Real time medical simulation

Outline

1. Introduction : Why real time ?

2. Continuum mechanics and Finite

Element Method

3. Dynamical system : Simulate virtual

organs

4. Collision detection and response

5. Parallel Computing

3

Interactive simulations

Sequence of images discrete in time

Images of the mechanical state of deformable structure

interpolate between images

Fast computation time :

Computation of the deformation

Collision detection

Contact solution

Rendering

User interactions

Need stable algorithm

Unpredictable motion

Deformation

Collision detection

Contact solving

Rendering

Interactions

Sim

ula

tio

n L

oo

p

4

External Forces

Motion

&

Deformation

Physics

computation

___________

Deformable

Rigid

Fluid

No contact

Timet t + dt

Feedback

___________

Visual

Haptic

Tactile

Collision

Detection

___________

Are the objects

in contact?

Conta

ct

Collision

Response

___________

Penalty method

Other Methods

Motion

&

Deformation

Physics

Computation

___________

Reuse the

same physical

modelContact Forces

Contact Forces

Simulation loop

5

Real time : definition

Additional constraints :

A frame rate of at least 60 FPS is necessary to have a visual fluidity

Less than 20 ms to process all the computations (computation of the deformation,

collision detection, contact force response, rendering, …)

Interactive simulation :

A simulation which is not slower than real time but fast enough to allow user interactions

At least 10 FPS

Simulation with haptic feedback

Haptic requires at least 1Khz to simulate a smooth force feedback, ant stiff interactions

Les than 1 ms to update haptic forces

Accuracy ? It’s relatively easy to simulate very fast with very simplified models

6

Strategy for real time

Behavior model

Simple mechanics law (mass spring models, shape matching, linear elasticity, ...)

FEM mesh

Influence directly both : the accuracy and the number of equations solved during a

time step

Temporal integration

Implicit vs. Explicit

Direct/Iterative solvers

Collision detection and contact response

Advanced modulation (friction, unilateral constraints,…)

Code optimizations and parallelism

High parallel architectures (GPU, CPU…)

Find a tradeoff between accuracy and computation time

Outline



Element Method

3. Image and meshes : From patient

data to medical simulations


organs

5. SOFA : A framework for interactive

medical simualtions

8

Continuum mechanics : basics

Assume a continuity of the matter (true at macroscopic scale, wrong at atomic

scale)

𝛷

Strain tensor

𝝈 Constituve law

휀 = 𝑓(𝜎)

𝜎 = 𝑓−1(휀)

Mechanical model :

• Co-rotational

• Neo Hookean

• MooneyRivlin

• Costa

• ….

No analytical solution for complex structures !

• Strain tensor : measure of the deformation– Cauchy-Green , Green-Lagrange, Euler-Almansi, Biot, Hill

• Stress tensor : measure of the stress

– Cauchy, first Piola Kirchhoff, second Piola Kirchhoff

• Constitutive law : measure of the deformation

– Hooke law, non linear material

9

Equations formulation

Continuum mechanics

Assumption of continuity of matter

Based on physics law

Define a deformation model

Solution based on finite element method

Discretization in small elements

Integrate and solve equations by elements

Approximation which depends on the size of elements

Solution of a dynamic system :

Time evolution of deformable bodies

Involves acceleration, velocity, and position

s Constitutive law

Strain

Str

ess

D

M a = f (p,v)

10

Discretization

Method = receipt how to

discretize the domain and solution

reformulate the differential equations

Digest of current methods

mass-spring damper (MSD)

finite-difference methods (FDM)

boundary element methods (BEM)

finite element method (FEM)

meshless (meshfree) methods

11

Real-time soft tissue deformation :: basic principles

Trade-off between accuracy and computation time

Many possible approaches but often only physically plausible

Physics-based does not mean it is accurate enough for medical training

Most common methods

Mass-spring systems

Finite Element Method

Meshless techniques

Main difference is in the computation of the internal forces

And to some extent the choice of the time integration scheme

12

A simple case: mass-spring models

Example: simulated springs

Particle

mass m, position, velocity

Spring

Rest length l0

Stiffness k

Force f = k(l – l0) with l=(x2 –x1)

Acceleration: a=f/m

Simplest case

One particle, one spring

Theoretical solution

13

Finite Element Method (FEM)

First appeared in 40s and 50s in civil and aeronautics engineering

Weak formulation of the problem

Multiplication by test function and integration over domain

Discretization of the into elements

Discretization of domain into geometric elements (e.g. simplexes)

Discretization of the equations using shape functions (for the solution and test func.)

