Adaptive finite element modeling for 2D electromagnetic ...€¦ · Adaptive finite element...

Adaptive finite element modeling for 2D electromagnetic problems using

unstructured grids

Kerry KeyScripps Institution of Oceanography

La Jolla, USA

with thanks to:

Chester WeissSandia National Laboratories

Albuquerque, USA

Yuguo LiScripps Institution of Oceanography

La Jolla, USA

Outline• Motivation

• Methodology

• 2D MT Verification: Segmented Slab Model

• A “Real” 2D MT Example

• 2.5 CSEM Example

• Conclusions

Marine Magnetotellurics (MT) andControlled-Source Electromagnetics (CSEM)

Techniques Employed:

Unstructured Finite Elements: Grid of irregular triangular elements.

Adaptive Finite Element Method: FE solution computed iteratively, using successively refined grids until desired solution accuracy obtained. Relies on usage of an a posteriori error estimator.

Advantages of Unstructured Grids

•Complex structures easier to grid than with rectangular elements.

•Readily handles seafloor topography, which can strongly influence MT responses (e.g. Schwalenberg and Edwards, 2004)

•Accommodates both tiny and large structures in same grid

•Efficient use of nodes---fine gridding doesn’t have to extend to model sides.

Advantages of Adaptive Finite Elements

•Adaptive method offers asymptotically exact solution (just keep on iterating)

•Code users often not FE experts and adaptive methods offer the chance of ensuring an accurate solution.

•For a given solution accuracy, AFE is more efficient use of nodes than brute-force fine gridding of entire domain.

Adaptive Grid Refinement

1.Compute FE solution on coarse grid2.Estimate a posteriori error for each element3.Refine “bad” elements4.Repeat 1–3 until solution accuracy achieved

Algorithm:

Finite Element Solution of Elliptic PDE

!" · (k"u) + qu = f in ! # R2PDE:

Weak form solved using linear finite elements:

uh : finite element approximation of u

gradient of finite element solution. Piecewise constant with each element.

A Posteriori Error Estimation

Gradient Recovery Method:

•Piecewise constant gradient with each element is poor approximation of true gradient

•Post-processing methods offer improved or recovered gradient

•Cheap error indicator for each element is normed difference of recovered gradient and piecewise constant gradient.

2 4 6 8 10 12

!0.5

0

0.5

2 4 6 8 10 12

!0.5

0

0.5

2 4 6 8 10 12

!0.5

0

0.5

2 4 6 8 10 12

!0.5

0

0.5

Gradient Recovery, 1D Example

Function

FEApprox.

FE Gradient

Recovered Gradient

Gradient Recovery in 2D

Piecewise Constant Gradient:

Recovered Gradient:

Basic Error Estimator (BEE):

Bank and Xu (2003)

Comparison of Gradients

Recovered (Smoothed)Grad x

Piecewise Constant Grad x Grad z Grad z

• Basic error estimator effective for global refinement.

• However, for MT and CSEM solution accuracy required only at receiver locations.

• More efficient approach is Dual/Adjoint Method

Dual Error Weighting (DEW)(Ovall, 2004)

Dual Error Weighting (DEW):

Primary Problem:

Dual Problem:

Adjoint/Dual:

Dual Functional:

2D MT Example and Verification:Segmented Slab Model

Quasi-analytic solution available from Weaver et al. (1985,1986)

TE Mode Convergence

TE Mode Comparison

Salt sill

!300

!200

!100

0

100

200

De

pth

(km

)

143 vertices, 260 triangles

37147 vertices, 74150 triangles

-200-400-600 200 400 600 0Position (km)

33163 vertices, 66272 triangles

Starting Grid

18th GridBasic Error Estimator

18th GridDual Error Weighting

Site Locations

TE Mode Comparison

TM Mode Convergence

TM Mode Comparison

Salt sill

0

100

200Depth

(km

)

-200-400-600 200 400 600 0Position (km)

71717 vertices, 143250 triangles

28084 vertices, 53631 triangles

20th GridBasic Error Estimator

20th GridDual Error Weighting

Convergence Rate for Various Amounts of Refinement

Salt sill

10!1 100 101 102

1

10

Time (s)

)%( tifsi

M5%15%25%35%

Accuracy of 100s Period Grid at other Periods for TE Mode

Salt sill10!1

100

Mis

fit

(%)

MaxRMS

100

101

102

103

104

10!1

100

Period (s)

Mis

fit

(%)

Ex

Hy

A “Real” Example:Sigsbee Escarpment MT Model

0 5 10 15 20 25 30 35 40 45 50

!10

!5

0

5

10

Position (km)

Dep

th (k

m)

SIGSBEE2.1: 1789 vertices, 3526 triangles

Air

Sea

Sediments

Salt and basement

Hydrates

Salt sill

Salt diapirs

Starting Grid

0 5 10 15 20 25 30 35 40 45 50

!10

!8

!6

!4

!2

0

2

4

6

8

10

Position (km)

Dep

th (k

m)

