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Introduction to

Computational 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

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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

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Intro Detection Geometry Discretization Solution FEA Examples

Outline

Mathematical foundation

contact geometry;

optimization methods.

Inside the Finite Element programme

contact detection;

contact discretization;

account of contact.

Contact problem resolution with FEM: guide for engineer

master-slave approach;

boundary conditions;

spurious cases.

Content

demonstrative;

simple;

general.

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 3/77

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Intro Detection Geometry Discretization Solution FEA Examples

Plan

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

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 4/77

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Intro Detection Geometry Discretization Solution FEA Examples

Finite Element Method

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

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

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 5/77

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Intro Detection Geometry Discretization Solution FEA Examples

Finite Element Method

FEM a powerfull and multi-purpose tool for linear and non-linear dynamic and static continuum mechanical and multi-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).

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 5/77

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Intro Detection Geometry Discretization Solution FEA Examples

Finite Element Method

Finite element analysis of contact problems

assembled components; Disk-blade contact

T.Dick, G.Cailletaud Centre des Matériaux

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 5/77

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Intro Detection Geometry Discretization Solution FEA Examples

Finite Element Method

Finite element analysis of contact problems

assembled components;

bearings; Rolling bearing

F.Massi et al.

LaMCoS, INSA-Lyon et al.

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 5/77

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Intro Detection Geometry Discretization Solution FEA Examples

Finite Element Method

Finite element analysis of contact problems

assembled components;

bearings;

rolling contact;

Tire rolling noise simulation

M.Brinkmeiera, U.Nackenhorst et al.

University of Hannover et al.

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 5/77

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Intro Detection Geometry Discretization Solution FEA Examples

Finite Element Method

Finite element analysis of contact problems

assembled components;

bearings;

rolling contact;

forming processes;

Deep drawing

G.Rousselier et al.

Centre des Matériaux et al.

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 5/77

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Intro Detection Geometry Discretization Solution FEA Examples

Finite Element Method

Finite element analysis of contact problems

assembled components;

bearings;

rolling contact;

forming processes;

geomechanical contact;

Post seismic relaxation

J.D.Garaud, L.Fleitout, G.Cailletaud Centre des Matériaux, ENS

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 5/77

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Intro Detection Geometry Discretization Solution FEA Examples

Finite Element Method

Finite element analysis of contact problems

assembled components;

bearings;

rolling contact;

forming processes;

geomechanical contact;

crash tests;

Crash-test

O.Klyavin, A.Michailov, A.Borovkov St Petersbourg State University

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 5/77

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Intro Detection Geometry Discretization Solution FEA Examples

Finite Element Method

Finite element analysis of contact problems

assembled components;

bearings;

rolling contact;

forming processes;

geomechanical contact;

crash tests;

human joints;

Knee simulation

E.Peña, B.Calvoa et al.

University of Zaragoza

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 5/77

(14)

Intro Detection Geometry Discretization Solution FEA Examples

Finite Element Method

Finite element analysis of contact problems

assembled components;

bearings;

rolling contact;

forming processes;

geomechanical contact;

crash tests;

human joints;

and many others.

Knee simulation

E.Peña, B.Calvoa et al.

University of Zaragoza

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 5/77

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Intro Detection Geometry Discretization Solution FEA Examples

They trust in the FEA

Leading international industrial companies

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 6/77

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Intro Detection Geometry Discretization Solution FEA Examples

Mechanical problem

From 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.

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 7/77

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Intro Detection Geometry Discretization Solution FEA Examples

Mechanical problem

From 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.

For example: Brinell hardness test

Scheme of the Brinell hardness test and dierent types of impression

[Harry Chandler, Hardness testing]

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 7/77

(18)

Intro Detection Geometry Discretization Solution FEA Examples

Mechanical problem

From a real life problem to an engineering problem

Need to determine:

problematic:

strength.

geometry;

sphere + half-space.

loads:

mechanical surface quasi-static.

material:

rigid + elasto-visco-plastic.

scale:

macro.

For example: Brinell hardness test

Scheme of the Brinell hardness test and dierent types of impression

[Harry Chandler, Hardness testing]

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 7/77

(19)

Intro Detection Geometry Discretization Solution FEA Examples

Mechanical problem

From an engineering problem to a nite element model

Problem:

problematic:

strength/life-time/fracture;

vibration/buckling;

thermo-electro-mechanical.

