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Concepts of Contact Formulation
Algorithm in CAE
Contact Non-linearity-Class-Share
KI TECHNOLOGIES,
CONTACT NONLINEARITY
MODELING SERIES.
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FEA tools offer several different formulations to enforce compatibility at a
contact interface, a contact is enforced numerically based on three rules :
No penetration between the bodies.
A proper balance between the applied and the contact forces in case of
a static problem.
Conservation of momentum in case of dynamic problems such as impact.
These are the most important requirements for any contact formulation
to capture the interaction accurately.
In general, there are two most popular formulations that meet these criteria.
They are a Penalty formulation And a Lagrange formulation. Apart from these methods,
Pure Penalty and Augmented Lagrange are penalty-based methods, and they are
expected to produce small, non-zero penetration
Penalty Formulation : Think in terms of springs
Penalty formulation treats contact as a stiff spring that resists bodies from penetrating
each other.
This resistance is modeled in the form of contact stiffness which is derived as a
function of several factors such as
Material properties of both the bodies.
The geometry of both surfaces.
Kinematics of both the bodies.
Contact stiffness is used in both the normal and tangential directions.
Penalty formulation allows for a small amount of penetration, delta u, between the two
bodies.
Contact stiffness, K, is calculated, and restoring force, F, is calculated to prevent this
penetration.
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This restoring force is nothing but the contact force.
Contact stiffness should be large enough to provide restoring force enough to reduce
the penetration to close to zero.
Penalty formulation is used for calculating both contact pressure and also frictional
stresses.
If un is the contact gap, and Δu1 and Δu2 are the sliding distances in lateral directions,
then the contact traction vector is defined by
Some finite amount of penetration, D > 0, is required mathematically to maintain
equilibrium. However, physically contacting bodies do not interpenetrate (D = 0).
Tip: As long as the penetration does not lead to a change in the contact region,
The penetration will not influence the contact pressure and Stress underneath the
contact element.
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The normal Contact Stiffness knormal is the most important parameter affecting both
accuracy and convergence behavior.
A large value of stiffness gives better accuracy, but the problem may become more
difficult to converge.
If the contact stiffness is too large, the model may oscillate, with contacting
surfaces bouncing off of each other.
There is no additional DOF.
There is no over constraining problem
Iterative solvers are applicable – large models are doable!
Tip:
Always use Penalty if:
Symmetric contact or self-contact is used.
Multiple parts share the same contact zone
3D large model(> 300.000 DOFs), use PCG solver.
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Augmented Lagrange
Like penalty with intelligence.
Penetration will be a limited internal algorithm.
Use this if penetration matters.
For nonlinear solid body contact of faces, Pure Penalty or Augmented Lagrange
formulations can be used:
Both of these are penalty-based contact formulations
Here, for a Fnormal, there is a stiffness knormal. The higher the contact stiffness, the
lower the penetration x penetration, as shown in the figure below
Ideally, for an infinite knormal, one would get zero penetration. This is not numerically
possible with penalty-based methods, but as long as x penetration is small or negligible,
the solution results will be accurate.
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The main difference between Pure Penalty and Augmented Lagrange methods is that
the latter augments the contact force (pressure) calculations:
Because of the extra term, the augmented Lagrange method is less sensitive to the
magnitude of the contact stiffness knormal.
Add an internally calculated term lamda.
Lagrange Formulation
Lagrange formulation treats contact as a constraint.
Unlike penalty formulation, it does not require contact stiffness.
Instead enforces contact penetration to be zero.
This is the constraint that it uses in calculations
The contact traction vector is included as part of the total reaction force vector.
Contact gaps, and sliding distance are also solved as additional degrees of freedom.
The equations of motion are then solved by imposing the constraint on contact gap,
un and sliding distances Δu1 and Δu2 .
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Another available option is Lagrange multiplier Formulation:
The Normal Lagrange Formulation adds an extra degree of freedom (contact
pressure) to satisfy contact compatibility. Consequently, instead of resolving
contact force as contact stiffness and penetration, contact force (contact pressure)
is solved explicitly as an extra DOF.
Fnormal = DOF
Enforces nearly-penetration with pressure DOF zero/nearly zero Does not require a
normal contact stiffness (zero elastic slip).
Requires Direct Solver, which can be more computationally expensive.
The contact formulations discussed in this lesson have both advantages and
disadvantages compared to each other.
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MPC (Multi Point Constraints)
Full bonded (linear – doesn’t separate)
Rigid Beams instead of springs
Handled well by the solver (better than CERIG)
Can slightly stiffen surfaces
For the specific case of “Bonded” and “No Separation” type of contact between two
faces, a multi-point constraint (MPC) formulation is available.
MPC internally adds constraint equations to “tie” the displacements between
contacting surfaces.
This approach is not penalty-based or Lagrange multiplier-based. It is a direct,
efficient way of relating surfaces of contact regions that are bonded.
Large-deformation effects also are supported with MPC-based bonded contact.
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Chattering is an issue that often occurs with the Normal Lagrange method
If no penetration is allowed (left), then the contact status is either open or closed
(a step function). This can sometimes make convergence more difficult because
contact points may oscillate between open/closed status. This is called chattering
If some slight penetration is allowed (right), it can make it easier to converge since
contact is no longer a step change.
The table below summarizes some pros (+) and cons (-) with different contact
formulations:
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