Improvement of a multigrid solver for 3D EM diffusion

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Improvement of a multigrid solver for 3D EM
diffusion

• Research proposal final thesis Applied
  Mathematics, specialism CSE (Computational
  Science and Engineering)

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Tom JönsthövelTU Delft januari 6, 2006Tutors
Kees Oosterlee (EWI, TUD)Wim Mulder (SIEP,
Shell)
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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

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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)

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(3D) Electromagnetic Diffusion
2D example
EM source
receivers
Oil/Gas?
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(3D) Electromagnetic Diffusion
Amundsen, Johansen Røsten (2004) A Sea Bed
Logging (SBL) calibration survey over the Troll
gas field
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Mathematical model
Maxwell equations in presence of current source
With,
Ohms law
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Maxwell equations
Eliminate the magnetic field from the equation
?
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Maxwell equations
Transform equation from time to frequency
domain
With ? angular frequency. Now,
In practice
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Maxwell equations
PEC boundary conditions (Perfectly Electrically
Conduction)
domain
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Discretization model equationsStep 1

• Choose discretization
• Finite Integration Technique (Clemens/Weiland
  01)
• Finite volume generalisation of Yees scheme
  (1966)
• Error analysis for constant-coeffients (Monk
  Sülli 1994)
• 2nd order accuracy for electric/magnetic field
  components

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Discretization model equationsStep 2
Placement EM field components (Yees scheme)
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Discretization model equationsStep 3
Next steps discretize all components of main
equation
ii)
iii)
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Discretization model equationsii)
1st 2nd
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Discretization curl
Stokes
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Discretization curl
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Introduce (discreet) residu
Goal solve for r 0 How? Multigrid solver
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Overview Multigrid
Idea use BIM for solving Axb 1. the error
exex-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
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Basis MG

• Pre-smoothing
• Coarse grid correction
• Restriction
• Compute approximation solution of defect
  equation
• - Direct/iterative solver
• - New cycle on coarser grid
• Prolongation
• Post-smoothing

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Basis MG

• Important choices
• Coarser grids
• Restriction operator residu from fine to coarse
• Prolongation operator correction from coarse to
  fine
• Smoother

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MG Components
Coarser grids
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MG Components
Restriction Full weighting
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MG Components
Prolongation Linear/bilinear interpolation, is
transpose of restriction
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MG Components
Smoother Pointwise smoother Symmetric GS-LEX
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Introduce Test probleem

• Artificial eigenvalues problem
• On the domain 0,2p3.
• This defines the source term Js.
• Convergence 108

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Stretching
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Problems MG Solver
s010 S/m, s11 S/m
cells equidistant equidistant equidistant equidistant Stretched (4) Stretched (4) Stretched (4) Stretched (4)
cells hmax MG MG bi hmax MG MG bi
16 0.39 7 6 6 0.45 8 6 6
32 0.20 8 7 7 0.26 11 8 8
64 0.098 8 7 7 0.17 12 14 14
128 0.049 8 6 6 0.13 81 32 32
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Anisotropy
2D anisotropic elliptical equations
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Anisotropy
Discretization in stencil notation
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Anisotropy
Error averaging with GS-LEX
If e?0,
No smoothing effect in x-direction
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Anisotropy and stretched grid
2D elliptical equations
Simple stretching
Hence
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Anisotropy

• Two possible improvements MG solver
• Semi coarsening
• Line-smoother

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Semicoarsening
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Line-smoother
Solve all unknowns on line in direction
anisotropy simultaneously. Reason Errors become
smooth if strong connected unknows are updated
collectively
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Preview results
Combination line-smoother and semi-coarsening
gives good results Factor 5 less MG iterations
needed
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Summary

• Oil/Gas reservoir?
• EM diffusion method ? Maxwell equations
• Multigrid solver ? Problems when gridstretching
  used
• Improvements
• Line-Smoother
• Semi Coarsening
• Results are obtained ? more research for
  improvement and generalisation, mathematical
  soundness

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Questions?