Numerical modeling of dynamic rupture propagation

Transcription

1 IX. Slovenská geofyzikálna konferencia, jún 2011 Fakulta matematiky, fyziky a informatiky UK, Bratislava Numerical modeling of dynamic rupture propagation Martin Galis Peter Moczo Jozef Kristek Christian Pelties Comenius University Bratislava, Slovakia Slovak Academy of Sciences, Bratislava, Slovakia Ludwig-Maximiliams-Universität München, Germany

2 introduction FEM (Finite-Element Method) TSN (Traction at Split Nodes) approach and its implementation in FEM

3 introduction FEM (Finite-Element Method) TSN (Traction at Split Nodes) approach and its implementation in FEM ADER-DGM (Arbitrarily high-order DERivative Discontinuous Galerkin Method) implementation of the frictional boundary conditions in ADER-DGM

4 introduction FEM (Finite-Element Method) TSN (Traction at Split Nodes) approach and its implementation in FEM ADER-DGM (Arbitrarily high-order DERivative Discontinuous Galerkin Method) implementation of the frictional boundary conditions in ADER-DGM 3D numerical comparison FEM-ASA vs ADER-DGM effect of the numerical nucleation zone on the rupture propagation

5 FEM (Finite-Element Method)

6 displacement formulation of equation of motion - D-EqM ( ) ρu = σ, σ = λδ u, + μ( u, + u, ) i ij j ij ij k k i j j i

7 displacement formulation of equation of motion - D-EqM ( ) ρu = σ, σ = λδ u, + μ( u, + u, ) i ij j ij ij k k i j j i choose an element with n nodes e.g. tetrahedron with 4 nodes node

8 displacement formulation of equation of motion - D-EqM ( ) ρu = σ, σ = λδ u, + μ( u, + u, ) i ij j ij ij k k i j j i choose an element with n nodes e.g. tetrahedron with 4 nodes node choose shape functions and approximation to displacement in the element k k ( ) ( ) ( ) u x, t = s x U t ; k = 1,..., n i j j i k unique displacements U i at nodes continuity of displacement u i at a contact of elements

9 multiply D-EqM by the shape functions k ρus = σ, s ; k= 1,..., n i ij j k

10 multiply D-EqM by the shape functions k ρus = σ, s ; k= 1,..., n i ij j k integrate over an element Ω e i k ρusdv = Ω e σ, ij j k sdv

11 multiply D-EqM by the shape functions ρus = σ, s ; k= 1,..., n i k ij j k integrate over an element Ω e i k ρusdv = Ω e σ, ij j k sdv integrate the r.h.s. by parts e i k k k e ij j e Ω Ω Ω i ρusdv = σ s, dv + Ts ds

12 multiply D-EqM by the shape functions ρus = σ, s ; k= 1,..., n i k ij j k integrate over an element Ω e i k ρusdv = Ω e σ, ij j k sdv integrate the r.h.s. by parts e i k k k e ij j e Ω Ω Ω i ρusdv = σ s, dv + Ts ds e e e M u = r + surface-traction boundary term local mass matrix local vector of nodal displacements local vector of restoring forces

13 e e e M u = r + surface-traction boundary term assemble all elements covering volume Ω closed by surface Ω= ΩFS Ω Ω Dirichlet M u = r + 0 global mass matrix global vector of nodal displacements global vector of restoring forces

14 e e e M u = r + surface-traction boundary term Ω e k Ts ds i assemble all elements covering volume Ω closed by surface Ω= ΩFS Ω Ω Dirichlet M u = r + 0 global mass matrix global vector of nodal displacements global vector of restoring force the theoretical boundary term vanishes - at a contact of two elements due to traction continuity - at the free surface due to zero traction

15 e e e M u = r + surface-traction boundary term Ω e k Ts ds i assemble all elements covering volume Ω closed by surface Ω= ΩFS Ω Ω Dirichlet M u = r + 0 global mass matrix global vector of nodal displacements global vector of restoring force the theoretical boundary term vanishes - at a contact of two elements due to traction continuity - at the free surface due to zero traction in fact, however, the final discretization does not give - the traction continuity at a contact of two elements - zero traction at the free surface and thus the zero boundary term in the global equation they are just low-order approximated

