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References Subduction Zone Simulations with Rate-and-State Friction E. Pipping 1 , R. Kornhuber 1 , M. Rosenau 2 , O. Oncken 2 1 Mathematisches Institut, Freie Universitt Berlin, 2 Geologische Systeme: Lithosphrendynamik, GeoForschungsZentrum

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References

Subduction Zone Simulations with Rate-and-State Friction

  • E. Pipping1, R. Kornhuber1, M. Rosenau2, O. Oncken2

1Mathematisches Institut, Freie Universität Berlin, 2Geologische Systeme: Lithosphärendynamik, GeoForschungsZentrum Potsdam

1st of July 2015

Subduction Zone Simulations with Rate-and-State Friction

  • E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken
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References

Overview / motivation

Setting: Single, pre-existing fault. Small deformations. No pore fluids, no temperature dependence.

Subduction Zone Simulations with Rate-and-State Friction

  • E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken
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References

Overview / motivation

Setting: Single, pre-existing fault. Small deformations. No pore fluids, no temperature dependence. Aim: Simulate complete seismic cycles with rate-and-state friction. Fully dynamic, no quasistatic or quasidynamic approximation.

Subduction Zone Simulations with Rate-and-State Friction

  • E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken
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References

Overview / motivation

Setting: Single, pre-existing fault. Small deformations. No pore fluids, no temperature dependence. Aim: Simulate complete seismic cycles with rate-and-state friction. Fully dynamic, no quasistatic or quasidynamic approximation. Approach: Rothe’s method (spatial discretisation: FEM)

Subduction Zone Simulations with Rate-and-State Friction

  • E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken
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References

Overview / motivation

Setting: Single, pre-existing fault. Small deformations. No pore fluids, no temperature dependence. Aim: Simulate complete seismic cycles with rate-and-state friction. Fully dynamic, no quasistatic or quasidynamic approximation. Approach: Rothe’s method (spatial discretisation: FEM)

time stepping scheme? step size restriction rate/state coupling resolve coupling? explicit or semi-implicit fully implicit

Subduction Zone Simulations with Rate-and-State Friction

  • E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken
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References

Overview / motivation

Setting: Single, pre-existing fault. Small deformations. No pore fluids, no temperature dependence. Aim: Simulate complete seismic cycles with rate-and-state friction. Fully dynamic, no quasistatic or quasidynamic approximation. Approach: Rothe’s method (spatial discretisation: FEM)

time stepping scheme? step size restriction rate/state coupling resolve coupling? explicit or semi-implicit fully implicit less efficient approaches

  • ur approach

no yes

Subduction Zone Simulations with Rate-and-State Friction

  • E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken
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References

Rate-and-state friction

Rate-and-state friction laws by Dieterich/Ruina (1983), µ(V , α) = µ∗ + a log V V∗ + b log θV∗ L

α

, ˙ θ(θ, V ) =

  • 1 − θV

L

ageing law − θV

L log θV L

slip law

Subduction Zone Simulations with Rate-and-State Friction

  • E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken
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References

Rate-and-state friction

Rate-and-state friction laws by Dieterich/Ruina (1983), µ(V , α) = µ∗ + a log V V∗ + b log θV∗ L

α

, ˙ θ(θ, V ) =

  • 1 − θV

L

ageing law − θV

L log θV L

slip law = a log V V∗ exp µ∗ + bα a

  • Subduction Zone Simulations with Rate-and-State Friction
  • E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken
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References

Rate-and-state friction

Rate-and-state friction laws by Dieterich/Ruina (1983), µ(V , α) = µ∗ + a log V V∗ + b log θV∗ L

α

, ˙ θ(θ, V ) =

  • 1 − θV

L

ageing law − θV

L log θV L

slip law = a log V V∗ exp µ∗ + bα a

  • Regularisation by Rice/Ben-Zion (1996)

≈ a sinh−1 V 2V∗ exp µ∗ + bα a

  • ≥ 0

Subduction Zone Simulations with Rate-and-State Friction

  • E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken
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SLIDE 10

References

Rate-and-state friction

Rate-and-state friction laws by Dieterich/Ruina (1983), µ(V , α) = µ∗ + a log V V∗ + b log θV∗ L

α

, ˙ θ(θ, V ) =

  • 1 − θV

L

ageing law − θV

L log θV L

slip law = a log V V∗ exp µ∗ + bα a

  • Regularisation by Rice/Ben-Zion (1996)

≈ a sinh−1 V 2V∗ exp µ∗ + bα a

  • ≥ 0

Another regularisation ≈ µ∗ + a log V V∗ + 1

  • + bα

Subduction Zone Simulations with Rate-and-State Friction

  • E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken
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References

Rate-and-state friction

Rate-and-state friction laws by Dieterich/Ruina (1983), µ(V , α) = µ∗ + a log V V∗ + b log θV∗ L

