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
Intro Detection Geometry Discretization Solution FEA Examples
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.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 3/77
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
Intro Detection Geometry Discretization Solution FEA 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.]
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 5/77
Intro Detection Geometry Discretization Solution FEA 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.]
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
Intro Detection Geometry Discretization Solution FEA Examples
Finite Element Method
Finite element analysis of contact problems
assembled components; Disk-blade contact
T.Dick, G.CailletaudCentre des Matériaux
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 5/77
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
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
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
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.CailletaudCentre des Matériaux, ENS
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 5/77
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.BorovkovSt Petersbourg State University
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 5/77
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
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
Intro Detection Geometry Discretization Solution FEA Examples
They trust in the FEALeading international industrial companies
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 6/77
Intro Detection Geometry Discretization Solution FEA Examples
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.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 7/77
Intro Detection Geometry Discretization Solution FEA Examples
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.
For example: Brinell hardnesstest
Scheme of the Brinell hardness test anddierent types of impression[Harry Chandler, Hardness testing]
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 7/77
Intro Detection Geometry Discretization Solution FEA Examples
Mechanical problemFrom 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 hardnesstest
Scheme of the Brinell hardness test anddierent types of impression[Harry Chandler, Hardness testing]
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 7/77
Intro Detection Geometry Discretization Solution FEA Examples
Mechanical problemFrom 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
Intro Detection Geometry Discretization Solution FEA Examples
Mechanical problemFrom 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
Intro Detection Geometry Discretization Solution FEA Examples
Mechanical problemFrom 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
Intro Detection Geometry Discretization Solution FEA Examples
Mechanical problemFrom 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
Intro Detection Geometry Discretization Solution FEA Examples
Mechanical problemFrom 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
Intro Detection Geometry Discretization Solution FEA Examples
Mechanical problemFrom 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
Intro Detection Geometry Discretization Solution FEA Examples
Mechanical problemFrom 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
Intro Detection Geometry Discretization Solution FEA Examples
Mechanical problemFrom engineering problem to a nite element model
Problem:problematic:
strength.
geometry:
sphere + half-space.
loads:
mechanical surfacequasi-static.
material:
scale:
FE model:analysis type:
stress-strain state.
nite element mesh:
sphere + large block.
analysis type and BC:
mechanical surfacequasi-static.
material model:
microstructure:
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 9/77
Intro Detection Geometry Discretization Solution FEA Examples
Mechanical problemFrom engineering problem to a nite element model
Problem:problematic:
strength.
geometry:
sphere + half-space.
loads:
mechanical surfacequasi-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 surfacequasi-static.
material model:
rigida + elasto-visco-plasticmodel.
microstructure:
arigid in FEA: much more harder than another solid,
special boundary conditions or geometricalrepresentation.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 9/77
Intro Detection Geometry Discretization Solution FEA Examples
Mechanical problemFrom engineering problem to a nite element model
Problem:problematic:
strength.
geometry:
sphere + half-space.
loads:
mechanical surfacequasi-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 surfacequasi-static.
material model:
rigida + elasto-visco-plasticmodel.
microstructure:
homogeneous > RVE.
arigid in FEA: much more harder than another solid,
special boundary conditions or geometricalrepresentation.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 9/77
Intro Detection Geometry Discretization Solution FEA Examples
Mechanical problemFrom 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
Intro Detection Geometry Discretization Solution FEA Examples
Mechanical problemFrom 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
Intro Detection Geometry Discretization Solution FEA Examples
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;
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 10/77
Intro Detection Geometry Discretization Solution FEA Examples
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;
Axisymmetry
Axisymmetry of geometryaxisymmetry of load
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 10/77
Intro Detection Geometry Discretization Solution FEA Examples
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;
Mirror symmetry
Axisymmetry of geometrymirror symmetry of load
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 10/77
Intro Detection Geometry Discretization Solution FEA Examples
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;
No symmetry
Axisymmetry of geometryno symmetry of load
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 10/77
Intro Detection Geometry Discretization Solution FEA Examples
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;
Plain strain
Mirror symmetry of geometrymirror symmetry of load
very long structure or xed edges
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 10/77
Intro Detection Geometry Discretization Solution FEA Examples
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;
Plain stress
Mirror symmetry of geometrymirror symmetry of load
very thin structure
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 10/77
Intro Detection Geometry Discretization Solution FEA Examples
Symmetry and plane problemsHow 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
Intro Detection Geometry Discretization Solution FEA Examples
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.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 11/77
Intro Detection Geometry Discretization Solution FEA Examples
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
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 12/77
Intro Detection Geometry Discretization Solution FEA Examples
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
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 12/77
Intro Detection Geometry Discretization Solution FEA Examples
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
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 12/77
Intro Detection Geometry Discretization Solution FEA Examples
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
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 12/77
Intro Detection Geometry Discretization Solution FEA Examples
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
What are the contact boundaryconditions?
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 12/77
Intro Detection Geometry Discretization Solution FEA Examples
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
What are the contact boundaryconditions?
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 12/77
Intro Detection Geometry Discretization Solution FEA Examples
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.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 13/77
Intro Detection Geometry Discretization Solution FEA Examples
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.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 13/77
Intro Detection Geometry Discretization Solution FEA Examples
Account of contactCoulomb's friction
Coulomb's friction conditions
|gt | (|σt |+ µσn) = 0; |σt | ≤ −µσn; gt = |gt |σt
|σt |, σt = σ · t
Contact boundary conditions ∼ unknown Neumann (force) boundaryconditions depending on the geometry.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 14/77
Intro Detection Geometry Discretization Solution FEA Examples
Account of contactCoulomb's friction
Coulomb's friction conditions
|gt | (|σt |+ µσn) = 0; |σt | ≤ −µσn; gt = |gt |σt
|σt |, σt = σ · t
Contact boundary conditions ∼ unknown Neumann (force) boundaryconditions depending on the geometry.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 14/77
Intro Detection Geometry Discretization Solution FEA Examples
Account of contactCoulomb's friction
Coulomb's friction conditions
|gt | (|σt |+ µσn) = 0; |σt | ≤ −µσn; gt = |gt |σt
|σt |, σt = σ · t
Contact boundary conditions ∼ unknown Neumann (force) boundaryconditions depending on the geometry.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 14/77
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 discretizationlarge deformation/large sliding
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 16/77
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
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
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 17/77
Intro Detection Geometry Discretization Solution FEA Examples
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
Finite Element Method
Smoothed ParticleHydrodynamics
Toruscylinder impact problem[B. Yang, T.A. Laursen, 2006]
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 18/77
Intro Detection Geometry Discretization Solution FEA Examples
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
Finite Element Method
Smoothed ParticleHydrodynamics
Buckling with self-contact[T. Belytschko, W.K. Liu, B. Moran, 2000]
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 18/77
Intro Detection Geometry Discretization Solution FEA Examples
Local contact detectionBasic 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
Intro Detection Geometry Discretization Solution FEA Examples
Local contact detectionBasic ideas
How to detect contact?
What penetrates and where?
