discrete and hybrid models: applications to concrete damage...dem and concrete lattice models hybrid...

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

Discrete and hybrid models: Applications toconcrete damage

Václav �milauer

2 July 2007

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

About me

PhD student, �nanced by Frenchministry of research this year.

Enrolled both in Prague (MilanJirásek) and Grenoble (LaurentDaudeville) − �doctorat enco-tutelle�.

Work focusing on Yade:open-source platform for

numerical calculations, since

2003;

continuing development,

funded mostly by laboratory

3S-R in Grenoble.

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

About me

http://yade.wikia.com

PhD student, �nanced by Frenchministry of research this year.

Enrolled both in Prague (MilanJirásek) and Grenoble (LaurentDaudeville) − �doctorat enco-tutelle�.

Work focusing on Yade:open-source platform for

numerical calculations, since

2003;

continuing development,

funded mostly by laboratory

3S-R in Grenoble.

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

1 Introduction

2 Discrete element methodDEM intricaciesDEM and concrete

3 Lattice models

4 Hybrid models

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

1 Introduction

2 Discrete element methodDEM intricaciesDEM and concrete

3 Lattice models

4 Hybrid models

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

1 Introduction

2 Discrete element methodDEM intricaciesDEM and concrete

3 Lattice models

4 Hybrid models

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

1 Introduction

2 Discrete element methodDEM intricaciesDEM and concrete

3 Lattice models

4 Hybrid models

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

(Primarily) continuous models

Problem formulated in terms ofdi�erential equations �continuum mechanics.

Displacement function u, foundby numerical solution ofboundary value problem.

Discontinuities in u are anextension of the method.

Strain unde�ned at discontinuity,�awkward� (sophisticated)methods.

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

(Primarily) continuous models

Problem formulated in terms ofdi�erential equations �continuum mechanics.

Displacement function u, foundby numerical solution ofboundary value problem.

Discontinuities in u are anextension of the method.

Strain unde�ned at discontinuity,�awkward� (sophisticated)methods.

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

(Primarily) discrete models

Local equations determine globalbehavior numerically − elementinteractions.

No integration necessary, morecomputationally intensive.

Discontinuity description trivial.

Continuity (cohesion) by linkingelements.

Discrete element method (DEM).

Lattice models.

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

(Primarily) discrete models

Local equations determine globalbehavior numerically − elementinteractions.

No integration necessary, morecomputationally intensive.

Discontinuity description trivial.

Continuity (cohesion) by linkingelements.

Discrete element method (DEM).

Lattice models.

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

(Primarily) discrete models

Local equations determine globalbehavior numerically − elementinteractions.

No integration necessary, morecomputationally intensive.

Discontinuity description trivial.

Continuity (cohesion) by linkingelements.

Discrete element method (DEM).

Lattice models.

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

DEM background

elements are rigid bodies, motiongoverned by Newton's laws

explicit integration in time

�smooth� (pinball) vs.�non-smooth� (overlaps) DEM

mechanics of granular media −

Cundall, 1971 (�distinct elementmethod� in 2D, sphericalelements)

molecular dynamics of gas(1980s)

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

DEM background

elements are rigid bodies, motiongoverned by Newton's laws

explicit integration in time

�smooth� (pinball) vs.�non-smooth� (overlaps) DEM

mechanics of granular media −

Cundall, 1971 (�distinct elementmethod� in 2D, sphericalelements)

molecular dynamics of gas(1980s)

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

DEM background

elements are rigid bodies, motiongoverned by Newton's laws

explicit integration in time

�smooth� (pinball) vs.�non-smooth� (overlaps) DEM

mechanics of granular media −

Cundall, 1971 (�distinct elementmethod� in 2D, sphericalelements)

molecular dynamics of gas(1980s)

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

DEM simple example

How to calculate spheres fallingthrough a funnel?

Known element constants (m, I,. . . )

and variables at t = ti (x, o, v,ω, state parameters).

Solve for variables att = ti+1 = ti + ∆t.

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

DEM simple example

How to calculate spheres fallingthrough a funnel?

Known element constants (m, I,. . . )

and variables at t = ti (x, o, v,ω, state parameters).

Solve for variables att = ti+1 = ti + ∆t.

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

DEM simple example

How to calculate spheres fallingthrough a funnel?

Known element constants (m, I,. . . )

and variables at t = ti (x, o, v,ω, state parameters).

Solve for variables att = ti+1 = ti + ∆t.

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

DEM simple example

How to calculate spheres fallingthrough a funnel?

Known element constants (m, I,. . . )

and variables at t = ti (x, o, v,ω, state parameters).

Solve for variables att = ti+1 = ti + ∆t.

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

DEM iteration

1 Calculate forces:independent �elds,

inter-element links,

element collisions.

2 Calculate acceleration fromforces.

3 Integrate over ∆t, t = ti + ∆t.

4 (Adjust ∆t.)

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

DEM iteration

1 Calculate forces:independent �elds,

inter-element links,

element collisions.

2 Calculate acceleration fromforces.

3 Integrate over ∆t, t = ti + ∆t.

4 (Adjust ∆t.)

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

DEM iteration

1 Calculate forces:independent �elds,

inter-element links,

element collisions.

2 Calculate acceleration fromforces.

3 Integrate over ∆t, t = ti + ∆t.

4 (Adjust ∆t.)

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

DEM iteration

1 Calculate forces:independent �elds,

inter-element links,

element collisions.

2 Calculate acceleration fromforces.

3 Integrate over ∆t, t = ti + ∆t.

4 (Adjust ∆t.)

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

DEM iteration

1 Calculate forces:independent �elds,

inter-element links,

element collisions.

2 Calculate acceleration fromforces.

3 Integrate over ∆t, t = ti + ∆t.

4 (Adjust ∆t.)

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

DEM iteration

1 Calculate forces:independent �elds,

inter-element links,

element collisions.

2 Calculate acceleration fromforces.

3 Integrate over ∆t, t = ti + ∆t.

4 (Adjust ∆t.)

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

Collision detection

Big research issue in appliedmathematics, trivial approach isO(n2). Better approach:

1 Replace each element by itsAABB (Axis-Aligned BoundingBox).

2 Sort x, y, z min-max arraysindependently.

3 Overlaps on all coordinates arecollision candidates.

4 Candidates tested geometricallyfor collision.

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

Collision detection

Big research issue in appliedmathematics, trivial approach isO(n2). Better approach:

1 Replace each element by itsAABB (Axis-Aligned BoundingBox).

2 Sort x, y, z min-max arraysindependently.

3 Overlaps on all coordinates arecollision candidates.

4 Candidates tested geometricallyfor collision.

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

Collision detection

Big research issue in appliedmathematics, trivial approach isO(n2). Better approach:

1 Replace each element by itsAABB (Axis-Aligned BoundingBox).

2 Sort x, y, z min-max arraysindependently.

3 Overlaps on all coordinates arecollision candidates.

4 Candidates tested geometricallyfor collision.

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

Collision detection

~n

P

~t

d

Geometry:

P, n, t, d for spheres (trivial).

Complicated for other shapes(e.g. tetrahedra: C, V , I).

Combinations: sphere withtetrahedron, parallelepiped, . . .

Forces are yet to be found.

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

Collision detection

~n

P

~t

d

Geometry:

P, n, t, d for spheres (trivial).

Complicated for other shapes(e.g. tetrahedra: C, V , I).

Combinations: sphere withtetrahedron, parallelepiped, . . .

Forces are yet to be found.

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

Collision detection

~n

P

~t

d

Geometry:

P, n, t, d for spheres (trivial).

Complicated for other shapes(e.g. tetrahedra: C, V , I).

Combinations: sphere withtetrahedron, parallelepiped, . . .

Forces are yet to be found.

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

Physical laws

ks

kn

The most simple model:

Fn = knd, ∆Fs = ks∆ut

(incremental).

Fracture with Coulomb criterionmax Fs = Fn tanφ.

How to determine kn, ks frommacroscopic characteristics?

Simplistically for sphere F = Ku,F = σS = E(1 − d/2r)πr2

(d� r)

Really used formulas: coe�cientswithout physical meaning.→ Model calibration.

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

Physical laws

ks

kn

The most simple model:

Fn = knd, ∆Fs = ks∆ut

(incremental).

Fracture with Coulomb criterionmax Fs = Fn tanφ.

How to determine kn, ks frommacroscopic characteristics?

Simplistically for sphere F = Ku,F = σS = E(1 − d/2r)πr2

(d� r)

Really used formulas: coe�cientswithout physical meaning.→ Model calibration.

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

Mesh generation

Requirements: isotropy, highcoordination number andcompacity, size distribution.

Regular packing leads toanisotropic behavior.

Dynamic methods: gravity,growing spheres. Slow.

Jean-François Durier: Geometricmethod based on tetrahedralmesh:

Leverages existing FEM

meshers − arbitrary shapes.

Very fast with excellent

parameters.

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

Mesh generation

Requirements: isotropy, highcoordination number andcompacity, size distribution.

Regular packing leads toanisotropic behavior.

Dynamic methods: gravity,growing spheres. Slow.

Jean-François Durier: Geometricmethod based on tetrahedralmesh:

Leverages existing FEM

meshers − arbitrary shapes.

Very fast with excellent

parameters.

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

DEM concrete fracture (Hentz)

Modélisation d'une Structure enBéton Armé Soumise à un Choc par laMéthode des Éléments Discrets (PhDthesis of Sebastian Hentz, 2003)

Dropping reinforced concretecube on reinforced concrete slab.

Concrete elements not �physical�(matrix, inclusions).

Reinforcement modelled byspecial elements (includingplasti�cation).

Parameters calibrated on basicsetups, not ex post − predictivevalue.

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

DEM concrete fracture, results

80+110 thousand elements(reinforcement + concrete)

Comparison with instrumentedexperiment at di�erent limitstates.

+ Good results (e.g. displacements±10%).

− Long calculation

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

Lattice overview

Replaces continuum byarrangement of 1D elements(trusses, rods, beams).

Nodes may carry inertia mass(dynamic) or not.

