Computational Contact Mechanics Yastrebov

Introduction toComputational Contact Mechanics

Part I. Basics

Vladislav A. Yastrebov

Centre des Matériaux, MINES ParisTech, CNRS UMR 7633

WEMESURF short course on contact mechanics and tribology

Paris, France, 21-24 June 2010

Intro Detection Geometry Discretization Solution FEA Examples

Preface

To whom the course is aimed?

Developpers and users.

What is the aim?

Accurate contact modeling, correct interpretation, etc.

FEM - Finite Element Method, FEA - Finite Element Analysis

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 2/77

Outline

Mathematical foundationcontact geometry;optimization methods.

Inside the Finite Element programmecontact detection;contact discretization;account of contact.

Contact problem resolution with FEM: guide for engineermaster-slave approach;boundary conditions;spurious cases.

Contentdemonstrative;simple;general.

1 Introduction

2 Contact detection

3 Contact geometry

4 Contact discretization methods

5 Solution of contact problem

6 Finite Element Analysis of contact problems

7 Numerical examples

Finite Element Method

FEM a powerfull and multi-purpose tool for linear andnon-linear dynamic and static continuum mechanical andmulti-physic problems

[Rousselier et al.] [Klyavin et al.] [Brinkmeiera et al.]

FEM a powerfull and multi-purpose tool for linear andnon-linear dynamic and static continuum mechanical andmulti-physic problems

[Rousselier et al.] [Klyavin et al.] [Brinkmeiera et al.]

Contact problems are not continuum and they require:

new rigorous mathematical basis: geometry,optimization, non-smooth analysis;particular treatment of the nite element algorithms;smart use of the Finite Element Analysis (FEA).

Finite element analysis of contact problems

assembled components; Disk-blade contact

T.Dick, G.CailletaudCentre des Matériaux

assembled components;

bearings;Rolling bearing

F.Massi et al.LaMCoS, INSA-Lyon et al.

bearings;

rolling contact;

Tire rolling noise simulation

M.Brinkmeiera, U.Nackenhorst et al.University of Hannover et al.

bearings;

rolling contact;

forming processes;

Deep drawing

G.Rousselier et al.Centre des Matériaux et al.

bearings;

rolling contact;

forming processes;

geomechanical contact;

Post seismic relaxation

J.D.Garaud, L.Fleitout, G.CailletaudCentre des Matériaux, ENS

bearings;

rolling contact;

forming processes;

crash tests;

Crash-test

O.Klyavin, A.Michailov, A.BorovkovSt Petersbourg State University

bearings;

rolling contact;

forming processes;

crash tests;

human joints;

Knee simulation

E.Peña, B.Calvoa et al.University of Zaragoza

bearings;

rolling contact;

forming processes;

crash tests;

human joints;

and many others.

Knee simulation

E.Peña, B.Calvoa et al.University of Zaragoza

They trust in the FEALeading international industrial companies

Mechanical problemFrom a real life problem to an engineering problem

Need to determine:the problematic:

strength/life-time/fracture;vibration/buckling;thermo-electro-mechanical.

relevant geometry;

relevant loads:

static/quasi-static/dynamic;mechanical/thermic;volume/surface.

relevant material:

rigid/elastic/plastic/visco-plastic;brittle/ductile.

relevant scale:

macro/meso/micro.

Need to determine:the problematic:

relevant geometry;

relevant loads:

relevant material:

relevant scale:

macro/meso/micro.

For example: Brinell hardnesstest

Scheme of the Brinell hardness test anddierent types of impression[Harry Chandler, Hardness testing]

Need to determine:problematic:

strength.

geometry;

sphere + half-space.

loads:

mechanical surface quasi-static.

material:

rigid + elasto-visco-plastic.

scale:

macro.

For example: Brinell hardnesstest

Scheme of the Brinell hardness test anddierent types of impression[Harry Chandler, Hardness testing]

Mechanical problemFrom an engineering problem to a nite element model

Problem:problematic:

FE model:analysis type:

stress-strain state;eigen values;coupled physics.

geometry;

nite element mesh;

geometry;

loads:

nite element mesh;

analysis type and BC:

geometry;

loads:

material:

nite element mesh;

material model:

geometry;

loads:

material:

scale:

macro/meso/micro.

nite element mesh;

material model:

microstructure:

RVE/microstructure/.

Mechanical problemFrom engineering problem to a nite element model

strength.

geometry:

loads:

material:

scale:

stress-strain state.

nite element mesh:

material model:

microstructure:

strength.

geometry:

loads:

material:

scale:

nite element mesh:

sphere + large block.

material model:

microstructure:

strength.

geometry:

loads:

mechanical surfacequasi-static.

material:

scale:

nite element mesh:

material model:

microstructure:

strength.

geometry:

loads:

material:

scale:

nite element mesh:

material model:

rigida + elasto-visco-plasticmodel.

microstructure:

arigid in FEA: much more harder than another solid,

special boundary conditions or geometricalrepresentation.

strength.

geometry:

loads:

material:

scale:

macro.

nite element mesh:

material model:

rigida + elasto-visco-plasticmodel.

microstructure:

homogeneous > RVE.

arigid in FEA: much more harder than another solid,

special boundary conditions or geometricalrepresentation.

Problem:problematic

geometry

material

FE model:analysis type

nite element mesh

analysis type and BC

material model

microstructure

Problem:problematic

geometry

material

FE model:analysis type

nite element mesh

analysis type and BC

material model

microstructure

Finite element mesh : )

Symmetry and plane problemsHow to solve problems faster

Main ideas:symmetry

geometry AND loading;

3D to 2D:

axisymmetry/planestrain/plane stress;

3D to smaller 3D:

half/quarter/sector;

Main ideas:symmetry

3D to 2D:

3D to smaller 3D:

Axisymmetry

Axisymmetry of geometryaxisymmetry of load

Main ideas:symmetry

3D to 2D:

3D to smaller 3D:

Mirror symmetry

Axisymmetry of geometrymirror symmetry of load

Main ideas:symmetry

3D to 2D:

3D to smaller 3D:

No symmetry

Axisymmetry of geometryno symmetry of load

Main ideas:symmetry

3D to 2D:

3D to smaller 3D:

Plain strain

Mirror symmetry of geometrymirror symmetry of load

very long structure or xed edges

Main ideas:symmetry

3D to 2D:

3D to smaller 3D:

Plain stress

Mirror symmetry of geometrymirror symmetry of load

very thin structure

Full 3D mesh Axisymmetric 2D mesh

Finite element meshBasics

In general the nite element mesh should

full the required solution precision;

correctly represent relevant geometry;

not be enormous;

be ne where strain is large;

be rough where strain is small;

avoid too oblong elements.