α

, ˙ θ(θ, V ) =

  • 1 − θV

L

ageing law − θV

L log θV L

slip law = a log V V∗ exp µ∗ + bα a

  • Regularisation by Rice/Ben-Zion (1996)

≈ a sinh−1 V 2V∗ exp µ∗ + bα a

  • ≥ 0

Another regularisation ≈ µ∗ + a log V V∗ + 1

  • + bα

Common assumption: Constant normal stress

Subduction Zone Simulations with Rate-and-State Friction

  • E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken
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References

Prototypical one-body problem

ΓN ΓN Ω ΓD ΓC

σ(u) = Bε(u) + Aε(˙ u) in Ω (linear viscoelasticity) ∇ · σ(u) + b = ρ¨ u in Ω (momentum balance) ˙ un = 0

  • n ΓC

(bilateral contact) σt = −λ˙ u, |σt| = λ|˙ u| = |σn|µ(|˙ u|, α) + C

  • n ΓC

with λ = 0 for ˙ u = 0 . . .

  • n ΓN,D

˙ α = ˙ α(|˙ u|, α)

  • n ΓC

(family of ODEs)

Subduction Zone Simulations with Rate-and-State Friction

  • E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken
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References

Towards spatial discretisation: Weak formulation

We get

  • Ω

ρ¨ u(v − ˙ u) +

  • Ω

Bε(˙ u): ε(v − ˙ u) +

  • Ω

Aε(u): ε(v − ˙ u) +

  • ΓC

φ(v, α) ≥

  • ΓC

φ(˙ u, α) + ℓ(v − ˙ u) for every v ∈ H with H = {v ∈ H1(Ω)d : v = 0 on ΓD, vn = 0 on ΓC}

  • r briefly

0 ∈ M¨ u + C ˙ u + Au + ∂Φ( · , α)(˙ u) − ℓ ⊂ H∗ and ˙ α = ˙ α(|˙ u|, α) a.e. on ΓC

Subduction Zone Simulations with Rate-and-State Friction

  • E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken
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References

Implicit time discretisation

Starting point: 0 ∈ M¨ u + C ˙ u + Au + ∂Φ( · , α)(˙ u) − ℓ ˙ α = ˙ α(|˙ u|, α) (S)

Subduction Zone Simulations with Rate-and-State Friction

  • E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken
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References

Implicit time discretisation

Starting point: 0 ∈ M¨ u + C ˙ u + Au + ∂Φ( · , α)(˙ u) − ℓ ˙ α = ˙ α(|˙ u|, α) After collocation (rate), approximation of ˙ u on [tn−1, tn] (state): 0 ∈ M¨ un + C ˙ un + Aun + ∂Φ( · , αn)(˙ un) − ℓn ˙ α = ˙ α(|˙ un−λ|, α) with ˙ un−λ = λ˙ un−1 + (1 − λ)˙ un (0 ≤ λ < 1) (S)

Subduction Zone Simulations with Rate-and-State Friction

  • E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken
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References

Implicit time discretisation

Starting point: 0 ∈ M¨ u + C ˙ u + Au + ∂Φ( · , α)(˙ u) − ℓ ˙ α = ˙ α(|˙ u|, α) After collocation (rate), approximation of ˙ u on [tn−1, tn] (state): 0 ∈ M¨ un + C ˙ un + Aun + ∂Φ( · , αn)(˙ un) − ℓn ˙ α = ˙ α(|˙ un−λ|, α) with ˙ un−λ = λ˙ un−1 + (1 − λ)˙ un (0 ≤ λ < 1) After time discretisation (rate), determining the flow operator (state) 0 ∈ λM τ M + C + τ λA A

  • ˙

un + ∂Φ( · , αn)(˙ un) − ℓn − . . . (R) αn = Ψτ(|˙ un−λ|, αn−1) (S)

Subduction Zone Simulations with Rate-and-State Friction

  • E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken
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References

Implicit time discretisation

Starting point: 0 ∈ M¨ u + C ˙ u + Au + ∂Φ( · , α)(˙ u) − ℓ ˙ α = ˙ α(|˙ u|, α) After collocation (rate), approximation of ˙ u on [tn−1, tn] (state): 0 ∈ M¨ un + C ˙ un + Aun + ∂Φ( · , αn)(˙ un) − ℓn ˙ α = ˙ α(|˙ un−λ|, α) with ˙ un−λ = λ˙ un−1 + (1 − λ)˙ un (0 ≤ λ < 1) After time discretisation (rate), determining the flow operator (state) 0 ∈ λM τ M + C + τ λA A

  • ˙

un + ∂Φ( · , αn)(˙ un) − ℓn − . . . (R) αn = Ψτ(|˙ un−λ|, αn−1) (S) Structure: (R) Positive rate effect convex minimisation! (S) Trivial.

