Computer graphics & visualization Soft Body Simulation

computer graphics & visualization

Simulation and Animation

Soft Body Simulation


Simulation and Animation – SS07Jens Krüger – Computer Graphics and Visualization Group

MotivationApplications of Deformable Objects

• Entertainment– Movies

• Pixar: Shrek, Nemo, The Incredibles…– Computer games

• Virtual simulator– Medicine

• Surgery, therapy, registration – Computational science

• Buildings, bridges



Deformation Models• The first physically realistic model was introduced to

computer graphics in 1987• Mathematically deformation can be described as a

time-varying vector valued function F(x, t)

x is the material coordinateF is the world coordinateu is the displacement vector



Dynamic SimulationsLagrange Equation of Motion:

External forces are in equilibrium with internal forces composed of elastic force, damping force and acceleration force.

uMuCuKF K : Stiffness

C : Damping

M : Mass



Mass-Spring systemsBody is modeled by masses linearly linked by springs with freely rotation springs.



Mass-spring simulation• Spring Link

– Edge = spring– Vertex = mass– Hooke´s law– Scalar Spring stiffness D

l

lllDFi 0

0l

l

1F 2F



Mass-spring simulation• Volume preservation

– Introduce volume forces– Scalar Volume stiffness Dv

FiFv

Fi

Fi



Time Integration• Verlet Integration (per vertex) – explicit scheme

– Easy & fast

• Numerical accuracy is critical– Timesteps have to be small (explicit scheme)

Faster accumulation of rounding errors– Damping necessary



Algorithm

Forever// Force calculationfor every vertex

calculate forceendfor

// Time integrationfor every vertex

update positionendfor

endfor

Positions

Forces



Mass-Spring systems• Body is modeled by masses

linearly linked by springs• Advantage:

– Simple

• Disadvantages:– Realistic physical properties

difficult to model– Unstable Explicit Integration

for stiff materials



Excursion

• Mass- Spring On the GPU



Data structures

• How to store per-vertex adjacency information?– Incident Edges

• Calculation of volume forces problematic

– Incident tetrahedra• Calculation of spring and volume forces

– Store tetrahedra (edges) directly(not only indices into a shared edge/tetrahedra list)

• Memory consuming, but avoids additional dependency level

• Disadvantages– Tetrahedra are stored multiple times– #Neighbors is not constant



Data structures (cont.)



Data structures (cont.)• Sort vertices according

to their valence

• Stack of textures thatgets smaller to the top

• Reduces memory overhead

• Further optimization– Early z-test



Force calculationfor every point p0

for every tetrahedron t incident on p0

// via four texture fetchesget center vertex coordinate p0get corner indices i1-3, get element stiffness esget rest spring lengths l1-3, get rest volume v

// via three dependent fetchesget coordinates of p1-3 through i1-3

calculate force on p0add to total force on p0

endforendfor



Results



Finite Stuff…

• FDM and FEM



Finite difference method FDMRemember FDM from CFD



• Discretization – Layout of grid points on a grid

• Location of discrete points across the domain

– Arbitrary grids can be employed• Structured or unstructured grids

– Implicit or explicit representation of topology (adjacency information)• Uniform grids: uniform spacing of grid points in x and y

y

x

x

y Pij

Pij+1

Pij-1

Pi+1j

Pi+1j+1

Pi+1j-1

Pi-1j

Pi-1j+1

Pi-1j-1

y = x

CFD – Computational Fluid Dynamics



• Finite differences– Approximate partial derivatives by finite differences

between points

– Derived by considering the Taylor expansion

sdifferencecentralxOx

uu

sdifferencebackwardxOx

uu

sdifferenceforwardxOx

uu

x

u

jiji

jiij

ijji

ij

)(2

)(

)(

211

11

11




• Finite differences for higher order and mixed partial derivatives

22

11

2

2

)()(

2xO

x

uuu

x

u jiijji

ij

22111,111112

)(,)(4

yxOyx

uuuu

yx

u jijijiji

ij




• Difference equations - example– The 2D wave equation

02

2

2

22

2

2

y

u

x

uc

t

u

042 11112

2

11

yx

uuuuuc

t

uuu tij

tij

tij

tji

tji

tij

tij

tij

Partial Differential Equation

Discretization on a 2D Cartesian grids yields Difference Equation




Finite Elements method (FEM)?



FEM• FEM is a method used for finding approximate solution

of partial differential equations (PDE)• Method: split the domain into a (possibly large) but

finite number of elements (triangles, rectangles, tetrahedra, hexahedra, etc.)

• Good for– complex domains (cars, planes, parts)– Dynamic domains (soft bodies, moving boundaries)– when the desired precision varies over the entire domain



FEM1. Split into the domain into a finite number of elements2. Define shape functions (Interpolations- funktionen) in

elements (e.g. barycentric interpolation)3. Approximate PDE solution between the sample points by the

shape function with the sample points as variables.

4. Compute partial derivative based on interpolated solution

i

ii upNpu )()(

ii

i ux

pN

x

u )( e.g.



FEM5. Multiply PDE with “test function”6. Spacial Integration (over p) over partial derivatives within

finite element7. Result: element equation (not dependent on p anymore)8. Combine element equations into a single global equation

system (via shared sample points)9. Solve this (possibly non linear) system



FEM vs. FDM• FDM is an approximation to the differential equation; the finite element

method is an approximation to its solution. • FEM is its able to handle complex geometries (and boundaries), FDM in its

basic form is restricted to handle rectangular shapes and simple alterations thereof

• FDM can be very easy to implement. • FDM can be seen as a special case of the FEM approach• there are reasons to consider the mathematical foundation of the FEM more

sound, for instance, because the quality of the approximation between grid points is poor in FDM.

• The quality of a FEM approximation is often higher than in the corresponding FDM approach, but this is extremely problem dependent and several examples to the contrary can be provided



Mass-spring systems• Pros

– Fast– No pre-process– GPU Implementation

• Cons– Volume not considered

• Physical laws violated!– Material Properties hard to

choose• Spring constants vs. elastic modulus

– Stable simulation requires very small time steps

• performance gets worse than expected• almost impossible for stiff material



To the next level …

• Finite element soft bodies



Elasticity theory

• : Total energy• u(x) : Displacement field• f(x) : External (surface) forces• g(x) : Volumetric forces (e.g. gravity)• (u) : Linearized notation strain tensor E• D : Material law (Hooke‘s law)

dsufdxugdxuDuu TTT )()(

2

1)(

uMuCuKF FuTKuuT



Elasticity theory: Terms

• 2nd Term: (Distant) Volume Forcese.g. Gravity, Linear/Angular Momentum, magnetic/electric fields

• 3rd Term: (Direct) Surface Forcese.g. push/pull, collision/contact


2

1)(



Now consider only the first part …

Elastic Energy:


2

1)(



Elasticity theory cont.Strain (Dehnung) Tensor E

describes elongations ofinfinitesimal small volumeelement along coordinates axes

x

y

z

Eyy

Eyx

EyzExy

Exx

Exz

Ezy

Ezz

Ezx



Elasticity theory cont.Stress (Spannung) Tensor internal forces acting on coordinate planes

x

y

z

yy

yx

yz

xy

xx

xz

zy

zz

zx



Elasticity theory cont.Material Law • Strain Tensor E and Stress Tensor are

coupled via Hooke‘s law

• , : Lamé coefficients“elongation along axes”

“deformation”



Elasticity theory cont.Lamé coefficients

– derived from elasticity modulus Em

and Poisson‘s ratio (Querkontraktion)

)1(2

)21)(1(

m

m

E

E



Elastic Modulus Em …Material Elastic Modulus GPa

Rubber (small strain) 0.01-0.1Low density polyethylene 0.2Polypropylene 37377,00Bacteriophage capsids 39142,00Polyethylene terephthalate 38385,00Polystyrene 38414,00Nylon 39266,00Oak wood (along grain) 11,00High-strength concrete (under compression) 30,00Magnesium metal (Mg) 45,00Aluminium alloy 69,00Glass (all types) 72,00Brass and bronze 103-124Titanium (Ti) 105-120Carbon fiber reinforced plastic (unidirectional, along grain) 44105,00Wrought iron and steel 190-210Tungsten (W) 400-410Silicon carbide (SiC) 450,00Tungsten carbide (WC) 450-650Single carbon nanotube [1] 1,000+Diamond (C) 1,050-1,200

… is a measure of the stiffness of a given material, not of a given part.



Poisson‘s ratio…material poisson's ratio

rubber 0.50saturated clay 0.40-0.50

magnesium 0.35titanium 0.34copper 0.33

aluminium-alloy 0.33

clay 0.30-0.45stainless steel 0.30-0.31

steel 0.27-0.30glass 0.24

cast iron 0.21-0.26sand 0.20-0.45

concrete 0.20Foam 0.10 to 0.40Cork ca. 0.00

auxetics negative

ll

dd



Auxetic Materials



Linearized Notation• E is a symmetric tensor

using only upper triangular part is sufficient

23

13

12

33

22

11

23

13

12

33

22

11

2

2

2

2

2

2

)(

E

E

E

E

E

E

uD



Intuition: Elastic Energy

• We have

you can think if this integral as


2

1)(

dxuDuT )()(

Forceextensionconstant springdistance SeeDe



Strain Formulations• Cauchy strain (linear, constant matrix)

– BroNielsen ’96

• Corotated Cauchy strain (linear, dynamic matrix)– Rankin ’88, Müller ’02, Hauth ’04

• Green strain (non-linear, dynamic matrix)



Linear approximation of Strain• Cauchy Strain

– ignores non linear part– looses rotational invariance



Linear approximation of StrainLinear vs. non-linear vs. co-rotational strain



Back to FEMSo far:• considered continuous elasticity

Now:• discretize to a finite number of cells/elements

Remember:• We are looking primarily for u (displacement

vector)



FEMConsider the continuous u as:

• a set of supporting points (FEM vertices)• a continuum in between the supporting points

generated from the interpolation of u at the supporting points

big difference to finite differences here!!



FEM Elements:



Considering a single elementCan use tetrahedron barycentric interpolation

PBN

P

N

N

N

N

PPPP

i

B

1

4

3

2

1

4321

1P

3P

2P

4PP



Remember this slide?1. Split into the domain into a finite number of elements2. Define shape functions in elements3. Approximate PDE solution between the sample points by the

shape function with the sample points as variables.

4. Compute partial derivative based on interpolated solution

i

ii upNpu )()(

ii

i ux

pN

x

u )( e.g.



Tet - FEM

PBN

P

N

N

N

N

PPPP

i

B

1

4

3

2

1

4321 i

ii upNpu )()(

ii

i ux

pN

x

u )(



How to solve for the deformations? is stationary (does not change over time)

thus it does not change over u either derivative for u must be zero

Consider only the first term in the lecture,other terms are processes analogously.


2

1)(



Solving the equation…Compute partial derivatives of u

dxuDuT )()(2

1

T

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

2

3

3

2

1

3

3

1

1

2

2

1

3

3

2

2

1

1 ,,,,,

i

ii upNpu )()(



Solving the equation…Due to the linearity of ε can be rewritten as

e

T

Bu

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

2

3

3

2

1

3

3

1

1

2

2

1

3

3

2

2

1

1 ,,,,,

i

ii upNpu )()(



Rewriting the equation

ee

eT

TeTTe

T

uK

udxDBB

xAxAxx

dxDBuBuu

dxuDuu

)(

2 )(2

1

)()(2

1



We did it…

2

2

2 dt

uuuM

dt

uuCuKf

uMuCuKF

dtttdttdttdttdttdtt

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Computer graphics & visualization Soft Body Simulation