In contact problems the nite element mesh should

not allow corners at master surface;

be very ne and precise on both contacting surfaces;

use carefully quadratic elements.

Boundary conditions

Two solids Ω1 and Ω2.

Volumetric forces Fv : e.g. inertia mu.

Neumann (static, force) boundaryconditions: distributed andconcentrated loads.

Dirichlet (kinematic, displacement)boundary conditions: displacements.

In each solid we full div(σ) + Fv = 0

Boundary conditions

What are the contact boundaryconditions?

Boundary conditions

What are the contact boundaryconditions?

Account of contactSignorini conditions

Signorini conditions of nonpenetration gn > 0 and non-adhesionσn ≤ 0

gnσn = 0, gn ≥ 0, σ ≤ 0, σn = σ · n

Contact boundary conditions ∼ unknown Neumann (force) boundaryconditions depending on the geometry.

Account of contactSignorini conditions

Signorini conditions of nonpenetration gn > 0 and non-adhesionσn ≤ 0

gnσn = 0, gn ≥ 0, σ ≤ 0, σn = σ · n

Account of contactCoulomb's friction

Coulomb's friction conditions

|gt | (|σt |+ µσn) = 0; |σt | ≤ −µσn; gt = |gt |σt

|σt |, σt = σ · t

Contact detection

Two FE meshes penetrate each other

What penetrates?

Nodes, lines, elements?

Into what it penetrates?

Into elements, under surface?

How do we detect such penetration?

Contact detection

What penetrates?

Contact detection

What penetrates?

Contact detection

What penetrates?

Contact detection

What penetrates?

Zoom on penetration

Contact detection

What penetrates?

Volume intersection

Contact detection

What penetrates?

Surface-in-volume 1

Contact detection

What penetrates?

Surface-in-volume 2

Contact detection

What penetrates?

Nodes-in-volume 1

Contact detection

What penetrates?

Nodes-in-volume 2

Contact detection

What penetrates?

Nodes-to-surface 1

Contact detection

What penetrates?

Nodes-to-surface 2

Contact discretization

What is elementary contact contrubutor?

Local contacting unit?

Node-to-node

Node-to-segment/Node-to-edge/Node-to-surface

Gauss point to surface

Node-to-node discretization:small deformation/small sliding

Node-to-node

Large deformation/large sliding

Node-to-node

Node-to-segment discretizationlarge deformation/large sliding

Node-to-node

Contact domain method:large deformation/large sliding

Node-to-node

Conception of the contact element

Contact domain method:large deformation/large sliding

1 Introduction

2 Contact detection

3 Contact geometry

Spatial search and local detection

Spatial searchDetection of contacting solids:

multibody systems

Discrete Element Method

Molecular dynamics

Example of DEM application[Williams,O'Connor,1999]

Local contact detectionDetection of contacting nodes andsurfaces of two discretized solids

Smoothed ParticleHydrodynamics

Toruscylinder impact problem[B. Yang, T.A. Laursen, 2006]

Spatial search and local detection

Spatial searchDetection of contacting solids:

multibody systems

Discrete Element Method

Molecular dynamics

3rd body layer modeling[V.-D. Nguyen, J. Fortin et al., 2009]

Local contact detectionDetection of contacting nodes andsurfaces of two discretized solids

Smoothed ParticleHydrodynamics

Buckling with self-contact[T. Belytschko, W.K. Liu, B. Moran, 2000]

Local contact detectionBasic ideas

How to detect contact?

What penetrates and where?

What/where approach

What/where approach, for example,

What? Slave nodesWhere? Under master surface

Assymetry of contacting surfaces

New paradigm

BUT! We need to detect contact before anypenetration occures!

New paradigm

Contact detection idea

In general the detection consists in checking ifslave nodes are close enough

to the master surface

New paradigm

Contact detection idea

In general the detection consists in checking ifslave nodes are close enougha

to the master surface

aWhat does it mean close enough?

Maximal detection distance

Maximal detection distance concept dmax

If a slave node is closer than dmax to the master surface, then this slavenode is considered as possibly contacting at current load step and acontact element is to be created.

Solid with slave nodes and solid with a master surface.

External detection zone dist < dmax and gn ≥ 0

Internal detection zone dist < dmax and gn < 0

Contact detection zone dist < dmax

Verciation if slave nodes are in the detection zone

No slave nodes in the detection zone

Solid with slave nodes and solid with a master surface.

Contact detection zone dist < dmax

Verciation if slave nodes are in the detection zone

Some slave nodes are in the detection zone. One node is missed!

Create contact elements with detected slave nodes.

How to choose dmax?

Dangerous solution

Make the user responsible for this choice

Practical solutions: choose accordingly to

master surface mesh;

boundary conditions (maximal variation of displacement ofcontacting surface nodes during one iteration/increment)

use both criterions.

Discretized master surface

Master surface contsist of many elementary master surfaces.

The aim is to nd for each slave node the associated master surface.

Triangles - slave nodes and circles - master nodes which are connected bymaster surfaces.

Contact detection zone.What does it consist of?

Zoom on the master contact surface andthe associated detection zone.

Contact detection zone is nothing but the region, where each point iscloser to the master surface than dmax

Each contact element consists of a slave node and of the master surfaceonto which it projects, i.e. the closest master surface.

Closest points denition

How to dene the closest point ρ∗ onto the master surface Γm for agiven slave node rs?

