The Development of a 3D Self-Adaptive Goal-Oriented hp-finite Element Software for Simulations of Resistivity Logging Instruments

Transcription

1 A Progress Report on: The Development of a 3D Self-Adaptive Goal-Oriented hp-finite Element Software for Simulations of Resistivity Logging Instruments David Pardo (dzubiaur@yahoo.es), L. Demkowicz, C. Torres-Verdin, M. Paszynski, J. Kurtz Collaborators: Science Department of Baker-Hughes, L.E. Garcia-Castillo, W. Rachowicz December 13, 2005 Department of Petroleum and Geosystems Engineering, and Institute for Computational Engineering and Sciences (ICES)

2 OVERVIEW 1. Motivation 2. Numerical Methodology hp-finite Elements (Exponential convergence) Automatic Goal-Oriented Refinements (in the quantity of interest) 3. Current Stage of the 3D High Performance FE Software 4. Preliminary Results 5. Conclusions and Future Work

3 MOTIVATION Induction Instrument in a Deviated Well MATERIAL I LOGGING INSTRUMENT MATERIAL II MATERIAL II MATERIAL III MATERIAL III MATERIAL IV BOREHOLE Goal: Determine EM field at the receiver antennas. 2

4 MOTIVATION Through Casing Resistivity Tools MATERIAL I BOREHOLE STEEL CASING MATERIAL II MATERIAL II MATERIAL III ELECTRODES MATERIAL III MATERIAL IV Goal: Determine EM field at the receiver antennas. 3

5 MOTIVATION Electrode Problem 100 m 0.1m 99.9m BACKGROUND: 1 Ohm. m 99 m RECEIVER Loop antenna in a homogeneous media at DC. TRANSMITTER 1 m Computational domain: 100 m x 100 m. CROSS SECTION OF ANTENNAS: 0.05 m x 0.05 m 100 m 4

6 THE hp-finite ELEMENT METHOD The h-finite Element Method 1. Convergence limited by the polynomial degree, and large material contrasts. 2. Optimal h-grids do NOT converge exponentially in real applications. x z y 3. They may lock (100% error). The p-finite Element Method 1. Exponential convergence feasible for analytical ( nice ) solutions. 2. Optimal p-grids do NOT converge exponentially in real applications. x z y 3. If initial h-grid is not adequate, the p-method will fail miserably. The hp-finite Element Method 1. Exponential convergence feasible for ALL solutions. 2. Optimal hp-grids DO converge exponentially in real applications. x z y 3. If initial hp-grid is not adequate, results will still be great. 5

7 GOAL-ORIENTED ADAPTIVITY Mathematical Formulation (Goal-Oriented Adaptivity) DIRECT PROBLEM - Ψ - 2D Cross-Section DUAL PROBLEM - G - 2D Cross-Section Representation Formula for the Error in the Quantity of Interest: L(Ψ)=b(Ψ,G) = Ω σ Ψ GdV 6

8 GOAL-ORIENTED ADAPTIVITY Mathematical Formulation (Goal-Oriented Adaptivity) We define: e = Ψ Ψ hp, ǫ = G G hp. Ψ exact solution of direct problem G exact solution of dual problem Upper Bound for the Error in the Quantity of Interest: L(e) = b(e, G) = b(e, ǫ) = σ e ǫ dv Ω ALGORITHM I: σ e ǫ dv K K ALGORITHM II: σ ( e) 2 dv σ ( ǫ) 2 dv K K K 7

9 SELF-ADAPTIVE GOAL-ORIENTED hp-fem Algorithm for Goal-Oriented Adaptivity - STEP I - Solve Direct and Dual Problems on Grid hp Solve Direct and Dual Problems on Grid h/2, p+1 Use the fine grid solution to estimate the coarse grid error function. Apply the fully automatic goal-oriented hp-adaptive algorithm. Next optimal hp-grid: 8

10 SELF-ADAPTIVE GOAL-ORIENTED hp-fem Algorithm for Goal-Oriented Adaptivity - STEP II - Solve Direct and Dual Problems on Grid hp Solve Direct and Dual Problems on Grid h/2, p+1 Use the fine grid solution to estimate the coarse grid error function. Apply the fully automatic goal-oriented hp-adaptive algorithm. Next optimal hp-grid: 9

