An efficient formulation for the simulation of elastic wave propagation in 1-dimensional collinding bodies

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AN EFFICIENT FORMULATION FOR THE

SIMULATION OF ELASTIC WAVE PROPAGATION

IN 1-DIMENSIONAL COLLIDING BODIES

A. Depouhon, E. Detournay, V. Deno¨el

Universit´e de Li`ege (Belgium) University of Minnesota (USA)

ICOVP

Lisbon, September 9, 2013

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PERCUSSIVE DRILLING: DOWN-THE-HOLE HAMMER

F T Piston Bit Bilinear bit/rock interaction F p Hammer casing 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

M ˙v + Cv + Ku = fext+ fcon+ fbri

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CONTACT HANDLING: PENALTY METHOD

• Regularizes unilateral constraints by introducing numerical stiffness at

interface

Body 1 Body 2

Contact stiffness

λcon= f ([g]₋), [g]₋= min(0, g)

• Quadratic contact potential possibly leads to energetic instability

Time Gap Time Fo rc e

• Stores energy during persistent contact

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CONTACT HANDLING: PENALTY METHOD

interface

Body 1 Body 2

Contact stiffness

λcon= f ([g]₋), [g]₋= min(0, g)

0 5 10 15 20 10−1 100 101 102 -vf vf 0 -vf vf H0 g

Elastic bouncing bar benchmark

Newmark + linear contact force

Time No rm alize d en er gy _Conservative Exact Dissipative

• Stores energy during persistent contact

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CONTACT HANDLING: PENALTY METHOD

interface

Body 1 Body 2

Contact stiffness

λcon= f ([g]₋), [g]₋= min(0, g)

• Stores energy during persistent contact, Armero & Pet˝ocz (1998)

Midpoint + nonlinear contact force

N o rm a lize d e n e rg y Time 0 10 15 20 99.0% 99.5% 100% 5 Exact ArPe98 3 / 14

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EVENT-DRIVEN INTEGRATION SCHEME

Concept Event 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 behavior

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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−1 ˜ vn−1     Γ0=     K 2M_∆t + C K 0 2I −∆tI 0 −∆tI 0 χ ∆t I I −χ∆tI −χ∆tK 0 χ ∆t K M     `n=     2(fn+ fn+1) 0 0 0     Γ1=     0 2M ∆t − C 0 0 2I 0 0 0 I 0 0 0 0 M 0 0    

• Also possible to reduce system to displacement-based formulation

0 = Z1un+1+ Z2un+ Z3vn− hn → vn+1= H1un+1+ H2un+ H3vn

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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. UpdateIn

5. Assess convergence: If g∗_I

n

< tol : reset constraints, proceed with integration Else : update bracket [tL, tR],

proceed with event detection

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EVENT LOCALIZATION

1. 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− tL

gR− gL

gL

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

C0=       0 gL 1 0 Tg_L0 1 gR 1 1 Tg_R0 −2 −1 2 −1 0       , C1=       1 1 1 1 0       t∗= T min Λ∈(0,1) Λ(C0, C1), T = tR− tL 7 / 14

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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 32 0 5 10 15 0 5 10 15 -0.6 -0.4 -0.2 0.0 Deformation 0 3 5 11 13 19 21 27 29 32 Axial de fo rm at ion He ight H ei ght -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

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BOUNCING BAR – BENCHMARKING

Problem data

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

Error measure: mechanical energy after 1 period

E = 1₂(uTKu + vTMv)t =16τW 100 101 102 103 0 2000 4000 6000 8000 10000 12000 14000 Total number of iterations 100 101 102 103 0 5 10 15 20

Contact stiffness ratio

CPU Time [s] Bisection Linear interp. Hermite interp. 9 / 14

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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

100 101 102 103 0 2000 4000 6000 8000 10000 12000 14000 Total number of iterations 100 101 102 103 0 5 10 15 20

Contact stiffness ratio

CPU Time [s] Bisection Linear interp. Hermite interp. 9 / 14

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DTH SYSTEM

Problem data

• 5 events: contact + BRI

• Linear interpolation + bisection

• Uniform velocity prior to impact (no defo.)

Piston Bit Contact Interface 1 Contact Interface 2 F p

Abit= 6525 mm2 Lbit= 357 mm Esteel= 210 GPa ρsteel= 7850 · 10−6g/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 V elo ci ty [m m /m s] Time [ms] Piston Bit 11 / 14

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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 12 / 14

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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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An efficient formulation for the simulation of elastic wave propagation in 1-dimensional collinding bodies