FE model:

analysis type:

stress-strain state;

eigen values;

coupled physics.

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 8/77

(20)

Intro Detection Geometry Discretization Solution FEA Examples

Mechanical problem

From an engineering problem to a nite element model

Problem:

problematic:

strength/life-time/fracture;

vibration/buckling;

thermo-electro-mechanical.

geometry;

FE model:

analysis type:

stress-strain state;

eigen values;

coupled physics.

nite element mesh;

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 8/77

(21)

Intro Detection Geometry Discretization Solution FEA Examples

Mechanical problem

From an engineering problem to a nite element model

Problem:

problematic:

strength/life-time/fracture;

vibration/buckling;

thermo-electro-mechanical.

geometry;

loads:

static/quasi-static/dynamic;

mechanical/thermic;

volume/surface.

FE model:

analysis type:

stress-strain state;

eigen values;

coupled physics.

nite element mesh;

analysis type and BC:

static/quasi-static/dynamic;

mechanical/thermic;

volume/surface.

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 8/77

(22)

Intro Detection Geometry Discretization Solution FEA Examples

Mechanical problem

From an engineering problem to a nite element model

Problem:

problematic:

strength/life-time/fracture;

vibration/buckling;

thermo-electro-mechanical.

geometry;

loads:

static/quasi-static/dynamic;

mechanical/thermic;

volume/surface.

material:

rigid/elastic/plastic/visco- plastic;

brittle/ductile.

FE model:

analysis type:

stress-strain state;

eigen values;

coupled physics.

nite element mesh;

analysis type and BC:

static/quasi-static/dynamic;

mechanical/thermic;

volume/surface.

material model:

rigid/elastic/plastic/visco- plastic;

brittle/ductile.

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 8/77

(23)

Intro Detection Geometry Discretization Solution FEA Examples

Mechanical problem

From an engineering problem to a nite element model

Problem:

problematic:

strength/life-time/fracture;

vibration/buckling;

thermo-electro-mechanical.

geometry;

loads:

static/quasi-static/dynamic;

mechanical/thermic;

volume/surface.

material:

rigid/elastic/plastic/visco- plastic;

brittle/ductile.

scale:

macro/meso/micro.

FE model:

analysis type:

stress-strain state;

eigen values;

coupled physics.

nite element mesh;

analysis type and BC:

static/quasi-static/dynamic;

mechanical/thermic;

volume/surface.

material model:

rigid/elastic/plastic/visco- plastic;

brittle/ductile.

microstructure:

RVE/microstructure/.

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 8/77

(24)

Intro Detection Geometry Discretization Solution FEA Examples

Mechanical problem

From engineering problem to a nite element model

Problem:

problematic:

strength.

geometry:

loads:

material:

scale:

FE model:

analysis type:

stress-strain state.

nite element mesh:

analysis type and BC:

material model:

microstructure:

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 9/77

(25)

Intro Detection Geometry Discretization Solution FEA Examples

Mechanical problem

From engineering problem to a nite element model

Problem:

problematic:

strength.

geometry:

sphere + half-space.

loads:

material:

scale:

FE model:

analysis type:

stress-strain state.

nite element mesh:

sphere + large block.

analysis type and BC:

material model:

microstructure:

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 9/77

(26)

Intro Detection Geometry Discretization Solution FEA Examples

Mechanical problem

From engineering problem to a nite element model

Problem:

problematic:

strength.

geometry:

sphere + half-space.

loads:

mechanical surface quasi-static.

material:

scale:

FE model:

analysis type:

stress-strain state.

nite element mesh:

sphere + large block.

analysis type and BC:

mechanical surface quasi-static.

material model:

microstructure:

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 9/77

(27)

Intro Detection Geometry Discretization Solution FEA Examples

Mechanical problem

From engineering problem to a nite element model

Problem:

problematic:

strength.

geometry:

sphere + half-space.

loads:

mechanical surface quasi-static.

material:

rigid + elasto-visco-plastic.

scale:

FE model:

analysis type:

stress-strain state.

nite element mesh:

sphere + large block.

analysis type and BC:

mechanical surface quasi-static.

material model:

rigid

a

+ elasto-visco-plastic model.

microstructure:

arigid in FEA: much more harder than another solid, special boundary conditions or geometrical

representation.

