Landers fault data
provided by Yann Klinger (IPGP)
dynamics in complex geometries
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 ˆ n w± = niσiz ± Zvz
Ω
, i = x, yby interaction of incident wave with a non-linear function
w− w+ a q 2 = Ω qTqdxdy = Ω (ρ 2v 2 z + 1 2Gσizσiz)dxdy Mechanical energy
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
Strict stability
Strict stability is shown by integrating to long times. The scheme is strictly stable since the error is bounded in time (below)
Convergence
Rate of convergence measured using the l-2 norm on the entire computa-tional domain (unstructured and structured region). Method of manufactured solutions are used to construct a known exact solution.
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
963116
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 numerical scheme leads to strict stability, meaning that the discrete model dissipates energy slighly faster than the continuous model. Characteristic formulation of the fault b.c
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
then discretized using an unstructured grid, with grid
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 are used to prove strict stability by obtaining a discrete energy rate estimate. ∂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 finite volume alone.
Landers fault meshed and coupled with a structured grid close-up 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 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 Log10 Hybrid 2nd Hybrid 3rd Hybrid 4th 2nd 2.12 2.07 2.03 2.00 Hybrid # 1 2 3 4 # : Refinement