What/where approach, for example,
What? Slave nodesWhere? Under master surface
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 19/77
Intro Detection Geometry Discretization Solution FEA Examples
Local contact detectionBasic ideas
How to detect contact?
What penetrates and where?
What/where approach, for example,
What? Slave nodesWhere? Under master surface
Assymetry of contacting surfaces
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 19/77
Intro Detection Geometry Discretization Solution FEA Examples
Local contact detectionBasic ideas
How to detect contact?
What penetrates and where?
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!
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 19/77
Intro Detection Geometry Discretization Solution FEA Examples
Local contact detectionBasic ideas
How to detect contact?
What penetrates and where?
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!
Contact detection idea
In general the detection consists in checking ifslave nodes are close enough
to the master surface
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 19/77
Intro Detection Geometry Discretization Solution FEA Examples
Local contact detectionBasic ideas
How to detect contact?
What penetrates and where?
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!
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?
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 19/77
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 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.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 20/77
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 slavenode is considered as possibly contacting at current load step and acontact element is to be created.
External detection zone dist < dmax and gn ≥ 0
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 20/77
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 slavenode is considered as possibly contacting at current load step and acontact element is to be created.
Internal detection zone dist < dmax and gn < 0
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 20/77
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 slavenode is considered as possibly contacting at current load step and acontact 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
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 slavenode is considered as possibly contacting at current load step and acontact 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
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 slavenode is considered as possibly contacting at current load step and acontact 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
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 slavenode is considered as possibly contacting at current load step and acontact 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
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 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.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 20/77
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 slavenode is considered as possibly contacting at current load step and acontact 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
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 slavenode is considered as possibly contacting at current load step and acontact 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
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 slavenode is considered as possibly contacting at current load step and acontact 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
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 slavenode is considered as possibly contacting at current load step and acontact 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
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 slavenode is considered as possibly contacting at current load step and acontact 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 ofcontacting surface nodes during one iteration/increment)
use both criterions.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 20/77
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 bymaster surfaces.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 21/77
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 bymaster surfaces.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 21/77
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
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 andthe associated detection zone.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 21/77
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 iscloser to the master surface than dmax
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 21/77
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 surfaceonto which it projects, i.e. the closest master surface.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 21/77
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 surfaceonto which it projects, i.e. the closest master surface.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 21/77
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 surfaceonto which it projects, i.e. the closest master surface.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 21/77
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 surfaceonto which it projects, i.e. the closest master surface.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 21/77
Intro Detection Geometry Discretization Solution FEA Examples
Closest points denition
How to dene the closest point ρ∗ onto the master surface Γm for agiven slave node rs?
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 22/77
Intro Detection Geometry Discretization Solution FEA Examples
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|
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 22/77
Intro Detection Geometry Discretization Solution FEA Examples
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
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 22/77
Intro Detection Geometry Discretization Solution FEA Examples
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
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
∣∣∣∣ξ∗
= 0
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 22/77
Intro Detection Geometry Discretization Solution FEA Examples
Closest points denition
How to dene the closest point ρ∗ onto the master surface Γm for agiven slave node rs?
minξ∈[0;1]F (rs , ξ)⇔ (rs − ρ(ξ∗)) · ∂ρ
∂ξ
∣∣∣∣ξ∗
= 0
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 23/77
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
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
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(Γm)
(rs − ρ(ξ∗)) · ∂ρ
∂ξ
˛ξ∗
= 0
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.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 24/77
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(Γm)
(rs − ρ(ξ∗)) · ∂ρ
∂ξ
˛ξ∗
= 0
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.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 24/77
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(Γm)
(rs − ρ(ξ∗)) · ∂ρ
∂ξ
˛ξ∗
= 0
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.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 24/77
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(Γm)
(rs − ρ(ξ∗)) · ∂ρ
∂ξ
˛ξ∗
= 0
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.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 24/77
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(Γm)
(rs − ρ(ξ∗)) · ∂ρ
∂ξ
˛ξ∗
= 0
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.
For more details see Contact geometry section
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 24/77
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
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 25/77
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-slavechoice /
Correct master-slavechoice ,
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 26/77
Intro Detection Geometry Discretization Solution FEA Examples
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)
slow;
innite looping;
not stable;
+ more accurate.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 27/77
Intro Detection Geometry Discretization Solution FEA Examples
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)
slow;
innite looping;
not stable;
+ more accurate.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 27/77
Intro Detection Geometry Discretization Solution FEA Examples
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.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 28/77
Intro Detection Geometry Discretization Solution FEA Examples
Contact detection techniques
All-to-all detection Elaborated techniques
Example: two curved contacting surfaces 1 million of slave nodes against 1 million of master segments.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 28/77
Intro Detection Geometry Discretization Solution FEA Examples
Contact detection techniques
All-to-all detection Elaborated techniques
Example: two curved contacting surfaces 1 million of slave nodes against 1 million of master segments.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 28/77
Intro Detection Geometry Discretization Solution FEA Examples
Contact detection techniques
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(!)
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 28/77
Intro Detection Geometry Discretization Solution FEA Examples
Contact detection techniques
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(!)
Bounding box+grid detection
1 minute
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 28/77
Intro Detection Geometry Discretization Solution FEA Examples
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.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 29/77
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 30/77
Intro Detection Geometry Discretization Solution FEA Examples
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.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 31/77
Intro Detection Geometry Discretization Solution FEA Examples
Challenges in contact geometry
Challenges
requirement of smooth surface.
non-uniqueness of projection;
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 32/77
Intro Detection Geometry Discretization Solution FEA Examples
Challenges in contact geometry
Challenges
assymetry of contacting surfacesa;
requirement of smooth surface.
non-uniqueness of projection;
asomething penetrates into something, something slides over something
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 32/77
Intro Detection Geometry Discretization Solution FEA Examples
Challenges in contact geometry
Challenges
assymetry of contacting surfacesa;
requirement of smooth surface.
non-uniqueness of projection;
asomething penetrates into something, something slides over something
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 32/77
Intro Detection Geometry Discretization Solution FEA Examples
Challenges in contact geometry
Challenges
assymetry of contacting surfacesa;
requirement of smooth surface.
non-uniqueness of projection;
asomething penetrates into something, something slides over something
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 32/77
Intro Detection Geometry Discretization Solution FEA Examples
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
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 33/77
Intro Detection Geometry Discretization Solution FEA Examples
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
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 33/77
Intro Detection Geometry Discretization Solution FEA Examples
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
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 33/77
Intro Detection Geometry Discretization Solution FEA Examples
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
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 33/77
Intro Detection Geometry Discretization Solution FEA Examples
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
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 33/77
Intro Detection Geometry Discretization Solution FEA Examples
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
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 33/77
Intro Detection Geometry Discretization Solution FEA Examples
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.
Deformation twinnings in coating[Forest, 2000]
Escaping dislocations[Fivel, 2009]
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 33/77
Intro Detection Geometry Discretization Solution FEA Examples
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.
Deformation twinnings in coating[Forest, 2000]
Escaping dislocations[Fivel, 2009]
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 33/77
Intro Detection Geometry Discretization Solution FEA Examples
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.
