The Finite Element Method for the Analysis of Non-Linear and Dynamic Systems Prof. Dr. Eleni Chatzi, Dr. J.P. Escall´ on Lecture ST2 - 03 December, 2015 Institute of Structural Engineering Method of Finite Elements II 1
The Finite Element Method for the Analysis ofNon-Linear and Dynamic Systems
Prof. Dr. Eleni Chatzi, Dr. J.P. Escallon
Lecture ST2 - 03 December, 2015
Institute of Structural Engineering Method of Finite Elements II 1
NL FE Special Considerations - The Contact Problem
What is Contact?
Physically, contact stress is transmitted between two bodieswhen they touch.
Numerically, contact is a severely discontinuous form ofnon-linearity.
Difficulties
Complex non-linear behaviour = contact between two or morebodies.
Relative sliding of the surfaces has to be evaluated iteratively.
Deformable-to-deformable body contact generates non-lineartime-dependent BC.
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The Contact Problem
Contact Discretization
Node-to-Surface (Implicit only)
Surface-to-Surface (Implicit only)
Node-to-Face (Explicit only)
Edge-to-Edge
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The Contact Problem
Node-to-Surface (strict master/slave formulation)
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The Contact Problem
Node-to-Surface
Contact is enforced between a slave node and master surface facetslocal to the node:
The opening/penetration distance is measured along the normalto the master surface
A nodal area is assigned to each slave node to convert contactforces to contact stresses
The more refined surface should act as the slave surface
The stiffer body should be the master
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The Contact Problem
Surface-to-Surface
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The Contact Problem
Surface-to-Surface
Contact is enforced between the slave node and a larger number ofmaster surface facets around it:
The opening/penetration distance is measured along the slavesurface facet normal
Sliding is measured perpendicular to the slave normal
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The Contact Problem
Contact Discretization comparison: Abaqus/Explicit
Undetected penetrations of master nodes into the slave surface donot occur with surface-to-surface discretization:
Source: Abaqus Analysis User’s Manual
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The Contact Problem
Contact discretizations in Abaqus/Explicit
Node-to-Face: No distinction is made between master/slavesurfaces as in Node-to-Surface (i.e., contact is enforcedeverywhere).
Edge-Edge: It is very effective in enforcing contact that cannotbe detected as penetrations of nodes into faces.
Source: Abaqus Analysis User’s Manual
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The Contact Problem
Surface Description
Discrete Surface: Discontinuities in the surface normal directionat surface facet boundaries can contribute to convergencedifficulties.
Smooth Surface: Surface smoothing is used to reduce thediscretization error asociated with faceted representations ofcurved surfaces.
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The Contact Problem
Hard Contact Enforcement Methods
Direct Enforcement Method: Strict enforcement ofpressure-penetration relationship using Lagrange multipliermethod (only implicit).
Penalty method: approximate enforcement using penaltystiffness.
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The Contact Problem
Direct Enforcement
Variational formulation for a steady-state analysis without contact:
Π =1
2UTKU − UTF (1)
Contact Constraint:Ui = U∗
i (2)
Variational formulation for a steady-state analysis with contactenforcement using Lagrange multiplier method:
Π∗ =1
2UTKU − UTF + λ(Ui − U∗
i ) (3)
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The Contact Problem
Direct Enforcement
Equilibrium Condition δΠ∗ = 0:
δΠ∗ = δUTKU − δUTF + λδUi + δλ(Ui − U∗i ) = 0 (4)
The above relationship can be written as:
KU + λei = F (5)
eTi U = U∗i (6)
In matrix form: [K eieTi 0
]×[Uλ
]=
[FU∗i
](7)
λ is the vector of Lagrange multiplier degrees of freedom (constraintforces) *One per constraint.
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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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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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The Contact Problem
Penalty Method
The right hand side of the potential function Π is amended in thefollowing manner:
Π∗ =1
2UTKU − UTF +
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∗ = δUTKU − δUTF + α(Ui − U∗i )δUi = 0 (11)
(K + αeieTi )U = F + αU∗
i ei (12)
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The Contact Problem
Penalty Method
The Penalty method corresponds to having a spring to bring backthe penetrating node to the surface.
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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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The Contact Problem
Strain-free adjustments of initial overclosures
Within the penetration tolerance, all initial overclosures aretreated with strain-free adjustments.
Initial overclosures can be due to pre-processing errors ordiscretization of curved surfaces.
