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