2D surface smoothing with Beziercurves
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 33/77
Intro Detection Geometry Discretization Solution FEA Examples
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.
3D surface smoothing with Beziersurface
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 33/77
Intro Detection Geometry Discretization Solution FEA Examples
Account of geometrical quantitiesFirst order variations
Contact in the weak form
Geometrical quantities: normal gap gn = gn(x) and tangential slidinggt = gt(x
0, 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
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 34/77
Intro Detection Geometry Discretization Solution FEA Examples
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
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 35/77
Intro Detection Geometry Discretization Solution FEA Examples
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)
Geometrical quantities
Normal gap gn = (rs − ρ) · n
Tangential sliding gtdt = δρ(ξ)
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Intro Detection Geometry Discretization Solution FEA Examples
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)))
Geometrical quantities
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)))
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 36/77
Intro Detection Geometry Discretization Solution FEA Examples
Continuum geometrical formulationCovariant and contravariant bases, fundamental surface tensors
Covariant surface basis∂ρ∂ξ
=n
∂ρ∂ξ1
; ∂ρ∂ξ2
oContravariant surface basis
∂ρ∂ξ
=n
∂ρ∂ξ1
; ∂ρ∂ξ2
oBasis change
∂ρ∂ξ
= A−1 ∂ρ∂ξ
∂ρ
∂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
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 37/77
Intro Detection Geometry Discretization Solution FEA Examples
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 ·(
∂ρ
∂ξ· (δrs − δρ) + gnn · δ
∂ρ
∂ξ
)
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 38/77
Intro Detection Geometry Discretization Solution FEA Examples
Continuum geometrical formulationSecond order variations
Second order variation of the normal gap δgn
∆δgn =− n ·„
δ∂ρ
∂ξ·∆ξ + ∆
∂ρ
∂ξ· δξ + δξ ·
∂2ρ
∂ξ2·∆ξ
«+
+ gnδξ ·H · A−1 ·H ·∆ξ+
+ gn
„n · δ
∂ρ
∂ξ
«· A−1 ·
„∆
∂ρ
∂ξ· n
« (1)
Second order variation of the surface parameter δξ
∆δξ = (gnH− A)−1 ·»
∂ρ
∂ξ·
„δ∂ρ
∂ξ·∆ξ + ∆
∂ρ
∂ξ· δξ + δξ ·
∂2ρ
∂ξ2·∆ξ
«− gnn ·
„δ∂2ρ
∂ξ2·∆ξ + ∆
∂2ρ
∂ξ2· δξ + δξ ·
∂3ρ
∂ξ3·∆ξ
«+
∆
∂ρ
∂ξ· n + H ·∆ξ
ff·
„I n · (δρ− δrs)+ gnA
−1 ·∂ρ
∂ξ·
δ∂ρ
∂ξ+
∂2ρ
∂ξ2· δξ
ff«+
δ∂ρ
∂ξ· n + H · δξ
ff·
„I n · (∆ρ−∆rs)+ gnA
−1 ·∂ρ
∂ξ·
∆
∂ρ
∂ξ+
∂2ρ
∂ξ2·∆ξ
ff«–(2)
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Intro Detection Geometry Discretization Solution FEA Examples
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 + ∆
∂ρ
∂ξ· δξ + δξ · ∂2ρ
∂ξ2·∆ξ
)
∆δξ = −A−1 ·»
∂ρ
∂ξ·
„δ∂ρ
∂ξ·∆ξ + ∆
∂ρ
∂ξ· δξ + δξ ·
∂2ρ
∂ξ2·∆ξ
«+
+
∆
∂ρ
∂ξ· n + H ·∆ξ
ff· (I n · (δρ− δrs))+
+
δ∂ρ
∂ξ· n + H · δξ
ff· (I n · (∆ρ−∆rs))
– (3)
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 40/77
Intro Detection Geometry Discretization Solution FEA Examples
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 ]
T ;
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
]T
[X (t)] = [Φ′i ]T
[X ]
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 41/77
Intro Detection Geometry Discretization Solution FEA Examples
Discretized contact geometryFrom continuum formulation to the Finite Element Method II
First variations of geometrical quantities
δgn =
n−φ1n
...−φNn
T
·
δx0δx1...
δxN
= [∇gn]T · δ [X ] (4)
δξi = cij
∂ρ∂ξj
− ∂ρ∂ξj
φ1 + gnnφ1,j
...
− ∂ρ∂ξj
φN + gnnφN,j
T
·
δx0δx1...
δxN
= [∇ξi ]T · δ [X ] (5)
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 42/77
Intro Detection Geometry Discretization Solution FEA Examples
Discretized contact geometryFrom continuum formulation to the Finite Element Method II
Second order variation of the normal gap ∆δgn
∆δgn =δ [X ]T ·n−n
ˆΦ′i
˜⊗ [∇ξi ]
T − [∇ξi ]⊗ˆΦ′i
˜Tn
−“hij − gnhika
kmhmj
”[∇ξi ]⊗ [∇ξj ]
T
+gnaijn
ˆΦ′i
˜⊗
hΦ′j
iTn
ff·∆ [X ] =
= δ [X ]T · [∇∇gn] ·∆ [X ]
(6)
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 43/77
Intro Detection Geometry Discretization Solution FEA Examples
Discretized contact geometryFrom continuum formulation to the Finite Element Method II
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
iT
+
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 ] ⊗
„n
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 ]
”⊗
∂ρ
∂ξm
hΦ′jiT!
+ gnakm
0@ ∂ρ
∂ξm·
∂2ρ
∂ξj∂ξl
1A„[∇ξl ] ⊗„n
hΦ′k
iT+ hks [∇ξs ]T
«+
+“n
hΦ′k
i+ hks [∇ξs ]
”⊗ [∇ξl ]
T”io
· ∆ [X ] =
= δ [X ]T ·ˆ∇∇ξi
˜· ∆ [X ]
(7)
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 44/77
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
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.
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Intro Detection Geometry Discretization Solution FEA Examples
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.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 47/77
Intro Detection Geometry Discretization Solution FEA Examples
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.
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Intro Detection Geometry Discretization Solution FEA Examples
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
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Intro Detection Geometry Discretization Solution FEA Examples
NTN Node-to-nodeDenition of the normal
Denition of the normal for the node-to-nodediscretization
NTN contact element detection
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Intro Detection Geometry Discretization Solution FEA Examples
NTN Node-to-nodeDenition of the normal
Denition of the normal for the node-to-nodediscretization
Master-slave discretization
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 48/77
Intro Detection Geometry Discretization Solution FEA Examples
NTN Node-to-nodeDenition of the normal
Denition of the normal for the node-to-nodediscretization
Normals on the master surface
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 48/77
Intro Detection Geometry Discretization Solution FEA Examples
NTN Node-to-nodeDenition of the normal
Denition of the normal for the node-to-nodediscretization
Denition of the normals at master nodes as an average of thenormals of adjacent segments
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 48/77
Intro Detection Geometry Discretization Solution FEA Examples
NTN Node-to-nodeDenition of the normal
Denition of the normal for the node-to-nodediscretization
Denition of the normals at master nodes
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Intro Detection Geometry Discretization Solution FEA Examples
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
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Intro Detection Geometry Discretization Solution FEA Examples
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.
