Coupled High-Order Finite Difference and Unstructured Finite Volume Methods for Earthquake Rupture Dynamics in Complex Geometries

Landers fault data provided by Yann Klinger (IPGP)

Ossian O’Reilly (osox9609@student.uu.se)

(1)

, Jeremy E. Kozdon

(2)

, Eric M. Dunham

(2)

, Jan Nordström

(3)

(1) Department of Information Technology, Uppsala University; (2) Department of Geophysics, Stanford University; (3) Department of Mathematics, Linköping University.

Introduction

Unstructured grid methods are well suited for earthquake problems in complex geom

-tors and the SAT penalty method (simultaneous approximation term) it is possible to

method alone. Another advantage of the SBP and SAT method is that it is possible to prove strict stability, meaning that the semi-discrete solution dissipates energy at a slightly faster rate than the continuous solution so that the error remains bounded in time, which is par-ticulary useful for long time computations.

Conclusions

long time integrations and complex geometries in linear elastodynamics . Characteristic formulation of non-linear boundary conditions has been used

purely velocity dependent friction laws. velocity,

stresses, density,

shear modulus,

shear wave speed, shear impedance. rigid elastic y x -izing governing equations with respect to

SBP-SAT References

Carpenter et al., Journal of Computational Physics, 1999

Nordström & Gustafsson, , 2003.

Nordström, , 2007.

Linear elastic anti-plane problem with non-planar fault

Linear elastic anti-plane problem derived from momentum conservation and Hooke’s law

ρ∂q t = Ax ∂q ∂x + Ay ∂q ∂y , q = ρ 2vz ρ √ 2Gσxz ρ √ 2Gσyz T ρ∂vz ∂t = ∂σiz ∂xi ∂σiz ∂t = G ∂vz ∂xi Ax =  c0 cs 0 0s 0 0 0 0  , Ay =  0 0 c0 0 0s cs 0 0   wave fault ⇒ b C ˆ n w± = niσiz ± Zvz

Ω

and energy dissipation rate

exterior boundaries d q 2 dt = _∂Ω\C vzσiznids − _C V τ ds ≤ 0       V = vz(xf, yf, t) τ = _−niσiz(xf, yf, t) τ = F (V )

Fault friction applied along the curve where

C : (xf(r), yf(r)), r : [a, b] (xf(a), yf(a)) and (xf(b), yf(b)) endpoints.

outward unit normal

area: ∆VA

Time stability

Grid refinement study

Rate of convergence is measured using the L norm on the entire computa- -tional domain (unstructured and structured region)

v = [q(1), q(2)]T

N1 number of nodes on the coarsest grids,

( unstructured grid + structured grid) discrete solution.

µ = log10 u − v1 2/log10 u − v2 2 log10(√N1/√N2)

Estimated rate of convergence

u piece-wise continuous solution

B : (xB, yB)

Characteristic boundary treatment

Friction law enforced weakly using the SAT penalty method

(simultaneous approximation term) where the boundary solutions are penalized for not satisfying the boundary conditions. Properly choosen penalty parameters for a SBP scheme leads to strict stability, which

means that the discrete model dissipates energy slighly faster than

the continuous model. A characteristic formulation of the fault boundary condition is neccessary to avoid stiffness.

Kronecker product between matrices,

projects penalities so that only grid points on the fault are penalized.

First derivative approximation with respect to x,

The penalty parameter depends on the discretization,

Node-centered Finite volume method

Governing equations are integrated and transformed by Green’s theorem

An unstructured grid is used, with the data stored at the nodes. Control volumes are formed around each node using the midpoints and the centre of gravities of the neighboring triangles.

for the interior node

boundary node , (shown to the right)

1 2 1 2 3 4 ∆yB ρ Ω ∂vz ∂t dxdy = _Ω ∂σiz ∂xi dxdy = ∂Ω σiznids 3 4 5 6

Other spatial derivatives are treated in a similar manner.

The resulting scheme yields summation-by-parts (SBP) operators, which mimic integration by parts in a discrete sense.

∂vz(xA, yA, t) ∂x → 1 2∆VA 6 i=1 (vz)i∆yi ∂vz(xB, yB, t) ∂x → (vz)B∆yB 2∆VB + 1 2∆VB 4 i=1 (vz)i∆yi A σiz Ω ∂σiz ∂t dxdy = G _Ω ∂vz ∂xi dxdy = G ∂Ω vznids A : (xA, yA), (shown above)

Methods are coupled at an interface with co-located nodes for accuracy. Interface conditions are weakly enforced using the SAT penalty method.

Projects penalties so that only grids point on the interface are penalized Maps to the organization of at the interface

Penalty term for the unstructured method

Structured grid, Interface Unstructured grid, E(2)_I Tc(2) q(2) _{− E}(2)_I∗ q(1) slip velocity, frictional resistance,

purely velocity dependent friction law.

w−(xf, yf, t) = W−(w+(xf, yf, t)) q(1) q(2) E(2)_I∗ E(2)_I 5 A

Control volume for node A

∆y2

B

The highest rate of convergence for the hybrid scheme is the convergence rate of the unstructured method. The hybrid method is both more accurate and efficient than the finite volume method alone.

Complex geometry treated by an unstructured mesh, which is coupled to some surrounding structured grids.

ˆ n ˆ n 3rd 2.56 2.16 2.04 1.99 4th 2.73 2.20 2.01 1.94 FVM* 2.04 2.04 2.02 2.00 µ

FVM : Finite volume method

close-up

Coupled High-Order Finite Difference and Unstructured Finite Volume Methods

for Earthquake Rupture Dynamics in Complex Geometries

∂

.

First derivative approximation,

and the

.

ˆ n,

Friction law transformed to characteristic form, reflected wave is determined by interaction of incident wave with a non-linear function

. . 2 , . . -. t ∂ . , .

Time stability is verified numerically by computing the error for long time integrations. As shown below the error remains bounded in time for all of the hybrid schemes.

. , , , , , , , , , , Error 1 _Nodes 2.5 Hybrid 2nd Hybrid 3rd Hybrid 4th FVM 2nd -2.5 0.5 Log₁₀ Log₁₀ Place holder 160 80 0 1 0 -1.5 t Error Log_{10 Hybrid 2nd} Hybrid 3rd Hybrid 4th 2nd 2.12 2.07 2.03 2.00 Hybrid # 1 2 3 4 # : Refinement