Nonlinear Finite Element Methods and Computational Contact...

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Alexander Popp ECCOMAS Advanced Course 2018 - Computational Structural Dynamics - Prague, Czech Republic, June 4-8, 2018 Nonlinear Finite Element Methods and Computational Contact Mechanics Institute for Mathematics and Computer-Based Simulation (IMCS) University of the Bundeswehr Munich (UniBw M), Germany

Transcript of Nonlinear Finite Element Methods and Computational Contact...

Page 1: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Alexander Popp

ECCOMAS Advanced Course 2018- Computational Structural Dynamics - Prague, Czech Republic, June 4-8, 2018

Nonlinear Finite Element Methods and Computational

Contact Mechanics

Institute for Mathematics and Computer-Based Simulation (IMCS)University of the Bundeswehr Munich (UniBw M), Germany

Page 2: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Unit 22 - CONTACT PROBLEMS 1 (Friday, 10:20-11:20)

• history and main challenges • very simple FEM example • contact kinematics • contact constraint enforcement • weak formulations for FEM

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Page 3: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Agenda

ɠ MOTIVATION

ɡ MOTIVATION

ɢ METHODS 1

ɣ METHODS 2

ɤ METHODS 3

Main Challenges, History and Examples

A Simple FEM Contact Example

Contact Kinematics

Contact Constraint Enforcement

Weak Formulations with Contact for FEM

!3

Page 4: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Wear

Multiphysics

Thermomechanics

Complex Materials

Lubrication

FE Discretization

Contact Algorithms Linear Solvers Parallel Computing

PHYS

ICS

NU

MER

ICS

Contact-related challenges

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Page 5: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Contact Mechanics – A (very) brief history…

1882

Pioneering work by H. Hertz on the “Contact of elastic bodies“, until today the basis for analytical solutions

Before 1970s

Analytical solution techniques for more complex (linear elastic) contacts, e.g. Timoshenko and Goodier (1970),e.g. Johnson (1985)

Since 1970s

First papers on finite element methods applied to contact problems, e.g. Francavilla and Zienkiewicz (1975),e.g. Hughes et al. (1976)

Since 1980s

Abundant amount of literature on linear and complex nonlinear contact scenarios within the FEM,e.g. Wriggers et al. (1990), e.g. Laursen and Simo (1992)

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Page 6: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Contact Mechanics – A (very) brief history…

Since 1980sNode-to-segment (NTS) approach is the most popular discretization approach for FEM-based contact

Late 1980sSegment-to-segment (STS) approach is suggested, but remains unpopular, e.g. Simo, Wriggers and Taylor (1985), e.g. Papadpoulos and Taylor (1992)

Since 2000s Mortar methods first applied to (simple linear elastic) contact, consistent extension of the STS approach, e.g. Ben Belgacem (1998), e.g. McDevitt and Laursen (2000)

Today

Mortar methods increasingly popular, generally accepted as being superior to classical NTS or GPTS discretization, particularly for complex contact scenarios, emerging techniques, e.g. Nitsche-based contact

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Page 7: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Contact interface laws - Frictional contact

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[Courtesy: P. Wriggers]

Page 8: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Contact interface laws - Frictionless contact

[Courtesy: P. Wriggers]

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Page 9: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Contact - Numerical modeling

[Courtesy: P. Wriggers]

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Page 10: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Contact - Numerical modeling

!10

Page 11: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Interface mechanics - Wear modeling

[Courtesy: S. Stupkiewicz]

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Page 12: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Interface mechanics - Wear modeling

[Courtesy: S. Stupkiewicz]

!12

Page 13: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Beam-to-beam contact

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reaction torque:

Page 14: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Current research topics

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[Courtesy: J.-F. Molinari]

Page 15: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Current research topics

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[Courtesy: J. Rojek]

Page 16: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Agenda

ɠ MOTIVATION

ɡ MOTIVATION

ɢ METHODS 1

ɣ METHODS 2

ɤ METHODS 3

!16

Main Challenges, History and Examples

A Simple FEM Contact Example

Contact Kinematics

Contact Constraint Enforcement

Weak Formulations with Contact for FEM

Page 17: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

FEM analysis of the contact of two bars• geometrically linear problem (small deformation kinematics) • materially linear problem (linear elasticity / Hooke’s law) • contact nevertheless introduces nonlinearity into the system • potential energy of the system:

