hp-adaptive finite elements and multi-target goal-oriented ... · first international workshop on...

First International Workshop on Multiphysics, Multiscale, and Optimization Problems

hp-Adaptive Finite Elements andMulti- Target Goal-Oriented Adaptivity

D. Pardo, I. Gomez-RevueltoBasque Center for Applied Mathematics (BCAM)

and IKERBASQUE

TEAM MEMBERS:D. Pardo (Research Professor)I. Gomez-Revuelto (Visiting Professor)A.-G. Saint-Guirons (Postdoctoral Fellow)M. Grigoroscuta (Postdoctoral Fellow)D. Lasa (Ph.D. Student)

MAIN COLLABORATORS:M. J. Nam, M. Paszynski,

L.E. Garcıa-Castillo, V. Calo,C. Torres-Verdın, L. Demkowicz,

H. Barucq, P. Queralt.

7 June 2010

D. Pardo, I. Gomez-Revuelto For more info, visit: www.bcamath.org/research/mip

overview

1. Motivation: Oil-Industry Applications.

2. hp Finite Element Methods.

3. hp Adaptivity and Goal-Oriented Adaptivity.

4. hp Goal-Oriented Adaptivity with Multiple Goals.

5. Conclusions.

1


motivation and objectives

Seismic Measurements

Figure from the USGS Science Center for Coastal and Marine Geology

2



Marine Controlled-Source Electromagnetics (CSEM)

Figure from the UCSD Institute of Oceanography

3



Deviated Wells (Forward Problem)Dip Angle

Invasion

Anisotropy

Different Sources(Triaxial Induction)

Eccentric LoggingInstruments

Laterolog

Through-Casing

Induction-LWD

Induction-Wireline

Goal: To find the EM fields at the receiver antennas.

4



• Acoustic domain: borehole fluid ( cf , ρf )

iωp + c2fρf∇ · v = 0

iωρfv +∇p = 0

• Elastic tool, casing, formation ( Vp, Vs, ρs)

0 = ∇ · σ + ρsω2u

σ = λI∇ · u + µ(∇u +∇Tu)

λ = ρs(V2

p − 2V 2s ), µ = ρsV

2s

• Coupling:nf · ∇p = ρfω2nf · u

ns · σ = −pns

• Acoustic source:nf · ∇p = 1

5



Radar Cross Section (RCS)Analysis

EDGE DIFRACTION

MULTIPLEREFLECTIONS

WAVEGUIDEMODES

SURFACE WAVES

Goal: Determine the RCS of a plane.

Waveguide Design

Goal: Determine electric fieldintensity at the ports.

Modeling of a Logging While

Drilling (LWD) electromagnetic

measuring device

SIDE VIEW

T1

T2

R2

R1LAYERSDIFFERENT MATERIAL

EM FIELD

Goal: Determine EM field atthe receiver antennas.

6


hp-finite elements

The h-Finite Element Method1. Convergence limited by the polynomial degree, and large material

contrasts.

2. Optimal h-grids do NOT converge exponentially in real applications.

3. They may “lock” (100 % error).

The p-Finite Element Method1. Exponential convergence feasible for analytical (“nice”) solutions.

2. Optimal p-grids do NOT converge exponentially in real applications.

3. If initial h-grid is not adequate, the p-method will fail miserably.

The hp-Finite Element Method1. Exponential convergence feasible for ALL solutions.

2. Optimal hp-grids DO converge exponentially in real applications.

3. If initial hp-grid is not adequate, results will still be great.

7


hp-finite elementsWaveguide filled with a dielectric

h-Adaptivity

hp-Adaptivity

10000 10000010

−1

100

101

102

Number of Unknowns

Rel

ativ

e E

nerg

y−N

orm

Err

or (

in %

)

hp−adaptivityp=2, h−adaptivityp=1, h−adaptivity

dd

8


hp-finite elements3D waveguide with a meniscus-shaped biological cultive

Final hp-grid

9


automatic hp-adaptivity

Fully automatic hp-adaptive strategy

-global hp-refinement

9


9

-hp-adaptivity

10


automatic hp-adaptivity

Energy norm based fully automatic hp-adaptive strategyCoarse grids Fine grids

(hp) (h/2, p + 1)


9


9

SOL. METHOD ON FINE GRIDS:A TWO GRID SOLVER

11


automatic goal-oriented hp-adaptivity

Motivation (Goal-Oriented Adaptivity)

Test Problem

Resistivity = 1 Ohm mInfinite Domain

Frequency = 2 Mhz

50 meters

Receiver (R)Transmitter (T)

12





Frequency = 2 Mhz

50 meters


• Solution decays exponentially.

