-adaptive hp a fully automatic goal-oriented finite element … · 2011. 2. 10. · c....

Progress Report (Baker-Atlas)

A Fully Automatic Goal-Oriented hp-AdaptiveFinite Element Strategy for

Simulations of Resistivity Logging Instruments.

David Pardo ([email protected]),C. Torres-Verdin, L. Demkowicz, L. Tabarovsky, A. Bespalov

Collaborators: Science Department of Baker-Atlas, J. Kurtz, M. Paszynski

February 22, 2005

Institute for Computational Engineering and Sciences (ICES)The University of Texas at Austin

OVERVIEW

1. Overview2. Mathematical Formulation of Electrodynamic Problems

• Maxwell’s Equations and Boundary Conditions

• Variational Formulation

• Axial Symmetry

3. Error Estimation: Towards Certified Solutions• Control of Modeling and Discretization Errors

• Automatic Built-in Error Estimation

• Benchmarking Problems

4. Numerical Results

5. Conclusions and Future Work

The University of Texas at Austin

David Pardo 22 Feb 2005

MAXWELL’S EQUATIONS (FREQUENCY DOMAIN)

Time Harmonic Maxwell’s Equations:

∇× E = −jωµH

∇× H = (σ + jω²)E + Jimp

Reduced Wave Equation:E-Formulation H-Formulation

∇×

(

1

µ∇ × E

)

−(ω2²−jωσ)E = −jωJ imp ; ∇×

(

1

σ + jω²∇ × H

)

+jωµH = ∇×1

σ + jω²J imp

Boundary Conditions (BC):

• Perfect Electric Conductor Surface:

n × E = 0 ; n · H = 0

• Idealized Antennas (Impressed Surface Electric Current):

n ×1

µ∇× E = −jωJimpS ; n × H = J

impS

The University of Texas at Austin2

High Performance Finite Element Software



Variational formulation

The reduced wave equation in Ω,

E-Formulation: ∇ ×(

1

µ∇ × E

)

− (ω2² − jωσ)E = −jωJ imp

H-Formulation: ∇ ×(

1

σ + jω²∇ × H

)

+ jωµH = ∇ ×1

σ + jω²J imp

Variational formulation:

E-Formulation:

Find E ∈ HD(curl; Ω) such that:∫

Ω

1

µ(∇ × E)(∇ × F̄) dV −

∫

Ω

(ω2² − jωσ)E · F̄ dV =

−jω

∫

Ω

Jimp · F̄ dV + jω

∫

ΓN

JimpS · F̄ dS ∀ F ∈ HD(curl; Ω)

H-Formulation:

Find H ∈ H̃S + HD(curl; Ω) with J̃impS = n × H|S and such that:

∫

Ω

1

σ + jω²(∇ × H)(∇ × F̄) dV + jω

∫

Ω

µH · F̄ dV =∫

Ω

∇ × (1

σ + jω²Jimp) · F̄ dV ∀ F ∈ HD(curl; Ω)





Variational formulation in cylindrical coordinatesUsing cylindrical coordinates (ρ, φ, z):

Eφ-Formulation:

Find Eφ ∈ H̃1D(Ω) such that:∫

Ω

1

µ(∂Eφ

∂z

∂F̄φ

∂z+

1

ρ2∂(ρEφ)

∂ρ

∂(ρF̄φ)

∂ρ) dV −

∫

Ω

k2Eφ · F̄φ dV =

−jω

∫

Ω

J impφ · F̄φ dV + jω

∫

ΓN

J impφ,S · F̄φ dS ∀ Fφ ∈ H̃1

D(Ω) .

Eρ,z-Formulation:

Find E = (Eρ, 0, Ez) ∈ H̃D(curl; Ω) such that:∫

Ω

1

µ(∂Eρ

∂z

∂F̄ρ

∂z+

∂Ez

∂ρ

∂F̄z

∂ρ) − k2

∫

Ω

EρF̄ρ + EzF̄z dV = −jω

∫

Ω

J impρ

F̄ρ + Jimpz

F̄z dV +

jω

∫

ΓN

J impρ,S F̄ρ + Jimpz,S F̄z dS ∀ F = (Fρ, 0, Fz) ∈ H̃D(curl; Ω) .

