Continuation Methods for Performing Stability Analysis of Large-Scale Applications

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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 DOEs 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.
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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

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Example of MultiplicityExothermic 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

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Examples of Hysteresis / Turning Point
Bifurcations (Eigenvalue l0)
Capillary Condensation
Flow in CVD Reactor
Yeast Cell-Cycle Control
Buckling of Garden Hose
Block Copolymer Self-Assembly
PropanePropylene Combustion
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Examples of Hopf Bifurcations (Eigenvalue l0wi
)

• Vortex Shedding

• Rising Bubble

Ober and Shadid
Theodoropoulos and Kevrekidis
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Eigensolver via ARPACK
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LOCA has been Targeted to Existing Large-Scale
Application Codes
Assumption Application code uses Newtons 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

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Bordering Algorithms meet these Requirements
Bordering Algorithm

• but 4 solves of J per Newton Iteration are used
  to drive J singular!

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Bordering Algorithm for Hopf tracking
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LOCAThe Library of Continuation Algorithms
LOCA Algorithms
LOCA Interface
Arclength continuation
Turning point (fold) tracking
Pitchfork tracking
Phase transition tracking
rSQP optimization hooks (Biegler, CMU)
Residual fill (R)
Jacobian Matrix solve (J-1b)
Mat-Vec multiply (Jb)
Set parameters (l, g)
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LOCAThe Library of Continuation Algorithms
LOCA Algorithms
LOCA Interface
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-1b)
Mat-Vec multiply (Jb)
Set parameters (l, g)
Fill mass matrix (M)
Shifted Matrix Solve (Js M)
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LOCAThe Library of Continuation Algorithms
LOCA Algorithms
LOCA Interface
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-1b)
Mat-Vec multiply (Jb)
Set parameters (l, g)
Fill mass matrix (M)
Shifted Matrix Solve (Js M)
Complex matrix solve (JiwM)
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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

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At Pr1.0, Two Pitchfork Bifurcations Located
with Eigensolver
3D Flow
2D Flow
No Flow
5 Coupled PDEs, 50x50x20 Mesh 275K Unknowns
Eigenvector at Pitchfork
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Three Flow Regimes Delineated by Bifurcation
Tracking Algorithms
Codimension 2 Bifurcation Near (Pr0.027, Ra2050)
Eigenvectors at Hopf
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Rayleigh-Benard Problem used to Demonstrate
Scalability of Algorithms
275K Unknowns 128 Procs
Steady Solve 5 Minutes
Eigenvalue Calculation (5) 10-20 Minutes
Pitchfork Tracking 25 Minutes
Hopf Tracking 80 Minutes (p200)
Scalability Continuation 16M Eigensolver
16M Turning Point 1M Pitchfork
1M Hopf 0.7M
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CVD Reactor Design and Scale-upTracking of
instability leads to design rule
Good Flow
Bad Flow
Design rule for location of instability signaling
onset of bad flow
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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
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LOCATramonto Capillary condensation phase
transitions studied in porous media
Phase diagram
Density Contours
Tramonto Frink and Salinger, JCP 1999,2000,2002
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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
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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
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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.