lagrangian finite element methodslagrangian …...straightforward method to deal with visco-elastic...
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Lagrangian Finite Element MethodsLagrangian Finite Element Methodsfor modelling long-term tectonic processes
Boris KausBoris Kaus ETH Zurich & USC, Los Angeles
Yolanda Deubelbeiss (ETH), Clare Steedman (USC),
Thorsten Becker (USC), Stefan Schmalholz (ETH)
UC Davis, 07.2008
OutlineOutline
Visco-elasto-plastic vs. viscous formulations.
2D & 3D L i FEM d2D & 3D Lagrangian FEM codes.
Code accuracyCode accuracy.
Issue with free surface & possible ways around.Issue with free surface & possible ways around.
Some applications.
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Governing equationsGoverning equationsmass ofon conservati 0
P
xv
i
i
∂∂
=∂∂
( ) energyofonconservatic
momentum ofon conservati
HTkDT
gxx
P
elijijij
ij
ij
i
−++⎟⎟⎞
⎜⎜⎛ ∂∂
=
=∂
∂+
∂∂
−
εεχτρ
ρτ
&&( )
ratestrain 21
energy ofon conservati cp
xv
xv
Hx
kxDt
i
j
j
iij
ijijijii
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂∂
=
++⎟⎟⎠
⎜⎜⎝ ∂∂
ε
εεχτρ
&
21
21
rheology plastic-elasto- visco
DtD
Gpl
ijij
ijij
plij
elij
visijij
ij
++=
++=⎠⎝
ετ
τη
ε
εεεε
&&
&&&&
) way that asuch (in otherwise,0
if ,0
22 DtG
yieldII
yieldIIpl
ij
⎞⎛⎩⎨⎧
=><
=ττ
ττε
η
&
e.g., Moresi et al., PEPI (2007)
( )
( )( ) density of dependence-T 1
viscositypowerlawfor e.g, exp
00
1
TTnRTQB n
n
II
−−=
⎟⎠⎞
⎜⎝⎛=
−
αρρ
εη &
07.2008 Geophysical Fluid Dynamics/www.gfd.ethz.ch 3
PEPI (2007)-> Possible to extend it to (elastic) compressible cases. Might be important for initiation of shear bands.
Rheology & viscoelastic discretizationRheology & viscoelastic discretizationrheology plastic-elasto- visco
21
21
++= plij
ijijij Dt
DG
ετ
τη
ε &&
11
rheology plastic-elasto- visco21
21 '
⎟⎞
⎜⎛ ∂∂⎟
⎞⎜⎛ ∂∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛+
∂
∂+= pl
ijijij
ijij
vvvv
tG
τ
εττ
τη
ε &&
21
21 e wher
rotationadvection
'⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−∂∂
−⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
−∂∂
+∂∂
= kji
k
k
i
k
j
j
kik
k
ijkij x
vxv
xv
xv
xv ττ
ττ
4444444 34444444 21321
21
21 ' +
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛+
Δ
−+= pl
ijold
ij
oldnewnew
ij tGijij
ijετ
τττ
ηε && Implicit time discretization
( ) ( )1
2 'Δ−+−=
⎠⎝
oldij
oldeff
plijijeff
new tijij
ττχεεητ &&
0,:G if ;1
1,1
==∞→Δ
+=
Δ+
= effeffeffeff tGtG
χηη
η
χηηη
Vi l ti f l ti i il t i
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=> Visco-elastic formulations are similar to viscous formulations, except for a few additional terms. See also Furuichi et al.,
J. Comp. Phys (2008)
What about plasticity?( ) CPφττ +=≤ PragerDruckertan ( )
( ) ( )pl
oldij
oldeff
plijijeff
newYieldII
ijijt
CP
ττχεεητ
φττ
Δ−+−=
+=≤'
itdfl
2
Prager-Drucker tan
&
&&
( )( )( )
oldij
oldeffij
vepeff
new
plij
ijijtττχεητ
ε
⎧
Δ−+= '2
:rewriteand for solve
&
( )( )YieldII
oldII
oldIIeffYield
vepeff II
tττ
ττηε
ττχτη >
⎪⎩
⎪⎨
⎧
≤
Δ−−= Initial
Initial
iter
'
if ,if
2with &
Yieldeff IIττη⎪⎩ ≤if ,
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Visco-elasto-plastic vs. viscous models( ) plastic-elasto-Visco 2 'old
ijold
effijvepeff
new tττχεητ & Δ−+= ( ) Viscous 2
p
ijnew
ijeffijeff
ij
ijij
εητ
χη
&=
Conservation of momentum:
( ) Viscous 2 iij gxx
P ρεη =∂∂
+∂∂
− &
Conservation of momentum:
( ) ( )( ) oplasticViscoelast 2 'oldij
oldeff
jiij
vepeff
ji
ji
tx
gxx
P
xx
ijττχρεη Δ−
∂∂
−=∂∂
+∂∂
−
∂∂
&
Adding elasticity & plasticity requires only small changes to any viscous flow solver
jji xxx ∂∂∂
changes to any viscous flow solver.
