Capturing Coulomb Friction within an Assembly of Thin Rods...Capturing Coulomb Friction within an Assembly of Thin Rods FlorenceBertails-Descoubes joint work with Gilles Daviet (Meche

Capturing Coulomb Frictionwithin an Assembly of Thin Rods

Florence Bertails-Descoubesjoint work with Gilles Daviet (Meche project), Florent Cadoux, and Vincent Acary

Inria Rhône-Alpes, BiPop research group

BiPop Spring SchoolJune 2010

Overview

We consider an assembly of thin elastic rods with

• Inelastic impacts between rods• Non-penetration constraints between rods• Contacts with dry friction

Goal:• Test various models for the dynamics of individual rods• Test various methods for resolving frictional contact• Analyze results in terms of robustness, efficiency and realism• Evaluate the impact of the rod model on the quality of results

In practice

Models:• Two families of rods models used

• Maximal-coordinates: Corde model• Reduced-coordinates: super-helix model

• Friction modeled with Coulomb’s law• Two families of solvers used

• Global: Alart and Curnier (1991), Cadoux (2009)• Local: Gauss-Seidel method

Evaluation:• Robustness evaluated as the quality of convergence• Efficiency evaluated as the mean comput. time per frame• Realism only evaluated through visual perception (for now)

In practice

• Not an exhaustive study yet (we’re on the way)• Still, some conclusions can be raised

Outline

1 Rods: models and contact formulation

2 Coulomb friction: notations and formulations

3 Frictional contact algorithms

4 Simulation results (not published yet - to be released !)

Outline





Generic discrete rod: notations

• Centerline r(s)

• Degrees of freedom (dofs): q ∈ Rm

• Affine kinematics relationship:

u(s) = r(s) = H(s) q+w with H(s) =∂r∂q (s)

• Equation of motion:M(q) q + f (t, q, q) = 0

• Discrete equations (e.g., v = qt+dt):

Mv + f = 0 and u(s) = H(s) v + w

In practice: two rod models used

• Corde model (Spillman et al. 2008)• Super-Helix model (Bertails, Audoly 2006)

Corde model (Spillman et al. 2008)

• Maximal-coordinates model (explicit centerline)• Mass-spring system with stretch, bending and twist energies• Rod sampled in N + 1 nodes• Spatially discrete centerline: ri = r(si ), i ∈ {1 . . .N + 1}• Orientations modeled with quaternions zi• Constraints to be enforced:

• unitary quaternions: ‖zi‖ = 1 → post-normalization• coupling between the discrete centerline and the quaternions→ modeled as soft constraints

• Two decoupled systems for resolving ri and zi


• Dofs: q = {ri}i , q∗ = {zi}i(m = 3 (N + 1) + 4N)

• Kinematics:r(s) = (1− λ) ri−1 + λ ri = H(s) qwith H(s) = [0, . . . , 0, (1− λ), λ, 0 . . . , 0]

• Equation of motion:M q + V q + G(q) = F (t, q, v)M and V diagonal, G(q) nonlinear

• Discrete equations (v = qt+dt)

Mv + f = 0 and u(s) = H(s) v

with M = M + dt V + dt2∇G sparse






Mv + f = 0 and u(s) = H(s) v







Mv + f = 0 and u(s) = H(s) v







Mv + f = 0 and u(s) = H(s) v







Mv + f = 0 and u(s) = H(s) v


Super-helix model (Bertails, Audoly 2006)

• Reduced-coordinates model (implicit centerline)• Inextensible model with bending and twist energies• Rod sampled in N elements• Spatially continuous centerline r(s) (exact calculus)• Discrete curvatures κ1

i , κ2i and twist τi , i ∈ {1 . . .N}

• No constraint to be enforced


• Dofs: q = {κκκi}i (m = 3N)• Kinematics:

r(s) = H(s) q + r∗(s)

H(s) =[∂r∂κκκ1

(s), . . . , ∂r∂κκκQ

(s), 0, . . . , 0]

• Equation of motion

M(q) q + νK q + K q = F (t, q, v)

with M dense and K diagonal.• Discrete equations: v = qt+dt

Mv + f = 0

with M = M + dt νK + dt2 K dense



r(s) = H(s) q + r∗(s)


(s), . . . , ∂r∂κκκQ

(s), 0, . . . , 0]


M(q) q + νK q + K q = F (t, q, v)


Mv + f = 0




r(s) = H(s) q + r∗(s)


(s), . . . , ∂r∂κκκQ

(s), 0, . . . , 0]


M(q) q + νK q + K q = F (t, q, v)


Mv + f = 0




r(s) = H(s) q + r∗(s)


(s), . . . , ∂r∂κκκQ

(s), 0, . . . , 0]


M(q) q + νK q + K q = F (t, q, v)


Mv + f = 0




r(s) = H(s) q + r∗(s)


(s), . . . , ∂r∂κκκQ

(s), 0, . . . , 0]


