Finite Element Solution of Contact Problems: From 1974 to 2004

Finite Element Solution of Contact Problems:From 1974 to 2004

by

Robert L. TaylorProfessor in the Graduate School

Department of Civil & Environmental Engineering

University of California at Berkeley

Advances in Computational Mechanics

Celebrating the 60th Birthday of Tom Hughes

7 April 2004

Remarks on FEM Contact Analysis

• This talk summarizes:

– Formulations to treat contact problems by FEM.

– Spatial approximation for contact and tied interfaces.

– Methods of solution using Lagrange multiplier, per-

turbed lagrangian, penalty, augmented lagrangian and

constraint elimination methods.

– Contact patch test requirement.

– Example solutions for various treatments.

Finite Element Contact Analysis

• Multiple body problems can have two states:

(a) No contact condition.

(b) Contact state.

1974-77: The Foundations: With TJRH!





1982-89: Node-to-Surface Contact

• Simplest form where nodes on one body do not interact atnodes on second body (Goudreau & Hallquist, 1982)

gn = x

2(s)− x

2(c)

cm

2

m1

s

Master Body

Slave Body

•A •

B

•C

• Note nodes of master body not in contact.

• Consistent tangent (Simo & Wriggers, 1985; Parische, 1989)


• Normals from FE mesh:

Slave

Master

n

Slave

Master

n

(a) Normal to Master. (b) Normal to Slave.

• Note master form involves 3 nodes & slave form 5 nodes.

• Slave normal gives smoother sliding.


• Example: Compare σ22 for treatments

(a) Node to node (b) Node to surface


• Difficulty: Corner Condition

m

s

Master Body

Slave Body

• Use two facets and modify functional to

Πc =2∑

i=1

{(λi + λki)n

Tci [xs − x(ξi)]−

1

κλ2

i

}


• Difficulty: Slave normal not perpendicular to master facet.

gn

c

4

3

5

1

2

T

n

Slave

Master

• For smooth sliding must ignore δξ term and let

δΠc =[

δλn, δxTs , δxT

α

] gn Ac

λn Ac nc

−λn Ac Nα nc

Gives slave force on 3-node only.

• Tangent unsymmetric.

1985: Solution Methods

• Methods for imposing contact constraint (Landers & T, 1985):

– Lagrange multiplier method.

– Perturbed lagrangian, penalty and perturbed tangent

methods (related).

– Augmented lagrangian method of Uzawa.

– Constraint elimination method.


g = x2(s)− x

2(m)

m

s

Master Body

Slave Body

• Lagrange multiplier form of solution

Πc = λ g = λ

[(X(s)

2 + u(s)2 )− (X(m)

2 + u(m)2 )

]

• Added variational equation

δΠc = δλ g +(δu

(s)2 − δu

(m)2

)λ =

[δu

(s)2 δu

(m)2 δλ

] λ

−λg

• λ - contact force to prevent penetration.


• Linearization of Πc produces tangent matrix 0 0 10 0 −11 −1 0

du(s)2

du(m)2

dλ

=

−λ

λ− g

• Added identical to any FE assembly process.

• Introduces new unknown (λ) for each contact pair.

• Equations indefinite (zero multiplier diagonal).


g = x2(s)− x

2(m)

m

s

Master Body

Slave Body

• Perturbed lagrangian form of solution

Πc = λ g −1

2κλ2

• Variational equation

δΠc =[

δu(s)2 δu

(m)2 δλ

] λ

−λ

g − 1κ λ

• Solution: λ = κ g where κ constraint parameter to prevent

penetration (penalty).


• Linearization produces tangent matrix 0 0 10 0 −11 −1 −1/κ

du(s)2

du(m)2

dλ

=

−λ

λ

− g + 1κ λ

• Eliminate λ and dλ gives reduced tangent[

κ −κ−κ κ

] du(s)2

du(m)2

=

{−κ g

κ g

}

• Final gap non-zero - depends on value of κ.


g = x2(s)− x

2(m)

m

s

Master Body

Slave Body

• Perturbed tangent form (Lim & T, 2001)

• Lagrange multiplier variational equation

δΠc =[

δu(s)2 δu

(m)2 δλ

] λ

−λg

• Linearized form for perturbed lagrangian form 0 0 1

0 0 −11 −1 −1/κ

du(s)2

du(m)2

dλ

=

−λ

λ− g

• Not consistently linearized form (non-zero diagonal for λ).


g = x2(s)− x

2(m)

m

s

Master Body

Slave Body

• Penalty function form

Π =1

2κ g2

where κ is penalty parameter.

• Matrix equation for nodes given by[κ −κ

−κ κ

] du(s)2

du(m)2

=

{−κ g

κ g

}

• N.B. Not always same as perturbed lagrangian form.

• Avoids indefinite equation of Lagrange multiplier approach.


g = x2(s)− x

2(m)

m

s

Master Body

Slave Body

• Augmented Lagrangian penalty form

• Mix of penalty and Lagrange multiplier[κ −κ

−κ κ

] du(s)2

du(m)2

=

{−λk − κ g

λk + κ g

}

• Update to ’Lagrange multiplier’ computed using

λk+1 = λk + κ g

• Update after each Newton iteration or in new loop.