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 9/77

(28)

Intro Detection Geometry Discretization Solution FEA Examples

Mechanical problem

From engineering problem to a nite element model

Problem:

problematic:

strength.

geometry:

sphere + half-space.

loads:

mechanical surface quasi-static.

material:

rigid + elasto-visco-plastic.

scale:

macro.

FE model:

analysis type:

stress-strain state.

nite element mesh:

sphere + large block.

analysis type and BC:

mechanical surface quasi-static.

material model:

rigid

a

+ elasto-visco-plastic model.

microstructure:

homogeneous > RVE.

arigid in FEA: much more harder than another solid, special boundary conditions or geometrical

representation.

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 9/77

(29)

Intro Detection Geometry Discretization Solution FEA Examples

Mechanical problem

From engineering problem to a nite element model

Problem:

problematic geometry loads material scale

FE model:

analysis type nite element mesh analysis type and BC material model microstructure

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 9/77

(30)

Intro Detection Geometry Discretization Solution FEA Examples

Mechanical problem

From engineering problem to a nite element model

Problem:

problematic geometry loads material scale

FE model:

analysis type nite element mesh analysis type and BC material model microstructure

Finite element mesh : )

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 9/77

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Intro Detection Geometry Discretization Solution FEA Examples

Symmetry and plane problems

How to solve problems faster

Main ideas:

symmetry

geometry AND loading;

3D to 2D:

axisymmetry/plane strain/plane stress;

3D to smaller 3D:

half/quarter/sector;

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 10/77

(32)

Intro Detection Geometry Discretization Solution FEA Examples

Symmetry and plane problems

How to solve problems faster

Main ideas:

symmetry

geometry AND loading;

3D to 2D:

axisymmetry/plane strain/plane stress;

3D to smaller 3D:

half/quarter/sector;

Axisymmetry

Axisymmetry of geometry axisymmetry of load

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 10/77

(33)

Intro Detection Geometry Discretization Solution FEA Examples

Symmetry and plane problems

How to solve problems faster

Main ideas:

symmetry

geometry AND loading;

3D to 2D:

axisymmetry/plane strain/plane stress;

3D to smaller 3D:

half/quarter/sector;

Mirror symmetry

Axisymmetry of geometry mirror symmetry of load

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 10/77

(34)

Intro Detection Geometry Discretization Solution FEA Examples

Symmetry and plane problems

How to solve problems faster

Main ideas:

symmetry

geometry AND loading;

3D to 2D:

axisymmetry/plane strain/plane stress;

3D to smaller 3D:

half/quarter/sector;

No symmetry

Axisymmetry of geometry no symmetry of load

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 10/77

(35)

Intro Detection Geometry Discretization Solution FEA Examples

Symmetry and plane problems

How to solve problems faster

Main ideas:

symmetry

geometry AND loading;

3D to 2D:

axisymmetry/plane strain/plane stress;

3D to smaller 3D:

half/quarter/sector;

Plain strain

Mirror symmetry of geometry mirror symmetry of load very long structure or xed edges

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 10/77

(36)

Intro Detection Geometry Discretization Solution FEA Examples

Symmetry and plane problems

How to solve problems faster

Main ideas:

symmetry

geometry AND loading;

3D to 2D:

axisymmetry/plane strain/plane stress;

3D to smaller 3D:

half/quarter/sector;

Plain stress

Mirror symmetry of geometry mirror symmetry of load

very thin structure

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 10/77

(37)

Intro Detection Geometry Discretization Solution FEA Examples

Symmetry and plane problems

How to solve problems faster

Full 3D mesh Axisymmetric 2D mesh

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 10/77

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Intro Detection Geometry Discretization Solution FEA Examples

Finite element mesh

Basics

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.

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 11/77

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Intro Detection Geometry Discretization Solution FEA Examples

Boundary conditions

Two solids Ω 1 and Ω 2 .

Volumetric forces F v : e.g. inertia m ¨ u.

Neumann (static, force) boundary conditions: distributed and concentrated loads.

Dirichlet (kinematic, displacement) boundary conditions: displacements.

In each solid we full div (σ) + F v = 0

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 12/77

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Intro Detection Geometry Discretization Solution FEA Examples

Boundary conditions

Two solids Ω 1 and Ω 2 .

Volumetric forces F v : e.g. inertia m ¨ u.

Neumann (static, force) boundary conditions: distributed and concentrated loads.

Dirichlet (kinematic, displacement) boundary conditions: displacements.