Assembly of algebraic system of equations

Results in large sparse matrix

Imposition of the discretized boundary conditions

Several methods (penalty, elimination, Lagrange multipliers)

Numerical solution of the algebraic system

14

Weak form:

Discretization:

Finite Element Method (FEM)

Problem:

Assembling:

15

Discretization methods

Discretization

The first step in creating a physical representation of an object

Choice of discretization method depends on

data dimension (1D, 2D, 3D, 4D...)

input domain geometry (CAD, surface mesh, intensity image)

output type of elements (beams, plates, tetrahedra, hexahedra)

other (available tools/techniques, available hardware, target size/quality/format)

16

Discretization methods

Wide range of algorithms

Tessellations (tiling of a plane)

Delaunay triangulations [DT] (no point of the triangulation lies inside any circumcircle of

any triangle of the triangulation

Voronoi diagrams (dual to DT)

17

Discretization quality

Quality of elements is a crucial in physics-based applications (vertex Jacobian)

degenerated elements result in numerical instability (singularity of the Jacobian)

Various measures of element quality:

smallest angle/largest angle (2D)

dihedral angle (3D)

determinant of vertex Jacobian

ratio of inscribed/circumscribed radii

others (edge ratio, Frobenius aspect etc)

18

Influence of the mesh

CGAL

Convert a segmented image in a tetrahedral

mesh

Mesh

Discrete representation of the domain

Relation between the degree of freedom

Mesh influence

Computation time (on equation is solved at each

node)

Accuracy (the FEM solution tends toward the

exact solution when the size of elements tends

toward zero).

Fast Accurate

Fast Accurate

Fast Accurate

19

Constitutive laws

Small displacements and small deformations :

Linear constitutive law

Simple and fast

Inflates for large deformations/displacements

Large displacements and small deformations :

Compensate geometrical non linearity

Take into account the rotations / linear deformation

Co-rotationnal model

Large displacements and large deformations :

Simulation of large deformation

Hyperelastic model :

MJED [Marchesseau et al. 2010]

TLED [Comas et al. 2008]

Difficult to solve in real time

Co-rotational

FEM

Linear

FEM

Outline



Element Method


organs



21

Time evolution of the system

Second Newton law : σ𝑓(𝑝, 𝑣) = 𝑀 𝑎

Static and dynamic

Dynamic equation ∶ 𝑀 ሷ𝑢 − 𝐶 ሶ𝑢 − 𝒦(𝑢) = F

M mass matrix (lumped)

C damping matrix

𝒦(p,v) stiffness function of internal forces

F external forces (gravity , boundary conditions, …)

u displacement

Time Integration : update the acceleration using velocities v and position pExplicit integration

Implicit Integration

(Static)

(Dynamic)

22

Simulation loop

Advancing through time

What appears to be a continuous motion is in fact a sequence of discrete steps

If “dt” is small enough, the motion seems continuous

For smooth visual, dt should be less than 0.04 s (25 FPS)

However, other considerations can have a big impact on the value of “dt”

Time integration

Computing the simulation state at the next time step t+dt given :

State at the current time t: positions xt and velocities vt at each point

Equations and constraints expressing the required mechanics

1. a = a(x,v,t)

2. c(x,v,t) >= 0

How to obtain a fast yet stable simulation?

the choice of the integration scheme is very important

it’s called “integration” as it allows to go from xt to xt or from xt to xt.. . .