Drawbacks:
/ does not pass1 Taylor's test.
1[G. Zavarise, L. De Lorenzis, 2009]A modied NTS algorithm passing thecontact patch test
Scheme of two non-matching meshes
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 49/77
Intro Detection Geometry Discretization Solution FEA Examples
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.
Drawbacks:
/ does not pass1 Taylor's test.
1[G. Zavarise, L. De Lorenzis, 2009]A modied NTS algorithm passing thecontact patch test
Detection of contact elements
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 49/77
Intro Detection Geometry Discretization Solution FEA Examples
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.
Drawbacks:
/ does not pass1 Taylor's test.
1[G. Zavarise, L. De Lorenzis, 2009]A modied NTS algorithm passing thecontact patch test
NTS contact elements
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Intro Detection Geometry Discretization Solution FEA Examples
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.
Drawbacks:
/ does not pass1 Taylor's test.
Remark: for a stable large deformationimplementation NTS should besupplemented with Node-to-Vertex andNode-to-Edge.
NTS contact elements
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Intro Detection Geometry Discretization Solution FEA Examples
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;
, large deformations and slip;
, mesh independent.
Scheme of two non-matching meshes
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Intro Detection Geometry Discretization Solution FEA Examples
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;
, large deformations and slip;
, mesh independent.
Drawbacks:
/ complicated segment denition;
/ only 2D version;
/ constant contact pressurewithin one segment.
Scheme of two non-matching meshes
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Intro Detection Geometry Discretization Solution FEA Examples
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;
, large deformations and slip;
, mesh independent.
Drawbacks:
/ complicated segment denition;
/ only 2D version;
/ constant contact pressurewithin one segment.
Projections of both surfaces
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Intro Detection Geometry Discretization Solution FEA Examples
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;
, large deformations and slip;
, mesh independent.
Drawbacks:
/ complicated segment denition;
/ only 2D version;
/ constant contact pressurewithin one segment.
Contact sub-elements
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 50/77
Intro Detection Geometry Discretization Solution FEA Examples
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;
, large deformations and slip;
, mesh independent.
Drawbacks:
/ complicated segment denition;
/ only 2D version;
/ constant contact pressurewithin one segment.
Integration line
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Intro Detection Geometry Discretization Solution FEA Examples
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;
, large deformations and slip;
, mesh independent.
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Intro Detection Geometry Discretization Solution FEA Examples
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;
, large deformations and slip;
, 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
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Intro Detection Geometry Discretization Solution FEA Examples
CDM Contact Domain Method
Contact domain method for discretization[Oliver, Hartmann et al. 2009]
Advantages:
, passes Taylor's test;
, continuous formulation ofcontact elements;
, large deformations and slip.
Scheme of two non-matching meshes
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Intro Detection Geometry Discretization Solution FEA Examples
CDM Contact Domain Method
Contact domain method for discretization[Oliver, Hartmann et al. 2009]
Advantages:
, passes Taylor's test;
, continuous formulation ofcontact elements;
, large deformations and slip.
Drawbacks:
/ requirements on the FE mesh in3D1;
/ not elaborated.
1Triangluation problems for arbitrarycontacting surfaces in 3D.
Scheme of two non-matching meshes
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 52/77
Intro Detection Geometry Discretization Solution FEA Examples
CDM Contact Domain Method
Contact domain method for discretization[Oliver, Hartmann et al. 2009]
Advantages:
, passes Taylor's test;
, continuous formulation ofcontact elements;
, large deformations and slip.
Drawbacks:
/ requirements on the FE mesh in3D1;
/ not elaborated.
1Triangluation problems for arbitrarycontacting surfaces in 3D.
Projections of both surfaces
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 52/77
Intro Detection Geometry Discretization Solution FEA Examples
CDM Contact Domain Method
Contact domain method for discretization[Oliver, Hartmann et al. 2009]
Advantages:
, passes Taylor's test;
, continuous formulation ofcontact elements;
, large deformations and slip.
Drawbacks:
/ requirements on the FE mesh in3D1;
/ not elaborated.
1Triangluation problems for arbitrarycontacting surfaces in 3D.
Contact domain construction
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 52/77
Intro Detection Geometry Discretization Solution FEA Examples
Smoothing technique
Smothing of the master surface withHermite polynoms;
P-slines;
Bézier curves;
etc.
Consequences
fulls requirements of C 1-smoothnessall along the master surface;
nonphysical edge eects;
complicated in 3D requires specialFE discretizations.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 53/77
Intro Detection Geometry Discretization Solution FEA Examples
Smoothing technique
Smothing of the master surface withHermite polynoms;
P-slines;
Bézier curves;
etc.
Consequences
fulls requirements of C 1-smoothnessall along the master surface;
nonphysical edge eects;
complicated in 3D requires specialFE discretizations.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 53/77
Intro Detection Geometry Discretization Solution FEA Examples
Smoothing technique
Smothing of the master surface withHermite polynoms;
P-slines;
Bézier curves;
etc.
Consequences
fulls requirements of C 1-smoothnessall along the master surface;
nonphysical edge eects;
complicated in 3D requires specialFE discretizations.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 53/77
Intro Detection Geometry Discretization Solution FEA Examples
Smoothing technique
Smothing of the master surface withHermite polynoms;
P-slines;
Bézier curves;
etc.
Consequences
fulls requirements of C 1-smoothnessall along the master surface;
nonphysical edge eects;
complicated in 3D requires specialFE discretizations.
Examples of NTS contactelements smoothed with
Bézier curves
Scheme of two non-matchingmeshes
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 53/77
Intro Detection Geometry Discretization Solution FEA Examples
Smoothing technique
Smothing of the master surface withHermite polynoms;
P-slines;
Bézier curves;
etc.
Consequences
fulls requirements of C 1-smoothnessall along the master surface;
nonphysical edge eects;
complicated in 3D requires specialFE discretizations.
Examples of NTS contactelements smoothed with
Bézier curves
Smoothing of the mastersurface
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 53/77
Intro Detection Geometry Discretization Solution FEA Examples
Smoothing technique
Smothing of the master surface withHermite polynoms;
P-slines;
Bézier curves;
etc.
Consequences
fulls requirements of C 1-smoothnessall along the master surface;
nonphysical edge eects;
complicated in 3D requires specialFE discretizations.
Examples of NTS contactelements smoothed with
Bézier curves
Contact detection
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 53/77
Intro Detection Geometry Discretization Solution FEA Examples
Smoothing technique
Smothing of the master surface withHermite polynoms;
P-slines;
Bézier curves;
etc.
Consequences
fulls requirements of C 1-smoothnessall along the master surface;
nonphysical edge eects;
complicated in 3D requires specialFE discretizations.
Examples of NTS contactelements smoothed with
Bézier curves
Contact element construction(edge contact element)
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 53/77
Intro Detection Geometry Discretization Solution FEA Examples
Smoothing technique
Smothing of the master surface withHermite polynoms;
P-slines;
Bézier curves;
etc.