Source: Abaqus Analysis User’s Manual
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The Contact Problem
Friction Models
Coulomb friction modelτcr = µp (13)
τcr = min(µp, τmax) (14)
Source: Abaqus Analysis User’s Manual
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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 ManualInstitute of Structural Engineering Method of Finite Elements II 21
The Contact Problem
Friction Enforcement
Lagrangian multiplier method
Penalty method
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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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The Contact Problem
Penalty method
It approximates stick with stiff elastic behaviour. InAbaqus/Explicit by default the same penalty stiffness used inhard contact is used for frictional constraints. On the contraryin Abaqus/Standard (Implicit) it depends on the elastic slip.
The elastic slip γcrit is calculated as: γcrit = Ff li, where Ff isthe slip tolerance, and li is the ”characteristic contact surfacelength”.
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The Contact Problem
Elastic behaviour (τeq < τcrit):
γeli (t+ ∆t) = γeli (t) + ∆γi (16)
τi = Gγeli =τcritγcrit
γeli =µp
γcritγeli (17)
dτi = Gdγi +τiτcrit
(µp+∂u
∂pp)dp (18)
The contributions from the contact pressure p are non-symmetric!
Plastic behaviour (τeq > τcrit):
∆γi = γeli (t+ ∆t) − γeli (t) + ∆γsli = γeli − γeli + ∆γsli (19)
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The Contact Problem
The shear stress at the end of the increment is evaluated with theelastic relationship:
τi(t+ ∆t) = τi = Gγeli =τcritγcrit
γeli (20)
Slip increment:
∆γsli =τiτcrit
∆γsleq (21)
Replacing γeli and ∆γsli in eq. 19 yields:
∆γi =τiτcrit
γcrit − γeli +τiτcrit
∆γsleq (22)
τi =γeli + ∆γiγcrit + ∆γsleq
τcrit (23)
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The Contact Problem
The critical stress equality yields:
τcrit = G(γpreq − ∆γsleq) =τcritγcrit
(γpreq − ∆γsleq) (24)
where γpreq is the ”equivalent elastic predictor strain”
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The Contact Problem
τcrit = G(γpreq − ∆γsleq) =τcritγcrit
(γpreq − ∆γsleq) (25)
∆γsleq = γpreq − γcrit (26)
Replacing eq. 26 into eq. 23 yields:
τi =γpri
γcrit + ∆γsleqτcrit =
γpriγpreq
τcrit = niτcrit (27)
where ni is the normalized slip direction.
The iterative solution scheme for τcrit as a function of the slip rate(γsleq = ∆γsleq/∆t) yields:
∆τi = (δij−ninj)τcritγpreq
dγj +ni(µ+p∂µ
∂p)dp+ninj
p
∆t
∂µ
∂γeqdγj (28)
The unsymmetric terms may have a strong effect on the speed ofconvergence of the Newton scheme!
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The Contact Problem
Contact Algorithm (Implicit calculations)
Newton-Raphson iterative scheme
(KT )in+1∆Ui+1n+1 = (Fint)
in+1 + (Fext)n+1 + (Rc(U
i+1n+1))
i+1n+1
U i+1n+1 = U i
n+1 + ∆U i+1n+1
where KT is the tangent stiffness matrix, i refers to the ongoingiteration with the Newton-Raphson process and n refers to theloading increment.
The contact forces vector Rc depends on U which influences boththe contact surface shape and the magnitude of the contact reaction.
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The Contact Problem
Inverting the tangent stiffness matrix yields:
∆U i+1n+1 = ((KT )in+1)
−1)((Fint)in+1 + (Fext)n+1)
...+ ((KT )in+1)−1)(Rc(U
i+1n+1))
i+1n+1
which may be written in a simpler way:
∆U i+1n+1 = (∆Ulib)
i+1n+1 + (∆Uc)
i+1n+1
with:
(∆Ulib)i+1n+1 = ((KT )in+1)
−1)((Fint)in+1 + (Fext)n+1)
(∆Uc)i+1n+1 = ((KT )in+1)
−1)(Rc(Ui+1n+1))
i+1n+1
The displacement is split into two parts, one independent from thecontact problem (prediction), and a term depending exclusively oncontact (correction).
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The Contact Problem
Source: A. Batailly et. al., 2013
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The Contact Problem
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The Contact Problem
Contact in Explicit Codes
Calculation is advanced explicitly element-by-element (No needto assembly a global stiffness matrix and no Newton iterationsare performed).
No convergence problems related to faceted representation ofcurves (smoothing is not relevant).
Contact forces do not depend on the displacements (no iterativeprocess is carried out).
Easier to deal with sliding friction because calculations areadvanced explicitly element-by-element.
Time step is very small and therefore is suitable to analyse shortcontact dynamic problems where friction plays an importantrole, i.e., impact.
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