23

Explicit integration

Principle : use velocity and acceleration at the beginning of the time-step

Example: forward Euler

at = M-1f

xt+h = xt + hvt with h = dt

vt+h = vt + hat

Main characteristics

Simple

Amplifies motion

Damping is required

Stiff systems require very small time steps

24

Implicit integration


Principle : use velocity and acceleration at the end of the time-step

Example: backward Euler

Solve

vt+h = vt + hat+h

xt+h = xt + hvt+h

Main characteristics

Complex (requires a solver)

Under-estimates motion

Introduces additional damping

Stable for stiff systems even with large time steps

Well-suited for soft bodies

25


Principle : compute the acceleration using the position and velocities at the

beginning of the time step

Main properties

Simple (Local solution)

Fast and parallelizable

Amplifies the motion (Need to add damping)

Stability not guarantee (Requires very small time step for stiff objects)

𝒇(𝑝𝑡 , 𝑣𝑡)𝑴𝑎𝑡+ℎ =𝑣𝑡+ℎ = 𝑣𝑡 + ℎ𝑎𝑡+ℎ𝑝𝑡+ℎ = 𝑝𝑡 + ℎ𝑣𝑡+ℎ

Forward Euler :

26


Critical time step

Conditionally stable = tc exists such that if h < tc then the simulation is stable

For linear behavior models tc = 𝐿

𝑠

L = charasteristic length

s = dilatational wave speed

For tetrahedral elements L = 3𝑉

𝑚𝑎𝑥𝐴with V the volume of the tetrahedra and maxA the largest

area of the 4 triangle of the faces of the tetrahedra

s = 𝐸 ∗1−𝑣

𝑟ℎ𝑜∗ 1 + 𝑣 ∗ (1 − 2𝑣) with rho the mass density, v Poisson coefficient et E le Young

modulus

27


Principle: Update the acceleration from position and velocity at the end of the

time step

Main properties

Complex (implies to solve a linear system)

Motion is under estimate (Numerical damping)

Stable with large time step and stiff objects (simulation of anatomical structures)

𝒇(𝑝𝑡+ℎ, 𝑣𝑡+ℎ)𝒇(𝑝𝑡 , 𝑣𝑡)

Problem : 𝑝𝑡+ℎand 𝑣𝑡+ℎ are unknown

𝑴𝑎𝑡+ℎ =𝑣𝑡+ℎ = 𝑣𝑡 + ℎ𝑎𝑡+ℎ𝑝𝑡+ℎ = 𝑝𝑡 + ℎ𝑣𝑡+ℎ

Backward Euler

28


Euler implicit :

𝑢𝑡+ℎ = 𝑢𝑡 + ℎ ሶ𝑢𝑡+ℎ ሶ𝑢𝑡+ℎ = ሶ𝑢𝑡 + ℎ ሷ𝑢𝑡+ℎ

Rearranging gives :

ሶ𝑢𝑡+ℎ =𝑢𝑡+ℎ − 𝑢𝑡

ℎ ሷ𝑢𝑡+ℎ =ሶ𝑢𝑡+ℎ − ሶ𝑢𝑡

ℎ

Dynamic equation :

𝑀 ሷ𝑢𝑡+ℎ − 𝐶 ሶ𝑢𝑡+ℎ −𝒦(𝑢𝑡+ℎ) = 𝑝

𝑀ሶ𝑢𝑡+ℎ − ሶ𝑢𝑡

ℎ− 𝐶

𝑢𝑡+ℎ − 𝑢𝑡ℎ

−𝒦(𝑢𝑡+ℎ) = 𝑝

𝑀𝑢𝑡+ℎℎ2

− 𝐶𝑢𝑡+ℎℎ

−𝒦 𝑢𝑡+ℎ = 𝑝 +𝑀ሶ𝑢𝑡ℎ+𝑀

𝑢𝑡ℎ2

− 𝐶𝑢𝑡ℎ

𝓢 𝑢𝑡+ℎ 𝓕

29


We want to find 𝑢𝑡+ℎ that equalizes the two members of the equation 𝓢 𝑢𝑡+ℎ = 𝓕

𝓢 𝑢𝑡+ℎ = 𝑀𝑢𝑡+ℎℎ2

− 𝐶𝑢𝑡+ℎℎ

−𝒦 𝑢𝑡+ℎ

𝓕 = 𝑝 +𝑀ሶ𝑢𝑡ℎ+𝑀

𝑢𝑡ℎ2

− 𝐶𝑢𝑡ℎ

For any input parameter 𝑥, we define the residual function

𝑅 𝑥 = 𝓢 𝑥 − 𝓕

The solution is given by 𝑥 such that 𝑅(𝑥) = 0.