• Augmented Lagrangian perturbed form

Πc = (λ + λk) g −1

2κλ2

• Mix with perturbed Lagrangian form: 0 0 10 0 −11 −1 −1/κ

du(s)2

du(m)2

dλ

=

− (λ + λk)

(λ + λk)− g + 1

κ λ

where Uzawa update λk+1 = λk + κ g used.

• Method of choice to program: Can reduce to all otherforms.

1988: Friction by Perturbed Lagrangian

• Friction - Coulomb model simplest form

λt ≤ µ | λn |

• Stick: If | λt |< µ | λn | set gt = 0

• Variational equation

δΠc = δλn(gn − 1κn

λn) + δgn λn + δλt(gt − 1κt

λt) + δgt λt

– Slip : If | λt |= µ | λn | gives gt 6= 0

δΠc = δλn(gn− 1κn

λn)+δgn λn+δλt(gt− 1κt

λt)+δgt [µ | λn | sign(gt)]

– Yields unsymmetric tangent(Ju & T, 1988).

– Augmented form can make symmetric - Laursen & Simo(1993).

1991: The Contact Patch Test

• Contact patch: Consistency (constant stress) & Stability

test (Papadopoulos & T, 1991).



• Example: Node-Surface: 1-pass

-1.03E+01

-1.02E+01

-1.02E+01

-1.01E+01

-1.01E+01

-1.00E+01

-9.95E+00

-9.89E+00

-9.84E+00

-9.78E+00

-9.73E+00

-9.67E+00

-1.03E+01

_________________ S T R E S S 2

(a) Mesh and loading (b) σ22 stress result

• Test fails for node to segment treatment.

• Two pass switch master/slave: Penalty passes (penetrate).Augmented lagrangian or perturbed tangent fails test.

FEM Node-to-Surface Contact Analysis

• All previous schemes have limitations:

– Valid only for low order elements (3-node triangles & 4-

node quadrilaterals in 2-D; 4-node tets to 8-node bricks

in 3-D).

– Fails contact patch test unless two-pass scheme used.

– Two-pass works in penalty form - fails stability test for

Lagrange multiplier.

2002-2004: Mortar Methods for Contact & Ties

• Contact interface requires addition of term

Πc =∫Γs

λT(x(m) − x(s)

)dΓ

where x(m) and x(s) deformed positions of master & slave.

• Integrals carried out by quadrature on sub-segments.

• Use interpolations

x(m) = Na(ξ) x̃(m)a (t) ; x(s) = Nb(ξ) x̃(s)

b (t) ; λ = N̂c(ξ) λ̃c(t)

where Na, Nb usual shape functions and N̂c multiplier ones.


• Options for N̂c:

– Delta function N̂c = δ(ξ − ξc) gives node to surface.

– Slave shape function N̂c = δca Na(ξ) gives standard mor-tar method.

– Dual of slave shape function (e.g., linear case)

N̂c = α1 N1 + α2 N2

gives dual mortar method.

• Dual mortar method has advantages.


• Two-dimensional mortar functions and duals

N1

N2

N3

N4

^ ^ ^ ^N

1N

2N

3N

4


• Two-dimensional mortar shape functions and duals

– Dual satisfies∫h

N̂aNb dΓ = δab

∫h

Na dΓ

– Assume linear functions

N̂a = α1 N1 + α2 N2

gives for N̂1

h

6

[2 11 2

] [α1α2

]=

h

2

[10

]Solution: α1 = 2, α2 = −1.

N1

N2

1

^N

1

^N

2

2

1


• Quadratic functions:

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

ξ − Coordinate

Nc −

Sh

ape

Fu

nct

ion

s

N1

N2

N3

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

1.5

ξ − Coordinate

Nc −

Du

al F

un

ctio

ns

N1

N2

N3

(a) Shape functions (b) Dual Functions

2002-2004: Mortar Methods for Contact & Ties• Substitution of Discretization in

Πc =∫Γ

λT(x(m) − x(s)

)dΓ

and integrating on slave facets

Πc =∑s

∫Γs

λ̃Tc N̂s

c (ξ)[Ns

a(ξ)x̃sa −Nm

b (ξm)x̃mb

]dΓ

where ξ for slave and ξm projected master point.

1

1

ξ

ξ

ξm

Slave

Master

2002-2004: Mortar Methods for Contact & Ties• Accurate quadrature uses segments∫

Γs

f(ξ) dξ =∑m

∫Γms

f̂(η) dη

with segment −1 ≤ η ≤ 1.

η2

η1

Slave Subsegments

• Gauss-Legendre quadrature needs 2-points/segment for linear-linear sides; 3-points/segment for quadratic-quadratic sides.


• Evaluation of segment integrals gives

Πs = λ̃Tc

[Gs

ca x̃sa −Gm

cb x̃mb

]where Πc =

∑s Πs and

Gsca =

∫Γs

N̂c(ξ)Na(ξ) dΓ I

Gmcb =

∫Γs

N̂c(ξ)Nb(ξm) dΓ I

where I are ndim× ndim.