Denition of the closest point ρ∗

ρ∗ ∈ Γm : ∀ρ ∈ Γm, |rs − ρ∗| ≤ |rs − ρ|

Functional for parametrized surface ρ = ρ(ξ):

F (rs , ξ) =1

2(rs − ρ(ξ))2

ρ∗ ∈ Γm : ∀ρ ∈ Γm, |rs − ρ∗| ≤ |rs − ρ|

Functional for parametrized surface ρ = ρ(ξ):

F (rs , ξ) =1

2(rs − ρ(ξ))2

In general case, nonlinear minimization problem:

minξ∈[0;1]F (rs , ξ)⇒ ξ∗ : ∀ξ ∈ [0; 1], |rs − ρ(ξ∗)| ≤ |rs − ρ(ξ)|

minξ∈[0;1]F (rs , ξ)⇔ (rs − ρ(ξ∗)) · ∂ρ

∣∣∣∣ξ∗

minξ∈[0;1]F (rs , ξ)⇔ (rs − ρ(ξ∗)) · ∂ρ

∣∣∣∣ξ∗

Existance/uniqueness of the closest point

Does the projection always exist?

Does the projection always exist? No!

Only if the master surface is smooth ρ(ξ) ∈ C 1(Γm)

(rs − ρ(ξ∗)) · ∂ρ

˛ξ∗

In the FEM the surface is not obligatory smooth, only continuousρ(ξ) ∈ C 0(Γm)

Even if a projection exists it can be non-unique.

(rs − ρ(ξ∗)) · ∂ρ

˛ξ∗

(rs − ρ(ξ∗)) · ∂ρ

˛ξ∗

(rs − ρ(ξ∗)) · ∂ρ

˛ξ∗

Does the projection always exist? No!Only if the master surface is smooth ρ(ξ) ∈ C 1(Γm)

(rs − ρ(ξ∗)) · ∂ρ

˛ξ∗

For more details see Contact geometry section

Blind spots

Blind spot problem

Blind spots gaps in the detection zone.

Do not miss slave nodes situated in blind spots.

Types of blind spots: external, internal, due to symmetry.

master projection zone

normals to master

blind spots

external

?internal

due to symmetry

s symmetry

Who is master, who is slave?Social inequality in contact problems

Master-slave denition

The choice of master and slave surfaces is not random.

Incorrect choice leads to meaningless solutions.

Initial FE mesh Incorrect master-slavechoice /

Correct master-slavechoice ,

Contact detection in global algorithm

At the beginning of eachincrement (loading step)

+ fast;

+ good convergence;

+ stable;

lack of accuracy.

At the beginning of eachiteration (convergence step)

innite looping;

not stable;

+ more accurate.

Contact detection in global algorithm

At the beginning of eachincrement (loading step)

+ fast;

+ good convergence;

+ stable;

lack of accuracy.

At the beginning of eachiteration (convergence step)

innite looping;

not stable;

+ more accurate.

Contact detection techniques

All-to-all detectionDirect all-to-all detection

Slave nodes to mastersegments.

Indirect all-to-all detection

Slave nodes to masternodes.Slave node to attachedmaster segments.

Advantages:

simple implementation.

Drawbacks:

time consumming O(N2);blind spots/passing by nodes.

Elaborated techniquesBounding boxes

Account only closeregions.

Regular grid methods:bucket [Benson].

Detect only into a celland in neighbouring cells

Sorting methods:heap sort, octree [Williams,O'Connor].

Data tree constructionand tree search.

Advantages:

relatively fast O(N),O(N logN)..

Drawbacks:

blind spots/passing by nodes.

All-to-all detection Elaborated techniques

Example: two curved contacting surfaces 1 million of slave nodes against 1 million of master segments.

Indirect all-to-all technique

187 hours(!)

Indirect all-to-all technique

187 hours(!)

Bounding box+grid detection

1 minute

SummaryContact detection

Global search/local contact detection;

slave-master or what-where approach;

conception of the maximal detection distance and its choice;

contact geometry and contact detection - closest point denition;

existance and uniqueness of the closest point;

from continuous and smooth to discretized C 0 surface;

attention - blind spots;

detect contact at the beginning of increment;

dierent detection techniques;

self-contact detection.

contact detection is strongly connectedwith contact geometry andcontact discretization.

1 Introduction

2 Contact detection

3 Contact geometry

Introduction

Geometry is a foundation for

master-slave approach;detection;discretization method;solution.

Geometrical quantities

penetration or normal gap gn - normal contact;tangential sliding ∆gt - frictional eects.

Challenges in contact geometry

Challenges

requirement of smooth surface.

non-uniqueness of projection;

Challenges

assymetry of contacting surfacesa;

asomething penetrates into something, something slides over something

Challenges

Smoothness of contact surface

Do real surfaces are smooth or not?

It depends on the scale.

The smaller scale, the higher roughness.

Fractal surface.

Surface deforms even without contact

macro deformation;

escaping dislocations and twins;

relaxation at nano-scale.

Finite element surface is not smooth

convergence problems;

unphysical oscilations;

remedy - special smoothing techniques.

