The Finite Element Method for the Analysis of Non-Linear ... · PDF fileThe Finite Element Method for the Analysis of Non-Linear and Dynamic Systems Prof. Dr. Eleni Chatzi, Dr. J.P

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.


The Contact Problem

Contact Discretization

Node-to-Surface (Implicit only)

Surface-to-Surface (Implicit only)

Node-to-Face (Explicit only)

Edge-to-Edge


The Contact Problem

Node-to-Surface (strict master/slave formulation)


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


The Contact Problem

Surface-to-Surface


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


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


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.



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.


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.


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)


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.


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


The Contact Problem

Direct Enforcement

ADVANTAGES:

Accuracy: The constraints are satisfied exactly

DISADVANTAGES:

Adds cost to the equation solver

Potential convergence problems


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)


The Contact Problem

Penalty Method

The Penalty method corresponds to having a spring to bring backthe penetrating node to the surface.


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


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.



The Contact Problem

Friction Models

Coulomb friction modelτcr = µp (13)

τcr = min(µp, τmax) (14)



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


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


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”.


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)


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)


The Contact Problem

The critical stress equality yields:

τcrit = G(γpreq − ∆γsleq) =τcritγcrit

(γpreq − ∆γsleq) (24)

where γpreq is the ”equivalent elastic predictor strain”


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!


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.


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).


The Contact Problem

Source: A. Batailly et. al., 2013


The Contact Problem


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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The Finite Element Method for the Analysis of Non-Linear ... · PDF fileThe Finite Element Method for the Analysis of Non-Linear and Dynamic Systems Prof. Dr. Eleni Chatzi, Dr. J.P