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.

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 libraryExpertise

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

Tmax

Reaction Rate

Examples of Hysteresis / Turning Point Bifurcations (Eigenvalue =0)

-2.6

-2.4

-2.2

-2

-1.8

-1.6

-1.4

0 0.1 0.2 0.3 0.4 0.5

mixedperp

free

ene

rgy

wA

Capillary Condensation Flow in CVD Reactor Yeast Cell-Cycle Control

Buckling of Garden Hose Block Copolymer Self-Assembly Propane&Propylene Combustion

Examples of Hopf Bifurcations (Eigenvalue =0+i )

• Vortex Shedding • Rising Bubble

Ober and Shadid Theodoropoulos and Kevrekidis

Eigensolver via ARPACK

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

Bordering Algorithms meet these Requirements

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

R 0=

Jn 0=

n 1=

Turning Point BifurcationJ 0 Rp

Jn x J Jpn

0 t0

xn

p

R–Jn–

1 n–

=

Full Newton Algorithm

Ja R–=

Jb R– p=

Jc Jn – xa J– n=

Jd Jn – xb Jp– n=

p 1 n– c– d =x a p b+=n c p d+=

Bordering Algorithm

Bordering Algorithm for Hopf tracking

f x 0=

Jy Mz+ 0=

Jz My– 0=

lty 1– 0=

ltz 0=

J 0 0 0f------

Jy x

-------------Mz

x--------------------+ J M Mz

Jy -------------

Mz --------------------+

Jz x

-------------My

x--------------------– M– J My–

Jz -------------

My --------------------–

0 lt

0 0 0

0 0 lt

0 0

xyz

f–Jy– Mz–

Jz– My+

1 lty–

ltz

=

J M– M J

g

h

Jy -------------

Jy x

-------------b Mz

--------------------- Mz

x---------------------b+ + +

Jz -------------

Jz x

-------------b My

---------------------– My

x---------------------b–+

=

Ja f–=

Jbf------–=

J M– M J

e

f

Jy x

-------------a Mz

x---------------------a+

Jz x

-------------a My

x---------------------a–

=

J M– M J

c

d

Mz

My–=

ltdl

te l

td l

tf l

tc–+

lthl

tc l

tgl

td–

--------------------------------------------=

1 lte l

tg+ +

ltc

------------------------------------–=

y y– e– g– c–=

z z– f– h– d–=

x a b+=

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-1b)

Mat-Vec multiply (Jb)

Set parameters ()

LOCA Algorithms LOCA Interface

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-1b)

Mat-Vec multiply (Jb)

Set parameters ()

Fill mass matrix (M)

Shifted Matrix Solve (J+M)

LOCA Algorithms LOCA Interface

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-1b)

Mat-Vec multiply (Jb)

Set parameters ()

Fill mass matrix (M)

Shifted Matrix Solve (J+M)

Complex matrix solve (J+iM)

LOCA Algorithms LOCA Interface

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 meshpartitioned for320 Processors

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

Three Flow Regimes Delineated by Bifurcation Tracking Algorithms

Codimension 2 BifurcationNear (Pr=0.027, Ra=2050) Eigenvectors at Hopf

Rayleigh-Benard Problem used to Demonstrate Scalability of Algorithms

ScalabilityScalabilityContinuation:

16MEigensolver: 16MTurning Point:

1MPitchfork:

1MHopf: 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

CVD Reactor Design and Scale-up:Tracking of instability leads to design rule

Ra 1.75Re0.5

1100Re---------+

=

Good Flow

Bad Flow

Design rule for location of instability signaling onset of ‘bad’ flow

Operability Window for Manufacturing Process Mapped with LOCA around GOMA

Slot Coating Application

Family of InstabilitiesFamily of Solutions w/ Instability

Steady Solution (GOMA)

back pressure

bac

k p

ress

ure

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

Summary: Powerful stability analysis tools have been developed for performing computational

design of large-scale applicationsGeneral 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

LOCAGood

Bad

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

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.