Multilevel Incomplete Factorizations for Non-Linear FE problems in Geomechanics

1. Due Giorni di Algebra Lineare Numerica Bologna, Marzo 6-7, 2008 Multilevel Incomplete Factorizations for Non-Linear FE problems in Geomechanics Carlo Janna, Massimiliano Ferronato and Giuseppe Gambolati DMMMSA – University of Padova Department of Mathematical Methods and Models for Scientific Applications

2. Outline • Introduction • Level structure of the matrix • Multilevel Incomplete Factorization (MIF) • Numerical results • Drawbacks and possible solutions • Conclusions

3. The Geomechanical problem } Fluid removal/injection from/to subsurface Prediction by numerical models Pore pressure and effective stress variation Environmental, geomechanical and geotechnical applications

4. The Geomechanical problem • Classical Biot’s consolidation theory: • The problem can be solved by decoupling: • The fluid-dynamic problem is solved first • The output of the fluid-dynamic part is used as input for the geomechanical problem

5. The Geomechanical problem • The structural part of the problem is solved by FE minimizing the total potential energy • The arising system of equations is non-linear because l and G are functions of the stress state

6. The Geomechanical problem • The presence of faults and fractures in the geological media is another source of non-linearity

7. = opposite of the distance Constrained Minimization Problem: Interface Elements and Penalty method The Geomechanical problem • Faults and fractures act as contact surfaces

8. The Geomechanical problem • The non-linear quasi static problem is solved by a Newton-like scheme that results in a sequence of linear systems: • Some considerations: • Only FE in the reservoir are subject to a relevant stress change • Surrounding FE can be considered to behave elastically • The faults presence involves only few FE

9. Dof linked to linear elements Dof linked to non-linear elements Dof linked to interface elements The Sparse Linear System • The linearized system can be reordered in such a way to show its natural 3-level block structure:

10. The Sparse Linear System • The system matrix A is Symmetric Positive Definite: use of PCG • The K1, B11, B12blocks do not change during a simulation • The K2, B22blocks change whenever a stress perturbation occurs in the reservoir • The Cblock change whenever the contact condition varies on the faults • The system is very ill-conditioned due to the penalty approach because ||C|| >> || K1||, ||K2||

11. The Multilevel Incomplete Factorization • Define a partial incomplete factorization of a matrix A: with:

12. The Multilevel Incomplete Factorization • The use of M-1 as a preconditioner requires the solution of: • The second step is performed in this way: • y2 can be found approximately by using again a partial incomplete factorization of S1

13. The Multilevel Incomplete Factorization • Advantages of the approach: • There is no need to perform the factorization of the whole matrix A at every non-linear iteration • It is possible to independently tune the fill-in degree of each level with 2 parameters ρi1, ρi2 • The unknows linked to the penalty block are kept toghether in a single level

14. Numerical Results 3D Geomechanical problem of faulted rocks discretized with FE and IE

15. Level 1 Level 2 Level 3 Numerical Results

16. Numerical Results

17. Numerical Results

18. Numerical Results Comparison between ILLT and MIF in terms of: • Number of Iterations • CPU Time • Memory occupation

19. MIF ILLT # Iterations 75 # Iterations 53 Prec. [s] 108.69 Liv. 1 [s] 23.17 T. CG [s] 36.43 Liv. 2 [s] 11.29 T. Tot. [s] 145.12 Liv. 3 [s] 17.04 3.92 Prec. [s] 51.50 T. CG [s] 21.26 T. Tot. [s] 72.76 2.68 3.29 Numerical Results • Performance in the solution of a single system

20. 2.68 1.89 3.29 2.35 Numerical Results Observations: • CPU time for Lev. 1 and Lev. 2 (34.46 s) can be made up for in a few non-linear iterations • Losing some performance (about 15%) the memory occupation can be further reduced

21. Numerical Results • Performance in a real whole simulation

22. Drawbacks • The original matrix is SPD, but an SPD multilevel incomplete factorization is not guaranteed to exist: • The factorization of the 11 block of the actual level may be indefinite • The Schur complement of the actual level may be indefinite

23. Possible solutions • The use of another solver instead of PCG, i.e. CR or SQMR, that does not require positive definiteness • Allowing for larger fill-in degrees • The implementation of special techniques to guarantee the positive definiteness of the factors

24. Possible solutions • Procedure of Ajiz & Jennings for the 11 block factorization Diagonal compensation to enforce

25. Possible solutions • Procedure of Tismenetsky for the Schur complement computation

26. Conclusions • The Multilevel Incomplete Factorization has proven to be a robust and reliable tool for the solution of non linear problems in geomechanics • Part of the preconditioner can be computed at the beginning of the simulation thus reducing the set-up phase during the non-linear iterations • Its level structure allows for a fine tuning of the fill-in degree and thus of the preconditioner quality

27. …and future work • The development of techniques to guarantee the positive definiteness of the preconditioner • Sensitivity analysis of the user-defined parameters • Application to other naturally multilevel problems (coupled problems such as coupled consolidation, flow & transport..)

28. Thank you for your attention DMMMSA – University of Padova Department of Mathematical Methods and Models for Scientific Applications