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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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Presentation on theme: "Improvement of a multigrid solver for 3D EM diffusion Research proposal final thesis Applied Mathematics, specialism CSE (Computational Science and Engineering)"— Presentation transcript:

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

2 Tom Jönsthövel TU Delft januari 6, 2006 Tutors: 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: Oil/Gas? EM source receivers

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 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)  2 nd order accuracy for electric/magnetic field components

12 Discretization model equations Step 2: Placement EM field components (Yee’s scheme):

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

14 Discretization model equations ii) 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=x ex -x apr 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 σ 0 =10 S/m, σ 1 =1 S/m cellsequidistantStretched (4%) h max MGbih max MGbi

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: 1.Semi coarsening 2.Line-smoother

33 Semicoarsening

34 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

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

36 Summary 1.Oil/Gas reservoir? 2.EM diffusion method  Maxwell equations 3.Multigrid solver  Problems when gridstretching used 4.Improvements: 1.Line-Smoother 2.Semi Coarsening 5.Results are obtained  more research for improvement and generalisation, mathematical soundness

37 Questions?


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