) ²

w1=1+3(p−q1

l)−2(

p−q1

l)²

w2=1−4(p−q2

l) ²

Quadratic shape function :

→ same methodology for triangles, quads, tetrahedra, hexahedra, etc.

Lagrange polynomial

q0

q2

w0 w1

w2

q2

∇w0=−3l

+4(p−q0

l²)

∇w1=3l−4 (

p−q1

l²)

∇w2=−8 (p− q2

l²) ²

u = p− p = ∑ wi(qi−q i)Linear mapping of material points :

Mapping of deformation gradient : F =∂ p∂ p

= ∑ (q i−q i)∇ wi+ I , dF = ∑ dq i∇ wi

, dp = ∑ widqi

→ not uniform inside an element ! → need several evaluations for energy integration

constant Jacobian

constant Jacobian

Natural neighbor interpolation

q0

q1

q2

p

q0

q1

q2

p

s1s0

s2h2

h1

h0a1

a2

a0

p=∑ w i q i

p=∑ w iq i

Voronoi diagram : division of space based on proximity criterion

Add each material point to node Voronoi digram

w i=ai

∑ ja j

w i=si /hi

∑ js j /h j

Sibson interpolant :

Laplace interpolant :

Weight derivatives are hard to compute → approximation using a background image and central differences

p− p=∑ wi(qi−q i)

[Sukumar]

Moving least squares

q0

q1

q2

p

q0

q1

q2

p

p− p=∑ wi(qi−q i)

θ( p−q0)

θ( p− qi)=Max (0,(1− p−qi²

r i²)3

)

Radial and compactly supporter kernel function :

for instance

Shape function : w i( p) = θ( p−qi) p ( p)TM ( p)−1 p(qi)

M ( p)=∑iθ( p−qi) p(qi) p (qi)

T , p(x)=[1x ]

As interpolant as possible given the relation :

With moment matrix and polynomial basis :

p− p = A ( p) p ( p)

dr

Each material point needs to be influenced by 4 nodes, otherwise M is singular

- Stress

- Strain energy

- Some common materials

- Soft tissue characterization

Material response

Linear Elastic MaterialLinear Elastic Material

Simplest Material behaviour

Only valid for small deformations (less than 5%)

Biological TissueBiological Tissue

complex phenomena arises

loading

Unloading

Hysteresis

Linear Domain

Slope =Young Modulus

Non-Linearity

Anisotropy

V0

V1V2

V3

Visco-elasticity

Stress tensor

: unit normal

: stress tensor

σ=[T (1)T (2)T (3)]=[

σ xx σ xy σ xzσ xy σ yy σ yzσ xz σ yz σzz

]

n

σ [N

m2]

t=σ .n

σ= [σ xxσ yyσ zzσ xyσ yzσ xz

]The stress tensor is symmetric due to the principle of conservation of angular momentum

Voigt Notation

Elastic potential energy

W (ε0)=∫0

ε0

σ :d ε

Area under stress/strain curve = energy density stored in the material

units : [N.m

m3]=[

Joules

m3]

: operator = Frobenius inner product σ :d ε=∑i∑ jσ ij d εij=tr (σ

T d ε)

Using Voigt notation : σ :d ε=[σxxσ yyσ zzσxyσ yzσ xz

]T

[d εxxd εyyd εzz2d εxy2d ε yz2 d εxz

]Density of internal forces: ∂W

∂ε=σ Stiffness : K=

∂ ²W∂ε ²

=∂σ∂ε

Linear materials

W=12

σT ε=12

εT H ε

Based on Hooke's law : σ=H ε

6x6 Hooke stiffness matrix

Strain energy density with Voigt notation :

Anisotropy (21 independent material constants)

H=[C11 C12 C13 C14 C15 C16

C22 C23 C24 C25 C26

C33 C34 C35 C36

C44 C45 C46

sym. C55 C56

C66

]

Linear materials

W=12

σT ε=12

εT H ε

Based on Hooke's law : σ=H ε

6x6 Hooke stiffness matrix

Strain energy density with Voigt notation :

Anisotropy (21 independent material constants)

Orthotropy (9)

H=[C11 C12 C13 0 0 0

C22 C23 0 0 0C33 0 0 0

C44 0 0sym. C55 0

C66

]

Linear materials

W=12

σT ε=12

εT H ε

Based on Hooke's law : σ=H ε

6x6 Hooke stiffness matrix

Strain energy density with Voigt notation :

Anisotropy (21 independent material constants)

Orthotropy (9)

Transversal isotropy (5)

