# Presentation: 168192

## 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-AC04-94AL85000.

- 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-1b) |
| Mat-Vec multiply (Jb) |
| Set parameter () |
| Fill mass matrix (M) |
| Complex matrix solve (J+iM) |
| Shifted Matrix Solve (J+M) |

- LOCA Algorithms

- LOCA Interface

## Slide 5

## 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  ******** ******

## 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**

**back pressure**

## LOCA+Tramonto: Capillary condensation phase transitions studied in porous media

- Phase diagram

- Density contours around random cylinders

##       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

**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**