Polished metal surface400x600 microns specimen

Fractal surface

Fractal surface.

macro deformation;

Fractal surface

Fractal surface.

macro deformation;

Fractal surface

Fractal surface.

macro deformation;

Fractal surface

Fractal surface.

macro deformation;

Fractal surface

Fractal surface.

macro deformation;

Fractal surface

Fractal surface.

macro deformation;

Deformation twinnings in coating[Forest, 2000]

Escaping dislocations[Fivel, 2009]

Fractal surface.

macro deformation;

Deformation twinnings in coating[Forest, 2000]

Escaping dislocations[Fivel, 2009]

Fractal surface.

macro deformation;

2D surface smoothing with Beziercurves

Fractal surface.

macro deformation;

3D surface smoothing with Beziersurface

Account of geometrical quantitiesFirst order variations

Contact in the weak form

Geometrical quantities: normal gap gn = gn(x) and tangential slidinggt = gt(x

Virtual work at contact surface δW cont = δWn + δWt

normal contact Wn(σn, gn) ⇒ δWn(σn, gn, δσn, δgn)frictional contact Wt(σt , gt) ⇒ δWt(σt , gt , δσt , δgt)

Resulting nonlinear equation R“δW int, δW ext, δW cont

”= 0

Need of analytical expressions

First order variation of the normal gap and tangential sliding

δgn =∂gn∂x

· δx δgt =∂gt∂x

· δx

Account of geometrical quantitiesSecond order variations

Linearization of the weak form

Resulting nonlinear equation R“δW int, δW ext, δW cont

”= 0

Linearization R(x) = 0⇒ R|x0

+ ∆R(x)|x0≈ 0

∆R(x) = ∆δW int + ∆δW ext + ∆δW cont

normal contact ∆δWn(σn, gn, δσn, δgn, ∆δσn, ∆δgn)frictional contact ∆δWt(σt , gt , δσt , δgt , ∆δσt , ∆δgt)

Need of analytical expressions

Second order variation of the normal gap and tangential sliding

∆δgn = ∆x · ∂2gn

∂x2· δx ∆δgt = ∆x · ∂2gt

∂x2· δx

Point and surfaceMaster-slave approach

Master-slave

Slave node penetrates under and slides over the master surface.

Slave node point rs

Master surface ρ(ξ)

Surface parametrization ξ = ξ1, ξ2Projection of the slave node

ρ(ξp)

Normal to the master surfacen(ξp)

Normal gap gn = (rs − ρ) · n

Tangential sliding gtdt = δρ(ξ)

Point and surfaceMaster-slave approach

Master-slave

Slave node penetrates under and slides over the master surface.

Slave node point rs = rs(t)

Master surface ρ(ξ) = ρ(t, ξ)

Surface parametrization ξ = ξ1, ξ2Projection of the slave node

ρ(ξp) = ρ(t, ξ(rs(t)))

Normal to the master surfacen(ξp) = n(t, ξ(rs(t)))

Normal gap gn(t) = (rs(t)− ρ(t, ξ(rs(t))) · n(t, ξ(rs(t)))

Tangential sliding gt(t, ξ(rs(t)))dt = δρ(t, ξ(rs(t)))

Continuum geometrical formulationCovariant and contravariant bases, fundamental surface tensors

Covariant surface basis∂ρ∂ξ

∂ρ∂ξ1

; ∂ρ∂ξ2

oContravariant surface basis

∂ρ∂ξ

∂ρ∂ξ1

; ∂ρ∂ξ2

oBasis change

∂ρ∂ξ

= A−1 ∂ρ∂ξ

∂ξi= aij ∂ρ

∂ξj

1st fundamental covariant surface tensorA ∼ aij = ∂ρ

∂ξi· ∂ρ

∂ξj

1st fundamental contravariant surface tensorA−1 ∼ aij = ∂ρ

∂ξi· ∂ρ

∂ξj

2nd fundamental surface tensorH ∼ hij = n · ∂2ρ

∂ξi∂ξj

Continuum geometrical formulationFirst order variations

First order variation of the normal gap δgn

δgn = n · (δrs − δρ)

First order variation of the surface parameter δξ

δξ = (A− gnH)−1 ·(

∂ξ· (δrs − δρ) + gnn · δ

Continuum geometrical formulationSecond order variations

Second order variation of the normal gap δgn

∆δgn =− n ·„

δ∂ρ

∂ξ·∆ξ + ∆

∂ξ· δξ + δξ ·

∂2ρ

∂ξ2·∆ξ

+ gnδξ ·H · A−1 ·H ·∆ξ+

„n · δ

«· A−1 ·

„∆

∂ξ· n

« (1)

Second order variation of the surface parameter δξ

∆δξ = (gnH− A)−1 ·»

∂ξ·

„δ∂ρ

∂ξ·∆ξ + ∆

∂ξ· δξ + δξ ·

∂2ρ

∂ξ2·∆ξ

«− gnn ·

„δ∂2ρ

∂ξ2·∆ξ + ∆

∂2ρ

∂ξ2· δξ + δξ ·

∂3ρ

∂ξ3·∆ξ

∂ξ· n + H ·∆ξ

„I n · (δρ− δrs)+ gnA

−1 ·∂ρ

∂ξ·

δ∂ρ

∂ξ+

∂2ρ

∂ξ2· δξ

δ∂ρ

∂ξ· n + H · δξ

„I n · (∆ρ−∆rs)+ gnA

−1 ·∂ρ

∂ξ·

∂ξ+

∂2ρ

∂ξ2·∆ξ

ff«–(2)

ApproximationSmall penetration

Approximation of zero penetration

Normal gap is small gn ≈ 0.

δgn = n · (δrs − δρ) δξ = A−1 · ∂ρ

∂ξ· (δrs − δρ)

∆δgn = −n ·(

δ∂ρ

∂ξ·∆ξ + ∆

∂ξ· δξ + δξ · ∂2ρ

∂ξ2·∆ξ

∆δξ = −A−1 ·»

∂ξ·

„δ∂ρ

∂ξ·∆ξ + ∆

∂ξ· δξ + δξ ·

∂2ρ

∂ξ2·∆ξ

∂ξ· n + H ·∆ξ

ff· (I n · (δρ− δrs))+

δ∂ρ

∂ξ· n + H · δξ

ff· (I n · (∆ρ−∆rs))

– (3)

Discretized contact geometryFrom continuum formulation to the Finite Element Method

Finite element method formalism

r =N∑i=1

φi (ξ)x i =N∑i=1

φi (ξ1, ξ2)x i

[X ] = [X (t)] = [x0(t), x1(t), . . . , xN(t)]T ;

[Φ] = [Φ(ξ)] = [0, φ1(ξ), . . . , φN(ξ)]T ;

[Φ′i ] =

[∂Φ(ξ)

∂ξi

]= [0, φ1,i , . . . , φN,i ]

rs = rs(t) = x0(t) = [S0]T [X ] , where [S0] = [1, 0, . . . , 0]T .

