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-AC04-94AL 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 library  Expertise

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 -1 b ) Mat-Vec multiply ( Jb ) Set parameter ( ) Fill mass matrix ( M ) Complex matrix solve ( J+ i  M ) Shifted Matrix Solve ( J+   M ) LOCA AlgorithmsLOCA 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   Turning Point Bifurcation Full Newton Algorithm Bordering Algorithm

Bordering Algorithm for Hopf tracking

LOCA around MPSalsa: 3D Rayleigh Benard Problem in 5x5x1 box Scalability Eigensolver: 16M Continuation: 16M Turning Point: 1M Pitchfork: 1M Hopf:0.7M

CVD Reactor Design and Scale-up: Tracking of instability leads to design rule 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 Instabilities Family of Solutions w/ Instability Steady Solution (GOMA) back pressure

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 rSQPExact State variables, 16 design variables, x 0 =y 0 =0 88 rSQP Iterations, f=1.5e-6, 30 sec / iter Re=10 FlowTransport 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