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

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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, Ohm’s 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 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

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

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

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Discretization model equations ii) 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 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

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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 [0,2π]3. This defines the source term Js. Convergence: 10–8

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Stretching

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Problems MG Solver σ 0 =10 S/m, σ 1 =1 S/m cellsequidistantStretched (4%) h max MGbih max MGbi

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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 ε → 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: 1.Semi coarsening 2.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 1,l,m 2,l,m Nx – 1,l,m

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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 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

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Questions?

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