Hφ-Formulation:

Find Hφ ∈ f1(Jimpρ,S , J

impz,S ) + H̃

1

D(Ω) such that:

∫

Ω

1

σ + jω²(∂Hφ

∂z

∂F̄φ

∂z+

1

ρ2∂(ρHφ)

∂ρ

∂(ρF̄φ)

∂ρ) dV − jω

∫

Ω

µHφ · F̄φ dV =

∫

Ω

1

σ + jω²(∂J imp

ρ

∂z−

∂J impz

∂ρ)F̄φdV ∀ Fφ ∈ H̃

1

D(Ω) .

Hρ,z-Formulation:

Find H = Hρ, 0, Hz) ∈ f2(Jimpφ,S ) + H̃D(curl; Ω) such that:

∫

Ω

1

σ + jω²(∂Hρ

∂z

∂F̄ρ

∂z+

∂Hz

∂ρ

∂F̄z

∂ρ) − jω

∫

Ω

µ(HρF̄ρ + HzF̄z) dV =

∫

Ω

1

σ + jω²(∂Jφ

∂zF̄ρ +

1

ρ

∂(ρJ impφ )

∂ρF̄z) dV ∀ F = (Fρ, 0, Fz) ∈ H̃D(curl; Ω) .




TOWARDS CERTIFIED SOLUTIONS

Sources of Error

Physical (Real) ProblemExact GeometryExact PhysicsExact InstrumentsNo Boundaries

No Errors in the Solution

−→

Mathematical ProblemApprox. GeometryMaxwell’s EquationsPerfect InstrumentsSome Boundaries

Errors Associated to:GeometryApproximationsCoupling PhysicsMaterial Coefficients

−→

Computational ProblemApprox. GeometryMaxwell’s EquationsPerfect InstrumentsBoundary Conditions

Errors Associated to:Mathematical ModelingGeometry and BC’sThe Code (Bugs)Discretization





Sources of Error


Solution is Not Exact Due toErrors in:

• Mathematical Modeling

• Geometry and BC’s

• The Code (Bugs)

• Discretization





Discretization Error

The self-adaptive goal-oriented hp-adaptive strategy provides a veryaccurate built-in discretization error estimator.

Coarse grid Fine grid(hp) (h/2, p + 1)

-

global hp-refinement

Discretization Error Estimate for Coarse Grid =Fine Grid Solution - Coarse Grid Solution





Sources of Error





• The Code (Bugs)

• Discretization





Avoiding Errors in the Code Using Benchmarking Examples

Solutions in a Homogeneous Lossy (1 Ω m) Media (2 Mhz)

Solenoid Antenna Toroid Antenna

10−80

10−60

10−40

10−20

100

0

5

10

15

20

25

30

35

40

45

50

55

Amplitude (V/m)

Dis

tanc

e in

z−a

xis

from

tran

smitt

er to

rece

iver

(in

m)

−180 −90 0 90 1800

5

10

15

20

25

30

35

40

45

50

55

Phase (degrees)

Dis

tanc

e in

z−a

xis

from

tran

smitt

er to

rece

iver

(in

m)

Analytical solutionNumerical solution


10−80

10−60

10−40

10−20

100

0

5

10

15

20

25

30

35

40

45

50

55

Amplitude (A/m)

Dis

tanc

e in

z−a

xis

from

tran

smitt

er to

rece

iver

(in

m)

−180 −90 0 90 1800

5

10

15

20

25

30

35

40

45

50

55

Phase (degrees)

Dis

tanc

e in

z−a

xis

from

tran

smitt

er to

rece

iver

(in

m)



Electric Field Magnetic Field





Sources of Error





• The Code (Bugs)

• Discretization





Avoiding Errors in the Code Using Benchmarking Examples

Solutions in a Homogeneous Lossy (1 Ω m) Media (2 Mhz) in Presence ofa Conductive Mandrel

Solenoid Antenna Toroid Antenna

10−10

10−5

100

0.5

1

1.5

2

2.5

3

3.5

Amplitude (V/m)

Dis

tan

ce

in

z−

axis

fro

m tra

nsm

itte

r to

re

ce

ive

r (in

m)

