lagrangian finite element methodslagrangian …...straightforward method to deal with visco-elastic...

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.

07.2008 Geophysical Fluid Dynamics/www.gfd.ethz.ch 2

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 &


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


=> 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 ,


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.

07.2008 Geophysical Fluid Dynamics/www.gfd.ethz.ch

Iterations for non-Newtonian viscosity & plasticity.6

FEM CODES


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


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.


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) .


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


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.


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.


~1 Mio dof

2D St k


• 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.


ACCURACY OF CODES IN THE PRESENCE OF STRONG JUMPS INPRESENCE OF STRONG JUMPS IN

VISCOSITY


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.


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


Deubelbeiss & Kaus (in press)

• If viscosity jumps occur inside element, element-wise averaging should be applied.

FREE SURFACE & MANTLE LITHOSPHERE INTERACTIONLITHOSPHERE INTERACTION


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”


with trying to find isostatic equilibrium

Simulations with 500 yrs timestep


After 10 timesteps



After 1th timestep



After 2nd timestep



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


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


fvGA =Δ− )(force

t

And he’s sober again…


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)


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.


challenging because of the drunken seaman problem.

Solutions exist, however.32

Dabrowski, Krotkiewski & Schmid, G-cubed (2008)


G cubed (2008)

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lagrangian finite element methodslagrangian …...straightforward method to deal with visco-elastic...

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