1. The Finite Element Method for the Analysis of
Non-Linear and Dynamic Systems
Prof. Dr. Eleni Chatzi, J.P. Escall´on
Lecture 11 - 17 December, 2013
Institute of Structural Engineering Method of Finite Elements II 1

2. NL FE Special Considerations - The Contact Problem
What is Contact?
Physically, contact stress is transmitted between two bodies
when they touch.
Numerically, contact is a severely discontinuous for of
non-linearity.
Difficulties
Complex non linear behaviour = contact between two or more
bodies.
Relative sliding of the surfaces has to be evaluated iteratively.
Deformable-to-deformable body contact generates non-linear
time-dependent BC.
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3. The Contact Problem
Contact Discretization
Node-to-Surface (Implicit only)
Surface-to-Surface (Implicit only)
Node-to-Face (Explicit only)
Edge-to-Edge
Institute of Structural Engineering Method of Finite Elements II 3

4. The Contact Problem
Node-to-Surface (strict master/slave formulation)
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5. The Contact Problem
Node-to-Surface
Contact is enforced between a slave node and master surface facets
local to the node:
The opening/penetration distance is measured along the normal
to the master surface
A nodal area is assigned to each slave node to convert contact
forces to contact stresses
The more refined surface should act as the slave surface
The stiffer body should be the master
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6. The Contact Problem
Surface-to-Surface
Institute of Structural Engineering Method of Finite Elements II 6

7. The Contact Problem
Surface-to-Surface
Contact is enforced between the slave node and a larger number of
master surface facets around it:
The opening/penetration distance is measured along the slave
surface facet normal
Sliding is measured perpendicular to the slave normal
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8. The Contact Problem
Contact Discretization comparison: Abaqus/Explicit
Undetected penetrations of master nodes into the slave surface do
not occur with surface-to-surface discretization:
Source: Abaqus Analysis User’s Manual
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9. The Contact Problem
Contact discretizations in Abaqus/Explicit
Node-to-Face: No distinction is made between master/slave
surfaces as in Node-to-Surface (i.e., contact is enforced
everywhere).
Edge-Edge: It is very effective in enforcing contact that cannot
be detected as penetrations of nodes into faces.
Source: Abaqus Analysis User’s Manual
Institute of Structural Engineering Method of Finite Elements II 9

10. The Contact Problem
Surface Description
Discrete Surface: Discontinuities in the surface normal direction
at surface facet boundaries can contribute to convergence
difficulties.
Smooth Surface: Surface smoothing is used to reduce the
discretization error asociated with faceted representations of
curved surfaces.
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11. The Contact Problem
Hard Contact Enforcement Methods
Direct Enforcement Method: Strict enforcement of
pressure-penetration relationship using Lagrange multiplier
method.
Penalty method: approximate enforcement using penalty
stiffness.
Institute of Structural Engineering Method of Finite Elements II 11

12. The Contact Problem
Direct Enforcement
Variational formulation for a steady-state analysis without contact:
Π =
1
2
UT
KU − UT
F (1)
Contact Constraint:
Ui = U∗
i (2)
Variational formulation for a steady-state analysis with contact
enforcement using Lagrange multiplier method:
Π∗
=
1
2
UT
KU − UT
F + λ(Ui − U∗
i ) (3)
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13. The Contact Problem
Direct Enforcement
Equilibrium Condition δΠ∗ = 0:
δΠ∗
= δUT
KU − δUT
F + λδUi + δλ(Ui − U∗
i ) = 0 (4)
The above relationship can be written as:
KU + λei = F (5)
eT
i U = U∗
i (6)
In matrix form:
K ei
eT
i 0
×
U
λ
=
F
U∗
i
(7)
λ is the vector of Lagrange multiplier degrees of freedom (constraint
forces) *One per constraint.
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14. The Contact Problem
Direct Enforcement
The contact virtual work contribution is:
δΠc
= λδUi + δλ(Ui − U∗
i ) (8)
This expression is written in Abaqus Theory Manual as:
δΠc
= δph + pδh (9)
where p is the Lagrangian multiplier, and h is the ”overclosure”.
Hard Contact
p=0 for h < 0 contact is open
h=0 for p = 0 contact is closed
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15. The Contact Problem
Direct Enforcement
ADVANTAGES:
Accuracy: The constraints are satisfied exactly
DISADVANTAGES:
Adds cost to the equation solver
Potential convergence problems
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16. The Contact Problem
Penalty Method
The right hand side of the potential function Π is amended in the
following manner:
Π∗
=
1
2
UT
KU − UT
F +
α
2
(Ui − U∗
i )2
(10)
I use a large α in order to make Ui = U∗
i , i.e., α >> max(kii)
Equilibrium condition δΠ∗ = 0:
δΠ∗
= δUT
KU − δUT
F + α(Ui − U∗
i )δUi = 0 (11)
(K + αeieT
i )U = F + αU∗
i ei (12)
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17. The Contact Problem
Penalty Method
The Penalty method corresponds to having a spring to bring back
the penetrating node to the surface.
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18. The Contact Problem
Penalty Method
ADVANTAGES:
Convergence rates significantly improve
Better equation solver performance
DISADVANTAGES:
Small amount of penetration (typically insignificant)
In some cases, the penalty stiffness needs to be adjusted
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19. The Contact Problem
Strain-free adjustments of initial overclosures
Within the penetration tolerance, all initial overclosures are
treated with strain-free adjustments.
Initial overclosures can be due to pre-processing errors or
discretization of curved surfaces.
Source: Abaqus Analysis User’s Manual
Institute of Structural Engineering Method of Finite Elements II 19

20. The Contact Problem
Friction Models
Coulomb friction model
τcr = µp (13)
τcr = min(µp, τmax) (14)
Source: Abaqus Analysis User’s Manual
Institute of Structural Engineering Method of Finite Elements II 20

