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Arc-Length Method for Frictional Contactwith a Criterion of Maximum Dissipation of Energy
Yoshihiro Kanno† Joao A.C. Martins‡
†The University of Tokyo (Japan)
‡Instituto Superior Tecnico (Portugal)
Frictional Contact with Maximum Dissipation – p.1/19
Frictional contact problems
f
u
contact / free
slip / stick
=⇒ limit / bifurcation points
u
f
0Frictional Contact with Maximum Dissipation – p.2/19
Existence results of contact problems
frictionlesscontinuation method [Miersemann & Mittelmann 89]
frictional & large disp.tangent stiffness [Wriggers 02]
arc-length method [Koo & Kwak 96]
approximation of friction law [Períc & Owen 92]
selection of paths (small deformation)min. of potential energy [Hilding 00]
Frictional Contact with Maximum Dissipation – p.3/19
Existence results of contact problems
frictionlesscontinuation method [Miersemann & Mittelmann 89]
frictional & large disp.tangent stiffness [Wriggers 02]
arc-length method [Koo & Kwak 96]
approximation of friction law [Períc & Owen 92]
selection of paths (small deformation)min. of potential energy [Hilding 00]
=⇒ arc-length method + path selection
=⇒ solve an optimization problem at each increment
Frictional Contact with Maximum Dissipation – p.3/19
Complementarity condition
x ≥ 0, y ≥ 0, x�y = 0
0 x
y
Generally & for short, we often write
0 ≤ (Ax) ⊥ (By) ≥ 0
Frictional Contact with Maximum Dissipation – p.4/19
MPEC
Mathematical Program with ComplementarityConstraints = MPEC
max f(x)
s.t. g(x) ≤ 0,
0 ≤ h(x) ⊥ w(x) ≥ 0
complementarity condition:
0 ≤ h(x) ⊥ w(x) ≥ 0
�h(x) ≥ 0, w(x) ≥ 0, h(x)�w(x) = 0
Frictional Contact with Maximum Dissipation – p.5/19
Contact problems (2D)
non-penetration condition :
un > 0 =⇒ rn = 0 : free
rn > 0 =⇒ un = 0 : contact
f
u >0n
r >0n
u t
u =0n
rt
r =r =0n t
Frictional Contact with Maximum Dissipation – p.6/19
Contact problems (2D)
non-penetration condition :
un > 0 =⇒ rn = 0 : free
rn > 0 =⇒ un = 0 : contact
f
u >0n
r >0n
u t
u =0n
rt
r =r =0n t
complementarity condition :
un ≥ 0, rn ≥ 0, unrn = 0.
Frictional Contact with Maximum Dissipation – p.6/19
Coulomb’s friction law
µrn ≥ |rt|
∆ut = −αrt,
{α > 0 (slip)
α = 0 (stick)
f
r >0n
rt
rt0
µrn
rt∆ut
µ : coefficient of friction
(rn, rt) : reaction (normal, tangential)
Frictional Contact with Maximum Dissipation – p.7/19
Coulomb’s friction law
complementarity condition:
µrn ≥ |rt|, λn ≥ |∆ut|(µrn, rt) · (λn,∆ut) = 0
f
r >0n
rt
rt0
µrn
rt∆ut
µ : coefficient of friction
(rn, rt) : reaction (normal, tangential)
Frictional Contact with Maximum Dissipation – p.7/19
Aims of the talk
quasi-static contact,Coulomb’s friction
limit points(smooth / nonsmooth)
avoid bifurcated unloadingpaths
u
f
0
Frictional Contact with Maximum Dissipation – p.8/19
Aims of the talk
quasi-static contact,Coulomb’s friction
=⇒ complementarityconditions
limit points(smooth / nonsmooth)
=⇒ arc-length method
avoid bifurcated unloadingpaths
=⇒ path with maximumdissipation of energy
u
f
0
Frictional Contact with Maximum Dissipation – p.8/19
Arc-length method
At each increment,
conventional arc-length methodu : displacementss : loading parameter
find (∆u(k),∆s(k))
satisfying ‖(∆u(k),∆s(k))‖ = S
by solving nonlinear equations
our methodfind (∆u(k),∆s(k)) and r(k)
satisfying ‖(∆u(k),∆s(k))‖ = S
by solving an MPEC
Frictional Contact with Maximum Dissipation – p.9/19
Choose the path with max. dissipation
system of equilibrium paths (♣):
φ(∆u,∆s) − r = 0 (equilibrium eqs.)
0 ≤ A∆u ⊥ Br ≥ 0 (non-penetration & friction)
hypersphere constraint onarc-length (♦):
‖(∆u,∆s)‖ = S
s ⎯S
u
(u ,s )(k)
(∆u ,∆s )∗ ∗
(k)
Frictional Contact with Maximum Dissipation – p.10/19
Choose the path with max. dissipation
system of equilibrium paths (♣):
φ(∆u,∆s) − r = 0 (equilibrium eqs.)