11 SELF-ADAPTIVE GOAL-ORIENTED hp-fem Algorithm for Goal-Oriented Adaptivity - STEP III - Solve Direct and Dual Problems on Grid hp Solve Direct and Dual Problems on Grid h/2, p+1 Use the fine grid solution to estimate the coarse grid error function. Apply the fully automatic goal-oriented hp-adaptive algorithm. Next optimal hp-grid: 10

12 SELF-ADAPTIVE GOAL-ORIENTED hp-fem Algorithm for Goal-Oriented Adaptivity - STEP IV - Solve Direct and Dual Problems on Grid hp Solve Direct and Dual Problems on Grid h/2, p+1 Use the fine grid solution to estimate the coarse grid error function. Apply the fully automatic goal-oriented hp-adaptive algorithm. Next optimal hp-grid: 11

13 CURRENT STAGE OF THE 3D hp-fe SOFTWARE 3Dhp-log Contains Several Packages: SOLVER OF LINEAR EQUATIONS ADAPTIVITY AUTOMATIC REFINEMENTS KERNEL GEOMETRY AND GRAPHICS parallel parallel version interacts with all packages Improvement needed PROBLEM SET UP 12

14 CURRENT STAGE OF THE 3D hp-fe SOFTWARE 3Dhp-log Contains Several Packages: METIS optimal ordering of elements MUMPS solver of linear equations interface solve1 solver of linear equations interface two_grid_hp iterative solver of linear equations ADAPTIVITY AUTOMATIC REFINEMENTS 3D hp kernel data structures mesh refinements GEOMETRY AND GRAPHICS parallel parallel version interacts with all packages External components Improvement needed PROBLEM SET UP 13

15 CURRENT STAGE OF THE 3D hp-fe SOFTWARE Importance of Ordering Elements (when using MUMPS) The time and memory used by the solver MUMPS depends upon the ordering of elements. The ordering of elements provided by METIS seems adequate. 14

16 CURRENT STAGE OF THE 3D hp-fe SOFTWARE Importance of Order of Approximation (when using MUMPS) For p=6, the solver MUMPS utilizes 50% more memory and time than for p=3. 15

17 CURRENT STAGE OF THE 3D hp-fe SOFTWARE Memory and Time Used by MUMPS The memory used by MUMPS depends linearly with respect to the number of unknowns. The time used by MUMPS has a quadratic dependance with respect to the number of unknowns. By using 16 processors, we expect to solve 2 million unknowns in 1 hour. 16

18 CURRENT STAGE OF THE 3D hp-fe SOFTWARE 3Dhp-log Contains Several Packages: METIS optimal ordering of elements MUMPS solver of linear equations interface solve1 solver of linear equations interface two_grid_hp iterative solver of linear equations goal_adapt goal oriented adaptivity h_adapt h adaptivity energy norm 3D hp kernel data structures mesh refinements GEOMETRY AND GRAPHICS hp_adapt hp adaptivity energy norm parallel parallel version interacts with all packages External components Improvement needed PROBLEM SET UP 17

19 CURRENT STAGE OF THE 3D hp-fe SOFTWARE Resources Needed by the Adaptive Algorithm The adaptive algorithm utilizes about half of the time used by the solver MUMPS. The amount of memory used by the adaptive algorithm is negligible, and results are not reported here. Since the final result is given by the final fine-grid solution, the adaptive algorithm does NOT need to be executed on the last iteration. For multiple logging instrument positions, the optimal grid may be reutilized without employing the adaptive algorithm. Resources needed by the adaptive algorithm are between 4% and 25% of the total resources needed by the 3D code (if MUMPS is used). 18

20 CURRENT STAGE OF THE 3D hp-fe SOFTWARE 3Dhp-log Contains Several Packages: METIS optimal ordering of elements MUMPS solver of linear equations interface solve1 solver of linear equations interface two_grid_hp iterative solver of linear equations goal_adapt goal oriented adaptivity h_adapt h adaptivity energy norm hp_adapt hp adaptivity energy norm 3D hp kernel data structures mesh refinements graph3d 3D graphics GMP Geometrical Modeling Package parallel parallel version interacts with all packages External components Improvement needed PROBLEM SET UP 19