𝑅 is still a non linear function.

We use the Newton Raphson algorithm to find the solution

30

Newton Rapshon

31


Defining 𝑥(𝑖+1) = 𝑥(𝑖) + 𝛥𝑥(𝑖), 𝑅 𝑥 is linearized as follows :

𝑅𝐿 𝑥(𝑖) + 𝛥𝑥(𝑖) = 𝑅𝐿 𝑥(𝑖) +𝜕𝑅

𝜕𝑥𝑥(𝑖) 𝛥𝑥(𝑖)

Value in 𝑥(𝑖) Tangent in 𝑥(𝑖)

At each iteration we want to find the intersection of the linearized residual function with 0

𝑅𝐿 𝑥(𝑖) +𝜕𝑅𝜕𝑥

𝑥(𝑖) 𝛥𝑥(𝑖) = 0 ⟺ 𝛥𝑥 𝑖 =𝜕𝑅𝜕𝑥

𝑥 𝑖−1

−𝑅𝐿 𝑥 𝑖

Displacement for the next iteration

Replacing in our original problem gives :

𝜕𝓢 𝑥 − 𝓕𝜕𝑥

𝑥 𝑖 𝛥𝑥 𝑖 = 𝓕 − 𝓢 𝑥 𝑖

𝜕𝓕𝜕𝑥

𝑥 𝑖 = 0with

⟺𝜕𝓢 𝑥𝜕𝑥


32


𝜕𝓢 𝑥𝜕𝑥


𝓢 𝑥 = 𝑀𝑥

ℎ2− 𝐶

𝑥

ℎ−𝒦 𝑥

𝓕 = 𝑝 +𝑀ሶ𝑢𝑡ℎ+𝑀

𝑢𝑡ℎ2

− 𝐶𝑢𝑡ℎ

Mechanical equation

Linearized problem

Combining equations gives :

𝑀

ℎ2−𝐶

ℎ−

𝜕𝒦 𝑥𝜕𝑥

𝑥 𝑖 𝑥(𝑖+1) − 𝑥(𝑖) = 𝓕 −𝑀

ℎ2−𝐶

ℎ𝑥 𝑖 +𝒦 𝑥 𝑖

𝛥𝑥(𝑖) = 𝑥(𝑖+1) − 𝑥(𝑖)Newton increment

𝑀

ℎ2−𝐶

ℎ−


𝑥 𝑖 𝑥(𝑖+1) = 𝓕 −𝑀

ℎ2−𝐶

ℎ𝑥 𝑖 +𝒦 𝑥 𝑖 +

𝑀

ℎ2−𝐶

ℎ−


𝑥 𝑖 𝑥(𝑖)

33


Usually we define the equations at acceleration level

𝑥(𝑖+1) = 𝑢𝑡 + ℎ ሶ𝑢𝑡 + ℎ2 ሷ𝑥(𝑖+1)

𝑥(𝑖+1)

= 𝑢𝑡 + ℎ ሶ𝑥(𝑖+1)

ሶ𝑥(𝑖+1)

= ሶ𝑢𝑡 + ℎ ሷ𝑥(𝑖+1)

Gathering and simplifying in the previous equation gives the final newton iteration :

𝑀 − ℎ 𝐶 − ℎ2𝜕𝒦 𝑥𝜕𝑥

𝑥 𝑖 ሷ𝑥(𝑖+1)

= 𝑝 +𝒦 𝑥 𝑖 + 𝐶 ሶ𝑢𝑡 +𝜕𝒦 𝑥𝜕𝑥

𝑥 𝑖 ℎ ሶ𝑢𝑡 + 𝑢𝑡 − 𝑥(𝑖)

The derivative of the stiffness function can be written as a matrix operator K

𝑀 − ℎ 𝐶 − ℎ2𝐾 ሷ𝑥(𝑖+1)

= 𝑝 +𝒦 𝑥 𝑖 + 𝐶 ሶ𝑢𝑡 + 𝐾 ℎ ሶ𝑢𝑡 + 𝑢𝑡 − 𝑥(𝑖)

Rayleigh damping• 𝐶 = 𝛼𝑀 − 𝛽𝐾

𝑀 − ℎ (𝛼𝑀 − 𝛽𝐾) − ℎ2𝐾 ሷ𝑥(𝑖+1)

= 𝑝 +𝒦 𝑥 𝑖 + 𝛼𝑀 − 𝛽 + ℎ 𝐾 ሶ𝑢𝑡 + 𝐾(𝑢𝑡 − 𝑥 𝑖 )

34

ሼ

𝑨 𝒙 = 𝒃

Semi implicit integration • To achieve real time computations we process only one iteration of the Newton

per time step

• Guarantee the stability but not the equilibrium of forces

• Converge toward the correct solution (at equilibrium without contacts)

• The newton algorithm is then initialize with 𝑥(𝑖) = 𝑢𝑡

𝑀 − ℎ 𝐶 − ℎ2𝐾 ሷ𝑥(𝑖+1)

= 𝑝 +𝒦 𝑥 𝑖 + 𝐶 ሶ𝑢𝑡 + ℎ 𝐾 ሶ𝑢𝑡

Implicit integration : Real time

35

1

4567

0 1 2 3 4 5 6 7

123

0 b c db e fc g h i jd f h k l m

i n o

a

j l o p q rm q s t

r t u

01

2 3

4

5

6

7

A is :• Directly obtained from the mesh structure• Sparse• Symmetric

• Positive definite

Direct solvers : • Direct inverse (dense)

• Factorization :

• A = LU • A = 𝐿𝐿𝑡 (cholesky)

• A = 𝐿𝐷𝐿𝑡Gauss elimination

Iterative solvers : • Refine a solution. Converge toward the correct

solution

• Gauss-Seidel• Jacobi

• Conjugate Gradient

Accurate / slow, expensive in memory

Approximation/ fast

Implicit integration : Real time

36

1

4567

0 1 2 3 4 5 6 7

123

0 b c db e fc g h i jd f h k l m

i n o

a

j l o p q rm q s t

r t u

01

2 3

4

5

6

7

0 1 2 3 0 1 3 0 2 3 4 5 0 1 2 3 5 6 2 4 5 2 3 4 5 6 7 3 5 6 7 5 6 7

0 4 7 12 18 21 27 31 34

a b c d b e f c g h i j d f h k l m i n o j l o p q r m q s t r t u

Row Pointer

Column Pointer

Values

Compressed Row Sparse format

37

Suppose we want to solve the following system of linear equations

Ax = b

for the vector x where the known n-by-n matrix A is symmetric (i.e. AT = A), positive

definite (i.e. xTAx > 0 for all non-zero vectors x in Rn), and real, and b is known as

well.

We say that two non-zero vectors u and v are conjugate (with respect to A) if

Since A is symmetric and positive definite, the left-hand side defines an inner

product

Two vectors are conjugate if they are orthogonal with respect to this inner

product. Being conjugate is a symmetric relation: if u is conjugate to v, then v is

conjugate to u.

Implicit integration : Conjugate Gradient

38

Suppose that is a set of n mutually conjugate

directions. Then is a basis of , so within we can expand the solution of

and we see that

For any

(because are mutually conjugate)

Find a sequence of n conjugate directions, and then compute the coefficients .


39

Let rk be the residual at the kth step:

The CG algorithm relies on a Gram-Schmidt orthonormalization. This gives the following expression:

Following this direction, the next optimal

location is given by

The is chosen such that is

conjugated to . Initially, is


40

ODE Solver

Non linear Solver

linear Solver

Euler implicitEuler explicitRunge Kutta

Verlet

Newton-Raphson

Conjugate GradientCholesky SolverDirecte inverse

Implicit integration : Solverse

41

Summary on time integration schemes

Two main types

Explicit and implicit

Many methods (Euler, Runge Kutta, Leap Frog, Verlet, ...)

