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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.