Main additions go to the right-hand-side.
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Iterations for non-Newtonian viscosity & plasticity.6
FEM CODES
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Finite element implementationIsoparametric elements
Numerical integration (Gaussian integration pts).
Mixed velocity-pressure.
Typically quadratic shape func. for velocity & linearTypically quadratic shape func. for velocity & linear
discontinuous for pressure.
Stokes system of equations.
Solve with iterated penalty method if using direct solverSolve with iterated penalty method if using direct solver.
Uzawa iterations if using iterative/MG solvers (as in
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CitCOM).
8
Advantages of Lagrangian codesAdvantages of Lagrangian codesHigh accuracy @ low resolution.Straightforward method to deal with visco-elastic problems (advection of stress tensors is donestress tensors is done automatically).Self consistent free surfaceSelf-consistent free-surface.
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Kaus & Schmalholz (2006)
Disadvantages:Remeshing required for large deformations (expensive). g q g ( p )
Remeshing sampling bias: oscillations might occur if
d it fi ld h ti l th h t d idensity fields change continuously throughout domain
(e.g., in convection problems) .
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R hi & tRemeshing & tracersIn some cases with contourlines.Typically through passive tracers.
contain information about stresses, strains, material properties, ...p p
Tracer advection scheme:(a) Find natural coordinates of a tracer inside
an element. (expensive).(b) Advect FEM grid: tracers are automatically
advected with elements (free)advected with elements. (free).(c) Before remeshing: compute real
coordinates of tracer as well as properties such as stress tensors etc. (rel. cheap).
(d) After remeshing: compute new element properties from tracers & integration points
07.2008 Geophysical Fluid Dynamics/www.gfd.ethz.ch
properties from tracers & integration points of old mesh. (expensive).
11
Poliakov & Podladchikov,GJI,1992
Sl MSloMo• Maxwell viscoelastic + Mohr Coulomb frictional plasticity.
• Velocity-pressure formulation
• Written in Fortran90.
• Direct solver• Direct solver
• Quadratic shape functions for V, linear discontinuous for P.
• Quadrilateral (2D) or brick (3D) isoparametric elements.( ) ( ) p
• Iterated penalty method for incompressibility
• Implicit plasticity.
• Markers to track material properties.
• Remeshing for large deformations.
• True free surfaceKaus et al. (in press)
• True free surface.
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Buiter et al. (2006)
LaMEM – Lithosphere and Mantle Evolution model3D parallel finite element code.Based on PETSc, written in C3D brick elements (linear & quadratic).T b d t i l tiTracer-based material properties.Visco-(elasto-plastic).Direct iterati e & m ltigrid sol ersDirect, iterative & multigrid solvers.Lagrangian, Eulerian, ALE mode.Under developmentUnder development.
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~1 Mio dof
2D St k
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• Less than 500 lines of code.2D Stokes
MILAMIN_VEPPowered by MILAMIN.
Di t lDirect solver
Unstructured finite elements.
Quadratic Crouzieux-Raviart elements.
Lagrangian, Eulerian, ALE mode.
Free surface.