Consequences
fulls requirements of C 1-smoothnessall along the master surface;
nonphysical edge eects;
complicated in 3D requires specialFE discretizations.
Examples of NTS contactelements smoothed with
Bézier curves
Constructed smoothed contactelements
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 53/77
Intro Detection Geometry Discretization Solution FEA Examples
Smoothing technique
Smothing of the master surface withHermite polynoms;
P-slines;
Bézier curves;
etc.
Consequences
fulls requirements of C 1-smoothnessall along the master surface;
nonphysical edge eects;
complicated in 3D requires specialFE discretizations.
Examples of NTS contactelements smoothed with
Bézier curves
Ordinary (top) and smothed(bottom) NTS contact element
in 2D
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 53/77
Intro Detection Geometry Discretization Solution FEA Examples
Smoothing technique
Smothing of the master surface withHermite polynoms;
P-slines;
Bézier curves;
etc.
Consequences
fulls requirements of C 1-smoothnessall along the master surface;
nonphysical edge eects;
complicated in 3D requires specialFE discretizations.
Examples of NTS contactelements smoothed with
Bézier curves
Ordinary (top) and smothed(bottom) NTS contact element
in 3D
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 53/77
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 54/77
Intro Detection Geometry Discretization Solution FEA Examples
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
|gt | (|σt |+ µσn) = 0; |σt | ≤ −µσn; gt = |gt |σt
|σt |, σt = σ · t
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 55/77
Intro Detection Geometry Discretization Solution FEA Examples
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
|gt | (|σt |+ µσn) = 0; |σt | ≤ −µσn; gt = |gt |σt
|σt |, σt = σ · t
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 55/77
Intro Detection Geometry Discretization Solution FEA Examples
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
|gt | (|σt |+ µσn) = 0; |σt | ≤ −µσn; gt = |gt |σt
|σt |, σt = σ · t
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 55/77
Intro Detection Geometry Discretization Solution FEA Examples
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
|gt | (|σt |+ µσn) = 0; |σt | ≤ −µσn; gt = |gt |σt
|σt |, σt = σ · t
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 55/77
Intro Detection Geometry Discretization Solution FEA Examples
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
|gt | (|σt |+ µσn) = 0; |σt | ≤ −µσn; gt = |gt |σt
|σt |, σt = σ · t
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 55/77
Intro Detection Geometry Discretization Solution FEA Examples
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
Γc
σcnδgndΓ = 0
Variational inequality (σcnδgn ≥ 0)∫
Ω1,2
σ · ·δεdΩ ≥∫
Ω1,2
fv · δudΩ +
∫Γ1,2N
f0 · δudΓ
Variational equality∫Ω1,2
σ · ·δεdΩ−∫
Ω1,2
fv · δudΩ−∫
Γ1,2N
f0 · δudΓ + C = 0
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 56/77
Intro Detection Geometry Discretization Solution FEA Examples
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
Γc
σcnδgndΓ = 0
Variational inequality (σcnδgn ≥ 0)∫
Ω1,2
σ · ·δεdΩ ≥∫
Ω1,2
fv · δudΩ +
∫Γ1,2N
f0 · δudΓ
Variational equality∫Ω1,2
σ · ·δεdΩ−∫
Ω1,2
fv · δudΩ−∫
Γ1,2N
f0 · δudΓ + C = 0
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 56/77
Intro Detection Geometry Discretization Solution FEA Examples
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
Γc
σcnδgndΓ = 0
Variational inequality (σcnδgn ≥ 0)∫
Ω1,2
σ · ·δεdΩ ≥∫
Ω1,2
fv · δudΩ +
∫Γ1,2N
f0 · δudΓ
Variational equality∫Ω1,2
σ · ·δεdΩ−∫
Ω1,2
fv · δudΩ−∫
Γ1,2N
f0 · δudΓ + C = 0
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 56/77
Intro Detection Geometry Discretization Solution FEA Examples
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
Γc
σcnδgndΓ = 0
Variational inequality (σcnδgn ≥ 0)∫
Ω1,2
σ · ·δεdΩ ≥∫
Ω1,2
fv · δudΩ +
∫Γ1,2N
f0 · δudΓ
Variational equality∫Ω1,2
σ · ·δεdΩ−∫
Ω1,2
fv · δudΩ−∫
Γ1,2N
f0 · δudΓ + C = 0
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 56/77
Intro Detection Geometry Discretization Solution FEA Examples
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.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 57/77
Intro Detection Geometry Discretization Solution FEA Examples
Optimization methods
Function to minimize f (x) and constraint gi (x) ≥ 0, i = 1,N
Penalty method
Lagrange multipliers method
Augmented Lagrangian method
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 58/77
Intro Detection Geometry Discretization Solution FEA Examples
Optimization methods
Function to minimize f (x) and constraint gi (x) ≥ 0, i = 1,N
Penalty method
fp(x) = f (x) + r 〈g(x)〉2
∇fp(x) = ∇f (x) + 2r∇〈g(x)〉∇g(x) = 0
xr−→∞−−−−→ x∗
Lagrange multipliers method
Augmented Lagrangian method
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 58/77
Intro Detection Geometry Discretization Solution FEA Examples
Optimization methods
Function to minimize f (x) and constraint gi (x) ≥ 0, i = 1,N
Penalty method
Lagrange multipliers 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
Augmented Lagrangian method
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 58/77
Intro Detection Geometry Discretization Solution FEA Examples
Optimization methods
Function to minimize f (x) and constraint gi (x) ≥ 0, i = 1,N
Penalty method
Lagrange multipliers method
Augmented Lagrangian 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)
]= 0
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 58/77
Intro Detection Geometry Discretization Solution FEA Examples
Optimization methodsDemonstration
Function : f (x) = x2 + 2x + 1
Constrain : g(x) = x ≥ 0Solution : x∗ = 0
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 59/77
Intro Detection Geometry Discretization Solution FEA Examples
Optimization methodsDemonstration
Function : f (x) = x2 + 2x + 1
Constrain : g(x) = x ≥ 0Solution : x∗ = 0
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 59/77
Intro Detection Geometry Discretization Solution FEA Examples
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
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 60/77
Intro Detection Geometry Discretization Solution FEA Examples
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
r = 0
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 60/77
Intro Detection Geometry Discretization Solution FEA Examples
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
r = 1
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 60/77
Intro Detection Geometry Discretization Solution FEA Examples
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
r = 10
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 60/77
Intro Detection Geometry Discretization Solution FEA Examples
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
r = 50
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 60/77
Intro Detection Geometry Discretization Solution FEA Examples
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
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.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 60/77
Intro Detection Geometry Discretization Solution FEA Examples
Optimization methodsDemonstration :: Lagrange multipliers method
f (x) = x2 + 2x + 1, g(x) = x ≥ 0, x∗ = 0
Lagrange multipliers method
L(x , λ) = f (x) + λg(x) → Saddle point → minxmaxλL(x , λ)
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 61/77
Intro Detection Geometry Discretization Solution FEA Examples
Optimization methodsDemonstration :: Lagrange multipliers method
f (x) = x2 + 2x + 1, g(x) = x ≥ 0, x∗ = 0
Lagrange multipliers method
L(x , λ) = f (x) + λg(x) → Saddle point → minxmaxλL(x , λ)
Additional unknown λ
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 61/77
Intro Detection Geometry Discretization Solution FEA Examples
Optimization methodsDemonstration :: Lagrange multipliers method
f (x) = x2 + 2x + 1, g(x) = x ≥ 0, x∗ = 0
Lagrange multipliers method
L(x , λ) = f (x) + λg(x) → Saddle point → minxmaxλL(x , λ)
Additional unknown λ Lagrangian
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 61/77
Intro Detection Geometry Discretization Solution FEA Examples
Optimization methodsDemonstration :: Lagrange multipliers method
f (x) = x2 + 2x + 1, g(x) = x ≥ 0, x∗ = 0
Lagrange multipliers method
L(x , λ) = f (x) + λg(x) → Saddle point → minxmaxλL(x , λ)
Additional unknown λ Lagrangian
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 61/77
Intro Detection Geometry Discretization Solution FEA Examples
Optimization methodsDemonstration :: Lagrange multipliers method
f (x) = x2 + 2x + 1, g(x) = x ≥ 0, x∗ = 0
Lagrange multipliers method
L(x , λ) = f (x) + λg(x) → Saddle point → minxmaxλL(x , λ)
Lagrangian Lagrangian (isolines)
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 61/77
Intro Detection Geometry Discretization Solution FEA Examples
Optimization methodsDemonstration :: Lagrange multipliers method
f (x) = x2 + 2x + 1, g(x) = x ≥ 0, x∗ = 0
Lagrange multipliers method
L(x , λ) = f (x) + λg(x) → Saddle point → minxmaxλL(x , λ)
Advantages ,exact solution.
