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Solidness/bifurcation investigation gives subjective data about time advancement of nonlinear frameworks by registering groups of enduring state arrangements. ...

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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 research center worked 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 frameworks display insecurities, e.g.: Multiple enduring states Ignition Symmetry Breaking Onset of Oscillations Phase Transitions LOCA: Library of Continuation Algorithms We require calculations, programming, and experience to effect ASCI-and SciDAC-sized applications. These wonders must be comprehended keeping in mind the end goal to perform computational configuration and advancement. Built up strength/bifurcation investigation libraries exist: AUTO (Doedel) CONTENT (Kuznetsov) MATCONT (Govaerts) Stability/bifurcation examination gives subjective data about time advancement of nonlinear frameworks by processing groups of unfaltering state arrangements.

LOCA library became out of continuation code in MPSalsa Andy Salinger, John Shadid, Roger Pawlowski, Louis Romero, Rich Lehoucq, Ed Wilkes, Beth Burroughs, Nawaf Bou-Rabee LOCA 1.0 discharged April 2002 Written in C with wrapper capacities for connecting to application code ~200 downloads Complete revamp in C++ around NOX structure started September 2002, a portion of Trilinos discharge September 2003. History

r T max 1 3 1 Reaction Rate, r Second parameter, h LOCA: Library or Continuation Algorithms LOCA gives: Parameter Continuation : Tracks a group of unfaltering state arrangements with parameter Linear Stability Analysis : Calculates driving eigenvalues by means of Anasazi (Thornquist, Lehoucq) Bifurcation Tracking : Locates nonpartisan strength point (x,p) and tracks as an element of a second parameter

Pseudo Arc-length Continuation Solves for Solution and Parameter Simultaneously

Codimension 1 Bifurcations Turning Point Combustion Buckling of an Arch Buckling of a Beam Pattern development Cell separation (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 decisions for LOCA: Must work with iterative (surmised) direct solvers on conveyed memory machines Non-Invasive Implementation (e.g. network visually impaired) Should maintain a strategic distance from or limit: Requiring more subordinates Changing sparsity example of grid Increasing memory necessities

Pseudo Arc-length Continuation Bordering Algorithm Full Newton Algorithm Bordering Algorithms Meet these Requirements

Bordering Algorithms Meet these Requirements Full Newton Algorithm Turning Point Bifurcation … yet 4 explains of per Newton Iteration are utilized to drive solitary! Circumscribing Algorithm

Given starting speculation , step size Solve nonlinear conditions to discover 1 st point on bend while !stop Compute indicator Compute anticipated point Solve continuation conditions for utilizing as beginning theory If effective Postprocess (e.g., register eigenvalues, yield information) Increase step size Else Decrease step size Restore past arrangement End If or stop = genuine End while Abstraction of Continuation Process LOCA Stepper Predictor modules NOX + continuation/bifurcation bunches Step size modules

NOX Nonlinear Solver (Kolda, Pawlowski, Hooper, Shadid) NOX executes different strategies for comprehending Code to assess is exemplified in a Group . NOX solver strategies are nonexclusive , and actualized as far as gathering/vector conceptual interfaces: NOX solvers will work with any gathering/vector that executes these interfaces.

Super Vectors and Super Groups Idea : Given a vector to store and a gathering speaking to the conditions , manufacture an augmented ("super") aggregate speaking to, e.g., pseudo bend length continuation conditions: and a super vector to store the arrangement part and parameter segment . Super gatherings/vectors are non specific : All conceptual gathering/vector strategies for super gatherings/vectors actualized as far as techniques for the basic gatherings/vectors. Super gatherings are NOX bunches : Extended nonlinear conditions understood by most NOX solvers

Continuation Groups NOX::Abstract::Group LOCA::Continuation::ExtendedGroup LOCA::Continuation::NaturalGroup LOCA::Continuation::ArclengthGroup NOX::Abstract::Group LOCA::Continuation::AbstractGroup setParam() getParam() administrator = () computeDfDp() computeEigenvalues() printSolution() Mandatory Default usage accessible Optional Concrete gathering

Interfacing Application Codes to LOCA v2.0 Interfacing NOX to the application code is 90% of the work! For continuation defining moment following pitchfork following at exceptionally least should have the capacity to also set/recover parameter values, save complete condition of framework by duplicating bunch . For Hopf following, must execute a complex comprehend: Can over-burden numerous extra strategies if better procedures are accessible piece unravels particular grid illuminates assessing subordinates:

Single parameter continuation 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 backing for LAPACK (all intefaces) Epetra (all aside from Hopf) LOCA\'s Current Capabilities

Continuation Example: Chan Problem ChanProblemInterface.H ChanProblemInterface.C ChanContinuation.C ChanContinuation.txt

Turning Point Continuation Example ChanTPContinuation.C ChanTPContinuation.txt

Structural Mechanics Example: Salinas Bending a 1D Beam Example issue from Salinas test suite Original continuation keep running with 50 load steps NOX/LOCA interface composed by Russell Hooper Variable stride size calculation lessened to 37 load steps

Structural Mechanics Example: Salinas Snap-Through for a 2-Bar Truss

3D Rayleigh-B é nard Problem in 5 x 5 x 1 box (208K questions, 16 processors) MPSalsa (Shadid et al., SNL): Incompressible Navier-Stokes Heat and Mass Transfer, Reactions Unstrucured Finite Element (Galerkin/Least-Squares)

Improve vigor Better stride size control Improved bifurcation following calculations Debugging More elements for homotopy Incorporate Multi-vector bolster Multi-parameter continuation (Henderson, IBM) Constraint implementation Automatic separation Future Development