In each solid we full div (σ) + F v = 0

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 12/77

(41)

Intro Detection Geometry Discretization Solution FEA Examples

Boundary conditions

Two solids Ω 1 and Ω 2 .

Volumetric forces F v : e.g. inertia m ¨ u.

Neumann (static, force) boundary conditions: distributed and concentrated loads.

Dirichlet (kinematic, displacement) boundary conditions: displacements.

In each solid we full div (σ) + F v = 0

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 12/77

(42)

Intro Detection Geometry Discretization Solution FEA Examples

Boundary conditions

Two solids Ω 1 and Ω 2 .

Volumetric forces F v : e.g. inertia m ¨ u.

Neumann (static, force) boundary conditions: distributed and concentrated loads.

Dirichlet (kinematic, displacement) boundary conditions: displacements.

In each solid we full div (σ) + F v = 0

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 12/77

(43)

Intro Detection Geometry Discretization Solution FEA Examples

Boundary conditions

Two solids Ω 1 and Ω 2 .

Volumetric forces F v : e.g. inertia m ¨ u.

Neumann (static, force) boundary conditions: distributed and concentrated loads.

Dirichlet (kinematic, displacement) boundary conditions: displacements.

In each solid we full div (σ) + F v = 0 What are the contact boundary conditions?

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 12/77

(44)

Intro Detection Geometry Discretization Solution FEA Examples

Boundary conditions

Two solids Ω 1 and Ω 2 .

Volumetric forces F v : e.g. inertia m ¨ u.

Neumann (static, force) boundary conditions: distributed and concentrated loads.

Dirichlet (kinematic, displacement) boundary conditions: displacements.

In each solid we full div (σ) + F v = 0 What are the contact boundary conditions?

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 12/77

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Intro Detection Geometry Discretization Solution FEA Examples

Account of contact

Signorini conditions

Signorini conditions of nonpenetration g n > 0 and non-adhesion σ n ≤ 0

g n σ n = 0 , g n ≥ 0 , σ ≤ 0 , σ n = σ · n

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

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 13/77

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Intro Detection Geometry Discretization Solution FEA Examples

Account of contact

Signorini conditions

Signorini conditions of nonpenetration g n > 0 and non-adhesion σ n ≤ 0

g n σ n = 0 , g n ≥ 0 , σ ≤ 0 , σ n = σ · n

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

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 13/77

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Intro Detection Geometry Discretization Solution FEA Examples

Account of contact

Coulomb's friction

Coulomb's friction conditions

| g ˙ t | (|σ t | + µσ n ) = 0 ; |σ t | ≤ −µσ n ; g ˙ t = | g ˙ t | σ t

t | , σ t = σ · t Contact boundary conditions ∼ unknown Neumann (force) boundary conditions depending on the geometry.

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 14/77

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Intro Detection Geometry Discretization Solution FEA Examples

Account of contact

Coulomb's friction

Coulomb's friction conditions

| g ˙ t | (|σ t | + µσ n ) = 0 ; |σ t | ≤ −µσ n ; g ˙ t = | g ˙ t | σ t

t | , σ t = σ · t Contact boundary conditions ∼ unknown Neumann (force) boundary conditions depending on the geometry.

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 14/77

(49)

Intro Detection Geometry Discretization Solution FEA Examples

Account of contact

Coulomb's friction

Coulomb's friction conditions

| g ˙ t | (|σ t | + µσ n ) = 0 ; |σ t | ≤ −µσ n ; g ˙ t = | g ˙ t | σ t

t | , σ t = σ · t Contact boundary conditions ∼ unknown Neumann (force) boundary conditions depending on the geometry.

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 14/77

(50)

Intro Detection Geometry Discretization Solution FEA Examples

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?

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 15/77

(51)

Intro Detection Geometry Discretization Solution FEA Examples

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?

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 15/77

(52)

Intro Detection Geometry Discretization Solution FEA Examples

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?

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 15/77

(53)

Intro Detection Geometry Discretization Solution FEA Examples

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?

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 15/77

(54)

Intro Detection Geometry Discretization Solution FEA Examples

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?

Zoom on penetration

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 15/77

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Intro Detection Geometry Discretization Solution FEA Examples

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?

Volume intersection

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 15/77

(56)

Intro Detection Geometry Discretization Solution FEA Examples

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?