Drawbacks /Lagrangian is not smooth;
additional degrees of freedomincrease the problem.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 61/77
Intro Detection Geometry Discretization Solution FEA Examples
Optimization methodsDemonstration :: Augmented Lagrangian method
f (x) = x2 + 2x + 1, g(x) = x ≥ 0, x∗ = 0
Augmented Lagrangian method
L(x , λ) = f (x) + r 〈−g(x)〉2 + λg(x) → minxmaxλL(x , λ)
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 62/77
Intro Detection Geometry Discretization Solution FEA Examples
Optimization methodsDemonstration :: Augmented Lagrangian method
f (x) = x2 + 2x + 1, g(x) = x ≥ 0, x∗ = 0
Augmented Lagrangian method
L(x , λ) = f (x) + r 〈−g(x)〉2 + λg(x) → minxmaxλL(x , λ)
Additional unknown λ
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 62/77
Intro Detection Geometry Discretization Solution FEA Examples
Optimization methodsDemonstration :: Augmented Lagrangian method
f (x) = x2 + 2x + 1, g(x) = x ≥ 0, x∗ = 0
Augmented Lagrangian method
L(x , λ) = f (x) + r 〈−g(x)〉2 + λg(x) → minxmaxλL(x , λ)
Standard Lagrangian r = 0 Isolines of the standard Lagrangianr = 0
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 62/77
Intro Detection Geometry Discretization Solution FEA Examples
Optimization methodsDemonstration :: Augmented Lagrangian method
f (x) = x2 + 2x + 1, g(x) = x ≥ 0, x∗ = 0
Augmented Lagrangian method
L(x , λ) = f (x) + r 〈−g(x)〉2 + λg(x) → minxmaxλL(x , λ)
Augmented Lagrangian r = 1 Isolines of the augmentedLagrangian r = 1
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 62/77
Intro Detection Geometry Discretization Solution FEA Examples
Optimization methodsDemonstration :: Augmented Lagrangian method
f (x) = x2 + 2x + 1, g(x) = x ≥ 0, x∗ = 0
Augmented Lagrangian method
L(x , λ) = f (x) + r 〈−g(x)〉2 + λg(x) → minxmaxλL(x , λ)
Augmented Lagrangian r = 10 Isolines of the augmentedLagrangian r = 10
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 62/77
Intro Detection Geometry Discretization Solution FEA Examples
Optimization methodsDemonstration :: Augmented Lagrangian method
f (x) = x2 + 2x + 1, g(x) = x ≥ 0, x∗ = 0
Augmented Lagrangian method
L(x , λ) = f (x) + r 〈−g(x)〉2 + λg(x) → minxmaxλL(x , λ)
Augmented Lagrangian r = 50 Isolines of the augmentedLagrangian r = 50
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 62/77
Intro Detection Geometry Discretization Solution FEA Examples
Optimization methodsDemonstration :: Augmented Lagrangian method
f (x) = x2 + 2x + 1, g(x) = x ≥ 0, x∗ = 0
Augmented Lagrangian method
L(x , λ) = f (x) + r 〈−g(x)〉2 + λg(x) → minxmaxλL(x , λ)
Advantages ,exact solution;
smoothed functional.
Drawbacks /additional degrees of freedomincrease the problem.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 62/77
Intro Detection Geometry Discretization Solution FEA Examples
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)
]= 0
[∇x f (x)
0
]+
[−2r 〈−g(x)〉∇xg(x)
0
]+
[λ∇g(x)g(x)
]
Uzawa algorithm
λi+1 = λi − 2r 〈−g(x)〉
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 63/77
Intro Detection Geometry Discretization Solution FEA Examples
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)
]= 0
[∇x f (x)
0
]+
[−2r 〈−g(x)〉∇xg(x) + λ∇xg(x)
g(x)
]
Uzawa algorithm
λi+1 = λi − 2r 〈−g(x)〉
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 63/77
Intro Detection Geometry Discretization Solution FEA Examples
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)
]= 0
[∇x f (x)
0
]+
[−2r 〈−g(x)〉∇xg(x) + λ∇xg(x)
g(x)
]
Uzawa algorithm
λi+1 = λi − 2r 〈−g(x)〉
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 63/77
Intro Detection Geometry Discretization Solution FEA Examples
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)
]= 0
[∇x f (x)
0
]+
[(−2r 〈−g(x)〉+ λi )∇xg(x)
g(x)
]Uzawa algorithm
λi+1 = λi − 2r 〈−g(x)〉
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 63/77
Intro Detection Geometry Discretization Solution FEA Examples
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)
]= 0
[∇x f (x)
0
]+
[λi+1∇xg(x)
g(x)
]Uzawa algorithm
λi+1 = λi − 2r 〈−g(x)〉
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 63/77
Intro Detection Geometry Discretization Solution FEA Examples
Optimization methodsAugmented Lagrangian method + Uzawa algorithm
Augmented Lagrangian
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 /
,
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 63/77
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 64/77
Intro Detection Geometry Discretization Solution FEA Examples
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.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 65/77
Intro Detection Geometry Discretization Solution FEA Examples
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.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 65/77
Intro Detection Geometry Discretization Solution FEA Examples
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.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 65/77
Intro Detection Geometry Discretization Solution FEA Examples
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.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 65/77
Intro Detection Geometry Discretization Solution FEA Examples
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.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 65/77
Intro Detection Geometry Discretization Solution FEA Examples
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.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 65/77
Intro Detection Geometry Discretization Solution FEA Examples
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.