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Page 18: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

FEM analysis of the contact of two bars• contact constraint equation

• FEM discretization with 3 linear bar elements • three discrete (nodal) displacement DoFs: u1, u2 and u3

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Page 19: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

FEM analysis of the contact of two bars• discretized potential energy

• variation (PMTPE)

• discrete constraint equation

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Page 20: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

FEM analysis of the contact of two bars• case (1) - no contact:

• arbitrary virtual displacements

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Page 21: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

FEM analysis of the contact of two bars• case (2) - active contact: • use Lagrange multiplier method

• matrix form of mixed variational statement

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Page 22: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

FEM analysis of the contact of two bars• case (2) - active contact • arbitrary virtual displacements

• load-displacement behavior

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Page 23: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

FEM analysis of the contact of two bars• nonlinear solution / system behavior

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inactive (gap open)

active (gap closed)

Page 24: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Agenda

ɠ MOTIVATION

ɡ MOTIVATION

ɢ METHODS 1

ɣ METHODS 2

ɤ METHODS 3

!24

Main Challenges, History and Examples

A Simple FEM Contact Example

Contact Kinematics

Contact Constraint Enforcement

Weak Formulations with Contact for FEM

Page 25: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Contact kinematics• configurations and contact-related quantities

!25

Page 26: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Contact kinematics - Boundaries• decomposition of the boundaries

• Dirichlet, Neumann and potential contact boundaries • spatial counterparts • active / inactive parts of contact boundary a priori unknown

• for discretization (later) introduce classical notions of

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Page 27: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Contact kinematics - Normal direction• fundamental measure for proximity, contact and penetration = gap function

• several ways to find projection point and contact normal vector • most common and well-known = closest point projection (CPP)

• alternative = ray-tracing approach

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nc = ⌧ ⇠ ⇥ ⌧ ⌘

nc =x̂(2) � x(1)

kx̂(2) � x(1)k

Page 28: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Contact kinematics - Normal direction• CPP has some problems • e.g. non-differentiable distance functions (bottom) • e.g. non-uniqueness and missing regularity (right) • ray tracing approach often more robust

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taken from [P. Wriggers: Computational Contact Mechanics, Springer, 2006]

Page 29: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Contact kinematics - Tangential direction• fundamental measure for relative motion = relative tangential velocity

• only exact for perfect sliding and persistent contact, i.e. only for

• not objective / frame-indifferent • nevertheless, typically employed in state-of-the-art FEM formulations • algorithmic modifications are introduced to restore objectivity • relative velocity of two material points, i.e. not including change of CPP • decomposition with respect to local tangent vectors

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Page 30: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Contact kinematics - Tangential direction• classical approach = slip advected bases • objective / frame-indifferent by definition

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Page 31: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Contact kinematics - Tangential direction• classical approach = slip advected bases • objective / frame-indifferent by definition • also referred to as covariant approach

• consider perfect sliding, i.e. gap vector remains unchanged

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relative material velocity change of projection

Page 32: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Contact kinematics - Tangential direction• last term can be used for objective relative velocity

• with the slip velocities expressed as

• however, this involves some rather intricate nonlinear continuum mechanics,e.g. pull-back, push-forward, Lie-derivative, co- and contravariant bases

• bottom line: trade-off between guaranteed objectivity and simplicity

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Page 33: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Contact traction• surface traction on the current contact boundary • normal-tangential split

• balance of linear momentum

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contact pressure tangential traction

Page 34: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Tied contact• also known as bilateral contact or mesh tying • configurations and important quantities

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Page 35: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Tied contact• domain decomposition problem • artificial internal interface (from continuum perspective) • only really makes sense from a discretization perspective,

e.g. heterogeneous materials, non-matching finite element meshes

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Page 36: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Tied contact constraints• fundamental measure for mesh tying = relative displacement or gap vector

• typically, tied contact assumed in the initial configuration:—> equivalent form of gap vector

• remember gap function for unilateral contact

• tied contact constraint

!36

Page 37: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Unilateral contact• configurations and contact-related quantities

!37

Page 38: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Unilateral contact constraints• all kinematic and kinetic quantities already defined • Hertz-Signorini-Moreau conditions