Surface-in-volume 1

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 15/77

(57)

Intro Detection Geometry Discretization Solution FEA Examples

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?

Surface-in-volume 2

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 15/77

(58)

Intro Detection Geometry Discretization Solution FEA Examples

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?

Nodes-in-volume 1

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 15/77

(59)

Intro Detection Geometry Discretization Solution FEA Examples

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?

Nodes-in-volume 2

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 15/77

(60)

Intro Detection Geometry Discretization Solution FEA Examples

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?

Nodes-to-surface 1

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 15/77

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Intro Detection Geometry Discretization Solution FEA Examples

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?

Nodes-to-surface 2

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 15/77

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Intro Detection Geometry Discretization Solution FEA Examples

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

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 16/77

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Intro Detection Geometry Discretization Solution FEA Examples

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

Large deformation/large sliding

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 16/77

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Intro Detection Geometry Discretization Solution FEA Examples

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-segment discretization large deformation/large sliding

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 16/77

(65)

Intro Detection Geometry Discretization Solution FEA Examples

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

Contact domain method:

large deformation/large sliding

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 16/77

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Intro Detection Geometry Discretization Solution FEA Examples

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

Conception of the contact element

Node-to-node discretization:

small deformation/small sliding

Contact domain method:

large deformation/large sliding

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 16/77

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Intro Detection Geometry Discretization Solution FEA Examples

Plan

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

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Intro Detection Geometry Discretization Solution FEA Examples

Spatial search and local detection

Spatial search Detection of contacting solids:

multibody systems Discrete Element Method Molecular dynamics

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

Local contact detection Detection of contacting nodes and

surfaces of two discretized solids Finite Element Method Smoothed Particle Hydrodynamics

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

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 18/77

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Intro Detection Geometry Discretization Solution FEA Examples

Spatial search and local detection

Spatial search Detection of contacting solids:

multibody systems Discrete Element Method Molecular dynamics

3rd body layer modeling

[V.-D. Nguyen, J. Fortin et al., 2009]

Local contact detection Detection of contacting nodes and

surfaces of two discretized solids Finite Element Method Smoothed Particle Hydrodynamics

Buckling with self-contact

[T. Belytschko, W.K. Liu, B. Moran, 2000]

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Intro Detection Geometry Discretization Solution FEA Examples

Local contact detection

Basic ideas

How to detect contact?

What penetrates and where?

What/where approach

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 19/77

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Intro Detection Geometry Discretization Solution FEA Examples

Local contact detection

Basic ideas

How to detect contact?

What penetrates and where?

What/where approach, for example, What? Slave nodes

Where? Under master surface

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 19/77

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Intro Detection Geometry Discretization Solution FEA Examples

Local contact detection

Basic ideas

How to detect contact?

What penetrates and where?

What/where approach, for example, What? Slave nodes

Where? Under master surface Assymetry of contacting surfaces

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 19/77

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Intro Detection Geometry Discretization Solution FEA Examples

Local contact detection

Basic ideas

How to detect contact?

What penetrates and where?

What/where approach, for example, What? Slave nodes

Where? Under master surface Assymetry of contacting surfaces New paradigm

BUT! We need to detect contact before any penetration occures!

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 19/77

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Intro Detection Geometry Discretization Solution FEA Examples

Local contact detection

Basic ideas

How to detect contact?

What penetrates and where?

What/where approach, for example, What? Slave nodes

Where? Under master surface Assymetry of contacting surfaces New paradigm

BUT! We need to detect contact before any penetration occures!

Contact detection idea

In general the detection consists in checking if slave nodes are close enough

to the master surface

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 19/77

(75)

Intro Detection Geometry Discretization Solution FEA Examples

Local contact detection

Basic ideas

How to detect contact?

What penetrates and where?

What/where approach, for example, What? Slave nodes

Where? Under master surface Assymetry of contacting surfaces New paradigm

BUT! We need to detect contact before any penetration occures!

Contact detection idea

In general the detection consists in checking if slave nodes are close enough a

to the master surface

a

What does it mean close enough?

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 19/77

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Intro Detection Geometry Discretization Solution FEA Examples

Maximal detection distance

Maximal detection distance concept dmax

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

Solid with slave nodes and solid with a master surface.

