Transcript A multigrid solver for 3D EM diffusion

Improvement of a multigrid solver for
3D EM diffusion
Research proposal final thesis Applied Mathematics, specialism CSE
(Computational Science and Engineering)
Tom Jönsthövel
TU Delft januari 6, 2006
Tutors:
Kees Oosterlee (EWI, TUD)
Wim Mulder (SIEP, Shell)
Overview presentation
• Introduction:
• Practical application/problem statement
• Mathematical model
• Discretization model equations
• Short overview Multigrid
• Problems encoutered with MG solver
• Possible improvements on MG solver
• Summary
• Questions
Practical Application
Practical goal:
• Image structures that could host potential
reservoirs
• Providing evidence of presence of hydrocarbons
How?
Recovering the conductivity profile from
measurements of electric and magnetic fields:
* Oil/Gas are more resistive than surrounding
(gesteente)
(3D) Electromagnetic Diffusion
2D example:
EM source
receivers
Oil/Gas?
(3D) Electromagnetic Diffusion
Amundsen, Johansen & Røsten (2004):
A Sea Bed Logging (SBL) calibration survey over the Troll gas field
Mathematical model
Maxwell equations in presence of current source

x, tE
x, t0
t B
H
x, t
x, tJ c 
x, tJ s 
x, t
t D
With,
J c
x, t 

x
E
x, t Ohm’s law
D
x, t 
x
E
x, t
B
x, t 
x
H
x, t
Maxwell equations
Eliminate the magnetic field from the equation:
x1 Ex, t
x, t
t H

H
x, t
x, t
J c
x, tJ s 
x, t
t
t D
t

1

E



E




tt
t
tJs
r E 
Maxwell equations
Transform equation from time to frequency domain:

1
E
x, t 2 Ê
x, 
e it d

With  angular frequency. Now,


i
t 
2

tt  

1
i0 
i
Ê 
0 Ĵ s
r Ê i
In practice:

Maxwell equations
PEC boundary conditions:
(Perfectly Electrically Conduction)
n E0
n
n H0
domain
Discretization model equations
Step 1:
Choose discretization:
Finite Integration Technique (Clemens/Weiland ’01)
 Finite volume generalisation of Yee’s scheme (1966)
 Error analysis for constant-coeffients (Monk & Sülli 1994)
 2nd order accuracy for electric/magnetic field
components
Discretization model equations
Step 2:
Placement EM field components (Yee’s scheme):
E 3,k,l,m1
2
E 2,k,l1 ,m
2
z
E 1,k1 ,l,m
x
2
y
E 1,k1 ,l,m E 1 
x k1 , y l 
2
2
1
x k1 x k
x k1
x E 1 x, y l, z m dx
k
Discretization model equations
Step 3:
Next steps; discretize all components of main equation:
1
i0 
i
Ê 
0 Ĵ s
r Ê i


i) i
0 
Ê

with, 
i
ii) 1 Ê
r
0 Ĵ s
iii) i
Discretization model equations
1
ii) 
r Ê
1st H  μ 1  E
2nd   H
Discretization curl
1
H  μ E
E dr E 1 x, y, zdx E 2 x, y, zdy EndV
C
C
V
Stokes
Discretization curl
 H
Introduce (discreet) residu
1

r V i0 
Ê 
i0 J s
r E 
Goal: solve for r = 0
How? Multigrid solver
Overview Multigrid
Idea: use BIM for solving Ax=b
1. the error e=xex-xapr becomes smooth (not small)
2. Quantity smooth on fine grid  approx on
coarser grid (e.g. double mesh size)
Concl: error smooth after x relaxation sweeps 
approx error on coarser grid  Cheaper/Faster
Basis MG
m
m
m
m
m
1
m
m
xm

r
:b
A
x

Â
ê
r

x
x

ê
h
h
h
h
h
h
h
h
h
h
h
• Pre-smoothing
• Coarse grid correction:
Restriction
Compute approximation solution of defect equation
- Direct/iterative solver
- New cycle on coarser grid
Prolongation
• Post-smoothing
Basis MG
Important choices:
•Coarser grids
•Restriction operator: residu from fine to coarse
•Prolongation operator: correction from coarse to fine
•Smoother
MG Components
Coarser grids:
MG Components
Restriction:
Full weighting
MG Components
Prolongation:
Linear/bilinear interpolation, is transpose of restriction
MG Components
Smoother:
Pointwise smoother
Symmetric GS-LEX
Introduce Test probleem
• Artificial eigenvalues problem:
ψ  sin x sin y sin z, Ek  α k
ψ
,
xk
 z  π : σ  σ 0  σ1 ( x  1)( y  2)( z  π) 2

z  π : σ  σ0
• On the domain [0,2π]3.
• This defines the source term Js.
• Convergence: 10–8
Stretching
Problems MG Solver
σ0=10 S/m, σ1=1 S/m
cells
equidistant
hmax
Stretched (4%)
MG
bi hmax
MG
bi
16 0.39
7
6 0.45
8
6
32 0.20
8
7 0.26
11
8
64 0.098
8
7 0.17
12
14
128 0.049
8
6 0.13
81
32
Anisotropy
2D anisotropic elliptical equations:
2



u

u
f

x,
y
,


0,
1
, u u
x, y #
xx
yy
u f 
x, y
, 


0 1
Anisotropy
Discretization in stencil notation:
1
Ah 


1
h2
 
2
1 
1
h
Anisotropy
Error averaging with GS-LEX:
e mh 1 
x k , y l 
1


e mh 1 
x k1 , y l 
e mh 
x k1 , y l 
2
1
e mh 1 
x k , y l1 e mh 
x k , y l1 

If ε→0,
1
1
m
1
m


e m

x
,
y


e
x
,
y

e
x k , y l1 

k l
k l1
h
h
h 
2
No smoothing effect in x-direction
Anisotropy and stretched grid
2D elliptical equations:


x, y
,  
0, 12 , u u
x, y #
xx u 
yy u f 
u f 
x, y
, 


Simple stretching:
hx 
hy

Hence:
u
x k1 , y l 2u
x k , y l u
x k1 , y l 
2

u



O
h

xx
2
h
u
x k , y l1 2u
x k , y l u
x k , y l1 
2

u


O
h

yy
2
h
Anisotropy
Two possible improvements MG solver:
1. Semi coarsening
2. Line-smoother
Semicoarsening
Line-smoother
Solve all unknowns on line in direction
anisotropy simultaneously.
Reason: Errors become smooth if strong
connected unknows are updated
collectively
1,l,m
2,l,m
Nx – 1,l,m
Preview results
Combination line-smoother and semi-coarsening gives
good results
Factor 5 less MG iterations needed
Summary
1.
2.
3.
4.
Oil/Gas reservoir?
EM diffusion method  Maxwell equations
Multigrid solver  Problems when gridstretching used
Improvements:
1. Line-Smoother
2. Semi Coarsening
5. Results are obtained  more research for improvement
and generalisation, mathematical soundness
Questions?