H=[C11 C12 C13 0 0 0

C11 C13 0 0 0C33 0 0 0

C 44 0 0sym. C 44 0

C11−C12

2]

Linear materials

W=12

σT ε=12

εT H ε

Based on Hooke's law : σ=H ε

6x6 Hooke stiffness matrix

Strain energy density with Voigt notation :

Anisotropy (21 independent material constants)

Orthotropy (9)

Transversal isotropy (5)

Isotropy (2)

H= Ε(1+ν)(1−2ν)[

1−ν ν ν 0 0 01−ν ν 0 0 0

1−ν 0 0 0(1−2ν)/2 0 0

sym. (1−2 ν)/2 0(1−2ν)/2

]E : Young's modulus [N/m²]V : Poisson's ratio [0,0.5[

σ = 2με + 2λ tr (ε) IEquivalent to :with Lamé coefficients : λ = Εν

(1+ν)(1−2 ν), μ= Ε

2(1+ν)

Deformation gradient :

Strain (linear):

Strain energy density (Hooke):

Total energy (midpoint integration) :

Forces :

= spring !

ε =12

(FT+F)−I =

ll−1

q0 l q1 q0

lq1

F =d pd p

=llu

u

W ( p) =ε

TH ε

2= Ε

2(

ll−1)²

W = v Ε2(

ll−1) ²

f 1 = −∂W∂q1

= Εsl( l −l)u = k ( l −l)u , f 0 = −f 1

pp

First order edge elements

Tensor invariants

I1=tr (C)

I2=12

[tr (C )2−tr (C2

)]

I3=det (C )

I1 = det (F )−2 /3 I1

I2 = det (F )−4 /3 I2

More general constitutive laws for hyperelastic materials are based on invariants of the right Cauchy deformation tensor :

For incompressible materials, deviatoric invariants are used : W ( I1 , I2 , det (F))

W (I1 , I2 , I 3)

Linear material : linear (Cauchy) strain + linear (Hooke) stress- strain function

St Venant Kirchoff material : non-linear (Green-Lagrangian) strain + linear (Hooke) stress- strain function

Other Hyperelastic Material

• Neo-Hookean Model:

• Mooney-Rivlin :

• Fung Isotropic Model :

• Veronda-Westmann :

• Incompressibility :

W = C1( I 1−3) , C1=μ

2

W = C1( I 1−3)+C2(I 2−3) , C 1+C2=μ

2

W = a( I 1−3)+b [ec. (I 1−3)−1]

W = −C1C2

2(I 2−3)+C1 [eC 2( I1−3)

−1]

W =k2(det (F)−1)²

W =k2ln (det (F))²

Estimating material parameters

Complex for biological tissue :- Heterogeneous and anisotropic materials

- Tissue behavior changes between in-vivo and in-vitro

- Ethics clearance for performing experimental studies

- Effect of preconditioning

- Potential large variability across population

Soft Tissue Characterization

Different possible methods- In vitro rheology

- In vivo rheology- Elastometry

- Solving Inverse problems

Soft Tissue Characterization

In vitro rheology- can be performed in a laboratory.

Technique is mature

- Not realistic for soft tissue (perfusion, …)

Soft Tissue Characterization

In vivo rheology- can provide stress/strain relationships at several

locations

- Influence of boundary conditions not well understood

Source : Cimit, Boston USA

Soft Tissue Characterization

Elastometry (MR, Ultrasound)• mesure property inside any organ non

invasively • validation ? Only for linear elastic materials

Source Echosens, Paris

Soft Tissue Characterization

Inverse Problems• well-suited for surgery simulation

(computational approach)• require the geometry before and after

deformation

Continuum solid mechanics : Summary

Time integration

Spatial integration

∫E

∬q

Nodes

Strain

Deformation grad.

positions: q

J 1=Fq

forces : f ε=−∫σ δv

positions:F=Θ1(q)

positions:F

positions:ε=Θ2(F )

J 2=ε F

forces : f F=J 2T f ε

forces : f F

forces : f q=J1T f F

Law of motion

Kinematic interpolation

Deformation measure

Material law

Deformation discretization

Time discretization

Material discretization

Spatial integration

Mid-point rule :– Suppose constant energy inside integration domains

Exact for 1st order elements

W = ∫ΩW ( p)δ v ≈ ∑i

W ( pi )δ vi

pδ v

q0 l q1

W = vk2(

ll−1) ²

p

approximantexact

Spatial integration

Gauss-Legendre quadrature :– Suppose that energy is a polynomial function of

position inside integration domains

W = ∫ΩW ( p)δ v ≈ ∑i

ciW ( pi)δ vi

e.g. : bilinear interpolation and Cauchy strain measure

→ energy and force are quadratic

→ 4 points allows for exact integrationp0

δ v

p1

p2 p3W = ∑i

14

W ( pi)δ vi

l y2

l y2√3

For irregular domains : no exact quadrature rule..→ approximation based on a background mesh