ρ = ρ(t, ξp) = φi (ξp)x i =[Φ(ξp)

]T[X (t)] ,= [Φ]T [X ]

ρi =∂ρ

∂ξi=

∂ρ(t, ξ)

∂ξi

∣∣∣∣ξp

[∂Φ(ξ)

∂ξi

∣∣∣∣ξp

[X (t)] = [Φ′i ]T

Discretized contact geometryFrom continuum formulation to the Finite Element Method II

First variations of geometrical quantities

δgn =

n−φ1n

...−φNn

δx0δx1...

= [∇gn]T · δ [X ] (4)

δξi = cij

∂ρ∂ξj

− ∂ρ∂ξj

φ1 + gnnφ1,j

− ∂ρ∂ξj

φN + gnnφN,j

δx0δx1...

= [∇ξi ]T · δ [X ] (5)

Second order variation of the normal gap ∆δgn

∆δgn =δ [X ]T ·n−n

ˆΦ′i

˜⊗ [∇ξi ]

T − [∇ξi ]⊗ˆΦ′i

−“hij − gnhika

”[∇ξi ]⊗ [∇ξj ]

+gnaijn

ˆΦ′i

hΦ′j

ff·∆ [X ] =

= δ [X ]T · [∇∇gn] ·∆ [X ]

Second order variation of surface parameter ∆δξ

∆δξi = δ [X ]T ·

8<:−cij

24hΦ′k

i ∂ρ

∂ξj

⊗ [∇ξk ]T + [∇ξk ] ⊗∂ρ

∂ξj

hΦ′k

0@ ∂ρ

∂ξj

·∂2ρ

∂ξk∂ξm+ gnn ·

∂3ρ

∂ξk∂ξj∂ξm

1A [∇ξk ] ⊗ [∇ξm ]T

+ gnhΦ′′jk

in ⊗ [∇ξk ]T + gn [∇ξk ] ⊗ n

hΦ′′jk

iT− δkj

„[∇gn ] ⊗

hΦ′k

iT+ hks [∇ξs ]T

«+“h

Φ′k

in + hks [∇ξs ]

”⊗ [∇gn ]T

+ gnakm

hΦ′ji ∂ρ

∂ξm⊗„n

hΦ′k

iT+ hks [∇ξs ]T

«+“n

hΦ′k

i+ hks [∇ξs ]

”⊗

∂ξm

hΦ′jiT!

+ gnakm

0@ ∂ρ

∂ξm·

∂2ρ

∂ξj∂ξl

1A„[∇ξl ] ⊗„n

hΦ′k

iT+ hks [∇ξs ]T

hΦ′k

i+ hks [∇ξs ]

”⊗ [∇ξl ]

T”io

· ∆ [X ] =

= δ [X ]T ·ˆ∇∇ξi

˜· ∆ [X ]

1 Introduction

2 Contact detection

3 Contact geometry

Introduction

Discretization

Discretization of the contact area into elementary units responsiblefor the contact stress transmission from one contacting surface toanother.

Units:

surface nodes;

surface of elements;

Gauss points ontosurfaces;

Problem types:

small deformation -no slip;

large deformation -arbitrary slip.

Discretization method and assymetry:

methods which enforce assymetry ofsurfaces:

node-to-segment.

methods which reduce this assymetry:

segment-to-segment, mortar, Nitsche;contact domain method.

NTN Node-to-node

Node-to-node discretization[Francavilla & Zienkiewicz, 1975], [Oden, 1981], [Kikuchi & Oden, 1988]

Advantages:

, very simple;

, passes Taylor's test1.

1 Taylor's patch test requires that auniform contact stress transmittes correctlyfrom one contacting surface to another.See section Finite Element Analysis.

Scheme of two conforming meshes.Pairing nodes form NTN contact elements.

NTN Node-to-node

Node-to-node discretization[Francavilla & Zienkiewicz, 1975], [Oden, 1981], [Kikuchi & Oden, 1988]

Advantages:

, very simple;

, passes Taylor's test1.

Drawbacks:

/ small deformation;

/ small slip;

/ requires conforming FE meshes.

1 Taylor's patch test requires that auniform contact stress transmittes correctlyfrom one contacting surface to another.See section Finite Element Analysis.

Scheme of two conforming meshes.Pairing nodes form NTN contact elements.

NTN Node-to-nodeDenition of the normal

Denition of the normal for the node-to-nodediscretization

Two FE mesh with matching nodes in the contact zone

NTN contact element detection

Master-slave discretization

Normals on the master surface

Denition of the normals at master nodes as an average of thenormals of adjacent segments

Denition of the normals at master nodes

NTS Node-to-Segment/Node-to-Surface

Node-to-segment discretization[Hughes, 1977] [Hallquist, 1979] [Bathe & Chaudhary, 1985] [Wriggers et al., 1990]

Advantages:

, simple;

, large deformations and slip;

, mesh independent.

Scheme of two non-matching meshes

Advantages:

, simple;

, mesh independent.

Drawbacks:

/ does not pass1 Taylor's test.

1[G. Zavarise, L. De Lorenzis, 2009]A modied NTS algorithm passing thecontact patch test

Advantages:

, simple;

, mesh independent.

Drawbacks:

Detection of contact elements

Advantages:

, simple;

, mesh independent.

Drawbacks:

NTS contact elements

Advantages:

, simple;

, mesh independent.

Drawbacks:

Remark: for a stable large deformationimplementation NTS should besupplemented with Node-to-Vertex andNode-to-Edge.

NTS contact elements

Segment-to-segment

Segment-to-segment discretization[Simo et al., 1985], [Zavarise & Wriggers, 1998]

Advantages:

, avoids some spurious modes ofNTS;

, use of higher order shapefunctions;

, mesh independent.

Segment-to-segment

Advantages:

, mesh independent.

Drawbacks:

/ complicated segment denition;

/ only 2D version;

/ constant contact pressurewithin one segment.

Segment-to-segment

Advantages:

, mesh independent.