Example of contact problem solved inANSYS (no FE mesh presented)
Example of contact problem solved in ABAQUS (noFE mesh presented)
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 65/77
Intro Detection Geometry Discretization Solution FEA Examples
Convergence by meshBasics
One dimensional example on mesh renement
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 66/77
Intro Detection Geometry Discretization Solution FEA Examples
Convergence by meshBasics
One dimensional example on mesh renement
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 66/77
Intro Detection Geometry Discretization Solution FEA Examples
Convergence by meshBasics
One dimensional example on mesh renement
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 66/77
Intro Detection Geometry Discretization Solution FEA Examples
Convergence by meshBasics
One dimensional example on mesh renement
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 66/77
Intro Detection Geometry Discretization Solution FEA Examples
Convergence by meshBasics
One dimensional example on mesh renement
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 66/77
Intro Detection Geometry Discretization Solution FEA Examples
Convergence by meshBasics
One dimensional example on mesh renement
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 66/77
Intro Detection Geometry Discretization Solution FEA Examples
Convergence by meshBasics
One dimensional example on mesh renement
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 66/77
Intro Detection Geometry Discretization Solution FEA Examples
Convergence by meshBasics
One dimensional example on mesh renement
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 66/77
Intro Detection Geometry Discretization Solution FEA Examples
Convergence by meshBasics
One dimensional example on mesh renement
Smart meshing with linear elements
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 66/77
Intro Detection Geometry Discretization Solution FEA Examples
Convergence by meshBasics
One dimensional example on mesh renement
Smart meshing with quadratic elements
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 66/77
Intro Detection Geometry Discretization Solution FEA Examples
Convergence by meshBasics
One dimensional example on mesh renement
Case of singularity
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 66/77
Intro Detection Geometry Discretization Solution FEA Examples
Convergence by meshBasics
One dimensional example on mesh renement
Case of hidden maximum
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 66/77
Intro Detection Geometry Discretization Solution FEA Examples
Convergence by meshBasics
One dimensional example on mesh renement
Convergence by mesh
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 66/77
Intro Detection Geometry Discretization Solution FEA Examples
Convergence by meshBasics
One dimensional example on mesh renement
Convergence by mesh
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 66/77
Intro Detection Geometry Discretization Solution FEA Examples
Convergence by meshBasics
One dimensional example on mesh renement
No convergence by mesh (singularity)
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 66/77
Intro Detection Geometry Discretization Solution FEA Examples
Convergence by meshBasics
One dimensional example on mesh renement
Case of two maximums
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 66/77
Intro Detection Geometry Discretization Solution FEA Examples
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.
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 67/77
Intro Detection Geometry Discretization Solution FEA Examples
Boundary conditionsHow fast can we go?
General thinks
Contact problems are always nonlinear
Nonlinear problems requires slow change of boundary conditions
Resolution of nonlinear problem
Departure point R(x0, f0) = 0
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 67/77
Intro Detection Geometry Discretization Solution FEA Examples
Boundary conditionsHow fast can we go?
General thinks
Contact problems are always nonlinear
Nonlinear problems requires slow change of boundary conditions
Resolution of nonlinear problem
Change of boundary conditions R(x0, f1) 6= 0
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 67/77
Intro Detection Geometry Discretization Solution FEA Examples
Boundary conditionsHow fast can we go?
General thinks
Contact problems are always nonlinear
Nonlinear problems requires slow change of boundary conditions
Resolution of nonlinear problem
Newton-Raphson iterationsR(x0, f1) + ∂R
∂x
∣∣x0
δx = 0→ δx = − ∂R∂x
∣∣x0R(x0, f1)→ x1 = x0 + δx
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 67/77
Intro Detection Geometry Discretization Solution FEA Examples
Boundary conditionsHow fast can we go?
General thinks
Contact problems are always nonlinear
Nonlinear problems requires slow change of boundary conditions
Resolution of nonlinear problem
Newton-Raphson iterationsR(x1, f1) + ∂R
∂x
∣∣x1
δx = 0→ δx = − ∂R∂x
∣∣x1
R(x1, f1)→ x2 = x1 + δx
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 67/77
Intro Detection Geometry Discretization Solution FEA Examples
Boundary conditionsHow fast can we go?
General thinks
Contact problems are always nonlinear
Nonlinear problems requires slow change of boundary conditions
Resolution of nonlinear problem
Convergence ‖x i+1 − x i‖ ≤ ε→ x1 = x i+1
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 67/77
Intro Detection Geometry Discretization Solution FEA Examples
Boundary conditionsHow fast can we go?
General thinks
Contact problems are always nonlinear
Nonlinear problems requires slow change of boundary conditions
Innite looping
Departure point R(x0, f0) = 0
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 67/77
Intro Detection Geometry Discretization Solution FEA Examples
Boundary conditionsHow fast can we go?
General thinks
Contact problems are always nonlinear
Nonlinear problems requires slow change of boundary conditions
Innite looping
Too fast change of boundary conditions R(x0, f1) 6= 0
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 67/77
Intro Detection Geometry Discretization Solution FEA Examples
Boundary conditionsHow fast can we go?
General thinks
Contact problems are always nonlinear
Nonlinear problems requires slow change of boundary conditions
Innite looping
Newton-Raphson iterationsR(x0, f1) + ∂R
∂x
∣∣x0
δx = 0→ δx = − ∂R∂x
∣∣x0R(x0, f1)→ x1 = x0 + δx
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 67/77
Intro Detection Geometry Discretization Solution FEA Examples
Boundary conditionsHow fast can we go?
General thinks
Contact problems are always nonlinear
Nonlinear problems requires slow change of boundary conditions
Innite looping
Newton-Raphson iterationsR(x1, f1) + ∂R
∂x
∣∣x1
δx = 0→ δx = − ∂R∂x
∣∣x1
R(x1, f1)→ x2 = x1 + δx
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 67/77
Intro Detection Geometry Discretization Solution FEA Examples
Boundary conditionsHow fast can we go?
General thinks
Contact problems are always nonlinear
Nonlinear problems requires slow change of boundary conditions
Innite looping
Innite looping
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 67/77
Intro Detection Geometry Discretization Solution FEA Examples
Boundary conditionsHow fast can we go?
General thinks
Contact problems are always nonlinear
Nonlinear problems requires slow change of boundary conditions
Convergence to the wrong solution
Departure point R(x0, f0) = 0
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 67/77
Intro Detection Geometry Discretization Solution FEA Examples
Boundary conditionsHow fast can we go?
General thinks
Contact problems are always nonlinear
Nonlinear problems requires slow change of boundary conditions
Convergence to the wrong solution
Too fast change of boundary conditions R(x0, f1) 6= 0
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 67/77
Intro Detection Geometry Discretization Solution FEA Examples
Boundary conditionsHow fast can we go?