Analytical solution

Mandrel Resisitivity: 10−7



Mandrel Resisitivity: 1

−180 −90 0 90 1800.5

1

1.5

2

2.5

3

3.5

Phase (degrees)

Dis

tan

ce

in

z−

axis

fro

m tra

nsm

itte

r to

re

ce

ive

r (in

m)

Analytical solution





10−5

100

105

0.5

1

1.5

2

2.5

3

3.5

Amplitude (A/m)

Dis

tan

ce

in

z−

axis

fro

m tra

nsm

itte

r to

re

ce

ive

r (in

m)

−180 −90 0 90 1800.5

1

1.5

2

2.5

3

3.5

Phase (degrees)

Dis

tan

ce

in

z−

axis

fro

m tra

nsm

itte

r to

re

ce

ive

r (in

m)

Analytical solution





Analytical solution





Electric Field Magnetic Field





Summary





• The Code (Bugs)

• Discretization




THROUGH CASING RESISTIVITY INSTRUMENTS

10 cm.

50 c

m.

100

cm.

10000 Ohm m

1 Ohm m

100 Ohm m

100 Ohm m0.1

Ohm

m

z

25 c

m.

25 c

m.

150

cm.

CASI

NG 0

.000

001

Ohm

m

Axisymmetric 3D problem.

Five different materials.

Size of computational domain:SEVERAL MILES.

Material properties varying byup to TEN orders of magnitude(10000000000!!!).

Objective: DetermineSecond Difference of PotentialReceiving Electrodes.





Logging Through Casing (Benchmark Problem)Rock Formation: Homogeneous Media

10000100010010110

−9

10−8

10−7

10−6

Resistivity (Ohm m)

Sec

ond

Diff

eren

ce o

f Pot

entia

l (no

rmal

ized

) in

Vol

ts

Kaufmann Approx. FormulaExact 2nd DerivativeExact 2nd Diff. of Pot.Numerical 2nd Diff. of Pot.

The second vertical difference of the Electric Potential is proportional to the formationconductivity.





Final Log Obtained by Our Finite Element Software

0.001 0.01 0.1 1 10−1

−0.5

0

0.5

1

1.5

2

Second Difference of Potential (normalized) in micro Volts

Po

sitio

n o

f R

ece

ivin

g E

lectr

od

es (

z−

axis

) in

m

Electrodes at the borehole centerElectrodes at the borehole wallExact solution for one layer formation

0.0001 0.001 0.01 0.1 1−1

−0.5

0

0.5

1

1.5

2


Po

sitio

n o

f R

ece

ivin

g E

lectr

od

es (

z−

axis

) in

m

Electrodes at the borehole centerElectrodes at the borehole wallExact solution for one layer formation

Resistivity of casing = 10−6 Resistivity of casing = 10−7





Approximation Error

0 0.05 0.1 0.15 0.2 0.25−1.5

−1

−0.5

0

0.5

1

1.5

2

Relative Error of Second Difference of Potential (in %)

Po

sitio

n o

f R

ece

ivin

g E

lectr

od

es (

z−

axis

)

5500 6000 6500 7000 7500 8000 8500−1.5

−1

−0.5

0

0.5

1

1.5

2

Number of Unknowns

Po

sitio

n o

f R

ece

ivin

g E

lectr

od

es (

z−

axis

)





Damaged Casing

HALF THICKNESSCASING AREA

0.00001 Ohm m

0.000001 Ohm mCASING RESISTIVITY

LESS RESISTIVE AREA

0.1

m0.1

m

Imperfect Casing

0.2m

0.2m

1.3m

1.3 m

FO

RM

AT

ION

FO

RM

AT

ION

BOREHOLE

CASING

ELECTRODES





Damaged Casing

0.0001 0.01 1 100 10000−1

−0.5

0

0.5

1

1.5

2


Pos

ition

of R

ecei

ving

Ele

ctro

des

(z−a

xis)

in m

Perfect casingDamaged casingDamaged casing using a calibrated toolExact solution for one layer formation

In the presenceof damagedcasing, the useof calibratedinstruments isessential.





10 cm.

50 c

m.

100

cm.

10000 Ohm m

1 Ohm m

100 Ohm m

100 Ohm m0.1

Ohm

m

z

CASI

NG 0

.000

01 O

hm m

165

cm.

20 c

m.