Drawbacks:

/ only 2D version;

Projections of both surfaces

Segment-to-segment

Advantages:

, mesh independent.

Drawbacks:

/ only 2D version;

Contact sub-elements

Segment-to-segment

Advantages:

, mesh independent.

Drawbacks:

/ only 2D version;

Integration line

Mortar and Nitsche methods

Mortar and Nitsche discretizations[Bernadi et al., 1994][Wohlmuth, 2000][Puso&Laursen, 2003][Becker&Hansbo, 1999]

Advantages:

, passes Taylor's test;

, correct contact stressdistribution within contactelement;

, use of any order shapefunctions;

, mesh independent.

Mortar and Nitsche methods

Mortar and Nitsche discretizations[Bernadi et al., 1994][Wohlmuth, 2000][Puso&Laursen, 2003][Becker&Hansbo, 1999]

Advantages:

, correct contact stressdistribution within contactelement;

, use of any order shapefunctions;

, mesh independent.

Drawbacks:

/ very complicatedimplementation1;

/ stability problem for curvedsurfaces.

13D implementation is a nightmare, but

it's feasible.

T.A. Laursen about mortar method,ECCM, 2010

CDM Contact Domain Method

Contact domain method for discretization[Oliver, Hartmann et al. 2009]

Advantages:

, continuous formulation ofcontact elements;

, large deformations and slip.

Advantages:

Drawbacks:

/ requirements on the FE mesh in3D1;

/ not elaborated.

1Triangluation problems for arbitrarycontacting surfaces in 3D.

Advantages:

Drawbacks:

/ not elaborated.

Projections of both surfaces

Advantages:

Drawbacks:

/ not elaborated.

Contact domain construction

Smoothing technique

Smothing of the master surface withHermite polynoms;

P-slines;

Bézier curves;

Consequences

fulls requirements of C 1-smoothnessall along the master surface;

nonphysical edge eects;

complicated in 3D requires specialFE discretizations.

Smoothing technique

P-slines;

Bézier curves;

Consequences

Smoothing technique

P-slines;

Bézier curves;

Consequences

Smoothing technique

P-slines;

Bézier curves;

Consequences

Examples of NTS contactelements smoothed with

Bézier curves

Scheme of two non-matchingmeshes

Smoothing technique

P-slines;

Bézier curves;

Consequences

Bézier curves

Smoothing of the mastersurface

Smoothing technique

P-slines;

Bézier curves;

Consequences

Bézier curves

Contact detection

Smoothing technique

P-slines;

Bézier curves;

Consequences

Bézier curves

Contact element construction(edge contact element)

Smoothing technique

P-slines;

Bézier curves;

Consequences

Bézier curves

Constructed smoothed contactelements

Smoothing technique

P-slines;

Bézier curves;

Consequences

Bézier curves

Ordinary (top) and smothed(bottom) NTS contact element

Smoothing technique

P-slines;

Bézier curves;

Consequences

Bézier curves

Ordinary (top) and smothed(bottom) NTS contact element

1 Introduction

2 Contact detection

3 Contact geometry

IntroductionBoundary value problem with constraints

Continous formulation of boundary value problem

Partial dierential equation

∇ · σ + fv = 0 in Ω1,2

Neumann and Dirichlet boundary conditions

σ · n = f0 at Γ1,2N , u = u0 at Γ1,2D

Contact constraints: non-penetration and non-adhesion at Γc Signorini's conditions

gnσn = 0, gn ≥ 0, σ ≤ 0, σn = σ · n

Contact constraints: Coulomb's friction at Γc

|σt |, σt = σ · t

∇ · σ + fv = 0 in Ω1,2

σ · n = f0 at Γ1,2N , u = u0 at Γ1,2D

gnσn = 0, gn ≥ 0, σ ≤ 0, σn = σ · n

|σt |, σt = σ · t

∇ · σ + fv = 0 in Ω1,2

σ · n = f0 at Γ1,2N , u = u0 at Γ1,2D

gnσn = 0, gn ≥ 0, σ ≤ 0, σn = σ · n

|σt |, σt = σ · t

∇ · σ + fv = 0 in Ω1,2

σ · n = f0 at Γ1,2N , u = u0 at Γ1,2D

gnσn = 0, gn ≥ 0, σ ≤ 0, σn = σ · n

|σt |, σt = σ · t

∇ · σ + fv = 0 in Ω1,2

σ · n = f0 at Γ1,2N , u = u0 at Γ1,2D

gnσn = 0, gn ≥ 0, σ ≤ 0, σn = σ · n

|σt |, σt = σ · t

IntroductionWeak form with contact terms

Continous formulation of the weak form for contact problems

Weak formZΩ1,2

σ · ·δεdΩ−Z

Ω1,2fv · δudΩ−

ZΓ1,2N

f0 · δudΓ = 0

Contact term in the weak form, contact pressure σcn = σ · n at ΓcZ

Ω1,2σ · ·δεdΩ−

ZΩ1,2

fv · δudΩ−Z

Γ1,2N

f0 · δudΓ−Z

σcnδgndΓ = 0

Variational inequality (σcnδgn ≥ 0)∫

σ · ·δεdΩ ≥∫

fv · δudΩ +

∫Γ1,2N

f0 · δudΓ

Variational equality∫Ω1,2

σ · ·δεdΩ−∫

fv · δudΩ−∫

Γ1,2N

f0 · δudΓ + C = 0

Weak formZΩ1,2

σ · ·δεdΩ−Z

ZΓ1,2N

f0 · δudΓ = 0

ZΩ1,2

fv · δudΩ−Z

Γ1,2N

f0 · δudΓ−Z

σcnδgndΓ = 0

fv · δudΩ +

∫Γ1,2N

f0 · δudΓ

fv · δudΩ−∫

Γ1,2N

f0 · δudΓ + C = 0

Weak formZΩ1,2

σ · ·δεdΩ−Z

ZΓ1,2N

f0 · δudΓ = 0

ZΩ1,2

fv · δudΩ−Z

Γ1,2N

f0 · δudΓ−Z

σcnδgndΓ = 0

fv · δudΩ +

∫Γ1,2N

f0 · δudΓ

fv · δudΩ−∫

Γ1,2N

f0 · δudΓ + C = 0

Weak formZΩ1,2

σ · ·δεdΩ−Z

ZΓ1,2N

f0 · δudΓ = 0

ZΩ1,2

fv · δudΩ−Z

Γ1,2N

f0 · δudΓ−Z

σcnδgndΓ = 0

fv · δudΩ +

∫Γ1,2N

f0 · δudΓ

fv · δudΩ−∫

Γ1,2N

f0 · δudΓ + C = 0

Methods for contact resolution

Variational inequality[Duvaut & Lions, 1976], [Kikuchi &Oden, 1988]