M ll i l t l ti h lMaxwell visco-elasto-plastic rheology
Thermo-mechanical coupling
Tracer-based or contour-based material propertiesp p
Phase transitions.
Set of MATLAB routines.
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ACCURACY OF CODES IN THE PRESENCE OF STRONG JUMPS INPRESENCE OF STRONG JUMPS IN
VISCOSITY
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Accuracy of codes (1)
• Compared 3 finite difference codes & MILAMINMILAMIN.• Differences in accuracy between finite difference codes is small.• Averaging of material properties inside control• Averaging of material properties inside control volume is however extremely important.• Harmonic averaging gives the best results.
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Deubelbeiss & Kaus (in press)
A (2)Accuracy (2)
N l t i iNo element-wise averaging
ce D
iffer
encArithmetic averaging
Harmonic averaging Fin.
Body-fitted mesh
• Best results if body-fitted mesh is used
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Deubelbeiss & Kaus (in press)
• If viscosity jumps occur inside element, element-wise averaging should be applied.
FREE SURFACE & MANTLE LITHOSPHERE INTERACTIONLITHOSPHERE INTERACTION
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Mantle-lithosphere interaction & free surfaceStart with a lithospheric setup (Continent, subducting slab and oceanic plate).
We push the oceanic plate from the left
Ocean ContinentFree surface
Subducting plate
Asthenosphere
Initial topography is zero -> initially the model will respond
with trying to find “isostatic equilibrium”
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with trying to find isostatic equilibrium
Simulations with 500 yrs timestep
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After 10 timesteps
Simulations with 500 yrs timestep
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Simulations with 500 yrs timestep
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Simulations with 5000 yrs timestep
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After 1th timestep
Simulations with 5000 yrs timestep
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After 2nd timestep
Simulations with 5000 yrs timestep
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Simulations with 5000 yrs timestep
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Seem to happens in all implicit codes (e.g., talk tomorrow).
Wh t th bl ?What causes the problem?Density difference between rocks and air is ~10-100 times larger than the density difference between different rocklarger than the density difference between different rock types.Small changes in surface topography cause relative largeSmall changes in surface topography cause relative large changes in stress in mantle.The time-step results in a topography that is isostatically outThe time step results in a topography that is isostatically out of balance.The code attempts to recover isostatic balanceThe code attempts to recover isostatic balance.ρ~0 kg/m3(air)
ρ~3200 kg/m3 ρ~3250 kg/m3
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SSome solution strategies:Treat free surface movement fully implicit (e.g, Cuvelier et al. 1987).
Add nodal coordinates of free surface as additional dof’s.Stiffness matrix larger & asymmetric.Nonlinear system of equations.
Sub time-steppingDivide every time step in 40-50 smaller time stepsOnly update buoyancy forces during a sub-time step (affects rhs only).
Semi-Implicit Free Surface Algorithm (SIFSA)Add a (normal) force to surface elements, which is proportionalto future displacement ΔA.to future displacement ΔA.Can be formulated in an implicit manner.Simple addition to existing code!Allows 10-20 times larger time steps.g
GvfffAv
Δ=+=
force
force
t
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fvGA =Δ− )(force
t
And he’s sober again…
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Direct numerical simulation of 2-phase flowQuasi-instantaneous modelling of Direct numerical simulation of 2 phase flowQuasi instantaneous modelling of lithospheric deformation
Cartoon modelData Priestley etCartoon model Data
Numerical model setup
Priestley et al. (2008)
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PhD project of Yolanda Deubelbeiss
Lagrangian FEM codes are very flexible tools that can be Summary
used to solve a wide range of geodynamic problems.
Addi l ti it & l ti it i l ( l ti l ) llAdding elasticity & plasticity involves (relatively) small
changes to existing viscous FEM codes.
FEM codes can be more accurate then finite difference
methods but also less accurate if strong jumps inmethods, but also less accurate if strong jumps in
viscosity occur. Be careful!
Modelling lithosphere-mantle-surface processes is
challenging because of the drunken seaman problem.
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challenging because of the drunken seaman problem.
Solutions exist, however.32
Dabrowski, Krotkiewski & Schmid, G-cubed (2008)
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G cubed (2008)
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