Toroid Antennas.

Size of computational domain:SEVERAL MILES.

Different frequencies.

Material properties varying byup to NINE orders ofmagnitude (1000000000!!!).

Objective: DetermineFirst Difference of Electric andMagnetic Fields.





First Difference of Electric Field at Different Frequencies

10−15

10−12

10−9

10−6

−1

−0.5

0

0.5

1

1.5

2

Amplitude of Electric Field (V/m)

First Difference of E Field Vertical Derivative

Po

sitio

n o

f re

ce

ive

r a

nte

nn

a in

th

e z

−a

xis

(in

m)

−180 −90 0 90 180−1

−0.5

0

0.5

1

1.5

2

Phase (degrees)

Po

sitio

n o

f re

ce

ive

r a

nte

nn

a in

th

e z

−a

xis

(in

m)

Antennas located at the Borehole CENTER

0.1 Hz10 Hz1 Khz10 Khz100 Khz1 Mhz


10−15

10−12

10−9

10−6

−1

−0.5

0

0.5

1

1.5

2


First Difference of E Field Vertical Derivative

Po

sitio

n o

f re

ce

ive

r a

nte

nn

a in

th

e z

−a

xis

(in

m)

−180 −90 0 90 180−1

−0.5

0

0.5

1

1.5

2

Phase (degrees)

Po

sitio

n o

f re

ce

ive

r a

nte

nn

a in

th

e z

−a

xis

(in

m)

Antennas located at the Borehole WALL



Toroid antennas are more sensitive to the rock formation resistivity whenlocated on the borehole’s wall





First Difference of Magnetic Field at Different Frequencies

10−15

10−12

10−9

10−6

−1

−0.5

0

0.5

1

1.5

2

Amplitude of Magnetic Field (A/m)

First Difference of Magnetic Field (Normalized)

Po

sitio

n o

f re

ce

ive

r a

nte

nn

a in

th

e z

−a

xis

(in

m)

−180 −90 0 90 180−1

−0.5

0

0.5

1

1.5

2

Phase (degrees)

Po

sitio

n o

f re

ce

ive

r a

nte

nn

a in

th

e z

−a

xis

(in

m)

Antennas located at the Borehole CENTER



10−15

10−12

10−9

10−6

−1

−0.5

0

0.5

1

1.5

2


First Difference of Magnetic Field (Normalized)

Po

sitio

n o

f re

ce

ive

r a

nte

nn

a in

th

e z

−a

xis

(in

m)

−180 −90 0 90 180−1

−0.5

0

0.5

1

1.5

2

Phase (degrees)

Po

sitio

n o

f re

ce

ive

r a

nte

nn

a in

th

e z

−a

xis

(in

m)

Antennas located at the Borehole WALL







Electromagnetic Fields at Different Frequencies

0.001 0.01 0.1 1 10 100 100010

−12

10−11

10−10

10−9

10−8

10−7

10−6

Am

plitu

de

of E

lectr

om

ag

ne

tic F

ield

s (

A/m

an

d V

/m) First Difference of EM Fields

Frequency (in Khz)0.001 0.01 0.1 1 10 100 1000

−150

−100

−50

0

50

100

150

Ph

ase

(d

eg

ree

s)

Frequency (in Khz)

Magnetic FieldElectric Field (z−component)

Magnetic FieldElectric Field (z−component)

ElectromagneticFields are al-most constantfor frequenciesbelow 1 kHz. Asudden drop inthe amplitudeoccurs at fre-quencies above20 kHz.





� � ��

� � ��

� � ��

� ��

� ��

� ��

� � ��

� � ��

� � ��

� � ��

� � ��

� � ��

�

� ��

� � ��

5.5" Borehole radio ; 0.5" Casing ; 2" Cement

175m

=58

0’50

m=

165’

475m

=16

00’

0.00

001

Oh

m m

1 O

hm

m

2 O

hm

m

20 Ohm m

5 Ohm m

5 Ohm m


Five different materials.

Different positions of receivingantenna.

Toroidal Transmitterand receiver separated by upto half a mile.

Objective: DetermineFirst Difference of PotentialReceiving Electrodes.






