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Continuation Methods for Performing Stability Analysis of Large-Scale Applications

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.

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Continuation Methods for Performing Stability Analysis of Large-Scale Applications

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  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 Tmax Reaction Rate • 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

  4. Examples of Hysteresis / Turning Point Bifurcations (Eigenvalue l=0) Capillary Condensation Flow in CVD Reactor Yeast Cell-Cycle Control Buckling of Garden Hose Block Copolymer Self-Assembly Propane&Propylene Combustion

  5. Examples of Hopf Bifurcations (Eigenvalue l=0+wi) • Vortex Shedding • Rising Bubble Ober and Shadid Theodoropoulos and Kevrekidis

  6. Eigensolver via ARPACK

  7. LOCA has been Targeted to Existing Large-Scale Application Codes Assumption: Application code uses Newton’s method • 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

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

  9. Bordering Algorithm for Hopf tracking

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

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

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

  13. 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

  14. 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

  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 275K Unknowns: 128 Procs Scalability Continuation: 16M Eigensolver: 16M Turning Point: 1M Pitchfork: 1M Hopf: 0.7M

  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 Steady Solution (GOMA) Family of Instabilities Family of Solutions w/ Instability back pressure back pressure

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

  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 Bad LOCA Good

  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 Invert Cayley Transform v.1 Cayley Transform v.2 Lehoucq and Salinger, IJNMF, 2001.

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