Lecture 2. How to model: Numerical methods

Outline

• Brief overview and comparison of methods

• FEM LAPEX

• FEM SLIM3D

• Petrophysical modeling

• Supplementary: details for SLIM3D

Ax

T

xDt

DTC ijij

iip

)(

ijeff

ij

i

j

j

iij Dt

D

Gx

v

x

v

2

1

2

1)(

2

1

Dt

Dvg

xx

P ii

j

ij

i

Full set of equations

mass

momentum

energy

01

i

i

x

v

Dt

DT

Dt

DP

K

Final effective viscosity

IIPNLeff

/)(2

1

exp

LL L II

HB

RT

exp

n NN N II

HB

RT

))1(exp(P

IIPPP RT

HB

PcII 0 if

0 PcII if

Numerical methods

According to the type of parameterization in space:

FDM, FEM, FVM, SM, BEM etc.

According to the type of parameterization in time:

Explicit, Implicit

According to how mesh changes (if) within a deforming body:

Lagrangian, Eulerian, Arbitrary Lagrangian Eulerian (ALE)

Brief Comparison of Methods

Finite Difference Method (FDM) :

FDM approximates an operator (e.g., the derivative)

Finite Element Method (FEM) :

FEM uses exact operators but approximates the solution basis functions.

FD Staggered grid

Spectral Methods (SM):

Spectral methods use global basis functions to approximate a solution across the entire domain.

Finite Element Methods (FEM):

FEM use compact basis functions to approximate a solution on individual elements.

Brief Comparison of Methods

Explicit vrs. Implicit

),( tXFDt

DX

Explicit vrs. Implicit

),( tXFDt

DX

)2/),2/(()()(

ttttXFt

tXttX

Should be:

Explicit vrs. Implicit

))),(()()(

ttXFt

tXttX

),( tXFDt

DX

Explicit approximation:

Explicit vrs. Implicit

tttXFtXttX ))),(()()(

),( tXFDt

DX

Explicit approximation:

2-D Thermomechanical Modelling

Explicit finite element algorithm

Basic calculational cycle:m ·d /dt = V F F

- solution of full dynamic equation of motion- calculations in Lagrangian coordinates- remeshing when grid is too distorted- no problems with highly non-linear rheology

General model setupComplex visco-elasto-plastic rheologyT=0, =0xz = zzT or , 0 xz zz = , - Archim.force T/z = const T/x=00xz = T/x=00xz = VxVx

Governing equations:

tLAxTxtTC plasticityCoulombMohrordtdG xvKtp gxxptvijijiip ijijij ii ijijiiinert

)(2121

tectoi

iner

iineri

j

ij

i

FDt

Dv

Dt

Dvg

xx

p

,

Dynamic relaxation:

Modified FLAC = LAPEX(Babeyko et al, 2002)

mF

ij ij

ij ij

a = F /m

x, ij ij

ij

F

Explicit finite element method

15

Markers track material and history properties

Upper crust

Lower crust

Mantle

Benchmark: Rayleigh-Taylor instabilityvan Keken et al. (1997)

Sand-box benchmark movie

Friction angle 19°

Weaker layer

Sand-box benchmark movie

2D m odels

2.5D m odels

3D- m odels

Fully 3D m odels

/ x / x / x3 3 1,2

v 3 , / x 3 i j

C urrent s ta te .

no restric tions

v 3 , / x 3 x 1

x 2

x 1

x 2

x 3

x 1

x 2

x 3

x 1

x 2

x 3

v -i ve loc ity vec tor com ponen t s tress tenso r com ponen t, i j -

Lagrangian 2.5D FE

Eulerian FD

x1

x2

x3

Simplified 3D concept.

Explicit method vs. implicit• Advantages

– Easy to implement, small computational efforts per time step.

– No global matrices. Low memory requirements.

– Even highly nonlinear constitutive laws are always followed in a valid physical way and without additional iterations.

– Strightforward way to add new effects (melting, shear heating, . . . . )

– Easy to parallelize.

• Disadvantages– Small technical time-step (order of a year)

Implicit ALE FEM SLIM3D(Popov and Sobolev, 2008)

Physical background

Balance equations

1ˆ Elastic strain:

21

Viscous strain: 2

Plastic strain:

el vs plij ij ij ij

elij ij

vsij ij

eff

plij

ij

G

Q

Deformation mechanisms

Popov and Sobolev (2008)

Mohr-Coulomb

Momentum: 0

Energy:

iji

j

i

i

g zx

qDUr

Dt x

Numerical backgroundDiscretization by Finite Element

Method

Fast implicit time stepping

+ Newton-Raphson solver

Remapping of entire fields

by Particle-In-

Cell technique

Arbitrary Lagrangian-Eulerian kinematical

formulation

Popov and Sobolev (2008)

11

k k k ku u K r

r Residual Vector

rK Tangent Matrixu

Numerical benchmarks

Numerical benchmarks

Numerical benchmarks

Numerical benchmarks

Numerical benchmarks

Numerical benchmarks

Solving Stokes equations with code Rhea

(adaptive mesh refinement)

Burstedde et al.,2008-2010

Mesh refinement: octree discretization

Solving Stokes equations with code Rhea

Stadler et al., 2010

Open codes

CitComCU. A finite element E parallel code capable of modelling thermo-chemicalconvection in a 3-D domain appropriate for convection within the Earth’s mantle.Developed from CitCom (Moresi and Solomatov, 1995; Moresi et al., 1996).

CitComS.A finite element E code designed to solve thermal convection problems relevant to Earth’s mantle in 3-D spherical geometry, developed from CitCom by Zhong et al.(2000).

Ellipsis3D. A 3-D particle-in-cell E finite element solid modelling code for viscoelastoplastic materials, as described in O’Neill et al. (2006).

Gale. An Arbitrary Lagrangian Eulerian (ALE) code that solves problems related toorogenesis, rifting, and subduction with coupling to surface erosion models. Thisis an application of the Underworld platform listed below.

PyLith . A finite element code for the solution of viscoelastic/ plastic deformation that was designed for lithospheric modeling problems.

SNAC is a L explicit finite difference code for modelling a finitely deforming elasto-visco-plastic solid in 3D.

Available from http://milamin.org/.

MILAMIN. A finite element method implementation in MATLAB that is capable of modelling viscous flow with large number of degrees of freedom on a normal computer Dabrowski et al. (2008).

Available from CIG (http://www.geodynamics.org

Ax

T

xDt

DTC ijij

iip

)(

ijeff

ij

i

j

j

iij Dt

D

Gx

v

x

v

2

1

2

1)(

2

1

Dt

Dvg

xx

P ii

j

ij

i

Full set of equations

mass

momentum

energy

01

i

i

x

v

Dt

DT

Dt

DP

K

Final effective viscosity

IIPNLeff

/)(2

1

exp

LL L II

HB

RT

exp

n NN N II

HB

RT

))1(exp(P

IIPPP RT

HB

Ax

T

xDt

DTC IIII

iip

)(

ijeff

ij

i

j

j

iij Dt

D

Gx

v

x

v

2

1

2

1)(

2

1

Dt

DvgTP

xx

P ii

j

ij

i

),(

Full set of equations

mass

momentum

energy

01

i

i

x

v

Dt

DT

Dt

DP

K

Petrophysical modeling

Goals of the petrophysical modeling

To establish link between rock composition and its physical To establish link between rock composition and its physical properties.properties.

Direct problems:Direct problems:prediction of the density and seismic structure (also prediction of the density and seismic structure (also anisotropic)anisotropic)incorporation in the thermomechanical modelingincorporation in the thermomechanical modeling

Inverse problem:Inverse problem:interpretation of seismic velocities in terms of interpretation of seismic velocities in terms of

compositioncomposition

Petrophysical modeling

Internally-consistent dataset of thermodynamic Internally-consistent dataset of thermodynamic properties of minerals and solid solutionsproperties of minerals and solid solutions

(Holland and Powell ‘90, Sobolev and Babeyko ’94)

Gibbs free energy minimization algorithmGibbs free energy minimization algorithmAfter de Capitani and Brown ‘88

Equilibrium mineralogical composition of a rockEquilibrium mineralogical composition of a rockgiven chemical composition and PT-conditions

Density and elastic properties Density and elastic properties optionally with cracks and anisotropy

SiO2

Al2O3

Fe2O3

MgO + (P,T)CaOFeONa2OK2O

Density P-T diagram for average gabbro composition

1- granite, 2- granodiorite, 3- diorite, 4- gabbro, 5- olivine gabbro, 6- pyroxenite, 7,8,9- peridotites

Sobolev and Babeyko, 1994Vp- density diagramVp- density diagram

1- granite, 2- granodiorite, 3- diorite, 4- gabbro, 5- olivine gabbro, 6- pyroxenite, 7,8,9- peridotites

Sobolev and Babeyko, 1994Vp - Vp/Vs diagramVp - Vp/Vs diagram

CA back-arc

CA arc

Basic relationships which link P and S seismic velocities, temperature, A-parameter, attenuation etc

They were obtained from a generalization of different sorts of data: global seismic data, laboratory experiments, and numerical modeling.

The effect of anelasticity on seismic velocities

(Sobolev et al., 1997) 1,,, ),,()2/tan(21),,(),,( TPQTPVTPV SPSPSP

T

PTgAPTQ melt

S

)(exp),,(

1/)1(/ KSp QLQLQ

2

2

3

4

P

S

V

VL

where:P is pressure, T is the absolute temperature, ω is frequency, Tmelt is melting temperature, Qs and Qp are the S and P-

waves quality factors.

Values of parameters estimated from laboratory studies and global seismological researches: g=43.8, A=0.0034, α=0.20, Qk=479 (Sobolev et al. 1996)

Vp and temperatures in the mantle wedge

Vs and temperatures in the mantle wedge

Supplement: details for FEM SLIM3D (Popov and Sobolev, PEPI, 2008)

Finite element discretization

Time discretization and nonlinear solution

Objective stress integration

Linearization and tangent operator