Upload
rafer
View
54
Download
2
Tags:
Embed Size (px)
DESCRIPTION
Stability Analysis Algorithms for Large-Scale Applications. Andy Salinger, Roger Pawlowski, Ed Wilkes Louis Romero, Rich Lehoucq, John Shadid Sandia National Labs Albuquerque, New Mexico Computational Challenges in Dynamical Systems Fields Institute, Dec. 6, 2001. - PowerPoint PPT Presentation
Citation preview
Stability Analysis Algorithms for Large-Scale Applications
Andy Salinger, Roger Pawlowski, Ed Wilkes
Louis Romero, Rich Lehoucq, John Shadid
Sandia National Labs
Albuquerque, New Mexico
Computational Challenges in Dynamical Systems
Fields Institute, Dec. 6, 2001
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 Talk (Lift Talk)
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 track bifurcations of 1 Million unknown PDE discretizations.
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
The Targeting of Large-Scale Applications Codes Restricts the Choice of Algorithms
Requirements: Stability analysis algorithms must be scalable and relatively non-invasive:
• Must work with iterative (approximate) linear solvers
• Should avoid or limit: Requiring more derivatives Changing sparsity pattern of matrix Increasing memory requirements
Targeted Codes: Newton’s Method, Large-Scale, Parallel
• Navier-Stokes & Reaction-Diffusion, Free Surface Flows, Molecular Theory, Structural Mechanics, Circuit Simulation
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: ARPACK driver for Cayley transform
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: ARPACK driver for Cayley transform
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
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+=
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.
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
ScalabilityScalabilityEigensolver: 16MContinuation:
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:Buoyancy force can lead to undesirable flows
Chemical Vapor Deposition of Semiconductors: GaN, GaAs
Good and bad flows are found to coexist at certain values of (Ra, Re)
Good Flow
Bad Flow
30500 Unknowns
Good and bad flows are found to coexist at certain values of (Ra, Re)
30500 Unknowns
Tracking of bifurcation leads to design rule
Ra 1.75Re0.5
1100Re---------+
=
Ideal gas curves collapse onto Boussinesq for good choice of To
Boussinesq
Pawlowski, Salinger, Romero, Shadid 2001
Optimization Algorithms, such as rSQP, Need Same Calls as Bifurcation Algs
Collaboration with Biegler, CMU
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
Bifurcation Diagram (a.k.a. Adsorption Isotherm)Density contours aroundrandom cylinders
LOCA+Tramonto: Capillary condensation phase transitions studied in porous media
Liquid
Vapor
Partial Condensation
Phase diagram
Tramonto: Frink and Salinger, JCP 1999,2000,2002
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
Powerful analysis tools has been developed to study large-scale flow stability applications:
– General purpose algorithms in LOCA linked to massively parallel finite element codes.
– Bifurcations tracked for 1.0 Million unknown models– Singular formulation works semi-robustly
Future work : Support common linear solvers (e.g. Aztec, Trilinos, PetsC, LAPACK)Implement more invasive, non-singular (bordered) formulationsMultiparameter continuation (Henderson, IBM)New application codes, e.g. buckling of structures
J a
bt c