Variational equality1

optimization methods[Kikuchi & Oden, 1988], [Bertsekas,1984], [Luenberger, 1984], [Curnier& Alart, 1991], [Wriggers, 2006]mathematical programming methods[Conry & Siereg, 1971], [Klarbring,1986]

1Often used with so-called active set strategy, which determines which contactelements are active (in contact) and which are not.

Optimization methods

Function to minimize f (x) and constraint gi (x) ≥ 0, i = 1,N

Penalty method

Lagrange multipliers method

Augmented Lagrangian method

Penalty method

fp(x) = f (x) + r 〈g(x)〉2

∇fp(x) = ∇f (x) + 2r∇〈g(x)〉∇g(x) = 0

xr−→∞−−−−→ x∗

Penalty method

L(x,λ) = f (x) + λigi (x)

ming(x)≥0

f (x) −→ x∗ = x ←− minL(x,λ)

∇x,λL =

[∇xf (x) + λi∇xgi (x)

gi (x)

]= 0, λi ≤ 0

Penalty method

La(x,λ) = f (x) + λigi (x) + r 〈gi (x)〉2

ming(x)≥0

f (x) −→ x∗ = x ←− minLa(x,λ)

∇x,λLa =

[∇xf (x) + λi∇xgi (x) + 2r∇〈gi (x)〉∇gi (x)

gi (x)

Optimization methodsDemonstration

Function : f (x) = x2 + 2x + 1

Constrain : g(x) = x ≥ 0Solution : x∗ = 0

Optimization methodsDemonstration

Function : f (x) = x2 + 2x + 1

Constrain : g(x) = x ≥ 0Solution : x∗ = 0

Optimization methodsDemonstration :: penalty method

f (x) = x2 + 2x + 1, g(x) = x ≥ 0, x∗ = 0

Penalty method

fp(x) = f (x) + r 〈−g(x)〉2

f (x) = x2 + 2x + 1, g(x) = x ≥ 0, x∗ = 0

Penalty method

fp(x) = f (x) + r 〈−g(x)〉2

f (x) = x2 + 2x + 1, g(x) = x ≥ 0, x∗ = 0

Penalty method

fp(x) = f (x) + r 〈−g(x)〉2

f (x) = x2 + 2x + 1, g(x) = x ≥ 0, x∗ = 0

Penalty method

fp(x) = f (x) + r 〈−g(x)〉2

r = 10

f (x) = x2 + 2x + 1, g(x) = x ≥ 0, x∗ = 0

Penalty method

fp(x) = f (x) + r 〈−g(x)〉2

r = 50

f (x) = x2 + 2x + 1, g(x) = x ≥ 0, x∗ = 0

Penalty method

fp(x) = f (x) + r 〈−g(x)〉2

Advantages ,simple physical interpretation;

no additional degrees offreedom;

smooth functional.

Drawbacks /solution is not exact:

too small penalty →large penetration;too large penalty →ill-conditioning of the globalmatrix;

user has to choose penalty r

properly.

Optimization methodsDemonstration :: Lagrange multipliers method

f (x) = x2 + 2x + 1, g(x) = x ≥ 0, x∗ = 0

L(x , λ) = f (x) + λg(x) → Saddle point → minxmaxλL(x , λ)

f (x) = x2 + 2x + 1, g(x) = x ≥ 0, x∗ = 0

Additional unknown λ

f (x) = x2 + 2x + 1, g(x) = x ≥ 0, x∗ = 0

Additional unknown λ Lagrangian

f (x) = x2 + 2x + 1, g(x) = x ≥ 0, x∗ = 0

Additional unknown λ Lagrangian

f (x) = x2 + 2x + 1, g(x) = x ≥ 0, x∗ = 0

Lagrangian Lagrangian (isolines)

f (x) = x2 + 2x + 1, g(x) = x ≥ 0, x∗ = 0

Advantages ,exact solution.

Drawbacks /Lagrangian is not smooth;

additional degrees of freedomincrease the problem.

Optimization methodsDemonstration :: Augmented Lagrangian method

f (x) = x2 + 2x + 1, g(x) = x ≥ 0, x∗ = 0

L(x , λ) = f (x) + r 〈−g(x)〉2 + λg(x) → minxmaxλL(x , λ)

f (x) = x2 + 2x + 1, g(x) = x ≥ 0, x∗ = 0

Additional unknown λ

f (x) = x2 + 2x + 1, g(x) = x ≥ 0, x∗ = 0

Standard Lagrangian r = 0 Isolines of the standard Lagrangianr = 0

f (x) = x2 + 2x + 1, g(x) = x ≥ 0, x∗ = 0

Augmented Lagrangian r = 1 Isolines of the augmentedLagrangian r = 1

f (x) = x2 + 2x + 1, g(x) = x ≥ 0, x∗ = 0

Advantages ,exact solution;

smoothed functional.

Drawbacks /additional degrees of freedomincrease the problem.