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 20/77

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Intro Detection Geometry Discretization Solution FEA Examples

Maximal detection distance

Maximal detection distance concept dmax

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

External detection zone dist < dmax and g n ≥ 0

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 20/77

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Intro Detection Geometry Discretization Solution FEA Examples

Maximal detection distance

Maximal detection distance concept dmax

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

Internal detection zone dist < dmax and g n < 0

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 20/77

(79)

Intro Detection Geometry Discretization Solution FEA Examples

Maximal detection distance

Maximal detection distance concept dmax

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

Contact detection zone dist < dmax

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 20/77

(80)

Intro Detection Geometry Discretization Solution FEA Examples

Maximal detection distance

Maximal detection distance concept dmax

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

Verciation if slave nodes are in the detection zone

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 20/77

(81)

Intro Detection Geometry Discretization Solution FEA Examples

Maximal detection distance

Maximal detection distance concept dmax

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

No slave nodes in the detection zone

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 20/77

(82)

Intro Detection Geometry Discretization Solution FEA Examples

Maximal detection distance

Maximal detection distance concept dmax

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

No slave nodes in the detection zone

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 20/77

(83)

Intro Detection Geometry Discretization Solution FEA Examples

Maximal detection distance

Maximal detection distance concept dmax

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

Solid with slave nodes and solid with a master surface.

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 20/77

(84)

Intro Detection Geometry Discretization Solution FEA Examples

Maximal detection distance

Maximal detection distance concept dmax

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

Contact detection zone dist < dmax

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 20/77

(85)

Intro Detection Geometry Discretization Solution FEA Examples

Maximal detection distance

Maximal detection distance concept dmax

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

Verciation if slave nodes are in the detection zone

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 20/77

(86)

Intro Detection Geometry Discretization Solution FEA Examples

Maximal detection distance

Maximal detection distance concept dmax

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

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

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 20/77

(87)

Intro Detection Geometry Discretization Solution FEA Examples

Maximal detection distance

Maximal detection distance concept dmax

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

Create contact elements with detected slave nodes.

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 20/77

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Intro Detection Geometry Discretization Solution FEA Examples

Maximal detection distance

Maximal detection distance concept dmax

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

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 of contacting surface nodes during one iteration/increment) use both criterions.

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 20/77

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Intro Detection Geometry Discretization Solution FEA Examples

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 by master surfaces.

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

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Intro Detection Geometry Discretization Solution FEA Examples

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 by master surfaces.

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

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Intro Detection Geometry Discretization Solution FEA Examples

Discretized master surface

Master surface contsist of many elementary master surfaces.

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

Contact detection zone.

What does it consist of?

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

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Intro Detection Geometry Discretization Solution FEA Examples

Discretized master surface

Master surface contsist of many elementary master surfaces.

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

Zoom on the master contact surface and the associated detection zone.

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

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Intro Detection Geometry Discretization Solution FEA Examples

Discretized master surface

Master surface contsist of many elementary master surfaces.

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

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

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

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Intro Detection Geometry Discretization Solution FEA Examples

Discretized master surface

Master surface contsist of many elementary master surfaces.

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

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

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

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Intro Detection Geometry Discretization Solution FEA Examples

Discretized master surface

Master surface contsist of many elementary master surfaces.

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

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

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

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Intro Detection Geometry Discretization Solution FEA Examples

Discretized master surface

Master surface contsist of many elementary master surfaces.

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

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

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

(97)

Intro Detection Geometry Discretization Solution FEA Examples

Discretized master surface

Master surface contsist of many elementary master surfaces.

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

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

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

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Intro Detection Geometry Discretization Solution FEA Examples

Closest points denition

How to dene the closest point ρ onto the master surface Γ m for a given slave node r s ?

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 22/77

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Intro Detection Geometry Discretization Solution FEA Examples

Closest points denition

How to dene the closest point ρ onto the master surface Γ m for a given slave node r s ?

Denition of the closest point ρ

ρ ∈ Γ m : ∀ρ ∈ Γ m , | r s − ρ | ≤ | r s − ρ|

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 22/77

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Intro Detection Geometry Discretization Solution FEA Examples

Closest points denition

How to dene the closest point ρ onto the master surface Γ m for a given slave node r s ?

Denition of the closest point ρ

ρ ∈ Γ m : ∀ρ ∈ Γ m , | r s − ρ | ≤ | r s − ρ|

Functional for parametrized surface ρ = ρ(ξ):

F ( r s , ξ) = 1

2 ( r s − ρ(ξ)) 2

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 22/77

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Intro Detection Geometry Discretization Solution FEA Examples

Closest points denition

How to dene the closest point ρ onto the master surface Γ m for a given slave node r s ?