Magnetic Field Electric Field

10−12

10−11

10−10

10−9

−100

−80

−60

−40

−20

0

20

40

60

80

100Distance from Casing: 0 m


Po

sitio

n o

f re

ce

ive

r a

nte

nn

a in

th

e z

−a

xis

(in

m)

−180 −90 0 90 180−100

−80

−60

−40

−20

0

20

40

60

80

100

Phase (degrees)

Po

sitio

n o

f re

ce

ive

r a

nte

nn

a in

th

e z

−a

xis

(in

m)

0.1 Hz NO INV0.1 Hz 60m INV100 Hz NO INV100 Hz 60m INV


10−11

10−10

10−9

10−8

−100

−80

−60

−40

−20

0

20

40

60

80



Po

sitio

n o

f re

ce

ive

r a

nte

nn

a in

th

e z

−a

xis

(in

m)

−180 −90 0 90 180−100

−80

−60

−40

−20

0

20

40

60

80

100

Phase (degrees)

Po

sitio

n o

f re

ce

ive

r a

nte

nn

a in

th

e z

−a

xis

(in

m)



If the transmitter antenna is located on the surface of a cased well, thereceived EM signal within the borehole is too weak.







10−9

10−8

10−7

−100

−80

−60

−40

−20

0

20

40

60

80

100Frequency: 1 Hz


Po

sitio

n o

f re

ce

ive

r a

nte

nn

a in

th

e z

−a

xis

(in

m)

−180 −90 0 90 180−100

−80

−60

−40

−20

0

20

40

60

80

100

Phase (degrees)

Po

sitio

n o

f re

ce

ive

r a

nte

nn

a in

th

e z

−a

xis

(in

m)

50 m250 m500 m

50 m250 m500 m

10−12

10−10

10−8

−100

−80

−60

−40

−20

0

20

40

60

80

100Frequency: 1 Hz


Po

sitio

n o

f re

ce

ive

r a

nte

nn

a in

th

e z

−a

xis

(in

m)

−180 −90 0 90 180−100

−80

−60

−40

−20

0

20

40

60

80

100

Phase (degrees)

Po

sitio

n o

f re

ce

ive

r a

nte

nn

a in

th

e z

−a

xis

(in

m)

50 m250 m500 m

50 m250 m500 m

If the transmitter antenna is located on the surface of a cased well, thereceived cross-well EM signal is too weak.







10−11

10−10

10−9

10−8

−100

−80

−60

−40

−20

0

20

40

60

80



Po

sitio

n o

f re

ce

ive

r a

nte

nn

a in

th

e z

−a

xis

(in

m)

−180 −90 0 90 180−100

−80

−60

−40

−20

0

20

40

60

80

100

Phase (degrees)

Po

sitio

n o

f re

ce

ive

r a

nte

nn

a in

th

e z

−a

xis

(in

m)



10−12

10−11

10−10

−100

−80

−60

−40

−20

0

20

40

60

80



Po

sitio

n o

f re

ce

ive

r a

nte

nn

a in

th

e z

−a

xis

(in

m)

−180 −90 0 90 180−100

−80

−60

−40

−20

0

20

40

60

80

100

Phase (degrees)

Po

sitio

n o

f re

ce

ive

r a

nte

nn

a in

th

e z

−a

xis

(in

m)



If the transmitter antenna is located on the surface of a cased well, a 60meters layer of water cannot be detected by using cross-well EM signals.







10−8

10−7

10−6

−100

−80

−60

−40

−20

0

20

40

60

80



Po

sitio

n o

f re

ce

ive

r a

nte

nn

a in

th

e z

−a

xis

(in

m)

−180 −90 0 90 180−100

−80

−60

−40

−20

0

20

40

60

80

100

Phase (degrees)

Po

sitio

n o

f re

ce

ive

r a

nte

nn

a in

th

e z

−a

xis

(in

m)



10−9

10−8

10−7

−100

−80

−60

−40

−20

0

20

40

60

80



Po

sitio

n o

f re

ce

ive

r a

nte

nn

a in

th

e z

−a

xis

(in

m)

−180 −90 0 90 180−100

−80

−60

−40

−20

0

20

40

60

80

100

Phase (degrees)

Po

sitio

n o

f re

ce

ive

r a

nte

nn

a in

th

e z

−a

xis

(in

m)



If the transmitter antenna is located downhole a cased well, it is feasibleto perform meaningful cross-well EM measurements.




