Capturing Coulomb Friction within an Assembly of Thin Rods...Capturing Coulomb Friction within an...
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![Page 1: 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](https://reader033.fdocuments.in/reader033/viewer/2022041722/5e4f6b8fa02bd909fc6cecf0/html5/thumbnails/1.jpg)
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
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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
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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)
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In practice
• Not an exhaustive study yet (we’re on the way)• Still, some conclusions can be raised
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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 !)
![Page 6: 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](https://reader033.fdocuments.in/reader033/viewer/2022041722/5e4f6b8fa02bd909fc6cecf0/html5/thumbnails/6.jpg)
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 !)
![Page 7: 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](https://reader033.fdocuments.in/reader033/viewer/2022041722/5e4f6b8fa02bd909fc6cecf0/html5/thumbnails/7.jpg)
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
![Page 8: 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](https://reader033.fdocuments.in/reader033/viewer/2022041722/5e4f6b8fa02bd909fc6cecf0/html5/thumbnails/8.jpg)
In practice: two rod models used
• Corde model (Spillman et al. 2008)• Super-Helix model (Bertails, Audoly 2006)
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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
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Corde model (Spillman et al. 2008)
• 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
![Page 11: 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](https://reader033.fdocuments.in/reader033/viewer/2022041722/5e4f6b8fa02bd909fc6cecf0/html5/thumbnails/11.jpg)
Corde model (Spillman et al. 2008)
• 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
![Page 12: 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](https://reader033.fdocuments.in/reader033/viewer/2022041722/5e4f6b8fa02bd909fc6cecf0/html5/thumbnails/12.jpg)
Corde model (Spillman et al. 2008)
• 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
![Page 13: 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](https://reader033.fdocuments.in/reader033/viewer/2022041722/5e4f6b8fa02bd909fc6cecf0/html5/thumbnails/13.jpg)
Corde model (Spillman et al. 2008)
• 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
![Page 14: 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](https://reader033.fdocuments.in/reader033/viewer/2022041722/5e4f6b8fa02bd909fc6cecf0/html5/thumbnails/14.jpg)
Corde model (Spillman et al. 2008)
• 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
![Page 15: 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](https://reader033.fdocuments.in/reader033/viewer/2022041722/5e4f6b8fa02bd909fc6cecf0/html5/thumbnails/15.jpg)
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
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Super-helix model (Bertails, Audoly 2006)
• 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
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Super-helix model (Bertails, Audoly 2006)
• 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
![Page 18: 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](https://reader033.fdocuments.in/reader033/viewer/2022041722/5e4f6b8fa02bd909fc6cecf0/html5/thumbnails/18.jpg)
Super-helix model (Bertails, Audoly 2006)
• 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
![Page 19: 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](https://reader033.fdocuments.in/reader033/viewer/2022041722/5e4f6b8fa02bd909fc6cecf0/html5/thumbnails/19.jpg)
Super-helix model (Bertails, Audoly 2006)
• 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
![Page 20: 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](https://reader033.fdocuments.in/reader033/viewer/2022041722/5e4f6b8fa02bd909fc6cecf0/html5/thumbnails/20.jpg)
Super-helix model (Bertails, Audoly 2006)
• 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
![Page 21: 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](https://reader033.fdocuments.in/reader033/viewer/2022041722/5e4f6b8fa02bd909fc6cecf0/html5/thumbnails/21.jpg)
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
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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
![Page 23: 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](https://reader033.fdocuments.in/reader033/viewer/2022041722/5e4f6b8fa02bd909fc6cecf0/html5/thumbnails/23.jpg)
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
![Page 24: 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](https://reader033.fdocuments.in/reader033/viewer/2022041722/5e4f6b8fa02bd909fc6cecf0/html5/thumbnails/24.jpg)
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
![Page 25: 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](https://reader033.fdocuments.in/reader033/viewer/2022041722/5e4f6b8fa02bd909fc6cecf0/html5/thumbnails/25.jpg)
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
![Page 26: 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](https://reader033.fdocuments.in/reader033/viewer/2022041722/5e4f6b8fa02bd909fc6cecf0/html5/thumbnails/26.jpg)
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
![Page 27: 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](https://reader033.fdocuments.in/reader033/viewer/2022041722/5e4f6b8fa02bd909fc6cecf0/html5/thumbnails/27.jpg)
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)
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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)
![Page 29: 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](https://reader033.fdocuments.in/reader033/viewer/2022041722/5e4f6b8fa02bd909fc6cecf0/html5/thumbnails/29.jpg)
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
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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
![Page 31: 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](https://reader033.fdocuments.in/reader033/viewer/2022041722/5e4f6b8fa02bd909fc6cecf0/html5/thumbnails/31.jpg)
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
![Page 32: 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](https://reader033.fdocuments.in/reader033/viewer/2022041722/5e4f6b8fa02bd909fc6cecf0/html5/thumbnails/32.jpg)
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
![Page 33: 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](https://reader033.fdocuments.in/reader033/viewer/2022041722/5e4f6b8fa02bd909fc6cecf0/html5/thumbnails/33.jpg)
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 !)