• non-smooth, nonlinear contact law • multi-valued law at • (1) non-penetration condition • (2) negative pressure condition • (3) complementarity condition

• dynamics: persistency condition

!38

Page 39: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Tangential contact / friction• an entire scientific field: tribology • sometimes, friction ought to be maximized • sometimes, friction ought to be minimized

• governing micro-mechanical phenomena of dry frictionadhesion of the surfaces and ploughing of surface asperities

• we usually use Coulomb’s law here (purely phenomenological)

!39

Page 40: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Tangential contact constraints• simplest assumption: frictionless sliding

• most common approach for macroscopic friction = Coulomb’s law • purely phenomenological approach

• with the friction coefficient • (1) slip condition • (2) slip rule • (3) stick / slip separation • (4) complementarity condition

!40

Page 41: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Coulomb friction• two physically distinct states: stick and slip

• remember: purely phenomenological approach • modifications possible, i.e. rate and state dependent friction laws

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Page 42: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Coulomb friction• friction coefficient is NOT a material parameter

• friction is a system property (including B.C.) • still, Coulomb’s law with constant CoF is often a good engineering assumption

!42

[courtesy: J.-F. Molinari]

Page 43: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Agenda

ɠ MOTIVATION

ɡ MOTIVATION

ɢ METHODS 1

ɣ METHODS 2

ɤ METHODS 3

!43

Main Challenges, History and Examples

A Simple FEM Contact Example

Contact Kinematics

Contact Constraint Enforcement

Weak Formulations with Contact for FEM

Page 44: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Reminder - Weak formulation without contact• IBVP of nonlinear elastodynamics • strong form

• weak form

!44

Page 45: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Reminder - Tied and unilateral contact• tied contact constraints (last lecture)

• unilateral contact constraints - frictionless sliding (last lecture)

!45

Page 46: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

An optimization point of view• assume a conservative problem of nonlinear solid mechanics • solution is found by unconstrained minimization of energy functional

= principle of minimum of total potential energy (PMTPE)

• now, contact introduces additional constraints (look at tied contact first)

= vector-valued equality condition with nco constraints = constrained optimization (“real“ optimization) problem

• inequality constraints (unilateral contact) will follow later

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???

Page 47: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Optimization literature• tons of excellent literature on constrained optimization • just a very brief intro to the important aspects for contact here

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Page 48: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Lagrange multiplier method• introduce additional unknowns = discrete Lagrange multipliers • as many Lagrange multipliers as constraint equations, i.e. nco here

• if underlying potential is convex (= positive definite operator / unique minimum)then the solution is a saddle point of the extended potential

• total variation —> mixed variational formulation for stationary point

• solution is minimum w.r.t. the primary variable (here: displacements)and a maximum w.r.t. the dual variable (here: Lagrange multipliers)

!48

Page 49: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Lagrange multiplier method• corresponding discrete linearized problem statement

• right-hand sides are defined as discrete (nonlinear) residuals

• matrix blocks are defined as discrete stiffness blocks / linearizations

!49

Page 50: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Lagrange multiplier method( + ) exact satisfaction of given discrete constraints ( + ) sound mathematical / variational framework ( - ) additional variables / increased global system size ( - ) saddle point structure of resulting system

!50

Page 51: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Penalty method• remove constraints explicitly from the system • enforce constraints approximately by penalization of constraint violation

• penalty parameter is unphysical / user-defined / problem-specific, butin contrast to LM method we still have an unconstrained problem

• total variation —> purely primal variational formulation

• solution is still a minimum w.r.t. the primary variable (here: displacements),since constraint violation is associated with large artificial energy values

!51

Page 52: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Penalty method• corresponding discrete linearized problem statement

• right-hand sides are defined as discrete (nonlinear) residuals

• matrix blocks are defined as discrete stiffness blocks / linearizations

!52

Page 53: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Penalty method( + ) simplicity / no additional variables ( + ) formulation retains convexity / positive definiteness ( - ) exact solution only for , , not possible due to ill-conditioning ( - ) unphysical / user-defined / problem-specific penalty parameter

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Page 54: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Augmented Lagrange method• motivation: find optimal compromise between LM and penalty methods • mathematically: regularization of Lagrange multiplier functional

• extended functional contains a Lagrange multiplier and a penalty contribution,same solution point as LM but regularized saddle point system

• total variation —> mixed variational statement for stationary point

• solution is minimum w.r.t. the primary variable (here: displacements)and a maximum w.r.t. the dual variable (here: Lagrange multipliers)

!54

Page 55: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Augmented Lagrange method• two different algorithmic interpretations of Augmented Lagrange exist • infamous source of confusion in the literature!!!

