Multi-Scale Behaviour in the Geo-Science II: Introduction to Continuum Mechanics and

Geological Applicationsby

Hans Mühlhaus

The Australian Computational Earth Systems Simulator

(ACcESS)

Overview

Introduction1D relationships

Statics and kinematicsStress, Strain, Stretching, Spin, Objective Variables,Constitutive Relationships, stress equilibrium

Level set methodOutline, upwinding (Taylor Galerkin), two step methods, examples

Exercises

Displacement, Strain and Stretching

02

222 x

u

),( 022

022 txuxx

L0 (initial length)

u2 (displacement)L=L0+u2 (length after application

of force f2)

f2 (Force)

Strain:

x2

x1

Stretching:

2

222 x

vD

Where

t

txuv

),( 0

222

1D Force and Stress Equilibrium

31222 dxdxF

3212

22

,2

dxdxdxdt

ud

ttu

3212 dxdxdxg

x1

x2

312222

222 )(

2,22

dxdxdxx

F

2dx

1dx

The sum of all vertical forces must vanish for force equilibrium:

0))(( 32122

2

22,22 dxdxdxudt

dg

Constitutive Relationship(Hooke’s law)

2,222 Eu

Thus 0)( 22

2

222,2 udt

dgEu

1D Constitutive Relations and Balance Equations

More Constitutive Relationships:Newtonian Creep

2,222 v

0)( 2222,2 vdt

dgv

Insert into stress equilibrium:

Considering the definition of the material time derivative:

22,22,222,2/ gvvvv t

1D Constitutive Relations and Balance Equations

0)( 2,22,2, RqCvC t

22,22 dxqq

2q

x1

x2

C is a Concentration, R is a Reaction Term (e.g. Mass Source), qi Flux of Concentration

,R

Cdt

dC

x3

0)( 3212,22,2 dxdxdxRqCvC

Assumption:

2,2 Cq Thus: )()()( 2,2,2,2, CRCCvC t

Example 2: Heat Equation

)(1

)(1

)( 22222,2,2,22, HDc

kTc

TvTpp

Heat source (radioactive decay)

Heat capacity

Thermal conductivity

Change of concentration due to change of size

Governing Equations

0)(0)( ,222,22 iijij vgvf

332211

3,32,21,1,

babababa ii

iiijij

ijijjjp

jjt DkTc

TvT

,,,, )(1

)(

Einstein’s summation convention:

Stress Equilibrium:

Heat Equation:

Stretching:

)(2

1,,2,222 ijjiij vvDvD

3,3

2,33,22,2

1,33,11,22,11,1

)(2

1

)(2

1)(

2

1

vsymm

vvv

vvvvv

Dij

Governing equations

0))(( ,,,

izijlkjkiljlik gpv

ij

2,,,, )()( jjjjtp TTvTc

)(0

0TTe

Melt viscosity for magma with 0.65% water content

Here we use:

RT303.2

21041.2)(log10

with KTKPas 1000,025.0,104 018

0

1.0E+06

1.0E+07

1.0E+08

1.0E+09

1000 1050 1100 1150 1200

Temperature (Kelvin)

Me

lt v

isc

os

ity

(P

a s

)

exp(25291.5/T - 5.55)

4.0e8*exp(-(T-1000)*0.025)

Heat Equation

Stress Equilibrium

Satellite Image of Volcanic Event

22ndnd talk: Volcano modelling talk: Volcano modelling

Montserrat, West Indies

Intraplate: HotspotsIntraplate: Hotspots

• Anomalous areas of volcanism

• Mantle plumes– Ocean: low-viscosity

basaltic magmas, Hawaiian Islands

– Continental: high silica (high viscosity) rhyolites, Yellowstone

• Little information on magma source

VolcanoVolcano FactsFacts• 1511 known

eruptions in last 10000 years

• 238000 deaths in last 400 years

• Biggest eruption: Yellowstone, USA (2500km3)

• Potential problem: Vesuvius, Italy

Poorly understood natural phenomena with approximately 30 eruptions in any given year.

Volcanoes also produce many natural resources such as important minerals and metals.

Generic VolcanoGeneric Volcano

• Magma chamber at depth (5 – 60km)

• Plumbing from chamber to surface not well constrained

• 800 to 1200 degree C• Changes in regional stress,

earthquakes, can cause the volcano to erupt

• New eruption from exertion of magma forces, increased gas pressure or both

• Long term activity governed by rate of supply of new magma

• Different styles of volcanoes relate to different hazards

Physical properties of magmaPhysical properties of magma

• Magma = melt + crystals + gas.• Melt: Temperature 800-1300 оС, pressure 103 -10-1 MPa• Crystals: size 10-7-10-1 m, number density up to 1017 m-3,

fraction up to 95 %• Gas: H2O - 60-95%, CO2- 0-35%, mass fraction 0.1-7 %

• Melt viscosity 102 -1012 Pa•sBulk viscosity depends upon:•Chemical composition

- more SiO2 → higher viscosity•Temperature

- higher temperature → lower viscosity•Water content

- higher content → lower viscosity•Crystal content

- higher content → higher viscosity

Dome Growth StylesDome Growth Styles

Axisymmetrical lava dome Platy lava dome

Ross Griffiths & Jonathan Fink

Lobate lava dome Spiny lava dome

t=0

H

lava

lava

air air

)0,0(v )0,0(v

t>0

r=0 r=0r=R r=Rz=0 z=0

.constzz .constzz

D/2

D/2

The Level Set Method: Presentation• Implicit representation of the interface by the zero level set of a smooth function φ

• φ is usually a “signed” distance function

• At each time step, φ is updated solving the advection equation:

0

vt

The Level Set Method: Solving the advection equation (1/4)

• Explicit

• Implicit

• Taylor Galerkin

)( vt

)(vt

)(2

)(2

vvt

vt

Test:A gaussian is advected in a constant 1D velocity field.