•

|E(T )|

|E(R)|≈ 1060

• Results using energy-norm adaptivity:

– Energy-norm error: 0.001%

– Relative error in the quantity ofinterest > 1030 %.

Goal-oriented adaptivity is needed. Becker-Rannacher (1995,1996),Rannacher-Stuttmeier (1997), Cirak-Ramm (1998), Paraschivoiu-P atera (1998), Peraire-Patera(1998), Prudhomme-Oden (1999, 2001), Heuveline-Rannacher (2003), Sol in-Demkowicz (2004).

13





Frequency = 2 Mhz

50 meters


• Solution decays exponentially.

•

|E(T )|

|E(R)|≈ 1060

• Results using energy-norm adaptivity:

– Energy-norm error: 0.001%

– Relative error in the quantity ofinterest > 1030 %.

Goal-oriented adaptivity is needed. Becker-Rannacher (1995,1996),Rannacher-Stuttmeier (1997), Cirak-Ramm (1998), Paraschivoiu-P atera (1998), Peraire-Patera(1998), Prudhomme-Oden (1999, 2001), Heuveline-Rannacher (2003), Sol in-Demkowicz (2004).

14





Frequency = 2 Mhz

50 meters


10−80

10−60

10−40

10−20

1000

5

10

15

20

25

30

35

40

45

50

55

Amplitude (V/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

10

15

20

25

30

35

40

45

50

55

Phase (degrees)

Dis

tan

ce

in

z−

axis

fro

m tra

nsm

itte

r to

re

ce

ive

r (in

m)

Analytical solutionNumerical solution

Analytical solutionNumerical solution

Goal-oriented adaptivity is needed

15



Mathematical Formulation (Goal-Oriented Adaptivity)

We consider the following problem (in variational form):

Find L(u), where u ∈ V = H1(Ω) such that :∫

Ω

σ∇u · ∇v︸︷︷︸

b(u, v)

=

∫

Ω

f · v︸︷︷︸

F (v)

∀v ∈ V .

We define residual re(v) := b(e, v). We seek for a functional G ∈ V ′′ ∼ V

relating the residual and the error in the quantity of interest, that is , such thatG(re) := L(e). Functional G is the solution of the so-called dual problem:

Find G ∈ V such that :

b(u, G) = L(u) ∀u ∈ V .

Notice that, in particular, L(e) = b(e, G) ← (Representation Formula) .

16



Mathematical Formulation (Goal-Oriented Adaptivity)

DIRECT PROBLEM - u -2D Cross-Section

DUAL PROBLEM - G -2D Cross-Section

Representation Formula for the Error in the Quantity of Interest:

L(u) = b(u, G) =

∫

Ω

σ ∇u · ∇GdV

17



Summary on Goal-Oriented Adaptivity

Goal-Oriented adaptivity is a GENERALIZATION of Energy-no rmadaptivity.

• For a number of problems, goal-oriented adaptivity isexactly the same as energy-norm adaptivity.

• For other problems, the use of goal-oriented adaptivitybecomes ESSENTIAL.

To solve a problem with goal-oriented adaptivity is ALWAYSBETTER (or equal) than solving the problem with energy-normadaptivity.

18


multiple goals

Model Problem (Electromagnetism)

40 m

1 Ohm−m2 Mhz1 Tx9 Rx

40 m

ads

19


multiple goals

Formulations that do NOT work well

• We may define:

L(u) =

∫

RX1

u +

∫

RX2

u + ... +

∫

RXn

u

The above formula will only enable refinements in the proximity o f thereceiver closest to the transmitter, where the solution is larger tha n inthe rest of the receivers.