LOGGING INSTRUMENTS WITH A MANDREL

100 c

m50 c

m

100 Ohm m

10000 Ohm m

1 Ohm m

100 Ohm m

15 c

m100 c

m

0.0

00001 O

hm

m0.1

Oh

m m

Description of Antennas

10 c

m

10000 Ohm m

Borehole0.1 Ohm m

10000 Rel. Perme.

0.0

00001 O

hm

mM

an

dre

l

Radius=10.8 cm

Radius 7.6 cm

Magnetic Buffer

Goal: To Compute FirstDifference of Potentialon Receiving Antennas





Eφ (normalized) for a solenoid antenna

0.1 0.18 0.3−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3


Pos

ition

of r

ecei

ver a

nten

na in

the

z−ax

is (i

n m

)

−180 −90 0 90 180−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Phase (degrees)

Pos

ition

of r

ecei

ver a

nten

na in

the

z−ax

is (i

n m

)

100 Ohm m

1 Ohm m

10000 Ohm m

100 Ohm m

Frequency: 2 Mhz





First Difference of Eφ (normalized) for a solenoid antenna

0.05 0.075 0.1 0.125−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3


Pos

ition

of r

ecei

ver a

nten

na in

the

z−ax

is (i

n m

)

−180 −90 0 90 180−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Phase (degrees)

Pos

ition

of r

ecei

ver a

nten

na in

the

z−ax

is (i

n m

)

100 Ohm m

1 Ohm m

10000 Ohm m

100 Ohm m

Frequency: 2 Mhz





First Difference of Eφ (normalized) for a solenoid antenna

0.0003 0.0008−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3


Pos

ition

of r

ecei

ver

ante

nna

in th

e z−

axis

(in

m)

−180 −90 0 90 180−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Phase (degrees)

Pos

ition

of r

ecei

ver

ante

nna

in th

e z−

axis

(in

m)

100 Ohm m

1 Ohm m

10000 Ohm m

100 Ohm m

Frequency: 2 Mhz, NO MAGNETIC BUFFER





First Difference of Hφ (normalized) for a toroid antenna

10−6

10−4

10−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3


Pos

ition

of r

ecei

ver

ante

nna

in th

e z−

axis

(in

m)

−180 −90 0 90 180−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Phase (degrees)

Pos

ition

of r

ecei

ver

ante

nna

in th

e z−

axis

(in

m)

100 Ohm m

1 Ohm m

10000 Ohm m

100 Ohm m

Frequency: 2 Mhz





First Difference of Ez (normalized) for a toroid antenna

10−4

10−2

100

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3


Pos

ition

of r

ecei

ver a

nten

na in

the

z−ax

is (i

n m

)

−180 −90 0 90 180−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Phase (degrees)

Pos

ition

of r

ecei

ver a

nten

na in

the

z−ax

is (i

n m

)

100 Ohm m

1 Ohm m

10000 Ohm m

100 Ohm m

Frequency: 2 Mhz




CONCLUSIONS

• It is possible to simulate a variety of EM logging instrumentsby using the self-adaptive goal-oriented hp-FEM.

• For TCRT, preliminary simulations suggest to

1. Place antennas on the borehole’s wall,2. use calibrated instruments,3. use low frequencies (typically below 5 kHz),4. use downhole antennas for cross-well EM measurements, and5. avoid the use of cross-well EM measurements for assesment of

water injection.

• For LWD instruments, preliminary simulations suggest to

1. Use both solenoid and toroid antennas,2. use magnetic buffers to amplify EM signals, and3. compute first differences of EM fields.

Institute for Computational Engineering and Sciences



FUTURE WORK

Within the next 3 months

• Implementation of the goal-oriented self-adaptive algorithmfor 2D edge elements. It would allow us to solve 2.5 Dproblems.

• Parallel implementation of the 2D code (Maciek Paszynski).

Long term goals

• Solve 3D problems with casing at DC.

• Solve 3D problems with mandrel at AC.

• Invert coupled sonic and EM measurements in 2D.

The University of Texas at Austin

-adaptive hp a fully automatic goal-oriented finite element … · 2011. 2. 10. · c....

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