Stability Analysis Algorithms for Large-Scale Applications

1. Stability Analysis Algorithms for Large-Scale Applications Andy Salinger, Roger Pawlowski, Ed Wilkes Louis Romero, Rich Lehoucq, John Shadid Sandia National Labs Albuquerque, New Mexico Computational Challenges in Dynamical Systems Fields Institute, Dec. 6, 2001 Supported by DOE’s MICS and ASCI programs Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company,for the United States Department of Energy under contract DE-AC04-94AL85000.

2. Elevator Talk (Lift Talk) What are you working on these days? We’re developing a library of stability analysis algorithms that work with massively parallel engineering analysis codes. The main research issues are developing algorithms that are relatively non-invasive (easy to implement) and that work reasonably well with approximate iterative linear solvers. With this “LOCA” software, we’ve been able to track bifurcations of 1 Million unknown PDE discretizations.

3. Why Do We Need a Stability Analysis Capability? • Nonlinear systems exhibit instabilities, e.g • Multiple steady states • Ignition • Symmetry Breaking • Onset of Oscillations • Phase Transitions These phenomena must be understood in order to perform computational design and optimization. Current Applications: Reacting flows, Manufacturing processes, Microscopic fluids Potential Applications: Electronic circuits, structural mechanics (buckling) • Delivery of capability: • LOCA library • Expertise

4. The Targeting of Large-Scale Applications Codes Restricts the Choice of Algorithms • Targeted Codes: Newton’s Method, Large-Scale, Parallel • Navier-Stokes & Reaction-Diffusion, Free Surface Flows, Molecular Theory, Structural Mechanics, Circuit Simulation • Requirements: Stability analysis algorithms must be scalable and relatively non-invasive: • Must work with iterative (approximate) linear solvers • Should avoid or limit: • Requiring more derivatives • Changing sparsity pattern of matrix • Increasing memory requirements

5. LOCA:The Library of Continuation Algorithms LOCA Algorithms LOCA Interface

6. LOCA:The Library of Continuation Algorithms LOCA Algorithms LOCA Interface

7. LOCA:The Library of Continuation Algorithms LOCA Algorithms LOCA Interface

9. Full Newton Algorithm Turning Point Bifurcation Bordering Algorithm Q: Can General Bifurcation Algorithms Scale to ASCI-Sized Problems? • Large problem sizes require iterative linear solves • The less invasive bordering algorithms require inversion of matrices that are being driven singular  

10. Bordering Algorithm for Hopf tracking

11. Eigenvalue Approx with Arnoldi, ARPACK 3 Spectral Transformations have Different Strengths Complex Shift and Invert Cayley Transform v.1 Cayley Transform v.2 Lehoucq and Salinger, IJNMF, 2001.

12. Stability of Buoyancy-Driven Flow: 3D Rayleigh-Benard Problem in 5x5x1 box 200K node mesh partitioned for 320 Processors • MPSalsa(Shadid et al., SNL): • Incompressible Navier-Stokes • Heat and Mass Transfer, Reactions • Unstrucured Finite Element (Galerkin/Least-Squares) • Analytic, Sparse Jacobian • Fully Coupled Newton Method • GMRES with ILUT Preconditioner (Aztec package) • Distributed Memory Parallelism

13. At Pr=1.0, Two Pitchfork Bifurcations Located with Eigensolver 3D Flow 2D Flow No Flow 5 Coupled PDE’s, 50x50x20 Mesh: 275K Unknowns Eigenvector at Pitchfork

14. Three Flow Regimes Delineated by Bifurcation Tracking Algorithms Codimension 2 Bifurcation Near (Pr=0.027, Ra=2050) Eigenvectors at Hopf

15. Rayleigh-Benard Problem used to Demonstrate Scalability of Algorithms 275K Unknowns: 128 Procs Scalability Eigensolver: 16M Continuation: 16M Turning Point: 1M Pitchfork: 1M Hopf: 0.7M

16. CVD Reactor Design and Scale-up:Buoyancy force can lead to undesirable flows Chemical Vapor Deposition of Semiconductors: GaN, GaAs

17. Good and bad flows are found to coexist at certain values of (Ra, Re) Good Flow Bad Flow 30500 Unknowns

18. Good and bad flows are found to coexist at certain values of (Ra, Re) 30500 Unknowns

19. Ideal gas curves collapse onto Boussinesq for good choice of To Tracking of bifurcation leads to design rule Boussinesq Pawlowski, Salinger, Romero, Shadid 2001

20. Optimization Algorithms, such as rSQP, Need Same Calls as Bifurcation Algs Collaboration with Biegler, CMU

21. Operability Window for Manufacturing Process Mapped with LOCA around GOMA Slot Coating Application Steady Solution (GOMA) Family of Instabilities Family of Solutions w/ Instability back pressure back pressure

22. LOCA+Tramonto: Capillary condensation phase transitions studied in porous media Bifurcation Diagram (a.k.a. Adsorption Isotherm) Density contours around random cylinders

23. Liquid Vapor Partial Condensation LOCA+Tramonto: Capillary condensation phase transitions studied in porous media Phase diagram Tramonto: Frink and Salinger, JCP 1999,2000,2002

24. Counter-terrorism via PDE Optimization: Find fluxes at 16 surfaces to match data at 25 sensors 5 Fluxes Re=10 1 2 Flow Transport 6724 State variables, 16 design variables, x0=y0=0 88 rSQP Iterations, f=1.5e-6 , 30 sec / iter

25. Summary and Future Work Powerful analysis tools has been developed to study large-scale flow stability applications: • General purpose algorithms in LOCA linked to massively parallel finite element codes. • Bifurcations tracked for 1.0 Million unknown models • Singular formulation works semi-robustly Future work : • Support common linear solvers (e.g. Aztec, Trilinos, PetsC, LAPACK) • Implement more invasive, non-singular (bordered) formulations • Multiparameter continuation (Henderson, IBM) • New application codes, e.g. buckling of structures