• first variant: keep LM as unknowns, very similar to LM method (+) advantage: underlying potential is “convexified“

• second variant: Uzawa’s algorithm (+) advantage: no additional unknowns

- iterative procedure- penalty method as kernel- linear convergence rate only

!55

Page 56: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Treatment of inequality constraints• unilateral contact introduces inequality constraints

= vector-valued inequality condition with nco constraints = constrained optimization and active set problem

• Lagrange multiplier methodand variational inequality

• only variationsare admissible here

!56

Page 57: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Treatment of inequality constraints• variational inequality equivalent to set of optimality conditions

• active set strategies needed for inequality constraints • find active and inactive set of constraints

!57

Page 58: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Treatment of inequality constraints• active set strategies will be discussed in more detail later • simplest possible approach: iterative scheme

• solve system with equality constraints until convergence

• rather inefficient (nested iteration loops) • primal-dual active set strategy will be introduced later

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mathematicallyequivalent

Page 59: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Treatment of inequality constraints• look at penalty method for the sake of completeness • very straightforward approach: ad-hoc penalization of any active constraint

• Macaulay bracket

• total variation —> primal formulation with inequalities

!59

Page 60: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Agenda

ɠ MOTIVATION

ɡ MOTIVATION

ɢ METHODS 1

ɣ METHODS 2

ɤ METHODS 3

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Main Challenges, History and Examples

A Simple FEM Contact Example

Contact Kinematics

Contact Constraint Enforcement

Weak Formulations with Contact for FEM

Page 61: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Some general remarks• focus on Lagrange multiplier method in the following

(1) tied contact (2) frictional contact (with Coulomb friction) (3) unilateral contact (with frictionless sliding)

• penalty method can always be investigated by simply modifying theLagrange multiplier formulation accordingly

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e.g. mortar FEM for unilateral contact, replace Lagrange multiplier at each node

with penalty force law

Page 62: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Tied contact - Weak formulation• IBVP of nonlinear elastodynamics (here for two subdomains: i=1,2)

• typical solution and weighting spaces for displacements

!62

Page 63: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Tied contact - Weak formulation• interface Lagrange multipliers λ from dual trace space

• weak saddle point formulation with interface Lagrange multipliers λ

!63

Page 64: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Tied contact - Weak formulation• kinetic, internal and external virtual work contributions

• virtual work of Lagrange multipliers (i.e. interface tractions) and weak form of kinematic continuity constraints

!64

Page 65: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Frictional contact - Weak formulation• IBVP of nonlinear elastodynamics

• Hertz-Signorini-Moreau conditions and Coulomb’s law for frictional sliding

!65

Page 66: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Frictional contact - Weak formulation• again, typical solution and weighting spaces for displacements

• convex set of Lagrange multipliers [HÜEBER 2007]:

• final weak problem statement as variational inequality [WOHLMUTH 2011]

!66

Page 67: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Frictional contact - Weak formulation• kinetic, internal and external virtual work contributions

• virtual work of Lagrange multipliers (i.e. interface tractions) and weak form (variational inequality) of frictional contact constraints

!67

Page 68: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Unilateral contact - Weak formulation• IBVP of nonlinear elastodynamics

• Hertz-Signorini-Moreau conditions and frictionless sliding condition

!68

Page 69: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Unilateral contact - Weak formulation• again, typical solution and weighting spaces for displacements

• simplified convex set of Lagrange multipliers:

• final weak problem statement as variational inequality [WOHLMUTH 2011]

!69

Page 70: Nonlinear Finite Element Methods and Computational Contact ...shortcourse2018.it.cas.cz/im/data/my/2018_Lecture_22.pdf · First papers on finite element methods applied to contact

Technische Universität München

Alexander Popp, Institute for Mathematics and Computer-Based Simulation (IMCS), UniBw M

Unilateral contact - Weak formulation• kinetic, internal and external virtual work contributions

• virtual work of Lagrange multipliers (i.e. interface tractions) and weak form (variational inequality) of unilateral contact constraints

!70