1. Continuation Methods for Performing Stability Analysis of Large-Scale Applications LOCA: Library Of Continuation Algorithms Andy Salinger Roger Pawlowski, Louis Romero, Ed Wilkes Sandia National Labs Albuquerque, New Mexico 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. 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

3. Example of Multiplicity: Exothermic Chemical Reaction LOCA provides analysis tools to application code: Parameter Continuation (3 types): Tracks family of steady state solutions with parameter Eigensolver (3 Drivers for P_ARPACK): Calculates leading eigenvalues to determine linear stability (post-processing) Bifurcation Tracking (4 types): Locates neutral stability point (x,p) and tracks as a function of a second parameter T max Reaction Rate

4. Examples of Hysteresis / Turning Point Bifurcations (Eigenvalue =0) Capillary CondensationFlow in CVD ReactorYeast Cell-Cycle Control Buckling of Garden HoseBlock Copolymer Self-AssemblyPropane&Propylene Combustion

5. Examples of Hopf Bifurcations (Eigenvalue =0+  i ) Vortex Shedding Rising Bubble Ober and ShadidTheodoropoulos and Kevrekidis

6. Eigensolver via ARPACK

7. LOCA has been Targeted to Existing Large-Scale Application Codes Requirements for algorithms in LOCA 1.0: Must work with iterative (approximate) linear solvers on distributed memory machines Non-Invasive Implementation (matrix blind) Should avoid or limit:  Requiring more derivatives  Changing sparsity pattern of matrix  Increasing memory requirements Assumption: Application code uses Newton’s method

8. Bordering Algorithms meet these Requirements … but 4 solves of J per Newton Iteration are used to drive J singular! Turning Point Bifurcation Full Newton Algorithm Bordering Algorithm

9. Bordering Algorithm for Hopf tracking

10. LOCA: The Library of Continuation Algorithms Arclength continuation Turning point (fold) tracking Pitchfork tracking Phase transition tracking rSQP optimization hooks (Biegler, CMU) Residual fill ( R ) Jacobian Matrix solve ( J -1 b ) Mat-Vec multiply ( Jb ) Set parameters (  ) LOCA AlgorithmsLOCA Interface

11. LOCA: The Library of Continuation Algorithms Arclength continuation Turning point (fold) tracking Pitchfork tracking Phase transition tracking rSQP optimization hooks (Biegler, CMU) Eigensolver: Cayley transform driver for ARPACK Residual fill ( R ) Jacobian Matrix solve ( J -1 b ) Mat-Vec multiply ( Jb ) Set parameters (  ) Fill mass matrix ( M ) Shifted Matrix Solve ( J+   M ) LOCA AlgorithmsLOCA Interface

12. LOCA: The Library of Continuation Algorithms Arclength continuation Turning point (fold) tracking Pitchfork tracking Phase transition tracking rSQP optimization hooks (Biegler, CMU) Eigensolver: Cayley transform driver for ARPACK Hopf tracking Residual fill ( R ) Jacobian Matrix solve ( J -1 b ) Mat-Vec multiply ( Jb ) Set parameters (  ) Fill mass matrix ( M ) Shifted Matrix Solve ( J+   M ) Complex matrix solve ( J+ i  M ) LOCA AlgorithmsLOCA Interface

13. Stability of Buoyancy-Driven Flow: 3D Rayleigh-Benard Problem in 5x5x1 box 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 200K node mesh partitioned for 320 Processors

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

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

16. Rayleigh-Benard Problem used to Demonstrate Scalability of Algorithms Scalability Continuation: 16M Eigensolver: 16M Turning Point: 1M Pitchfork: 1M Hopf:0.7M Steady Solve 5 Minutes Eigenvalue Calculation (~5) 10-20 Minutes Pitchfork Tracking 25 Minutes Hopf Tracking 80 Minutes (p=200) 275K Unknowns: 128 Procs

17. CVD Reactor Design and Scale-up: Tracking of instability leads to design rule Good Flow Bad Flow Design rule for location of instability signaling onset of ‘bad’ flow

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

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

20. Summary: Powerful stability analysis tools have been developed for performing computational design of large-scale applications General purpose algorithms in LOCA linked to massively parallel codes that use Newton with iterative linear solves. Bifurcations tracked for 1.0 Million unknown models Singular (yet easy) formulations work semi-robustly LOCA Good Bad

21. Future Work Incorporate LOCA into Trilinos/NOX Do intelligent solves of nearly-singular matrices Multiparameter continuation (Henderson, IBM) New applications: Buckling of structures Electronic circuits www.cs.sandia.gov/LOCA

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