Basics

Discrete methods

Continuum based methods

Time integration

Contact handling

OutlineOutline

Kinetic energy

Nodes

Mass points

Time integration

Spatial integration

velocities : q

positions: q

forces : f p=− E p

∫E

∬q

velocities : p=J qpositions: p=Θ(q)

J= p q

forces : f q=JT f p

accelerations : q

accelerations : p=J q+dJ q

T =12pTM p p

T =12qTM q =

12qT (JTM p J ) q

M p = [. 0

.ρΔ v

0 .]

Simplifying the mass matrix

Approximation using midpoint rule :

Simplification using mass lumping:

M = [. . . .. . M ij .. . . .. . . .

] M ij = ∑ J iT J jρΔ v

M = [. 0

.M i

0 .]

Mass is moved to the nodes

M = ∫Ω

∂ p∂ q

T∂ p∂ q

ρ δ vGeneralized dense Mass matrix :

Temporal integration

Equilibrium = Solution of the variational problem:

Static solver (no mass, no velocity involved) :

– Explicit solution for small displacements :

– Iterative gradient descent

• Explicit scheme:

• Implicit scheme :

Find u such that is minimalE = ∫ΩW (u)δ v−∫

Ωρ f.u δ v−∫

∂ΩN t

N .uδ s

δ q=−∂E∂q

dt = fdt

q t+1=qt+f t dt

qt+1=qt+(fit+1+ fet)dt

→∂E∂q

= fi+fe = f = 0

f t+1≈ f t+K t δq→ ( I−K t .dt )(q t+1−qt)=f t dt

K=∂ fi∂ q

→ fi− fi0=K (q−q0) → K (q−q0)=−feConstant stiffness

Newtonian evolution → 1st order differential system:

Explicit schemes:

• Euler:

• Runge-Kutta: several evaluations to better extrapolate the new state [press92]

Semi-Implicit schemes:

• Symplectic Euler :

• Verlet: Use of previous states

→ Unstable for large time-step !!

Dynamic evolutionDynamic evolution

M q=f →δq=q dtδ q=q dt=M−1 f dt

qt+1=qt+qt dt

˙qt+1=qt+M−1 f t dt

˙qt+1=qt+M−1 f t dt

qt+1=qt+ ˙qt+1dt

Implicit schemes [terzopoulos87], [baraff98], [desbrun99], [volino01], [hauth01]

• First-order expansion of the force:

• Euler implicit

• Backward differential formulas (BDF) : Use of previous states

→ Unconditionally stable for any time-step

… But requires the inversion of a large sparse system

– Choleski decomposition + relaxation

– Iterative solvers: Conjugate gradient, Gauss Seidel

– Speed and accuracy can be improve through preconditioning

Dynamic evolutionDynamic evolution

(M−Bdt−K dt² )( ˙qt+1−qt) = (f t dt+K qt dt² )qt+1 = qt+ ˙qt+1dt

f t+1 ≈ f t +∂ f∂q

δ q +∂ f∂ q

δ q = f t + K δq + Bδ q

Biological TissueBiological Tissue

complex phenomena arises

loading

Unloading

Hysteresis

Linear Domain

Slope =Young Modulus

Non-Linearity

Anisotropy

V0

V1V2

V3

Visco-elasticity

Basics

Discrete methods

Continuum based methods

Time integration

Contact handling

OutlineOutline

Benjamin GILLES

Primitive intersection• Test triangle/triangle, triangle/points, sphere/points...• Acceleration using bounding volume hierarchies (BVH)

Volumetric detection• Distance fields• Image based

Simulation of articulated rigid bodiesCollision detectionCollision detection

Benjamin GILLES

Penalty methods• Acceleration-based : (stiff) springs [Moore88]

• Velocity-based: impulses [Mirtich94][Weinstein06]

• Position-based [Gascuel94][Lee00]

Constrained dynamics • Lagrange multipliers [Barraf94]

Acceleration, Velocity or Position based

Simulation of articulated rigid bodiesCollision responseCollision response

A. Nealen et al.Physically Based Deformable Models in Computer GraphicsEurographics 2005

ReferencesReferences

M. Teschner et al.Collision Detection for Deformable ObjectsComputer Graphics Forum 2005

SOFA opensource simulatorhttp://sofa-framework.org

Bullet physicshttp://bulletphysics.org/wordpress/