16 TSN (Traction at Split Nodes) approach and its implementation in FEM

17 elastic halfspaces H - and H + do not interact H - H traction-free surface traction-free surface

18 elastic halfspaces H - and H + do not interact H - H traction-free surface traction-free surface halfspaces coupled by a constraint surface traction n c T { } { } a F () t AT c = + ( n,) t m - + fault + + a F () t AT c + = ( n,) t m

19 enforcement of the frictional boundary condition on the fault if T () t S() t ct sh then c ct T () t = T () t no slip

20 enforcement of the frictional boundary condition on the fault if T () t S() t ct sh then c ct T () t = T () t no slip if T () t > S() t ct sh then T c f () t T = () t sh sh c ct T () t = T () t n n + m + m ct f Dv ( t + dt 2) = A T T dt + mm

21 we apply our adaptive smoothing algorithm (ASA) to the trial traction before it is used for updating the slip rate in order to reduce spurious high-frequency oscillations of the slip rate

22 ADER-DGM (Arbitrarily high-order DERivative Discontinuous Galerkin Method)

23 velocity-stress formulation of equation of motion - VS-EqM σ λv, δ μ( v, v, ) = 0 ij k k ij i j j i ρv i σ, = 0 ij j

24 velocity-stress formulation of equation of motion - VS-EqM σ λv, δ μ( v, v, ) = 0 ρv ij k k ij i j j i i σ, = 0 ij j define a vector of unknown variables ( σxx, σyy, σzz, σxy, σyz, σzx, x, y, z) Q = v v v T

25 velocity-stress formulation of equation of motion - VS-EqM σ λv, δ μ( v, v, ) = 0 ρv ij k k ij i j j i i σ, = 0 ij j define a vector of unknown variables ( σxx, σyy, σzz, σxy, σyz, σzx, x, y, z) Q = v v v T VS-EqMin thematrixform Q + A Q, + B Q, + C Q, = 0 p pq q x pq q y pq q z A, B, C space-dependent matrices include material properties

26 velocity-stress formulation of equation of motion - VS-EqM σ λv, δ μ( v, v, ) = 0 ρv ij k k ij i j j i i σ, = 0 ij j define a vector of unknown variables ( σxx, σyy, σzz, σxy, σyz, σzx, x, y, z) Q = v v v T VS-EqMin thematrixform Q + A Q, + B Q, + C Q, = 0 p pq q x pq q y pq q z A, B, C space-dependent matrices include material properties tetrahedral element (e.g.) ( Q ) = Qˆ ( t) Φ ( x ) h p pk k j polynomial basis functions of an optional degree

27 multiply VS-EqM by a test function and integrate over an element volume Ω e ( ) Q Φ dv A Q, + B Q, + C Q, Φ dv = 0 p k e pq q x pq q y pq q z k Ω

28 multiply VS-EqM by a test function and integrate over an element volume Ω e ( ) Q Φ dv A Q, + B Q, + C Q, Φ dv = 0 p k e pq q x pq q y pq q z k Ω integrate the 2nd integral by parts Ω e (,,, ) Q Φ dv Φ A + Φ B + Φ C Q dv p k e k x pq k y pq k z pq q Ω + Φ F ds = Ω e k p 0 numerical flux introduced Q h because may be discontinuous at an element boundary

29 multiply VS-EqM by a test function and integrate over an element volume Ω e ( ) Q Φ dv A Q, + B Q, + C Q, Φ dv = 0 p k e pq q x pq q y pq q z k Ω integrate the 2nd integral by parts Ω e (,,, ) Q Φ dv Φ A + Φ B + Φ C Q dv p k e k x pq k y pq k z pq q Ω + Φ F ds = Ω e k p 0 numerical flux introduced because Q h may be discontinuous at an element boundary Riemann problem an evolution physically continuous problem with initial discontinuous approximation of unknowns across an interface to find a flux such that continuity of particle velocity and traction at an element boundary is assured

