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 Universität Berlin, 2 Geologische Systeme: Lithosphärendynamik, GeoForschungsZentrum Potsdam 1st of July 2015 Subduction Zone Simulations with Rate-and-State Friction E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken
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
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
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
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? explicit or semi-implicit fully implicit rate/state step size coupling restriction resolve coupling? Subduction Zone Simulations with Rate-and-State Friction E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken
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? explicit or semi-implicit fully implicit rate/state step size coupling restriction less efficient resolve no approaches coupling? yes our approach Subduction Zone Simulations with Rate-and-State Friction E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken
References Rate-and-state friction Rate-and-state friction laws by Dieterich/Ruina (1983), � 1 − θ V ageing law µ ( V , α ) = µ ∗ + a log V + b log θ V ∗ ˙ L , θ ( θ, V ) = − θ V L log θ V V ∗ L slip law L � �� � α Subduction Zone Simulations with Rate-and-State Friction E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken
References Rate-and-state friction Rate-and-state friction laws by Dieterich/Ruina (1983), � 1 − θ V ageing law µ ( V , α ) = µ ∗ + a log V + b log θ V ∗ ˙ L , θ ( θ, V ) = − θ V L log θ V V ∗ L slip law L � �� � α � V � µ ∗ + b α �� = a log exp V ∗ a Subduction Zone Simulations with Rate-and-State Friction E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken
References Rate-and-state friction Rate-and-state friction laws by Dieterich/Ruina (1983), � 1 − θ V ageing law µ ( V , α ) = µ ∗ + a log V + b log θ V ∗ ˙ L , θ ( θ, V ) = − θ V L log θ V V ∗ L slip law L � �� � α � V � µ ∗ + b α �� = a log exp V ∗ a Regularisation by Rice/Ben-Zion (1996) � V � µ ∗ + b α �� ≈ a sinh − 1 exp ≥ 0 2 V ∗ a Subduction Zone Simulations with Rate-and-State Friction E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken
References Rate-and-state friction Rate-and-state friction laws by Dieterich/Ruina (1983), � 1 − θ V ageing law µ ( V , α ) = µ ∗ + a log V + b log θ V ∗ ˙ L , θ ( θ, V ) = − θ V L log θ V V ∗ L slip law L � �� � α � V � µ ∗ + b α �� = a log exp V ∗ a Regularisation by Rice/Ben-Zion (1996) � V � µ ∗ + b α �� ≈ a sinh − 1 exp ≥ 0 2 V ∗ a Another regularisation � V � ≈ µ ∗ + a log + 1 + b α V ∗ Subduction Zone Simulations with Rate-and-State Friction E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken
References Rate-and-state friction Rate-and-state friction laws by Dieterich/Ruina (1983), � 1 − θ V ageing law µ ( V , α ) = µ ∗ + a log V + b log θ V ∗ ˙ L , θ ( θ, V ) = − θ V L log θ V V ∗ L slip law L � �� � α � V � µ ∗ + b α �� = a log exp V ∗ a Regularisation by Rice/Ben-Zion (1996) � V � µ ∗ + b α �� ≈ a sinh − 1 exp ≥ 0 2 V ∗ a Another regularisation � V � ≈ µ ∗ + a log + 1 + b α V ∗ Common assumption: Constant normal stress Subduction Zone Simulations with Rate-and-State Friction E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken
References Prototypical one-body problem Γ D Ω Γ N Γ N Γ C σ ( u ) = B ε ( u ) + A ε (˙ u ) in Ω (linear viscoelasticity) ∇ · σ ( u ) + b = ρ ¨ in Ω (momentum balance) u u n = 0 ˙ on Γ C (bilateral contact) σ t = − λ ˙ u , | σ t | = λ | ˙ u | = | σ n | µ ( | ˙ u | , α ) + C on Γ C with λ = 0 for ˙ u = 0 on Γ N , D . . . α = ˙ ˙ α ( | ˙ u | , α ) on Γ C (family of ODEs) Subduction Zone Simulations with Rate-and-State Friction E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken
References Towards spatial discretisation: Weak formulation We get � � � � ρ ¨ u ( v − ˙ u ) + B ε (˙ u ): ε ( v − ˙ u ) + A ε ( u ): ε ( v − ˙ u ) + φ ( v , α ) Ω Ω Ω Γ C � ≥ φ (˙ u , α ) + ℓ ( v − ˙ u ) Γ C for every v ∈ H with H = { v ∈ H 1 (Ω) d : v = 0 on Γ D , v n = 0 on Γ C } or briefly 0 ∈ M ¨ u + C ˙ u + A u + ∂ Φ( · , α )(˙ u ) − ℓ ⊂ H ∗ and α = ˙ ˙ α ( | ˙ u | , α ) a.e. on Γ C Subduction Zone Simulations with Rate-and-State Friction E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken
References Implicit time discretisation Starting point: 0 ∈ M ¨ u + C ˙ u + A u + ∂ 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
References Implicit time discretisation Starting point: 0 ∈ M ¨ u + C ˙ u + A u + ∂ Φ( · , α )(˙ u ) − ℓ α = ˙ ˙ α ( | ˙ u | , α ) After collocation (rate), approximation of ˙ u on [ t n − 1 , t n ] (state): 0 ∈ M ¨ u n + C ˙ u n + A u n + ∂ Φ( · , α n )(˙ u n ) − ℓ n α = ˙ ˙ α ( | ˙ u n − λ | , α ) with ˙ u n − λ = λ ˙ u n − 1 + ( 1 − λ )˙ ( 0 ≤ λ < 1 ) u n (S) Subduction Zone Simulations with Rate-and-State Friction E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken
References Implicit time discretisation Starting point: 0 ∈ M ¨ u + C ˙ u + A u + ∂ Φ( · , α )(˙ u ) − ℓ α = ˙ ˙ α ( | ˙ u | , α ) After collocation (rate), approximation of ˙ u on [ t n − 1 , t n ] (state): 0 ∈ M ¨ u n + C ˙ u n + A u n + ∂ Φ( · , α n )(˙ u n ) − ℓ n α = ˙ ˙ α ( | ˙ u n − λ | , α ) with ˙ u n − λ = λ ˙ u n − 1 + ( 1 − λ )˙ ( 0 ≤ λ < 1 ) u n After time discretisation (rate), determining the flow operator (state) � λ M � τ M + C + τ 0 ∈ u n + ∂ Φ( · , α n )(˙ ˙ u n ) − ℓ n − . . . (R) A λ A α n = Ψ τ ( | ˙ u n − λ | , α n − 1 ) (S) Subduction Zone Simulations with Rate-and-State Friction E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken
References Implicit time discretisation Starting point: 0 ∈ M ¨ u + C ˙ u + A u + ∂ Φ( · , α )(˙ u ) − ℓ α = ˙ ˙ α ( | ˙ u | , α ) After collocation (rate), approximation of ˙ u on [ t n − 1 , t n ] (state): 0 ∈ M ¨ u n + C ˙ u n + A u n + ∂ Φ( · , α n )(˙ u n ) − ℓ n α = ˙ ˙ α ( | ˙ u n − λ | , α ) with ˙ u n − λ = λ ˙ u n − 1 + ( 1 − λ )˙ ( 0 ≤ λ < 1 ) u n After time discretisation (rate), determining the flow operator (state) � λ M � τ M + C + τ 0 ∈ u n + ∂ Φ( · , α n )(˙ ˙ u n ) − ℓ n − . . . (R) A λ A α n = Ψ τ ( | ˙ u n − λ | , α n − 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
References Rate/state coupling t c a p m o u ) | ∈ L 2 (Γ C ) | γ (˙ c , z t i h c s p i L ( S ) solve ODEs ( γ ) T : H → H ( R ) convex minimisation u ∈ H ˙ ( S ) ( γ ) trace map + norm ( R ) α ∈ L 2 (Γ C ) Subduction Zone Simulations with Rate-and-State Friction E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken
References Rate/state coupling t c a p m o u ) | ∈ L 2 (Γ C ) | γ (˙ c , z t i h c s p i L ( S ) solve ODEs ( γ ) T : H → H ( R ) convex minimisation u ∈ H ˙ ( S ) ( γ ) trace map + norm ( R ) α ∈ L 2 (Γ C ) 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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