The level set method …continued

)2 2

23

2+ O(Δ(+

t

Δt+

tΔt+=

Advection dominated pde’s need to require special treatment…..upwinding etcTaylor-Galerkin:

jji

i

2

jj

+

xvv

x

Δt+

xvΔt=

2

The level set method …continued

j

t

jt

t+t

xv

Δt=

2

2

2-step alternative to Taylor-Galerkin upwinding (very effective in the presence of diffusion terms….):

j

tt

jttt

xvΔt=

2

Formulation

Finley PDE:

jijijijjkijkjkijkjlkijkl XYvDvCvBvA ,,,,, )()(

Example : Momentum and Heat equation

0))

2(

0)(

,,,

/1

)(

,,,

,

jti

X

ijijtj

C

j

Y

t

tD

tt

Y

i

p

ijlk

A

ijkl

Tvvt

Tvt

T

t

T

RaTgpvE

jj

i

ijX

jijijkl

Davies, M., Gross, L., Mühlhaus, H.–B., 2004, Scripting High Performance Earth Systems Simulations on the SGI Altix 3700, Proc. 7th Intl Conf. on High Performance Computing and Grid in Asia Pacific Region, 244-251.

EScript

• for i in range(numDim):\par

• for j in range(numDim):\par• tau += stress[i,j] * stress[i,j]\par• tau = sqrt(0.5 * tau + small)\par• map["tau"] = tau\par• \par• # tau_Y\par• \par• # release memory\par• # power law\par• Xi_P1 = (tau / tau_0) ** (1 - n1)\par• Xi_P2 = (tau / tau_Ystep) ** (1 - n2)\par• Xi_P = Xi_P1 * Xi_P2 / (Xi_P1 + Xi_P2)\par• map["Xi_P"] = Xi_P\par• \par• # release memory\par• del tau_Ystep \par• \par

• # melting temperature\par

• T_M = T_M0 + gamma * p\par

• map["T_M"] = T_M\par• \par•

The Level Set Method: Solving the advection equation (2/4)

Taylor Galerkin:The gaussian keeps

its shape.

Implicit:The gaussian is deformed in the direction of the velocity field.

Level set applied to a cantilever beam

ijkksb

ijs

ijkkijij BD

2

1

3

1

3

1

2

1~2

1

3

1

3

1~2

1

Presentation of the test case:

Constitutive relationship:

Stress equilibrium:

2

7

6

10

10

10

=η

=η

=μ

medium

beam

beam

0, ijij g

Level set applied to a cantilever beam

Level set applied to a cantilever beam

Influence of contributions to stress rate:

Oldroyd stress rate:

termsrotco

kjkikjikij

advection

kijkij vvvt

.

,,,~

Level set applied to a cantilever beam

Accuracy of the method:

Conservation of volume

The Level Set Method: Solving the advection equation (3/4)

Previous test:No topological change in the

solution

Need for a new test with:

0x

v0

y

vand

New test: shearing flow

)cos()sin(

)sin()cos(

xyv

xyv

y

x

Mesh: 100x100Courant Number: 0.25•1000 steps forward•1000 steps with -v

The Level Set Method: Solving the advection equation (4/4)The shape gets “noisy”…

Problem:φ

looses its distance function property

Reinitialisation needed!

The Level Set Method: Reinitialisation (1/3)Idea:

Rebuild a “signed” distance function ψ from the distorted function φ

Requirements:• The interface must not be changed

• ψ must represent a distance function

Solution:Solve to steady state the equation:

Rewritten as:

00

1

)1)(( 0

sign

)( 0

signw

wwith

Interpretation:The “distance information” is carried by w, a unit vector

pointing away from the interface.

The Level Set Method: Reinitialisation (2/3)

1D2D

3D

The Level Set Method: Reinitialisation (3/3)

Same test as before, with

reinitialisation

The Level Set Method: Benchmarks (1/2)

Axisymmetrical case:A fluid is submitted to a centrifugal force

Interest:The analytical steady state is known (grey line)

Parameters:mesh:20x20, density of air: 0 kg/m3, density of fluid: 103 kg/m3

Results:

The Level Set Method: Benchmarks (2/2)

Rayleigh-Taylorinstability

Level set cont. : Merger of small and large bubbles

4

222

411

10

10,1800

10,3160

11,

121 )()( jjjjijij nnn

0

1 /

gradgradn

Parameters:

Surface tension:

10

121 )()( nnσσ graddiv

Calculation, includes inertia, implicit, CourantNumber=0.5, msh:30 by 458 node quad’s

Level set cont. : Merger of small and large bubbles

Level set cont. : Merger of small and large bubbles

Level set cont. : Merger of small and large bubbles

t=0

H

lava

lava

air air

)0,0(v )0,0(v

t>0

r=0 r=0r=R r=Rz=0 z=0

.constzz .constzz

D/2

D/2

Axisymmetric volcano

Axisymmetric volcano (T= 1000-1001)

ExercisesExercises

const

gh0

R

z

r

22r

A cylindrical container of radius R is filled initially to height h0 with an incompressible fluid of density and viscosity. The container is then rotated around his axis at a constant spin Determine the steady state position of the free surface of the fluid.

Exercise5Infinite vent: Hagen-Poiseulle flow

GrerrTr

T Trrt

2,,, )(

1

,,),( **2

0* Dxxt

cDtTTT p

)(2

1)( 0

2,

)(0

0 pgrver zrzTT

2,

)(0,,,

0)(1

rzTT

prr

pt ve

crT

rcT

Stress Equilibrium

Heat Equation

Dimensionless form

Here

0

420

4

)( DpgGr z

D

z

r