General thinks
Contact problems are always nonlinear
Nonlinear problems requires slow change of boundary conditions
Convergence to the wrong solution
Newton-Raphson iterationsR(x0, f1) + ∂R
∂x
∣∣x0
δx = 0→ δx = − ∂R∂x
∣∣x0R(x0, f1)→ x1 = x0 + δx
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 67/77
Intro Detection Geometry Discretization Solution FEA Examples
Boundary conditionsHow fast can we go?
General thinks
Contact problems are always nonlinear
Nonlinear problems requires slow change of boundary conditions
Convergence to the wrong solution
Newton-Raphson iterationsR(x1, f1) + ∂R
∂x
∣∣x1
δx = 0→ δx = − ∂R∂x
∣∣x1
R(x1, f1)→ x2 = x1 + δx
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 67/77
Intro Detection Geometry Discretization Solution FEA Examples
Boundary conditionsHow fast can we go?
General thinks
Contact problems are always nonlinear
Nonlinear problems requires slow change of boundary conditions
Convergence to the wrong solution
Solution. Correct?
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 67/77
Intro Detection Geometry Discretization Solution FEA Examples
Patch test
Methods passing Taylor's patch testmortar;
Nitsche;
node-to-node;
Contact domain method.
Possible schemes for contact patchtest
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 68/77
Intro Detection Geometry Discretization Solution FEA Examples
Patch test
Methods passing Taylor's patch testmortar;
Nitsche;
node-to-node;
Contact domain method.
Method not passing Taylor's patchtest
node-to-segment.
NTS not passing patch test -oscilation of contact pressure (top)Nitsche method passing patch test
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 68/77
Intro Detection Geometry Discretization Solution FEA Examples
Patch test
Methods passing Taylor's patch testmortar;
Nitsche;
node-to-node;
Contact domain method.
Method not passing Taylor's patchtest
node-to-segment.
But!
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
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 68/77
Intro Detection Geometry Discretization Solution FEA Examples
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 ,
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 69/77
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 70/77
Intro Detection Geometry Discretization Solution FEA Examples
Validation3D contact
Increments
x
y
z
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 71/77
Intro Detection Geometry Discretization Solution FEA Examples
Validation3D contact
Increments
x
y
z
x
y
z
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 71/77
Intro Detection Geometry Discretization Solution FEA Examples
Validation3D contact
Increments
x
y
z
x
y
z
x
y
z
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 71/77
Intro Detection Geometry Discretization Solution FEA Examples
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
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 72/77
Intro Detection Geometry Discretization Solution FEA Examples
ValidationShallow ironing
Results <Stress12>
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 72/77
Intro Detection Geometry Discretization Solution FEA Examples
ValidationShallow ironing
Results <Stress12>
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 72/77
Intro Detection Geometry Discretization Solution FEA Examples
ValidationShallow ironing
Results <Stress12>
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 72/77
Intro Detection Geometry Discretization Solution FEA Examples
ValidationShallow ironing
Results <Stress12>
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 72/77
Intro Detection Geometry Discretization Solution FEA Examples
ValidationShallow ironing
Results <Stress12>
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 72/77
Intro Detection Geometry Discretization Solution FEA Examples
ValidationShallow ironing
Results <Stress12>
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 72/77
Intro Detection Geometry Discretization Solution FEA Examples
ValidationShallow ironing
Comparison
K.A.Fischer, P. Wriggers [2006]
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 73/77
Intro Detection Geometry Discretization Solution FEA Examples
ValidationShallow ironing
Comparison
J. Oliver, S. Hartmann [2009]
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 73/77
Intro Detection Geometry Discretization Solution FEA Examples
ValidationShallow ironing
Comparison
Our results [2009]
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 73/77
Intro Detection Geometry Discretization Solution FEA Examples
ValidationShallow ironing
Comparison
Our results [2009]
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 73/77
Intro Detection Geometry Discretization Solution FEA Examples
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
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 74/77
Intro Detection Geometry Discretization Solution FEA Examples
ValidationKlang's problem
Finite element mesh
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 74/77
Intro Detection Geometry Discretization Solution FEA Examples
ValidationKlang's problem
Finite element mesh
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 74/77
Intro Detection Geometry Discretization Solution FEA Examples
ValidationKlang's problem
Results <Stress22>
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 75/77
Intro Detection Geometry Discretization Solution FEA Examples
ValidationKlang's problem
Results <Stress22>
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 75/77
Intro Detection Geometry Discretization Solution FEA Examples
ValidationKlang's problem
Results <Stress22>
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 75/77
Intro Detection Geometry Discretization Solution FEA Examples
ValidationKlang's problem
Results <Stress22>
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 75/77
Intro Detection Geometry Discretization Solution FEA Examples
ValidationKlang's problem
Results <Stress22>
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 75/77
Intro Detection Geometry Discretization Solution FEA Examples
ValidationKlang's problem
Results
Semianalytical, K.M.Klang [1979]. Simulation, P.Alart and A.Curnier [1990]
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 75/77
Intro Detection Geometry Discretization Solution FEA Examples
ValidationKlang's problem
Results
ZEBULON, 1 increment
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 75/77
Intro Detection Geometry Discretization Solution FEA Examples
ValidationKlang's problem
Results
ZEBULON, 5 increment
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 75/77
Intro Detection Geometry Discretization Solution FEA Examples
ValidationKlang's problem
Results
ZEBULON, 10 increment
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 75/77
Intro Detection Geometry Discretization Solution FEA Examples
ValidationKlang's problem
Results
ZEBULON, 25 increment
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 75/77
Intro Detection Geometry Discretization Solution FEA Examples
ValidationKlang's problem
Results
ZEBULON, 50 increment
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 75/77
Intro Detection Geometry Discretization Solution FEA Examples
ValidationKlang's problem
Results
ZEBULON, 100 increment
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 75/77
Intro Detection Geometry Discretization Solution FEA Examples
PerformanceDisk-blade contact
Disk-blade frictional contact, elasto-plastic material
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 76/77
Intro Detection Geometry Discretization Solution FEA Examples
PerformanceMulti contact
Multi plate frictionless contact
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 77/77
Intro Detection Geometry Discretization Solution FEA Examples
PerformanceMulti contact
Multi plate frictionless contact
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 77/77
Intro Detection Geometry Discretization Solution FEA Examples
PerformanceMulti contact
Multi plate frictionless contact
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 77/77
Intro Detection Geometry Discretization Solution FEA Examples
PerformanceMulti contact
Multi plate frictionless contact
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 77/77
Intro Detection Geometry Discretization Solution FEA Examples
PerformanceMulti contact
Multi plate frictionless contact
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 77/77
Intro Detection Geometry Discretization Solution FEA Examples
PerformanceMulti contact
Multi plate frictionless contact
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 77/77
Intro Detection Geometry Discretization Solution FEA Examples
PerformanceMulti contact
Multi plate frictionless contact
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 77/77
Intro Detection Geometry Discretization Solution FEA Examples
PerformanceMulti contact
Multi plate frictionless contact
V.A. Yastrebov | MINES ParisTech Computational Contact Mechanics Paris, 21-24 June 2010 77/77
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