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Continuation and Bifurcation Methods Using LOCAPowerPoint Presentation

Continuation and Bifurcation Methods Using LOCA

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### Continuation and Bifurcation Methods Using LOCA

Eric Phipps

Andy Salinger, Roger Pawlowski

Sandia National Laboratories

Trilinos Workshop at Copper Mountain

March 30, 2004

Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company,for the United States Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.

Why Do We Need Stability Analysis Algorithms for Large-Scale Applications?

Nonlinear systems exhibit instabilities, e.g.:

- Multiple steady states
- Ignition
- Symmetry Breaking
- Onset of Oscillations
- Phase Transitions

LOCA: Library of Continuation Algorithms

We need algorithms, software, and experience to impact ASCI- and SciDAC-sized applications.

These phenomena must be understood in order to perform computational design and optimization.

Established stability/bifurcation analysis libraries exist:

- AUTO (Doedel)
- CONTENT (Kuznetsov)
- MATCONT (Govaerts)

Stability/bifurcation analysis provides qualitative information about time evolution of nonlinear systems by computing families of steady-state solutions.

LOCA library grew out of continuation code in MPSalsa Applications?

Andy Salinger, John Shadid, Roger Pawlowski, Louis Romero, Rich Lehoucq, Ed Wilkes, Beth Burroughs, Nawaf Bou-Rabee

LOCA 1.0 released April 2002

Written in C with wrapper functions for linking to application code

~200 downloads

Complete rewrite in C++ around NOX framework began September 2002, part of Trilinos release September 2003.

Historyr Applications?

Tmax

1

3

1

Reaction Rate, r

Second parameter, h

LOCA: Library or Continuation AlgorithmsLOCA provides:

- Parameter Continuation: Tracks a family of steady state solutions with parameter
- Linear Stability Analysis: Calculates leading eigenvalues via Anasazi (Thornquist, Lehoucq)
- Bifurcation Tracking: Locates neutral stability point (x,p) and tracks as a function of a second parameter

Pseudo Arc-length Continuation Solves for Solution and Parameter Simultaneously

Codimension 1 Bifurcations Parameter Simultaneously

Turning Point

- Combustion
- Buckling of an Arch
- Buckling of a Beam
- Pattern formation
- Cell differentiation (morphogenesis)
- Vortex Shedding
- Predator-Prey models
- Puberty

Pitchfork

Hopf

LOCA Designed for Easy Linking to Existing Newton-based Applications

LOCA targets existing codes that are:

- Steady-State, Nonlinear
- Newton’s Method
- Large-Scale, Parallel

Algorithmic choices for LOCA:

- Must work with iterative (approximate) linear solvers on distributed memory machines
- Non-Invasive Implementation (e.g. matrix blind)
- Should avoid or limit:
- Requiring more derivatives
- Changing sparsity pattern of matrix
- Increasing memory requirements

Pseudo Arc-length Continuation Applications

Bordering Algorithm

Full Newton Algorithm

Bordering Algorithms Meet these RequirementsBordering Algorithms Meet these Requirements Applications

Full Newton Algorithm

Turning Point Bifurcation

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

Bordering Algorithm

Given initial guess , step size Applications

Solve nonlinear equations to find 1st point on curve

while !stop

Compute predictor

Compute predicted point

Solve continuation equations for using as initial guess

If successful

Postprocess (e.g., compute eigenvalues, output data)

Increase step size

Else

Decrease step size

Restore previous solution

End if

If or or

stop = true

End while

Abstraction of Continuation ProcessLOCA Stepper

Predictor modules

NOX +

continuation/

bifurcation

groups

Step size modules

NOX Nonlinear Solver Applications(Kolda, Pawlowski, Hooper, Shadid)

NOX implements various methods for solving

Code to evaluate is encapsulated in a Group.

NOX solver methods are generic, and implemented in terms of

group/vector abstract interfaces:

NOX solvers will work with any group/vector that implements these

interfaces.

Super Vectors and Super Groups Applications

Idea: Given a vector to store and a group representing the

equations , build an extended (“super”) group representing,

e.g., pseudo arc-length continuation equations:

and a super vector to store the solution component and parameter

component .

Super groups/vectors are generic:

All abstract group/vector methods for super groups/vectors

implemented in terms of methods of the underlying groups/vectors.

Super groups are NOX groups:

Extended nonlinear equations solved by most NOX solvers

Continuation Groups Applications

NOX::Abstract::Group

LOCA::Continuation::ExtendedGroup

LOCA::Continuation::NaturalGroup

LOCA::Continuation::ArclengthGroup

NOX::Abstract::Group

LOCA::Continuation::AbstractGroup

- setParam()
- getParam()
- operator = ()
- computeDfDp()
- computeEigenvalues()
- printSolution()

Mandatory

Default implementation

available

Optional

Concrete group

Interfacing Application Codes to LOCA v2.0 Applications

- Interfacing NOX to the application code is 90% of the work! For
- continuation
- turning point tracking
- pitchfork tracking
at very minimum must be able to additionally set/retrieve parameter

values, save complete state of system by copying group.

- For Hopf tracking, must implement a complex solve:
- Can overload many additional methods if better techniques
are available

- block solves
- singular matrix solves
- estimating derivatives:

Single parameter continuation Applications

Natural

Pseudo Arc-length

Bifurcations

Turning point

Pitchfork

Hopf

Predictors

Constant

Tangent

Secant

Random

Step size control

Constant

Adaptive

Artificial Homotopy

Generic interface to Anasazi

Native support for

LAPACK (all intefaces)

Epetra (all except Hopf)

LOCA’s Current CapabilitiesContinuation Example: Chan Problem Applications

ChanProblemInterface.H

ChanProblemInterface.C

ChanContinuation.C

ChanContinuation.txt

Structural Mechanics Example: Salinas Bending a 1D Beam Applications

Example problem from Salinas test suite

Original continuation run with 50 load steps

NOX/LOCA interface written by Russell Hooper

Variable step size algorithm reduced to 37 load steps

3D Rayleigh-B 2-Bar Trussénard Problem in 5x5x1 box(208K unknowns, 16 processors)

MPSalsa (Shadid et al., SNL):

- Incompressible Navier-Stokes
- Heat and Mass Transfer, Reactions
- Unstrucured Finite Element (Galerkin/Least-Squares)

Improve robustness 2-Bar Truss

Better step size control

Improved bifurcation tracking algorithms

Debugging

More features for homotopy

Incorporate

Multi-vector support

Multi-parameter continuation (Henderson, IBM)

Constraint enforcement

Automatic differentiation

Future Development
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