• For dual mortar: Gsca is diagonal.

• Conserve linear and angular momenta (Puso & Laursen, 2003).

2002-2004: Mortar Methods for Tied Surfaces

• Surface to surface tied interface - 4 node elements.

-1.65E+00

-1.50E+00

-1.35E+00

-1.20E+00

-1.05E+00

-9.00E-01

-7.50E-01

-6.00E-01

-4.50E-01

-3.00E-01

-1.50E-01

0.00E+00

-1.80E+00

_________________ DISPLACEMENT 2

-1.08E+01

-1.06E+01

-1.05E+01

-1.03E+01

-1.01E+01

-9.95E+00

-9.77E+00

-9.60E+00

-9.42E+00

-9.25E+00

-9.07E+00

-8.90E+00

-1.10E+01

_________________ S T R E S S 2

(a) Vertical displacement (b) Vertical stress


• Surface load on layer - 9 node elements

-7.85E-02

-7.13E-02

-6.42E-02

-5.71E-02

-4.99E-02

-4.28E-02

-3.57E-02

-2.85E-02

-2.14E-02

-1.43E-02

-7.13E-03

0.00E+00

-8.56E-02

_________________ DISPLACEMENT 2

-1.06E+01

-9.54E+00

-8.48E+00

-7.42E+00

-6.36E+00

-5.31E+00

-4.25E+00

-3.19E+00

-2.13E+00

-1.08E+00

-1.78E-02

1.04E+00

-1.17E+01

_________________ S T R E S S 2

(a) u2 displacement (b) σ22 stress


• Displacement for surface load on layer - 4 node elements

-8.89E-02

-8.05E-02

-7.20E-02

-6.36E-02

-5.51E-02

-4.66E-02

-3.82E-02

-2.97E-02

-2.13E-02

-1.28E-02

-4.39E-03

4.06E-03

-9.74E-02

_________________ DISPLACEMENT 2

-7.95E-02

-7.23E-02

-6.51E-02

-5.78E-02

-5.06E-02

-4.34E-02

-3.61E-02

-2.89E-02

-2.17E-02

-1.45E-02

-7.23E-03

0.00E+00

-8.68E-02

_________________ DISPLACEMENT 2

(a) Node to Surface (b) Mortar Method


• Stress for surface load on layer - 4 node elements

-9.35E+00

-8.49E+00

-7.63E+00

-6.77E+00

-5.91E+00

-5.05E+00

-4.19E+00

-3.33E+00

-2.47E+00

-1.61E+00

-7.53E-01

1.07E-01

-1.02E+01

_________________ S T R E S S 2

-9.34E+00

-8.48E+00

-7.62E+00

-6.76E+00

-5.90E+00

-5.05E+00

-4.19E+00

-3.33E+00

-2.47E+00

-1.61E+00

-7.49E-01

1.10E-01

-1.02E+01

_________________ S T R E S S 2

(a) Node to Surface (b) Mortar Method


• Displacement for surface load on layer - 4 node elements

-7.94E-02

-7.21E-02

-6.49E-02

-5.77E-02

-5.05E-02

-4.33E-02

-3.61E-02

-2.89E-02

-2.16E-02

-1.44E-02

-7.21E-03

0.00E+00

-8.66E-02

_________________ DISPLACEMENT 2

-7.95E-02

-7.23E-02

-6.51E-02

-5.78E-02

-5.06E-02

-4.34E-02

-3.61E-02

-2.89E-02

-2.17E-02

-1.45E-02

-7.23E-03

0.00E+00

-8.68E-02

_________________ DISPLACEMENT 2

(a) No Interface (b) Mortar Method


• Stress for surface load on layer - 4 node elements

-9.33E+00

-8.48E+00

-7.62E+00

-6.77E+00

-5.92E+00

-5.06E+00

-4.21E+00

-3.36E+00

-2.51E+00

-1.65E+00

-8.01E-01

5.23E-02

-1.02E+01

_________________ S T R E S S 2

-9.34E+00

-8.48E+00

-7.62E+00

-6.76E+00

-5.90E+00

-5.05E+00

-4.19E+00

-3.33E+00

-2.47E+00

-1.61E+00

-7.49E-01

1.10E-01

-1.02E+01

_________________ S T R E S S 2

(a) No Interface (b) Mortar Method


• Discussed only tied interface treatment.

• Implentation for full 2-d & 3-d contact in progress.

• Best to date observed is NIKE at LLNL by M.A. Puso.

• Gives good results without surface smoothing.

Summary: Contact Analysis

• Lecture discussed:

– Foundations for contact analysis, including transient be-

havior for impact/release. All extensions of contributions

by TJRH!

– Solution strategies for contact analysis (Lagrange multi-

plier, penalty, etc.)

– Contact patch test requirements – consistency & stability.

– Spatial discretization methods (node-node; node-surface;

surface-surface).

– Mortar (standard & dual) methods for surface-surface

treatment.

Happy 60th, Tom!

Date post:	18-Feb-2022
Category:	Documents
Upload:	others
View:	1 times
Download:	0 times

Finite Element Solution of Contact Problems: From 1974 to 2004

Documents