• We may define:

L(u) = max

∫

RX1

u,

∫

RX2

u, ...,

∫

RXn

u

This is a non-linear objective function, which is INCONSISTENT w ith themathematical theory.

20


multiple goals

Formulation

We define:

Lhp(u) =

∫

RX1

u

∫

RX1

uhp

+

∫

RX2

u

∫

RX2

uhp

+ ... +

∫

RXn

u

∫

RXn

uhp

=n∑

i=1

∫

RXi

u

∫

RXi

uhp

Note that definition of Lhp depends upon the solution uhp, and therefore, itcannot be defined “a priori”.

Operator Lhp is linear and continous.

21


multiple goals

Improved Formulation

We define:

Lhp(u) =n∑

i=1

R(eRXihp )

∫

RXi

u

∫

RXi

uhp

,

where R(eRXihp ) is an estimate of the relative error in the i-th quantity of

interest. From a practical point of view, this relative error can b e estimated byapproximating the exact solution with the solution in a globally re finedh/2, p + 1-grid.

The additional term enables save unneeded refinements at the rece iverswhere the error tolerance has already been achieved.

22


multiple goals

Goal: Approximate Solution at Rx 4.

1000 3000 6000

100

105

1010

Rel

ativ

e E

rror

(in

%)

Number of Unknowns

Rx1

Rx2

Rx3

Rx4

Rx5

Rx6

Rx7

Rx8

Rx9

asdfasdf

23


multiple goals

Final hp-grid

Goal: Rx 1 Goal: Rx 4

asdfasdf

24


multiple goals

Goal: Approximate the Sum of the Solution at All Receivers.

1000 4000 10000 30000

100

105

1010

Rel

ativ

e E

rror

(in

%)

Number of Unknowns

Rx1

Rx2

Rx3

Rx4

Rx5

Rx6

Rx7

Rx8

Rx9

asdfasdf

25


multiple goals

Goal: Approximate Solution at All Receivers.

1000 4000 10000 3000010

−2

100

102

104

106

108

1010

Rel

ativ

e E

rror

(in

%)

Number of Unknowns

Rx1

Rx2

Rx3

Rx4

Rx5

Rx6

Rx7

Rx8

Rx9

asdfasdf

26


multiple goals

Final hp-grid

Sum of Goals Multi-goal

asdfasdf

27


multiple goals

Final hp-grid (amplification)

Sum of Goals Multi-goal

asdfasdf

28


multiple goals

Final solution

Solution ( Ex) Solution (amplification)

asdfasdf

29


multiple goals

10

0 c

m5

0 c

m

100 Ohm m

10000 Ohm m

1 Ohm m

100 Ohm m

15

cm

10

0 c

m

0.0

00

00

1 O

hm

m0

.1 O

hm

mDescription of Antennas

10 c

m

10000 Ohm m

Borehole0.1 Ohm m

10000 Rel. Perme.

0.00

0001

Ohm

mM

andr

el

Radius=10.8 cm

Radius 7.6 cm

Magnetic Buffer

Goal: To Study the Effectof Invasion, Anistotropy, andMagnetic Permeability.

30


multiple goals

Axisymmetric Logging-While-Drilling (LWD) SimulationENERGY-NORM HP-ADAPTIVITY

p=1

p=2

p=3

p=4

p=5

p=6

p=7

p=8

-0.6851852 2.907408 -1.203704

1.018519 2Dhp90: A Fully automatic hp-adaptive Finite Element code

31


multiple goals

Axisymmetric Logging-While-Drilling (LWD) Simulation

GOAL-ORIENTED HP-ADAPTIVITY (Quadrilateral Elements)

p=1

p=2

p=3

p=4

p=5

p=6

p=7

p=8

-0.7222222 2.870370 -1.137037


32


multiple goals

Axisymmetric Logging-While-Drilling (LWD) Simulation

GOAL-ORIENTED HP-ADAPTIVITY (Zoom towards first receiver)

p=1

p=2

p=3

p=4

p=5

p=6

p=7

p=8

1.7160494E-02 0.1369136 0.3603704


33

multiple goals

First Vert. Diff. Hφ for different antennas

10−5

100

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Amplitude First Vert. Diff. Magnetic Field (A/m2)