Denition of the closest point ρ

ρ ∈ Γ m : ∀ρ ∈ Γ m , | r s − ρ | ≤ | r s − ρ|

Functional for parametrized surface ρ = ρ(ξ):

F ( r s , ξ) = 1

2 ( r s − ρ(ξ)) 2 In general case, nonlinear minimization problem:

min ξ∈[ 0 ; 1 ] F ( r s , ξ) ⇒ ξ : ∀ξ ∈ [ 0 ; 1 ], | r s − ρ(ξ )| ≤ | r s − ρ(ξ)|

min ξ∈[ 0 ; 1 ] F ( r s , ξ) ⇔ ( r s − ρ(ξ )) · ∂ρ

∂ξ ξ

= 0

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 22/77

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Intro Detection Geometry Discretization Solution FEA Examples

Closest points denition

How to dene the closest point ρ onto the master surface Γ m for a given slave node r s ?

min ξ∈[ 0 ; 1 ] F ( r s , ξ) ⇔ ( r s − ρ(ξ )) · ∂ρ

∂ξ ξ

= 0

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 23/77

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Intro Detection Geometry Discretization Solution FEA Examples

Existance/uniqueness of the closest point

Does the projection always exist?

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

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Intro Detection Geometry Discretization Solution FEA Examples

Existance/uniqueness of the closest point

Does the projection always exist? No!

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

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Intro Detection Geometry Discretization Solution FEA Examples

Existance/uniqueness of the closest point

Does the projection always exist? No!

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

1

m

) (r

s

− ρ(ξ

)) · ∂ρ

∂ξ

˛

˛

˛

˛

ξ

= 0

In the FEM the surface is not obligatory smooth, only continuous ρ(ξ) ∈ C

0

m

)

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

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

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Intro Detection Geometry Discretization Solution FEA Examples

Existance/uniqueness of the closest point

Does the projection always exist? No!

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

1

m

) (r

s

− ρ(ξ

)) · ∂ρ

∂ξ

˛

˛

˛

˛

ξ

= 0

In the FEM the surface is not obligatory smooth, only continuous ρ(ξ) ∈ C

0

m

)

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

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

(107)

Intro Detection Geometry Discretization Solution FEA Examples

Existance/uniqueness of the closest point

Does the projection always exist? No!

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

1

m

) (r

s

− ρ(ξ

)) · ∂ρ

∂ξ

˛

˛

˛

˛

ξ

= 0

In the FEM the surface is not obligatory smooth, only continuous ρ(ξ) ∈ C

0

m

)

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

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

(108)

Intro Detection Geometry Discretization Solution FEA Examples

Existance/uniqueness of the closest point

Does the projection always exist? No!

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

1

m

) (r

s

− ρ(ξ

)) · ∂ρ

∂ξ

˛

˛

˛

˛

ξ

= 0

In the FEM the surface is not obligatory smooth, only continuous ρ(ξ) ∈ C

0

m

)

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

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

(109)

Intro Detection Geometry Discretization Solution FEA Examples

Existance/uniqueness of the closest point

Does the projection always exist? No!

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

1

m

) ( r

s

− ρ(ξ

)) · ∂ρ

∂ξ

˛

˛

˛

˛

ξ

= 0

In the FEM the surface is not obligatory smooth, only continuous ρ(ξ) ∈ C

0

m

)

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

For more details see Contact geometry section

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

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Intro Detection Geometry Discretization Solution FEA Examples

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

s

s

s symmetry

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Intro Detection Geometry Discretization Solution FEA Examples

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-slave choice /

Correct master-slave choice ,

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 26/77

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Intro Detection Geometry Discretization Solution FEA Examples

Contact detection in global algorithm

At the beginning of each increment (loading step) + fast;

+ good convergence;

+ stable;

lack of accuracy.

At the beginning of each iteration (convergence step) slow;

innite looping;

not stable;

+ more accurate.

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Intro Detection Geometry Discretization Solution FEA Examples

Contact detection in global algorithm

At the beginning of each increment (loading step) + fast;

+ good convergence;

+ stable;

lack of accuracy.

At the beginning of each iteration (convergence step) slow;

innite looping;

not stable;

+ more accurate.

V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 27/77

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