21 CURRENT STAGE OF THE 3D hp-fe SOFTWARE Geometry and Graphics MATERIAL II LOGGING INSTRUMENT MATERIAL I MATERIAL II We are developing an interface program to automatically generate cylindrical-type grids with userprescribed angles. MATERIAL III MATERIAL III GMP utilizes isoparametric elements (i.e. approximation of the geometry coincides with that of the solution). After each refinement, an update on the geometry is performed. MATERIAL IV BOREHOLE A new graphics package shall be installed within this month. 20

22 CURRENT STAGE OF THE 3D hp-fe SOFTWARE 3Dhp-log Contains Several Packages: METIS optimal ordering of elements MUMPS solver of linear equations interface solve1 solver of linear equations interface two_grid_hp iterative solver of linear equations goal_adapt goal oriented adaptivity parallel parallel version interacts with all packages External components Improvement needed h_adapt h adaptivity energy norm hp_adapt hp adaptivity energy norm 3D hp kernel data structures mesh refinements laplace set up equation to solve problem set up problem graph3d 3D graphics GMP Geometrical Modeling Package generate_input set up geometry for the problem 21

23 PRELIMINARY RESULTS Electrode Problem 100 m 0.1m 99.9m BACKGROUND: 1 Ohm. m 99 m RECEIVER Loop antenna in a homogeneous media at DC. TRANSMITTER 1 m Computational domain: 100 m x 100 m. CROSS SECTION OF ANTENNAS: 0.05 m x 0.05 m 100 m 22

24 PRELIMINARY RESULTS Electrode Problem Final hp-grid Final solution 2D Solution: D Solution:

25 PRELIMINARY RESULTS Electrode Problem 24

26 PRELIMINARY RESULTS Fichera problem (unknown exact solution) OBJECTIVE: INTEGRAL OF SOLUTION OVER THIS BRICK z x y Equation: u = 0 Boundary Conditions: Neumann, Dirichlet z x y z x y Solution of Direct Problem Solution of Dual Problem 25

27 PRELIMINARY RESULTS Fichera problem (final hp-grids) Energy-norm: x z y x z y x z y Goal-oriented (algorithm I) Goal-oriented (algorithm II) 26

28 PRELIMINARY RESULTS Fichera problem (convergence history) Exponential Convergence in the Quantity of Interest 27

29 CONCLUSIONS AND FUTURE WORK The self-adaptive goal-oriented hp-adaptive strategy converges exponentially in terms of a user-prescribed quantity of interest vs. the CPU time. Preliminary results indicate that it shall be possible to simulate a variety of EM logging instruments in deviated wells by using the 3D self-adaptive goal-oriented hp-fem. The software is expected to be suitable for ALL kind of resistivity logging instruments in possibly cased wells. Cylindrical geometries can be accurately described by using higher-order elements. Department of Petroleum and Geosystems Engineering, and Institute for Computational Engineering and Sciences (ICES) 28

30 FUTURE WORK Tasks and Completion Date 3D DC CODE PHASE I: NEW adaptive package and solver. COMPLETED. PHASE II: Goal-Oriented Adaptivity. COMPLETED. PHASE III: Interface for Describing Logging Problems. COMPLETED. PHASE IV: New 3D Graphics Package. 20 Jan PHASE V: Parallel Solver (MUMPS). 1 Mar PHASE VI: Logging Examples (without casing). 10 Mar PHASE VII: Iterative Solver. 10 Jun 2006 PHASE VIII: Parallel Version of 3D code. 20 Jun 2006 PHASE IX: Through Casing Resistivity Instruments. 20 Jul D AC CODE PHASE X: AC CODE USING EDGE ELEMENTS. 1 April

31 APPENDIX: SHOULD WE USE CYLINDRICAL GRIDS? Simpler geometries. The possibility of using cylindrical grids MAIN ADVANTAGES Possibly less elements needed on the azimuthal direction. MAIN DISADVANTAGES Advantages mentioned above are not clear in the case of deviated wells. Extra boundary condition needed (Ψ(0) = Ψ(2π)). Integration becomes not exact. New partial differential equations need to be implemented. Continuous elements and Nedelec elements are based on cartesian geometries. At ρ = 0 degenerated elements may be needed. FOR DEVIATED WELLS, IT IS NOT CLEAR THAT THE USE OF CYLINDRICAL GRIDS (AS OPPOSED TO CARTESIAN GRIDS) BECOMES MORE ADEQUATE 30