Large-Scale Stability Analysis Algorithms

Andy Salinger, Roger Pawlowski, Ed Wilkes

Louis Romero, Rich Lehoucq, John Shadid

Sandia National Labs

Albuquerque, New Mexico

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.

Supported by DOE’s MICS and ASCI programs

Elevator Speech

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 directly track bifurcations of Million unknown incompressible flow applications.

What are you working on these days?

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

LOCA:The Library of Continuation Algorithms

Arc-length continuation

Turning point tracking

Pitchfork tracking

Hopf tracking

Phase transition tracking

Eigensolver: ARPACK driver for Cayley transform

rSQP optimization hooks (Biegler, CMU)

Residual fill (R)

Jacobian Matrix solve (J-1b)

Mat-Vec multiply (Jb)

Set parameter ()

Fill mass matrix (M)

Complex matrix solve (J+iM)

Shifted Matrix Solve (J+M)

LOCA Algorithms LOCA Interface

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

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 around MPSalsa:3D Rayleigh Benard Problem in 5x5x1 box

ScalabilityScalabilityEigensolver: 16MContinuation:

16MTurning Point:

1MPitchfork:

1MHopf: 0.7M

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

Density contours around random cylinders

Counter-terrorism via PDE Optimization: Find fluxes at 16 surfaces to match data at 25 sensors

rSQP Exact

1.954 2.0

0.032 0.0

-0.012 0.0

-0.014 0.0

-0.006 0.0

0.042 0.0

-0.017 0.0

0.003 0.0

0.002 0.0

-0.002 0.0

4.990 5.0

0.057 0.0

0.312 0.0

0.133 0.0

0.944 1.0

0.049 0.0

1

2

5

6724 State variables, 16 design variables, x0=y0=0

88 rSQP Iterations, f=1.5e-6 , 30 sec / iter

Re=10

Flow Transport

Fluxes

Summary and Future Work

A powerful analysis tool has been developed to study large-scale flow stability applications:

– General purpose algorithms in LOCA linked to general purpose finite element codes.

– Bifurcations tracked for 1.0 Million unknown models– Singular formulation works semi-robustly

Future work :– Implement more invasive non-singular formulations,

automated for supported linear solvers (e.g. Aztec, Trilinos, LAPACK, PetsC)

– More flow applications– New application codes– Periodic Orbit tracking