M(q) q + νK q + K q = F (t, q, v)


Mv + f = 0


Assembly of rods

• Ns rods interacting:• Self-contacts• Mutual contacts• External contacts

→ Total number of dofs = Nsm• We denote n the total number of contacts• Assumption: one contact involves at most two bodies A and B

Collision detection

• Piecewise linear approximation of r(s)(independent of the rod’s resolution)

• Bounding cylinders of radius ε• Detection of contact by computing theminimal distance between cylinders axes:→ if d < 2 ε, a contact is set active

• Output:• Locations sA

i and sBi of the contact i for

each body A, B• Normal ei of contact i

• Acceleration techniques• Constraints partitioning• Spatial hash map

Advantage: simple and fast (takes 1 % of the total comput. time)Drawback: may require small time steps to avoid missing contacts

Collision detection








Collision detection








Collision detection








Collision detection








Contact relative velocity

• Relative velocity ui ∈ R3 at contact i :

ui = uB(sBi )− uA(sA

i ) = Hi v + w i

v←[

vA

vB

], H i ←

[−HA(sA

i ) HB(sBi )], w i ← wB

i − wAi

NB: For self-contact, v = vA = vB, H i ← H(sBi )− H(sA

i )

• Relative velocity uuu ∈ R3 n for all contacts:

uuu = H v + w

where

H =

0 . . . 0 . . .X 0 . . . X . . . 0...

......

......

0 . . .X . . . 0 X . . . 0 . . . 0

∈M3n,Nsm(R)

Contact relative velocity

• Relative velocity ui ∈ R3 at contact i :

ui = uB(sBi )− uA(sA

i ) = Hi v + w i

v←[

vA

vB

], H i ←

[−HA(sA

i ) HB(sBi )], w i ← wB

i − wAi

NB: For self-contact, v = vA = vB, H i ← H(sBi )− H(sA

i )

• Relative velocity uuu ∈ R3 n for all contacts:

uuu = H v + w

where

H =

0 . . . 0 . . .X 0 . . . X . . . 0...

......

......

0 . . .X . . . 0 X . . . 0 . . . 0

∈M3n,Nsm(R)

Incremental problem

• Global system (without interactions):

M v + f = 0

→ unknowns: q and v

• Global system (with self/mutual frictional contact):M v + f = H>rrruuu = H v + w(uuu, rrr) satisfies the Coulomb’s law

(1)

→ unknowns: q, v, uuu and rrr

Incremental problem

• Global system (without interactions):

M v + f = 0

→ unknowns: q and v

• Global system (with self/mutual frictional contact):M v + f = H>rrruuu = H v + w(uuu, rrr) satisfies the Coulomb’s law

(1)


Elimination of v

• Let v = M−1 (H> rrr − f)

• Compact formulation in (uuu, rrr):{uuu = W rrr + q(uuu, rrr) satisfies the Coulomb’s law (2)

with W = H M−1 H> ∈M3n(R) and q = w−H M−1 f ∈ R3n

Elimination of v

• Let v = M−1 (H> rrr − f)

• Compact formulation in (uuu, rrr):{uuu = W rrr + q(uuu, rrr) satisfies the Coulomb’s law (2)

with W = H M−1 H> ∈M3n(R) and q = w−H M−1 f ∈ R3n

Outline





Coulomb’s law: disjonctive formulationLet µ ∈ R+ and Kµ be the second-order cone

Kµ = {‖rT‖ ≤ µrN} ⊂ R3rrN

rT

e

Coulomb’s law

(uuu, rrr) ∈ C(e, µ) ⇐⇒

either take off r = 0 et uN > 0or stick r ∈ int(Kµ) and u = 0or slide r ∈ ∂Kµ \ 0, uN = 0

and ∃α ≥ 0, uT = −α rT

r = 0 r ∈ ∂Kr ∈ K

uN > 0 u = 0 uN = 0

Coulomb’s law: functional formulation

IdeaExpress Coulomb’s law as f (u, r) = 0 with f a nonsmooth function

Example: Alart and Curnier formulation (1991)

fff AC (uuu, rrr) =

[f ACN (uuu, rrr)

fff ACT (uuu, rrr)

]=

[PR+(rN − ρNuN) − rNPBBB(0,µrN)(rrrT − ρTuuuT ) − rrrT

]

where ρN , ρT ∈ R∗+ and PK is the projection onto the convex K .