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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
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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)
![Page 36: 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](https://reader033.fdocuments.in/reader033/viewer/2022041722/5e4f6b8fa02bd909fc6cecf0/html5/thumbnails/36.jpg)
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)
![Page 37: 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](https://reader033.fdocuments.in/reader033/viewer/2022041722/5e4f6b8fa02bd909fc6cecf0/html5/thumbnails/37.jpg)
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)
![Page 38: 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](https://reader033.fdocuments.in/reader033/viewer/2022041722/5e4f6b8fa02bd909fc6cecf0/html5/thumbnails/38.jpg)
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)
![Page 39: 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](https://reader033.fdocuments.in/reader033/viewer/2022041722/5e4f6b8fa02bd909fc6cecf0/html5/thumbnails/39.jpg)
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)
![Page 40: 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](https://reader033.fdocuments.in/reader033/viewer/2022041722/5e4f6b8fa02bd909fc6cecf0/html5/thumbnails/40.jpg)
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)
![Page 41: 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](https://reader033.fdocuments.in/reader033/viewer/2022041722/5e4f6b8fa02bd909fc6cecf0/html5/thumbnails/41.jpg)
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 !)
![Page 42: 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](https://reader033.fdocuments.in/reader033/viewer/2022041722/5e4f6b8fa02bd909fc6cecf0/html5/thumbnails/42.jpg)
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
![Page 43: 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](https://reader033.fdocuments.in/reader033/viewer/2022041722/5e4f6b8fa02bd909fc6cecf0/html5/thumbnails/43.jpg)
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
→ unknowns: q, v, uuu and rrr
![Page 44: 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](https://reader033.fdocuments.in/reader033/viewer/2022041722/5e4f6b8fa02bd909fc6cecf0/html5/thumbnails/44.jpg)
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
![Page 45: 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](https://reader033.fdocuments.in/reader033/viewer/2022041722/5e4f6b8fa02bd909fc6cecf0/html5/thumbnails/45.jpg)
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
![Page 46: 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](https://reader033.fdocuments.in/reader033/viewer/2022041722/5e4f6b8fa02bd909fc6cecf0/html5/thumbnails/46.jpg)
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
![Page 47: 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](https://reader033.fdocuments.in/reader033/viewer/2022041722/5e4f6b8fa02bd909fc6cecf0/html5/thumbnails/47.jpg)
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
![Page 48: 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](https://reader033.fdocuments.in/reader033/viewer/2022041722/5e4f6b8fa02bd909fc6cecf0/html5/thumbnails/48.jpg)
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
![Page 49: 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](https://reader033.fdocuments.in/reader033/viewer/2022041722/5e4f6b8fa02bd909fc6cecf0/html5/thumbnails/49.jpg)
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
![Page 50: 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](https://reader033.fdocuments.in/reader033/viewer/2022041722/5e4f6b8fa02bd909fc6cecf0/html5/thumbnails/50.jpg)
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
![Page 51: 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](https://reader033.fdocuments.in/reader033/viewer/2022041722/5e4f6b8fa02bd909fc6cecf0/html5/thumbnails/51.jpg)
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
![Page 52: 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](https://reader033.fdocuments.in/reader033/viewer/2022041722/5e4f6b8fa02bd909fc6cecf0/html5/thumbnails/52.jpg)
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
![Page 53: 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](https://reader033.fdocuments.in/reader033/viewer/2022041722/5e4f6b8fa02bd909fc6cecf0/html5/thumbnails/53.jpg)
Cadoux’s method
• To solve the full incremental problem, incorporate (d):
F (s) = s with F i (s) := ‖uuuiT (s)‖
→ Fixed-point equation
![Page 54: 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](https://reader033.fdocuments.in/reader033/viewer/2022041722/5e4f6b8fa02bd909fc6cecf0/html5/thumbnails/54.jpg)
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
![Page 55: 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](https://reader033.fdocuments.in/reader033/viewer/2022041722/5e4f6b8fa02bd909fc6cecf0/html5/thumbnails/55.jpg)
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
![Page 56: 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](https://reader033.fdocuments.in/reader033/viewer/2022041722/5e4f6b8fa02bd909fc6cecf0/html5/thumbnails/56.jpg)
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 !)
![Page 57: 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](https://reader033.fdocuments.in/reader033/viewer/2022041722/5e4f6b8fa02bd909fc6cecf0/html5/thumbnails/57.jpg)
The End
Thank You for your attention !