Subduction Zone Simulations with Rate-and-State Friction

  • E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken
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References

Rate/state coupling

|γ(˙ u)| ∈ L2(ΓC) ˙ u ∈ H α ∈ L2(ΓC)

(S) (γ) L i p s c h i t z , c

  • m

p a c t (R)

T : H → H      (S) solve ODEs (R) convex minimisation (γ) trace map + norm

Subduction Zone Simulations with Rate-and-State Friction

  • E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken
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References

Rate/state coupling

|γ(˙ u)| ∈ L2(ΓC) ˙ u ∈ H α ∈ L2(ΓC)

(S) (γ) L i p s c h i t z , c

  • m

p a c t (R)

T : H → H      (S) solve ODEs (R) convex minimisation (γ) trace map + norm Analytic findings: Contraction if

  • Ageing law
  • Non-zero Viscosity
  • τ small enough

We then have: Existence, uniqueness, convergence ( algorithm!).

Subduction Zone Simulations with Rate-and-State Friction

  • E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken
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References

Rate/state coupling

|γ(˙ u)| ∈ L2(ΓC) ˙ u ∈ H α ∈ L2(ΓC)

(S) (γ) L i p s c h i t z , c

  • m

p a c t (R)

T : H → H      (S) solve ODEs (R) convex minimisation (γ) trace map + norm Analytic findings: Contraction if

  • Ageing law
  • Non-zero Viscosity
  • τ small enough

We then have: Existence, uniqueness, convergence ( algorithm!). More generally: Existence.

Subduction Zone Simulations with Rate-and-State Friction

  • E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken
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References

Application: a simplified subduction zone

Lower plate moves at a prescribed velocity, right end held fixed.

Subduction Zone Simulations with Rate-and-State Friction

  • E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken
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References

0.00 0.20 0.40 0.60

distance from trench [m] −20 µm −10 µm −5 µm −2.5 µm 0 µm 2.5 µm 5 µm 10 µm 20 µm

10−3 10−2 10−1

time step size [s]

2 4 6 8

fixed-point iterations

984 986 988 990 992 994 996 5 10 15 20

time [s] multigrid iterations Figure: Vertical surface displacement (relative to a time average).

Subduction Zone Simulations with Rate-and-State Friction

  • E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken
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References

Comparison: simulation and experiment

5 10 20 40 experiment simulation

recurrence time [s]

0.1 0.2 0.3 0.4

rupture width [m]

0.03 0.06 0.12

peak slip [mm] Figure: Tukey boxplots for recurrence time, rupture width, and peak slip.

Subduction Zone Simulations with Rate-and-State Friction

  • E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken
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References

Is complete resolution of the coupling efficient?

10−7 10−6 10−5 10−4 10−3 10−2 1.2 1.4 1.6 1.8 2 ·106 time-stepping tolerance fixed point tolerance multigrid iterations

Figure: Computational effort over the prescribed error tolerance ε.

Subduction Zone Simulations with Rate-and-State Friction

  • E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken
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References

Snapshots from a 3D simulation

0.2 0.4 −0.30 −0.20 −0.10 0.00 0.10 0.20 0.30

t0 ≈ 991 s depth [m]

0.2 0.4

t0 + 0.14 s

0.2 0.4

t0 + 0.21 s

0.2 0.4

t0 + 0.27 s 1 µm/s 2 µm/s 3 µm/s 5 µm/s 10 µm/s 20 µm/s 30 µm/s 50 µm/s 100 µm/s 200 µm/s 300 µm/s 500 µm/s 1000 µm/s

0.2 0.4

t0 + 0.34 s

0.2 0.4

t0 + 0.46 s

distance from trench [m]

Subduction Zone Simulations with Rate-and-State Friction

  • E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken
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References

Conclusion / outlook

Status quo

  • Robust, efficient solver.

Corner stone: nonlinear multigrid “TNNMG”.

  • Open source implementation in C++.

Foundation: DUNE framework

Outlook

  • 2-body problems
  • Plasticity
  • Parallelisation
  • Integration into multiscale fault network

Subduction Zone Simulations with Rate-and-State Friction

  • E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken
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References

Further reading

  • E. Pipping, O. Sander and R. Kornhuber. “Variational formulation
  • f rate- and state-dependent friction problems”. In: Zeitschrift für

Angewandte Mathematik und Mechanik. Journal of Applied Mathematics and Mechanics (2013). ISSN: 1521-4001. DOI: 10.1002/zamm.201300062.

  • E. Pipping. “Dynamic problems of rate-and-state friction in

viscoelasticity”. Dissertation. Freie Universität Berlin, 2014. URN: urn:nbn:de:kobv:188-fudissthesis000000098145-4.

  • E. Pipping, R. Kornhuber, M. Rosenau and O. Oncken. “Numerical

approximation of rate-and-state friction problems”. 2015. URL: http://publications.mi.fu-berlin.de/1538/. Forthcoming.

Subduction Zone Simulations with Rate-and-State Friction

  • E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken