AN EFFICIENT FORMULATION FOR THESIMULATION OF ELASTIC WAVE PROPAGATION

IN 1-DIMENSIONAL COLLIDING BODIES

A. Depouhon, E. Detournay, V. Denoel

Universite de Liege (Belgium)University of Minnesota (USA)

ICOVPLisbon, September 9, 2013

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PERCUSSIVE DRILLING: DOWN-THE-HOLE HAMMER

F

T

Piston Bit

F

p

Model specificities

• Linear elasticity

• Multibody with contact

• Bilinear bit/rock interaction (BRI) law

− Piecewise linear− Requires history variable

FE discretization → linear EOM for each body

Mv+Cv+Ku= f ext + f con + fbri

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CONTACT HANDLING: PENALTY METHOD

• Regularizes unilateral constraints by introducing numerical stiffness atinterface

Body 1 Body 2

Contact stiffness

λcon = f ([g ]−), [g ]− =min(0,g)

• Quadratic contact potential possibly leads to energetic instability

Time

Gap

Time

Force

• Stores energy during persistent contact

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CONTACT HANDLING: PENALTY METHOD

• Regularizes unilateral constraints by introducing numerical stiffness atinterface

Body 1 Body 2

Contact stiffness

λcon = f ([g ]−), [g ]− =min(0,g)

• Quadratic contact potential possibly leads to energetic instability

0 5 10 15 2010−1

100

101

102

-vfvf

0

-vfvfH0

g

Elastic bouncing bar benchmark

Newmark + linear contact force

Time

Nor

mal

ized

ener

gy

Conservative

Exact

Dissipative

• Stores energy during persistent contact

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CONTACT HANDLING: PENALTY METHOD

• Regularizes unilateral constraints by introducing numerical stiffness atinterface

Body 1 Body 2

Contact stiffness

λcon = f ([g ]−), [g ]− =min(0,g)

• Quadratic contact potential possibly leads to energetic instability

• Stores energy during persistent contact, Armero & Petocz (1998)

Midpoint + nonlinear contact force

Norm

alizedenergy

Time0 10 15 20

99.0%

99.5%

100%

5

Exact

ArPe98

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EVENT-DRIVEN INTEGRATION SCHEME

ConceptEvent Event

1. March EOM in time using any integration scheme

2. Detect occurrence of events related to BRI & contact status

3. Update EOM, go to 1

Pros & cons

+ Enables switch between BRI modes &correct handling of history variables

+ Maintains linearity of semi-discrete equations

+ Unilateral constraints become bilateral(easier)

+ Control of contact work

→ [g ]− = g

− Nonlinear robust event detection/localization required

− Possible troubles: chattering, zeno behavior4 / 14

DISSIPATIVE MIDPOINT SCHEME

Armero & Romero (2001)

• One-step, four stage, 2nd order accurate

• Spectral annihilation property (ρ∞ = 0 )

Γ0zn+1 = Γ1zn + `n

with (χ > 0)

zn =

un

vn

un−1vn−1

Γ0 =

K 2M

∆t +C K 02I −∆tI 0 −∆tI0 χ∆tI I −χ∆tI

−χ∆tK 0 χ∆tK M

`n =

2(fn + fn+1)

000

Γ1 =

0 2M

∆t −C 0 02I 0 0 0I 0 0 00 M 0 0

• Also possible to reduce system to displacement-based formulation

0= Z1un+1+Z2un +Z3vn −hn → vn+1 =H1un+1+H2un +H3vn5 / 14

EVENT DETECTION ALGORITHM

Key points

• Parallel search

• Constraint reset

Index set of active constraints

In = {i ∈ C : (gL) i · (gR) i < 0}

Algorithm

Given time bracket [tL, tR] and zL, zR, gL, gR, In

1. Locate earliest event ∀i ∈In (bisection, inverse interpolation, etc.)

2. Compute state vectors using the leftmost state and time step∆t∗ = t∗− tL: u∗, v∗

3. Compute event functions using updated state ∀i ∈ C : g∗

4. Update In

5. Assess convergence:

If∥∥g∗In

∥∥< tol : reset constraints, proceed with integration

Else : update bracket [tL, tR],proceed with event detection

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EVENT LOCALIZATION1. All procedures rely on the bracketing of an event after detection

through a sign change of the event function.2. Bisection is used if convergence is too slow.3. Iterators:

• Bisection

t∗ =1

2(tL+ tR)

• Inverse linear interpolation

t∗ = tL−tR− tLgR− gL

gL

• Barycentric hermite interpolation (Corless et al. (2007))

C0 =

0 gL1 0 Tg ′L

1 gR1 1 Tg ′R

−2 −1 2 −1 0

, C1 =

1

11

10

t∗ = T min

Λ∈(0,1)Λ(C0,C1), T = tR− tL

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ELASTIC BOUNCING BAR

Time ( )

0 3 5 11 13 19 21 27 29 32

0 3 5 11 13 19 21 27 29 320

5

10

15

0

5

10

15

-0.6-0.4-0.20.0

Deformation

0 3 5 11 13 19 21 27 29 32

Axial

deformation

Height

Height

-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2

(H0,L,c0,g) = (5,10,30,10)8 / 14

BOUNCING BAR – BENCHMARKING

Problem data

• Single event: gap at contact • 100 linear elements, CFL = 1

Error measure: mechanical energy after 1 period

E = 12(u

TKu+vTMv)t=16τW

10 0 10 1 10 2 10 30

2000

4000

6000

8000

10000

12000

14000

Tota

lnum

ber

ofite

ratio

ns

10 0 10 1 10 2 10 30

5

10

15

20

Contact stiffness ratio

CP

UTi

me

[s]

BisectionLinear interp.Hermite interp.

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BOUNCING BAR – BENCHMARKING

Learnings

• Contact stiffness strongly influences convergence of ED

• χ = 1/6: better convergence and lower error (∼ 3rd-order TDG)

• Bisection method not necessarily more expensive

10 0 10 1 10 2 10 30

2000

4000

6000

8000

10000

12000

14000

Tota

lnum

ber

ofite

ratio

ns

10 0 10 1 10 2 10 30

5

10

15

20

Contact stiffness ratio

CP

UTi

me

[s]

BisectionLinear interp.Hermite interp.

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DTH SYSTEM

Problem data

• 5 events: contact + BRI

• Linear interpolation + bisection

• Uniform velocity prior to impact (no defo.)

Piston Bit

ContactInterface 1

ContactInterface 2

F

p

Abit = 6525 mm2 Lbit = 357 mm Esteel = 210 GPa ρsteel = 7850 ·10−6 g/mm3

Apis = 6525 mm2 Lpis = 300 mm KR = 106 N/mm γ = 10

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DTH SYSTEM – CG VELOCITIES

Analysis

• Number of persistent contact phases varies with impact velocities

• Given nominal impact velocities design can ensure double impact

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−4

−2

0

2

4

6

8

Velocity[mm/ms]

Time [ms]

PistonBit

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DTH SYSTEM – PARAMETRIC ANALYSIS

Analysis

• Rebound velocities depend discontinuously on impact velocities

• Persistent contact phases play a critical role in post-impact velocities

Piston rebound velocity Bit rebound velocity

# contact @ P/B # contact @ B/R

1

2

3

4

5

1

0

-1

-2

-3

5 6 7 8 9 10

2

1

0

-1

-2

5 6 7 8 9 10

2

1

0

-1

-2

5 6 7 8 9 10

2

1

0

-1

-2

5 6 7 8 9 10

2

1

0

-1

-2

[m/s]

[1]

A

C

B

D

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SUMMARY

Framework for the handling of contact & events

• Contact is handled via penalty method

• Scheme is energetically stable

• Applies to

− Wave propagation in DTH systems (motivation) & chain systems− Linear structural dynamics

Benchmark 1 – Elastic bouncing bar

• Best performance/accuracy with dissipative midpoint for χ = 1/6

• � Influence of contact stiffness �

Benchmark 2 – DTH system

• Discontinuous dependence & persistent contact phases

→ Possible to model DTH systems with RBD? Tough ...

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CONTENTS

Motivation & IntroductionDTH percussive drillingContact handling: penalty methodEvent-driven integration scheme

Event-driven integrationDissipative midpoint schemeEvent detection

ApplicationsElastic bouncing barDTH hammer

Summary & work in progressSummary

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