Optimization methodsAugmented Lagrangian method + Uzawa algorithm

Augmented Lagrangian

L(x , λ) = f (x) + r 〈−g(x)〉2 + λg(x)

Necessary conditions of the solution[∇xL(x , λ)∇λL(x , λ)

[∇x f (x)

[−2r 〈−g(x)〉∇xg(x)

[λ∇g(x)g(x)

Uzawa algorithm

λi+1 = λi − 2r 〈−g(x)〉

L(x , λ) = f (x) + r 〈−g(x)〉2 + λg(x)

[∇x f (x)

[−2r 〈−g(x)〉∇xg(x) + λ∇xg(x)

Uzawa algorithm

λi+1 = λi − 2r 〈−g(x)〉

L(x , λ) = f (x) + r 〈−g(x)〉2 + λg(x)

[∇x f (x)

[−2r 〈−g(x)〉∇xg(x) + λ∇xg(x)

Uzawa algorithm

λi+1 = λi − 2r 〈−g(x)〉

L(x , λ) = f (x) + r 〈−g(x)〉2 + λg(x)

[∇x f (x)

[(−2r 〈−g(x)〉+ λi )∇xg(x)

]Uzawa algorithm

λi+1 = λi − 2r 〈−g(x)〉

L(x , λ) = f (x) + r 〈−g(x)〉2 + λg(x)

[∇x f (x)

[λi+1∇xg(x)

]Uzawa algorithm

λi+1 = λi − 2r 〈−g(x)〉

L(x , λ) = f (x) + r 〈−g(x)〉2 + λg(x)

Uzawa algorithm

λi+1 = λi − 2r 〈−g(x)〉

Advantages ,

exact solution;

smoothed functional;

no additional degrees offreedom.

Drawbacks /

1 Introduction

2 Contact detection

3 Contact geometry

Introduction

FEA requiresgood nite element mesh

represents the real geometry;ne enough to represent correctly stress-strain eld;rough enough to solve the problem in reasonable terms.

comprehension how close we are to the real solution;careful apposition of boundary conditions.

Introduction

Example of contact problem solved inANSYS (no FE mesh presented)

Example of contact problem solved in ABAQUS (noFE mesh presented)

Convergence by meshBasics

One dimensional example on mesh renement

Smart meshing with linear elements

Smart meshing with quadratic elements

Case of singularity

Case of hidden maximum

Convergence by mesh

No convergence by mesh (singularity)

Case of two maximums

Boundary conditionsHow fast can we go?

General thinks

Contact problems are always nonlinear

Nonlinear problems requires slow change of boundary conditions

innite looping;convergence to the wrong solution.

General thinks

Resolution of nonlinear problem

Departure point R(x0, f0) = 0

General thinks

Change of boundary conditions R(x0, f1) 6= 0

General thinks

Newton-Raphson iterationsR(x0, f1) + ∂R

∣∣x0

δx = 0→ δx = − ∂R∂x

∣∣x0R(x0, f1)→ x1 = x0 + δx

General thinks

∣∣x1

δx = 0→ δx = − ∂R∂x

∣∣x1

R(x1, f1)→ x2 = x1 + δx

General thinks

Convergence ‖x i+1 − x i‖ ≤ ε→ x1 = x i+1

General thinks

Innite looping

General thinks

Innite looping

Too fast change of boundary conditions R(x0, f1) 6= 0

General thinks

Innite looping

∣∣x0

δx = 0→ δx = − ∂R∂x

∣∣x0R(x0, f1)→ x1 = x0 + δx

General thinks

Innite looping

∣∣x1

δx = 0→ δx = − ∂R∂x

∣∣x1

R(x1, f1)→ x2 = x1 + δx

General thinks

Innite looping

General thinks

Convergence to the wrong solution

General thinks

Too fast change of boundary conditions R(x0, f1) 6= 0

General thinks

∣∣x0

δx = 0→ δx = − ∂R∂x

∣∣x0R(x0, f1)→ x1 = x0 + δx

General thinks

∣∣x1

δx = 0→ δx = − ∂R∂x

∣∣x1

R(x1, f1)→ x2 = x1 + δx

General thinks

Solution. Correct?

Patch test

Methods passing Taylor's patch testmortar;

Nitsche;

node-to-node;

Contact domain method.

Possible schemes for contact patchtest

Patch test

Nitsche;

node-to-node;

Method not passing Taylor's patchtest

node-to-segment.

NTS not passing patch test -oscilation of contact pressure (top)Nitsche method passing patch test

Patch test

Nitsche;

node-to-node;

Method not passing Taylor's patchtest

node-to-segment.

NTS passes the patch test intwo pass;

NTS passing patch test [G.Zavarise, L. De Lorenzis, 2009];

Revisiting Taylor's patch test[Criseld].

NTS not passing patch test -oscilation of contact pressure (top)Nitsche method passing patch test

Master-slave discretizationGeneral rules

Rule 1

Contacting surface with higher mesh density is always slave surface.

Rule 1

If the mesh densities are equal on two surfaces, master surface is thesurface which deforms less.

Initial FE mesh Incorrect master-slavechoice /

Correct master-slavechoice ,

1 Introduction

2 Contact detection

3 Contact geometry

Validation3D contact

Increments

ValidationShallow ironing

Finite element mesh Description

Plane strain

Ei = 68.96 · 108 Pa

Es = 68.96 · 107 Pa

νi = νs = 0.32

µ = 0.3

∆uv = 1mm/10 incr

∆uh = 10mm/500incr

NN = 3840

Results <Stress12>

Comparison

K.A.Fischer, P. Wriggers [2006]

Comparison

J. Oliver, S. Hartmann [2009]

Comparison

Our results [2009]

Comparison

Our results [2009]

ValidationKlang's problem

Finite element meshDescription

Plane stress

E = 2.1 · 1011 Pa

ν = 0.3

µ = 0.4

r = 5.999 cm

R = 6 cm

F = 18750 N

α = 120

NN = 2500

Finite element mesh

Results <Stress22>

Results

Semianalytical, K.M.Klang [1979]. Simulation, P.Alart and A.Curnier [1990]

Results

ZEBULON, 1 increment

Results

PerformanceDisk-blade contact

Disk-blade frictional contact, elasto-plastic material

PerformanceMulti contact

Multi plate frictionless contact

Computational Contact Mechanics Yastrebov

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