Irregular meshes, less sensitive todegenerate geometry.

Voronoï tesselation / Delaunaytriangulation.

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

Lattice beam model (Cusatis & co.)

Meso-scale lattice beam model(matrix, inclusions).

Constituive law with damage,fracture, plasticity.

Elaborate beam properties basedon geometry of the respectiveVoronoï cell.

Good match in tensile as well ascompressive (usual weak point oflattices) loading.

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

Method considerations

FEM

+ E�cient and easy for undamaged continuum.

− Di�cult discontinuity description.→ Undamaged zone.

Lattice

+ No collision detection necessary.

− No volumetric information.→ Fragmenting zone interior.

DEM

+ Compressive links created during simulation.

− Collisions: computationally expensive.→ Highly fragmented, collapsing zones.→ Colliding boundaries or detached zones.

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

Method considerations

FEM

+ E�cient and easy for undamaged continuum.

− Di�cult discontinuity description.→ Undamaged zone.

Lattice

+ No collision detection necessary.

− No volumetric information.→ Fragmenting zone interior.

DEM

+ Compressive links created during simulation.

− Collisions: computationally expensive.→ Highly fragmented, collapsing zones.→ Colliding boundaries or detached zones.

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

Method considerations

FEM

+ E�cient and easy for undamaged continuum.

− Di�cult discontinuity description.→ Undamaged zone.

Lattice

+ No collision detection necessary.

− No volumetric information.→ Fragmenting zone interior.

DEM

+ Compressive links created during simulation.

− Collisions: computationally expensive.→ Highly fragmented, collapsing zones.→ Colliding boundaries or detached zones.

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

Method considerations

FEM

+ E�cient and easy for undamaged continuum.

− Di�cult discontinuity description.→ Undamaged zone.

Lattice

+ No collision detection necessary.

− No volumetric information.→ Fragmenting zone interior.

DEM

+ Compressive links created during simulation.

− Collisions: computationally expensive.→ Highly fragmented, collapsing zones.→ Colliding boundaries or detached zones.

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

Method considerations

FEM

+ E�cient and easy for undamaged continuum.

− Di�cult discontinuity description.→ Undamaged zone.

Lattice

+ No collision detection necessary.

− No volumetric information.→ Fragmenting zone interior.

DEM

+ Compressive links created during simulation.

− Collisions: computationally expensive.→ Highly fragmented, collapsing zones.→ Colliding boundaries or detached zones.

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

Method considerations

FEM

+ E�cient and easy for undamaged continuum.

− Di�cult discontinuity description.→ Undamaged zone.

Lattice

+ No collision detection necessary.

− No volumetric information.→ Fragmenting zone interior.

DEM

+ Compressive links created during simulation.

− Collisions: computationally expensive.→ Highly fragmented, collapsing zones.→ Colliding boundaries or detached zones.

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

Method considerations

FEM

+ E�cient and easy for undamaged continuum.

− Di�cult discontinuity description.→ Undamaged zone.

Lattice

+ No collision detection necessary.

− No volumetric information.→ Fragmenting zone interior.

DEM

+ Compressive links created during simulation.

− Collisions: computationally expensive.→ Highly fragmented, collapsing zones.→ Colliding boundaries or detached zones.

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

FEM inside DEM

Particles in DEM are themselvesFEM domains.

Interest: reduces computationalexpenses wrt pure DEM forunfractures parts.

Allows for dynamical states − inpure FEM, leads to staticallyunder-determined states.

More di�cult collision detection(mesh − mesh).

For non-predetermined fracture,FEM→DEM transition (via crackmodeling and tracing) must beprovided.

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

FEM next to DEM

Part of domain expected to breakis DEM, the rest is FEM.

Reduces computation wrt pureDEM.

Must know fracturing (DEM)domain beforehand.

Parameters must be tuned tohave similar elastic behavior inboth domains.

The domain interface re�ectswaves (remedy: overlap zone −

E. Frangin).

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

DEM with lattice

Nodes are also DEM elements.

Or: boundary nodes are DEM(collision), insert equivalent DEMelement as needed.

Preserves volume when fractured;pure lattice collapses.

Discrete andhybridmodels:

Applicationsto concretedamage

Václav�milauer

Introduction

DiscreteelementmethodDEMintricacies

DEM andconcrete

Latticemodels

Hybridmodels

References

Sebastian Hentz, Modélisation d'une Structure en Béton ArméSoumise à un Choc par la Méthode des Éléments Discrets, PhDthesis, Grenoble 2003.

Janek Kozicki, Application of Discrete Models to Describe theFracture Process in Brittle Materials, PhD thesis, Gda«sk 2007.

Gianluca Cusatis &al., Con�nement-Shear Lattice CSL Modelfor Fracture Propagation in Concrete, Con�nement-Shear

Lattice CSL Model for Fracture Propagation in Concrete 2006,

7154−7171.

Ante Munjiza, Combined Finite-Discrete Element Method,Wiley, 2004.

Limin Sun &al., Application of extended distinct elementmethod with latticemodel to collapse analysis of RC bridges,Earthquake Engng Struct. Dyn. 2003, 1217�1236.