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Improvement of 3D EM Diffusion Multigrid Solver

This research proposal aims to improve a multigrid solver for 3D electromagnetic diffusion, with the goal of recovering conductivity profiles for potential reservoir detection in image structures. The proposal includes the mathematical model, discretization equations, an overview of multigrid, encountered problems, possible improvements, and a summary. Practical application and improvements in line-smoother and semi-coarsening methods are discussed. Results show promising outcomes, warranting further research for improvement and generalization. (500 characters)

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Improvement of 3D EM Diffusion Multigrid Solver

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  1. Improvement of a multigrid solver for 3D EM diffusion Research proposal final thesis Applied Mathematics, specialism CSE (Computational Science and Engineering)

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

  3. 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

  4. 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)

  5. (3D) Electromagnetic Diffusion 2D example: EM source receivers Oil/Gas?

  6. (3D) Electromagnetic Diffusion Amundsen, Johansen & Røsten (2004): A Sea Bed Logging (SBL) calibration survey over the Troll gas field

  7. Mathematical model Maxwell equations in presence of current source With, Ohm’s law

  8. Maxwell equations Eliminate the magnetic field from the equation: 

  9. Maxwell equations Transform equation from time to frequency domain: With  angular frequency. Now, In practice:

  10. Maxwell equations PEC boundary conditions: (Perfectly Electrically Conduction) domain

  11. Discretization model equationsStep 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

  12. Discretization model equationsStep 2: Placement EM field components (Yee’s scheme):

  13. i) with,  Discretization model equationsStep 3: Next steps; discretize all components of main equation: ii) iii)

  14. Discretization model equationsii) 1st 2nd

  15. Discretization curl Stokes

  16. Discretization curl

  17. Introduce (discreet) residu Goal: solve for r = 0 How? Multigrid solver

  18. 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

  19. 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

  20. Basis MG Important choices: • Coarser grids • Restriction operator: residu from fine to coarse • Prolongation operator: correction from coarse to fine • Smoother

  21. MG Components Coarser grids:

  22. MG Components Restriction: Full weighting

  23. MG Components Prolongation: Linear/bilinear interpolation, is transpose of restriction

  24. MG Components Smoother: Pointwise smoother Symmetric GS-LEX

  25. Introduce Test probleem • Artificial eigenvalues problem: • On the domain [0,2π]3. • This defines the source term Js. • Convergence: 10–8

  26. Stretching

  27. Problems MG Solver σ0=10 S/m, σ1=1 S/m

  28. Anisotropy 2D anisotropic elliptical equations:

  29. Anisotropy Discretization in stencil notation:

  30. Anisotropy Error averaging with GS-LEX: If ε→0, No smoothing effect in x-direction

  31. Anisotropy and stretched grid 2D elliptical equations: Simple stretching: Hence:

  32. Anisotropy Two possible improvements MG solver: Semi coarsening Line-smoother

  33. Semicoarsening

  34. 1,l,m 2,l,m Nx – 1,l,m Line-smoother Solve all unknowns on line in direction anisotropy simultaneously. Reason: Errors become smooth if strong connected unknows are updated collectively

  35. Preview results Combination line-smoother and semi-coarsening gives good results Factor 5 less MG iterations needed

  36. 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

  37. Questions?

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