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

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

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

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

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

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

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

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

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

10. The Sparse Linear System The system matrix A is Symmetric Positive Definite: use of PCG The system matrix A is Symmetric Positive Definite: use of PCG The K 1, B 11, B 12 blocks do not change during a simulation The K 1, B 11, B 12 blocks do not change during a simulation The K 2, B 22 blocks change whenever a stress perturbation occurs in the reservoir The K 2, B 22 blocks change whenever a stress perturbation occurs in the reservoir The C block change whenever the contact condition varies on the faults The C block change whenever the contact condition varies on the faults The system is very ill-conditioned due to the penalty approach because ||C|| >> || K 1 ||, ||K 2 || The system is very ill-conditioned due to the penalty approach because ||C|| >> || K 1 ||, ||K 2 ||

11. The Multilevel Incomplete Factorization Define a partial incomplete factorization of a matrix A: 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 use of M -1 as a preconditioner requires the solution of: The second step is performed in this way: The second step is performed in this way: y 2 can be found approximately by using again a partial incomplete factorization of S 1 y 2 can be found approximately by using again a partial incomplete factorization of S 1

13. The Multilevel Incomplete Factorization Advantages of the approach: Advantages of the approach: There is no need to perform the factorization of the whole matrix A at every non-linear iteration 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 ρ i1, ρ i2 It is possible to independently tune the fill-in degree of each level with 2 parameters ρ i1, ρ i2 The unknows linked to the penalty block are kept toghether in a single level 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. Numerical Results # of unknowns Level 1 435,207 Level 2 163,581 Level 3 20,014 Total618,802 Level 1 Level 2 Level 3

16. Numerical Results

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

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

20. Numerical Results CPU time for Lev. 1 and Lev. 2 (34.46 s) can be made up for in a few non-linear iterations 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 Losing some performance (about 15%) the memory occupation can be further reduced Observations: 3.29 2.68 2.351.89

21. ILLT CPU Prec 26h 57m 33s T. CG 3h 45m 42s Overhead 22m 45s Total CPU Time 31h 6m 0s MIF CPU Prec 4h 44m 10s T. CG 3h 41m 6s Overhead 21m 13s Total CPU Time 8h 46m 29s Numerical Results Performance in a real whole simulation 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 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 factorization of the 11 block of the actual level may be indefinite The Schur complement 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 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 Allowing for larger fill-in degrees The implementation of special techniques to guarantee the positive definiteness of the factors 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 Procedure of Ajiz & Jennings for the 11 block factorization Diagonal compensation to enforce

25. Possible solutions Procedure of Tismenetsky for the Schur complement computation 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 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 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 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 The development of techniques to guarantee the positive definiteness of the preconditioner Sensitivity analysis of the user-defined parameters Sensitivity analysis of the user-defined parameters Application to other naturally multilevel problems (coupled problems such as coupled consolidation, flow & transport..) 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