SIGSBEE2.1: 1789 vertices, 3526 triangles

MT Site Locations

0 5 10 15 20 25 30 35 40 45 50

!10

!8

!6

!4

!2

0

2

4

6

8

10

Position (km)

Dep

th (k

m)

SIGSBEE2.1: 1789 vertices, 3526 triangles

First Grid Error Indicator

0 5 10 15 20 25 30 35 40 45 50

!10

!8

!6

!4

!2

0

2

4

6

8

10

Position (km)

Dep

th (k

m)

SIGSBEE2.1.error.TM.9: 1789 vertices, 3526 triangles

!11

!10

!9

!8

!7

!6

!5

!4

0 5 10 15 20 25 30 35 40 45 50

!10

!8

!6

!4

!2

0

2

4

6

8

10

Position (km)

Dep

th (k

m)

SIGSBEE2.1.error.TE.9: 1789 vertices, 3526 triangles

!19

!18

!17

!16

!15

!14

!13

TEMode

TMMode

Grid 1

0 5 10 15 20 25 30 35 40 45 50

!10

!8

!6

!4

!2

0

2

4

6

8

10

Position (km)

Dep

th (k

m)

SIGSBEE2.1: 1789 vertices, 3526 triangles

Grid 3

0 5 10 15 20 25 30 35 40 45 50

!10

!8

!6

!4

!2

0

2

4

6

8

10

Position (km)

Dep

th (k

m)

SIGSBEE2.3: 3300 vertices, 6548 triangles

Grid 5

0 5 10 15 20 25 30 35 40 45 50

!10

!8

!6

!4

!2

0

2

4

6

8

10

Position (km)

Dep

th (k

m)

SIGSBEE2.5: 6040 vertices, 12028 triangles

Grid 7

0 5 10 15 20 25 30 35 40 45 50

!10

!8

!6

!4

!2

0

2

4

6

8

10

Position (km)

Dep

th (k

m)

SIGSBEE2.7: 11128 vertices, 22204 triangles

Grid 9 : Solution Converged

0 5 10 15 20 25 30 35 40 45 50

!10

!8

!6

!4

!2

0

2

4

6

8

10

Position (km)

Dep

th (k

m)

SIGSBEE2.9: 20445 vertices, 40828 triangles

Grid 9: Error Indicator in Model Center

Grid 9 MT Responses for three sites

10!1

100

101

App. R

es. (o

hm!

m)

100

101

102

103

104

0

20

40

60

80

Phase (

degre

es)

Period (s)

100

101

102

103

104

Period (s)

100

101

102

103

104

Period (s)

Controlled-Source EM Example

2.5 D Problem:

• 3D Source, 2D model

• Wavenumber Fourier transform along x-axis

• Solve coupled PDE for Ex and Hx

• Generally need about 30 wavenumbers logarithmically spaced

• Fourier transform back into spatial x.

Adaptive FE Implementation

1. Compute solution for a subset of wavenumbers (about 1 per decade).

2. Compute error indicator

3. Refine bad elements

4. Repeat 1-3 until solution converges

5. Compute solution on final grid for all wavenumbers, transform back to spatial x.

Algorithm:

0 5 10 15 20

0

2

4

6

8

10

12

Position (km)

Dep

th (k

m)

Complex.1.1.1: 760 vertices, 1472 triangles

!1

!0.5

0

0.5

1

1.5

2

2.5

3

Starting Grid50 sites, Inline Transmission, 0.1 Hz

reservoir salt

!30 !20 !10 0 10 20 30 40 50

!10

!5

0

5

10

15

20

25

30

Position (km)

Dep

th (k

m)

Complex.1.1.1: 760 vertices, 1472 triangles

!1

!0.5

0

0.5

1

1.5

2

2.5

3

Starting Grid (760 vertices)

0 5 10 15 20

0

2

4

&

'

10

12

Position (km)

3ep

th (k

m)

7omplex:1:1:1;< 545' =erti?es, 10'&; triangles

!1

!0:5

0

0:5

1

1:5

2

2:5

3

18th Refined Grid CPU Time about 15 minutes

about 5% refinement per iteration

Ex (kx) and Hx (kx) Responses at receivers for grids 1-16

Convergence Versus Receiver Position(relative difference between iterations)

Ex Hx

Convergence:Relative difference between grid iterations

0 5 10 1510!3

10!2

10!1

100

101

102

&rid +

Rel

ati1

e 2

i33er

en5e

6ax Ex6ean Ex6ax Hx6ean Hx

Receiver Responses

Hx

Ey

Ez

total field

scattered field

0 5 10 15 20

0

2

4

6

8

10

12

Position (km)

Dep

th (k

m)

Complex.1.1.1: 760 vertices, 1472 triangles

!1

!0.5

0

0.5

1

1.5

2

2.5

3

Receiver Responses

Ey

Conclusions

• Adaptive FE modeling is an useful tool for EM geophysics.

• Dual error weighting (DEW) far more efficient than basic gradient error estimator (BEE).

Future Work:

• Apply method to 3D FE codes.