0 ≤ A∆u ⊥ Br ≥ 0 (non-penetration & friction)
hypersphere constraint onarc-length (♦):
‖(∆u,∆s)‖ = S
s ⎯S
u
(u ,s )(k)
(∆u ,∆s )∗ ∗
(k)
prototype of incremental problem:
max −r(k)t · ∆ut (dissipation of energy)
s.t. (♣) & (♦)
Frictional Contact with Maximum Dissipation – p.10/19
MPEC
max (dissipation)−p(∆u,∆s)
s.t. (equilibrium eqs.),(complementarity conds.)
penaltyp= 1
2γk1‖(∆u, ∆s)‖2 ⇐= on arc-length
− γk2 (u(k), s(k)) · (∆u,∆s) ⇐= prevents ‘backward’ solution
u
s
0
(u ,s )(k) (k)
(∆u ,∆s )∗ ∗
(u ,s )(k) (k). .
Frictional Contact with Maximum Dissipation – p.11/19
MPEC
max (dissipation)−p(∆u,∆s)
s.t. (equilibrium eqs.),(complementarity conds.)
penaltyp= 1
2γk1‖(∆u, ∆s)‖2 ⇐= on arc-length
− γk2 (u(k), s(k)) · (∆u,∆s) ⇐= prevents ‘backward’ solution
MPEC hasconvex objective functionfeasible solutions
control ‖(∆u,∆s)‖ by choosing γk1
Frictional Contact with Maximum Dissipation – p.11/19
Characterization of solution (r(k)t = 0)
KKT conditions:(u(k+1)
s(k+1)
)=
(u(k)
s(k)
)+
γk2
γk1
(u(k)
s(k)
)+
1
γk1
nd∑l=1
ξl
(∂φl/∂∆u
∂φl/∂∆s
)
u
s
0
(u ,s )(k+1)
⎯⎯⎯γk1
γk2
⎯⎯⎯⎯dφ
d∆u( , )⎯⎯⎯⎯dφ
d∆s⎯⎯γk1
ξ
(u ,s )(k) (k). .
(u ,s )(k) (k)
(k+1)
(u ,s )(k) (k). .
Frictional Contact with Maximum Dissipation – p.12/19
Regularization of MPEC
regularization:(µrn, rt) · (λn,∆ut)= 0
⇓(µrn, rt) · (λn,∆ut)≤ ε, ε > 0
solve the sequence of regularized problemsby ε ↘ 0
by conventional SQPMatlab Optimization Toolbox (Ver. 2.1)
Frictional Contact with Maximum Dissipation – p.13/19
One-bar truss
analogous to [Mróz 02]
f
x
y
u1u2
d
l0
l0
k1
k2
rnrt
(a)
(b)
Frictional Contact with Maximum Dissipation – p.14/19
One-bar trusssmooth limit points
(C)→(D) : bifurcation points
0 0.2 0.4 0.6 0.8 1−4
−2
0
2
4
6
8
10
u1 / π
load
ing
para
met
er
(B) no contact
(C)
(D)
(E)
(F)
µ=0.3
(A)
Frictional Contact with Maximum Dissipation – p.15/19
One-bar trusssmooth limit points
(C)→(D) : bifurcation points
0 0.2 0.4 0.6 0.8 1−4
−2
0
2
4
6
8
10
u1 / π
load
ing
para
met
er
(B) no contact
(C)
(D)
(E)
(F)
µ=0
µ=0.3µ=0.1
(A)
Frictional Contact with Maximum Dissipation – p.15/19
Ex. of [Klarbring 90]
f(s)
x
y
y=5
x=−5
f0
0
(a)
−0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.140
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
(A)
(B)
(C)
x−coordinate of node (a)lo
adin
g pa
ram
eter
µ = 1.5
(B) : nonsmooth limit point
(B)→(C) : bifurcation points
Frictional Contact with Maximum Dissipation – p.16/19
Arch-type truss
x
y
f
0
H1
H2
(a)
(c)
(d)
(e) (f)
f
0 0.5 1 1.5 2 2.5 3 3.5 40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
(A) (B)(C)
(D) (E)
(F)
(G)
y−coordinate of node (a)
load
ing
para
met
er
(D) : smooth limit point
(E) : nonsmooth limit point
(D)→(E) : bifurcation points
Frictional Contact with Maximum Dissipation – p.17/19
Arch-type truss
0 0.5 1 1.5 2 2.5 3 3.5 40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
(A) (B)(C)
(D) (E)
(F)
(G)
y−coordinate of node (a)
load
ing
para
met
er
(D) : smooth limit point
(E) : nonsmooth limit point
(D)→(E) : bifurcation points
Frictional Contact with Maximum Dissipation – p.18/19
Conclusion
Arc-length methodwith
contact, the Coulomb frictionchoosing the path with the max. dissipationgeometrical nonlinearity
MPEC is solvedat each incrementby using regularization and SQP method
Trace equilibrium paths withsmooth / nonsmooth limit pointssuccessive bifurcation points
Frictional Contact with Maximum Dissipation – p.19/19