21. The Contact Problem
Friction Models
Additional features:
Friction coefficient dependence on slip rate
Friction coefficient dependence on contact pressure
Anisotropic friction
Source: Abaqus Analysis User’s Manual
Institute of Structural Engineering Method of Finite Elements II 21

22. The Contact Problem
Friction Enforcement
Lagrangian multiplier method
Penalty method
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23. The Contact Problem
Exact stick formulation
Lagrange multipliers are used to enforce exact sticking conditions.
The virtual work due to friction is evaluated as:
δΠ =
S
(τiδγi + ∆γiδqi)dS (15)
qi are the Lagrangian multipliers used to enforce exact stick
(∆γi = 0).
If τeq > τcrit the element passes from sticking to slipping.
If ∆γiτi(t) < 0 the element passes from slipping to sticking
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24. The Contact Problem
Penalty method
It approximates stick with stiff elastic behaviour. In
Abaqus/Explicit by default the same penalty stiffness used in
hard contact is used for frictional constraints. On the contrary
in Abaqus/Standard (Implicit) it depends on the elastic slip.
The elastic slip γcrit is calculated as: γcrit = Ff li, where Ff is
the slip tolerance, and li is the ”characteristic contact surface
length”.
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25. The Contact Problem
Elastic behaviour (τeq < τcrit):
γel
i (t + ∆t) = γel
i (t) + ∆γi (16)
τi = Gγel
i =
τcrit
γcrit
γel
i =
µp
γcrit
γel
i (17)
dτi = Gdγi +
τi
τcrit
(µp +
∂u
∂p
p)dp (18)
The contributions from the contact pressure p are non-symmetric!
Plastic behaviour (τeq > τcrit):
∆γi = γel
i (t + ∆t) − γel
i (t) + ∆γsl
i = γel
i − γel
i + ∆γsl
i (19)
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26. The Contact Problem
The shear stress at the end of the increment is evaluated with the
elastic relationship:
τi(t + ∆t) = τi = Gγel
i =
τcrit
γcrit
γel
i (20)
Slip increment:
∆γsl
i =
τi
τcrit
∆γsl
eq (21)
Replacing γel
i and ∆γsl
i in eq. 19 yields:
∆γi =
τi
τcrit
γcrit − γel
i +
τi
τcrit
∆γsl
eq (22)
τi =
γel
i + ∆γi
γcrit + ∆γsl
eq
τcrit (23)
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27. The Contact Problem
The critical stress equality yields:
τcrit = G(γpr
eq − ∆γsl
eq) =
τcrit
γcrit
(γpr
eq − ∆γsl
eq) (24)
where γpr
eq is the ”equivalent elastic predictor strain”
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28. The Contact Problem
τcrit = G(γpr
eq − ∆γsl
eq) =
τcrit
γcrit
(γpr
eq − ∆γsl
eq) (25)
∆γsl
eq = γpr
eq − γcrit (26)
Replacing eq. 26 into eq. 23 yields:
τi =
γpr
i
γcrit + ∆γsl
eq
τcrit =
γpr
i
γpr
eq
τcrit = niτcrit (27)
where ni is the normalized slip direction.
The iterative solution scheme for τcrit as a function of the slip rate
(˙γsl
eq = ∆γsl
eq/∆t) yields:
∆τi = (δij −ninj)
τcrit
γpr
eq
dγj +ni(µ+p
∂µ
∂p
)dp+ninj
p
∆t
∂µ
∂ ˙γeq
dγj (28)
The unsymmetric terms may have a strong effect on the speed of
convergence of the Newton scheme!
Institute of Structural Engineering Method of Finite Elements II 28

29. The Contact Problem
Contact Algorithm (Implicit calculations)
Newton-Raphson iterative scheme
(KT )i
n+1∆Ui+1
n+1 = (Fint)i
n+1 + (Fext)n+1 + (Rc(ui+1
n+1))i+1
n+1
Ui+1
n+1 = Ui
n+1 + ∆Ui+1
n+1
where KT is the tangent stiffness matrix, i refers to the ongoing
iteration with the Newton-Raphson process and n refers to the
loading increment.
The contact forces vector Rc depends on U which influences both
the contact surface shape and the magnitude of the contact reaction.
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30. The Contact Problem
Inverting the tangent stiffness matrix yields:
∆Ui+1
n+1 = ((KT )i
n+1)−1)((Fint)i
n+1 + (Fext)n+1)
... + ((KT )i
n+1)−1)(Rc(ui+1
n+1))i+1
n+1
which may be written in a simpler way:
∆Ui+1
n+1 = (∆Ulib)i+1
n+1 + (∆Uc)i+1
n+1
with:
(∆Ulib)i+1
n+1 = ((KT )i
n+1)−1)((Fint)i
n+1 + (Fext)n+1)
(∆Uc)i+1
n+1 = ((KT )i
n+1)−1)(Rc(Ui+1
n+1))i+1
n+1
The displacement is split into two parts, one independent from the
contact problem (prediction), and a term depending exclusively on
contact (correction).
Institute of Structural Engineering Method of Finite Elements II 30

31. The Contact Problem
Source: A. Batailly et. al., 2013
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32. The Contact Problem
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33. The Contact Problem
Contact in Explicit Codes
Calculation is advanced explicitly element-by-element (No need
to assembly a global stiffness matrix and no Newton iterations
are performed).
No convergence problems related to faceted representation of
curves (smoothing is not relevant).
Contact forces do not depend on the displacements (no iterative
process is carried out).
Easier to deal with sliding friction because calculations are
advanced explicitly element-by-element.
Time step is very small and therefore is suitable to analyse short
contact dynamic problems where friction plays an important
role, i.e., impact.
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