30 implementation of the frictional boundary conditions in ADER-DGM

31 imposing frictional boundary condition at S (the 2D example) frictional traction G = + imposed at S σ σ δσ xy xy xy

32 imposing frictional boundary condition at S (the 2D example) frictional traction G = + imposed at S σ σ δσ xy xy xy consequently G cs Q ( S ± ) = Q + 0, 0, δσ xy, 0, ± δσ xy μ and thus G cs G v y ± = vy ± ( σ xy σ xy) μ T

33 imposing frictional boundary condition at S (the 2D example) frictional traction G = + imposed at S σ σ δσ xy xy xy consequently G cs Q ( S ± ) = Q + 0, 0, δσ xy, 0, ± δσ xy μ and thus G cs G v y ± = vy ± ( σ xy σ xy) μ the slip rate is ( ) 2cs G y xy xy Δ v = σ σ μ T

34 imposing frictional boundary condition at S (the 2D example) frictional traction G = + imposed at S σ σ δσ xy xy xy consequently G cs Q ( S ± ) = Q + 0, 0, δσ xy, 0, ± δσ xy μ and thus G cs G v y ± = vy ± ( σ xy σ xy) μ the slip rate is ( ) 2cs G y xy xy Δ v = σ σ μ T all this means that the imposed frictional traction, G σ xy different from, instantly and locally generates an imposed slip rate parallel to the fault σ xy

35 the problem of the spurious high-frequency oscillations [m/s] [s] 7.5

36 the problem of the spurious high-frequency oscillations FEM + f ct mm Tsh = T Da A m + ( + m ) sh - collinearity n T + Dv = v v + fault ADER-DG σ G μ = σ + Δv 2c xy xy y s analogous to the analytical formulation in BIEM μ Δ v 2c s y - instantaneous (high-frequency) stress response to the slip rate + antiparallelism n T + Dv = v v + fault

37 3D numerical comparison FEM-ASA vs ADER-DGM planar fault receivers nucleation zone R1 R3 R2 7.5 km 6 km direction of initial shear traction friction law: linear slip weakening

38 3D numerical comparison FEM-ASA vs ADER-DGM planar fault receivers nucleation zone R1 R3 R2 7.5 km 6 km direction of initial shear traction friction law: linear slip weakening configuration sub-rayleigh supershear initial traction τ MPa 70.0 MPa static friction τ s MPa 73.5 MPa dynamic friction τ d MPa 63.0 MPa critical distance 0.4 m m strength par. S = ( τ τ ) ( τ τ ) s 0 0 d square nucleation zone, side length 3 km 3 km

39 3D numerical comparison FEM-ASA vs ADER-DGM order # of integ. points element size element type ADER-DG O m TET ADER-DG O m TET ADER-DG O m TET FEM-ASA O m HEX FEM-ASA O m HEX FEM-ASA O m HEX

40 sub-rayleigh rupture - slip-rate receiver R1 (antiplane) x-component receiver R2 (inplane) x-component receiver R3 x-component receiver R3 y-component

41 supershear rupture - slip-rate receiver R1 (antiplane) x-component receiver R2 (inplane) x-component receiver R3 x-component receiver R3 y-component

42 effect of the numerical nucleation zone on the rupture propagation simulations by FEM-ASA nucleation zones compared square with an abrupt change in traction for example SCEC ellipse with a smooth change in traction τ s + overshoot τ s τ 0 L

43 no rupture super-shear square nucleation zone sub-rayleigh 3.00 dimensionless size of nucleation zone L / L A strength parameter S = ( τ τ ) ( τ τ ) s 0 0 d

44 no rupture no rupture super-shear square nucleation zone super-shear elliptic nucleation zone sub-rayleigh sub-rayleigh 3.00 dimensionless size of nucleation zone L / L A strength parameter S = ( τ s τ0) ( τ0 τd )

45 no rupture no rupture super-shear square nucleation zone super-shear elliptic nucleation zone sub-rayleigh sub-rayleigh 3.00 dimensionless size of nucleation zone L / L A L L L D A / L / L A A L / U& R A strength parameter S = ( τ s τ0) ( τ0 τd )

46 thank you for your attention