All have benefits and drawbacks

The choice depends on the application

Mainly

explicit methods: conditionally stable, i.e. depends on time step dt

implicit methods: unconditionally stable, i.e. do not (entirely) depend on time step dt

Other considerations include energy loss or gain

42


A simulation is real-time if all the computations done in 1 time step are done in a

time equal to the time step.

The number of Frame Per Second (FPS) times dt must be equal to 1 second.

dt

t t+dt

Simulation

time

t+1sec

User time

(real)

t+1sect

43


Simulation too fast:

Slow down the simulation with a waiting loop

dt

t t+dt t+1sec

t+1sect

t+1sec

Simulation

time

User time

(real)

44


Simulation too slow :

Record the images of the simulation, and play them faster to reach real time.

Impossible when the user interact with the simulation !

dt

t t+dt t+1sec

t+1sect

Simulation

time

User time

(real)

45


Synchronize the simulated instruments with the simulated bodies

Slow down the motion of the instruments in the simulation

Difficult (impossible) to use for a surgeon

Delay when applying haptic forces

Real motion Simulation

Deformabl

eStructure

(slow)

46


Synchronize the simulated instruments with the user motion

The velocity of the instruments is faster than other deformable structures

Important error introduced during the interaction

Real motion Simulation

Structure

déformable

Deformabl

eStructure

(slow)

47


Crash test example :

The velocity has a huge impact on the resulting deformation

Delay in the simulation introduce important errors !

Outline



Element Method


organs



49

Collision Detection

Stable modeling of interactions between objects in real-time simulation

Determination of collision area

Computation of collision volume

Needs to be :

Fast for real-time simulation

Accurate for computation of contact forces

49

Collision volume

50

Collision Detection

Deciding what to test hat to test

Applying tests Applying tests

Determining whether a collision occurred

Determining when a collision occurred

Determining where a collision occurred

Broad-phase: select pairs for comparison

Goals:

Make selections quickly

Avoid selecting objects that can’t collide

Narrow-phase: compare individual pairs

Goals:

Accurate comparisons

Efficient comparisons

51

Collision Detection

Convex Polygon Test

Test point has to be on same side of all edge

Point/Sphere

Simplest 3D Bounding Volume

Center point and radius

Point in/out test: Calculate distance between test point and center point

If distance <= radius, point is inside

Axis-Aligned Bounding Boxes (AABB)

51

52

Mapping

Collision detection must be accurate

One sphere is not enough

Collision test are expensive

Use simplified mesh

Mapping:

Link between mechanical and collision mesh (both side)

Linear (or not) relation

Use barycentric coordinates

Area of a tetrahedron:

f the origin of the coordinate system is chosen to coincide with vertex d, then d = 0, so

53

Constraints

Penalty-based methods

Add springs between objects

Fast/simple

Object are coupled (slow)

Stiff contact are unstable (interpenetrations)

Augmented lagragian approach

Add new equations

Enforce non interpenetrations

Direct solver :

Coupled system (large)

Ill conditioned (lagrangian multipliers are not mechanical dofs)

