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-AC04-94AL 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 library  Expertise

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 -1 b ) Mat-Vec multiply ( Jb ) Set parameters (  ) LOCA AlgorithmsLOCA 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 -1 b ) Mat-Vec multiply ( Jb ) Set parameters (  ) Fill mass matrix ( M ) Shifted Matrix Solve ( J+   M ) LOCA AlgorithmsLOCA 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 -1 b ) Mat-Vec multiply ( Jb ) Set parameters (  ) Fill mass matrix ( M ) Shifted Matrix Solve ( J+   M ) Complex matrix solve ( J+ i  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

Eigenvalue Approx with Arnoldi, ARPACK 3 Spectral Transformations have Different Strengths Complex Shift and InvertCayley Transform v.1Cayley 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 mesh partitioned for 320 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 Bifurcation Near (Pr=0.027, Ra=2050) Eigenvectors at Hopf

Rayleigh-Benard Problem used to Demonstrate Scalability of Algorithms Scalability Eigensolver: 16M Continuation: 16M Turning Point: 1M Pitchfork: 1M Hopf:0.7M Steady Solve 5 Minutes Eigenvalue Calculation (~5) 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 Unknowns

Good and bad flows are found to coexist at certain values of (Ra, Re) Unknowns

Tracking of bifurcation leads to design rule Ideal gas curves collapse onto Boussinesq for good choice of T o 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 Instabilities Family of Solutions w/ Instability Steady Solution (GOMA) back pressure

LOCA+Tramonto: Capillary condensation phase transitions studied in porous media Bifurcation Diagram (a.k.a. Adsorption Isotherm) Density contours around random 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 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 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) formulations Multiparameter continuation (Henderson, IBM) New application codes, e.g. buckling of structures Ja btbt c