(uuu, rrr) ∈ C(e, µ) ⇐⇒ fff AC (uuu, rrr) = 0

Solving methods

• Newton algorithm (requires the computation of ∇f )• Fixed-point method (if a reformulation rrr = ggg(uuu, rrr) is possible)




fff AC (uuu, rrr) =

[f ACN (uuu, rrr)

fff ACT (uuu, rrr)

]=


]



Solving methods





fff AC (uuu, rrr) =

[f ACN (uuu, rrr)

fff ACT (uuu, rrr)

]=


]



Solving methods


Coulomb’s law: complementarity formulation

Idea (De Saxcé, 1998)Modify the velocity uuu → uuu so that uuu and rrr are complementary.

uuu := uuu + µ ‖uT‖ e ∈ K ∗µ(= K 1µ

)

(uuu, rrr) ∈ C(e, µ) ⇐⇒ K ∗µ 3 uuu ⊥ rrr ∈ Kµ

Solving methods

• Fixed-point method: rrr = PKµ(rrr − ρ uuu) where ρ ∈ R∗+• Newton algorithm (requires the computation of ∇PKµ)




)


Solving methods





)


Solving methods


Outline





Algorithms used

• Global methods:• Newton on the Alart and Curnier’s function• Cadoux’s approach

• Local methods (splitting Newton methods):• Splitting Alart and Curnier• Splitting De Saxcé projection

Incremental problem

• Global system with frictional contact:M v + f = H>rrruuu = H v + w(uuu, rrr) satisfies the Coulomb’s law

• Elimination of v:{uuu = W rrr + q(uuu, rrr) satisfies the Coulomb’s law

where W = H M−1 H> ∈M3n(R) and q = w−H M f ∈ R3n


Nonsmooth Newton on the Alart-Curnier function

• Formulation of the incremental problem{uuu = W rrr + qfff AC (uuu, rrr) = 0

⇔ fff AC (W rrr + q, rrr) = Φ(rrr) = 0

• (Damped) Newton iteration:

rrrk+1 = rrrk − αk G−1k Φ(rrrk) where Gk ∈ ∂Φ(rrrk)

• Natural stopping criterion:

12 ‖Φ(rrr)‖2 < ε







12 ‖Φ(rrr)‖2 < ε







12 ‖Φ(rrr)‖2 < ε







12 ‖Φ(rrr)‖2 < ε

Cadoux’s method

• Relies on De Saxcé’s change of variables:• For contact i :

uuui := uuui + µi ‖uuuT‖i ei ∈ K∗µi

• For all contacts:uuu := uuu + E s ∈ L∗

s = [‖uuuT‖1, . . . , ‖uuuT‖n]>, E = BDiag(µi ei) and L∗ =∏

i K∗µi

Cadoux’s method

• Formulation of the incremental problemMv + f = H>rrr (a)

uuu = Hv + w + Es (b)L∗ 3 uuu ⊥ rrr ∈ L (c)

s = [‖uuuT‖1, . . . , ‖uuuT‖n]> (d)

• Key of the approach:if s is fixed, then (a), (b), (c) are the optimality conditions ofa convex optimization problem subject to conical constraints

Cadoux’s method

• Formulation of the incremental problemMv + f = H>rrr (a)

uuu = Hv + w + Es (b)L∗ 3 uuu ⊥ rrr ∈ L (c)

s = [‖uuuT‖1, . . . , ‖uuuT‖n]> (d)

• Key of the approach:if s is fixed, then (a), (b), (c) are the optimality conditions ofa convex optimization problem subject to conical constraints

Cadoux’s method

• Primal problem{min 1

2v>M v + f> v (quadratic, strict. convex)Hv + w + E s ∈ L∗ (conical contraints)

• Dual problemmin 1

2rrr>W rrr + b> rrr (quadratic, convex)rrr ∈ L (conical constraints)W = H M−1 H>b = −H M−1 f + w + E s

Cadoux’s method

• Primal problem{min 1

2v>M v + f> v (quadratic, strict. convex)Hv + w + E s ∈ L∗ (conical contraints)

• Dual problemmin 1

2rrr>W rrr + b> rrr (quadratic, convex)rrr ∈ L (conical constraints)W = H M−1 H>b = −H M−1 f + w + E s

Cadoux’s method

• To solve the full incremental problem, incorporate (d):

F (s) = s with F i (s) := ‖uuuiT (s)‖

→ Fixed-point equation

Splitting algorithm

Gauss-Seidel iterative process

• Init rrr , uuu ←W rrr + q• While k ≤ Nitermax

1. For i = 1 . . . n (loop over the contacts)(a)uuui ← (W rrr + q)i(b) qi ← uuui −Wii rrr i(c)Find rrr i such that (Wii rrr i + qi ) ∈ C(ei , µi )

2. End for3. uuunew ←W rrrnew + q4. If (stopping criteria), break

• End while

Splitting algorithm: in practice

Solver used for single contacts

• Newton on the Alart and Curnier function• De Saxcé projection on the Coulomb’s friction cone

Stopping criteria

• For Alart-Curnier: 12 ‖Φ(rrr)‖ < ε

• For De Saxcé: ‖rrr i,k+1−rrr i,k‖‖rrr i,k‖ < ε ∀i

Outline





The End

Thank You for your attention !

Capturing Coulomb Friction within an Assembly of Thin Rods...Capturing Coulomb Friction within an Assembly of Thin Rods FlorenceBertails-Descoubes joint work with Gilles Daviet (Meche

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