53

B

A

A

B

J1

J1T

X1 B1

X2 B2

𝜆 𝛿

J2

J2T

𝐴 0 𝐽10 𝐵 𝐽2𝐽1𝑇 𝐽2

𝑇 0

𝑥1𝑥2𝜆

=𝑏1𝑏2𝛿

54

Validation of the method

Compare the positions with different approximations of the Compliance

The green shape represent the exact solution

The object is non homogeneous : Red tetrahedra are stiffer than blue ones

With Asynchronous update of the preconditioner, we get almost the correct

solution

The green part and the blue part overlap perfectly

54

dt=0,01

Update every 3 time steps on

average

dt=0,01

Complia

nce

Warping

dt=0,01

Update

every 50

time steps

dt=0,01

Update

every 20

time steps

55

Schur complement method

55

൞

𝐴𝑥1 + 𝐽1𝜆 = 𝑏1𝐵𝑥2 + 𝐽2𝜆 = 𝑏2𝐽1𝑇𝑥1 + 𝐽2

𝑇𝑥2 = 𝛿

൞𝑥1 = 𝐴−1𝑏1 − 𝐴−1𝐽1𝜆

𝑥2 = 𝐵−1𝑏2 − 𝐵−1𝐽2𝜆

1. Predictive motion– Solve 𝑦1 = 𝐴−1𝑏1 and 𝑦2 = 𝐵−1𝑏2– Objects are not coupled

𝐽1𝑇𝐴−1𝐽1 + 𝐽2

𝑇𝐵−1𝐽2 𝜆 = 𝛿 − 𝐽1𝑇𝐴−1𝑏1 − 𝐽2

𝑇𝐵−1𝑏2

2. Constraint definition– Use the free state (approximation/linearization)

– Defines 𝐽 and 𝛿

3. Compliance computation– 𝑊 = 𝐽1

𝑇𝐴−1𝐽1 + 𝐽2𝑇𝐵−1𝐽2

– Expensive (GPU solvers)

4. Contact force computation– Gauss Seidel (small)

– Status method

5. Corrective motion– Solve 𝑥1 = 𝑦1 −𝐴

−1𝐽1𝜆 and 𝑥2 = 𝑦2 −𝐴−1𝐽2𝜆

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Haptic with hyperelastic models (Iros 2015)

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• Asynchronous computation:1. Sparse factorization of the system is updated at low rate (10 – 20 Hz)

2. Haptic loop update the contact problem at high rate (1 Khz)

Simulation

Loop

(30-100 Hz)

Compliance

Loop

(1-10 Hz)

A

L,D

Haptic Loop

(500-1000 Hz)δW, H

Outline



Element Method


organs



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New parallel architectures

Difficult to increase the frequency of processors

Heat dissipation

Electric consumption

Parallelization of algorithms

Parallelization of a simulation is not a new idea :

Parallel processors, distributed processors, …

Real time simulation constraints :

A lot of very fast computations

Use GPU to achieve computations (GPGPU).

GPU is very different than CPU parallelization.

Run thousand of threads to exploit efficiently the GPU

GPU is seen as a specialized co-processeur :

Achieve high parallel tasks

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59

Multi-CPU using KAAPI

How to optimize the use of current hardware architectures (multi-core)

Change the control of

execution in the SOFA core

transparent to components

transparent to (most) solver

Main research contribution:

iterative extensions to task

graphs

A supported feature in

the next public release

5

9

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GPU architecture

GPU is seen as a specialized co-processeur :

Achieve high parallel tasks

Two level of parallelism :

A Grid contains several blocs

A Block contains several threads

GPU programing :

Local synchronisations very fast

Global synchronisations very slow

(Synchronisation with the CPU)

Run thousand of threads of threads to exploit

efficiently the GPU !60

CPU

Kernel 1

GPU

Grille 1

Block

(0,0)

Block

(1,0)

Blobk

(2,0)

Block

(0,1)

Block

(1,1)

Block

(1,2)

Grille 2Kernel 2

Block (1,1)

Thread

(0,0)Thread

(1,0)Thread

(2,0)Thread

(3,0)Thread

(4,0)

Thread

(0,1)

Thread

(1,1)

Thread

(2,1)

Thread

(3,1)

Thread

(4,1)

Thread

(0,2)

Thread

(1,2)

Thread

(2,2)

Thread

(3,2)

Thread

(4,3)

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Performances evaluation

Speedup between 5 to 25

Simulate 300 000 elements in real time (CG limited to 25 iterations)

Nombre

Number of elements

Speedup

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Parallel Computing for Interactive Applications

Main challenges

Leverage new multi-core architectures (CPUs and GPUs)

Large potential but difficult to exploit

New parallel architectures

Programming is still complex (no unique language, no standard, ...)

Simplify the development process (transparency)

Adaptive and multilevel computations

Develop dedicated algorithms (from serial to parallel)

Different levels of parallelization (simulation level, object level)

6

2

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Collision detection based Layered Depth Images

Use GPU to render the simulation scene along 3 directions

This generates a volumetric image

From this image we identify the intersection volumes

These volumes are used to compute

the collision response (forces)

6

3

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Real time medical simulation

Documents

Transcript of Real time medical simulation