Ver

tical

Pos

ition

of R

ecei

ving

Ant

enna

(m

)No Invasion

Vert. Dipoles (Ring)Toroid

−200 −100 0 100 200−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Phase (degrees)

Ver

tical

Pos

ition

of R

ecei

ving

Ant

enna

(m

)

Vert. Dipoles (Ring)Toroid

100 Ohm−m

10000 Ohm−m

1 Ohm−m

100 Ohm−m

100 Ohm−m

10000 Ohm−m

1 Ohm−m

100 Ohm−m

In LWD instruments, we obtain similar results using toroids or a ring of vert. dipoles

multiple goals

First Vert. Diff. Ez for a toroid antenna

10−4

10−2

100

102

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Amplitude First Vert. Diff. Ez (V/m2)

Ver

tical

Pos

ition

of R

ecei

ving

Ant

enna

(m

)

No Invasion

Toroid

−200 −100 0 100 200−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Phase (degrees)

Ver

tical

Pos

ition

of R

ecei

ving

Ant

enna

(m

)

Toroid

100 Ohm−m

10000 Ohm−m

1 Ohm−m

100 Ohm−m

100 Ohm−m

10000 Ohm−m

1 Ohm−m

100 Ohm−m

Toroids are adequate for identifying highly resistive layers


multiple goals

Verification of Sonic Measurements

• Formation thickness: 0.25m.

• Monopole and dipole source, central frequency 8603 Hz .

36


multiple goals

Monopole source — fast formation

P-wave S-wave DensityVp [m/s] 1/Vp [µs/ft] Vs [m/s] 1/Vs [µs/ft] ρ [kg/m3]

fluid 1524 200.0 − − 1100

fast form. 3048 100.0 1793 170.0 2200

tool 5860 52.0 3130 97.4 7800

37


multiple goals

Monopole source — fast formation — no tool

38


multiple goals

Monopole source — fast formation — with tool

39


multiple goals

Monopole source — fast formation


fluid 1524 200.0 − − 1100

fast form. 3048 100.0 1793 170.0 2200

tool 5860 52.0 3130 97.4 7800

40


multiple goals

Dipole source — fast formation — no tool

41


multiple goals

Dipole source — fast formation — with tool

42


multiple goals

Dipole source — fast formation


fluid 1524 200.0 − − 1100

fast form. 3048 100.0 1793 170.0 2200

tool 5860 52.0 3130 97.4 7800

43


multiple goals

Direct calculation of dispersion curves

Dispersion curves contain the information about the slowness of theformation.

• dispersion curves are smooth with respect to variations in frequency,

• it is enough to calculate results only for a few frequencies (below 50)

• further reduction of the number of needed frequencies to 10 possible when only Vp

(high frequencies) and Vs (low frequencies) are needed.

⇒ Dispersion curves are obtained directly from frequency domain r esults.

• no need to use in the simulations a Ricker wavelet (neither any other wavel et),

• the added stability of the problem,

• better performance of PML in frequency domain,

• smaller complexity of the problem.

44


multiple goals

Direct calculation of dispersion curves

45


multiple goals

Examples with layers

• Formation thickness: 0.25m.

• Monopole/dipole source: Ricker wavelet, central frequency 8603 Hz .

46


multiple goals

Layers (1) — monopole (waveforms)

47


multiple goals

Layers (2) — monopole (waveforms)

48


conclusions

1. hp-Adaptive Finite Elements provides superior accuracy andcan be used for simulation of multi-physics problems.

2. Goal-oriented adaptivity is useful in engineering probl emswhen a specific goal is pursued.

3. It is possible to design adaptive schemes for the case ofmultiple goals.

49

hp-adaptive finite elements and multi-target goal-oriented ... · first international workshop on...

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