Dept of Energy CSGF magazine 2007-8 -- DOE Lab Research

Sandia National Laboratories

From Flames to Fusion

(Page 2 of 3)

These huge algebraic systems must be attacked intelligently so parallel-processing computers can solve them efficiently. The methods Shadid and his fellow researchers have developed generally do that in two steps: applying a nonlinear solver; and applying iterative solvers with multigrid preconditioners.

Getting a Grid

In general, discretizing PDEs involves distributing points in a grid or mesh throughout the space being simulated, such as a chemical reaction chamber. The more points, the better the resolution and accuracy of the simulation — and the more demanding it is for computers to solve.

vMultigrid methods use a hierarchy of grids of varying resolutions. Because they use information from a sequence of grids rather than a single grid, these methods scale optimally — the number of iterations is constant and the work grows in direct proportion to the problem size. Collaborators Ray Tuminaro and Mazio Sala (from Swiss research institution ETH Zurich) have fully algebraic methods that accomplish this.

The preconditioning step is critical to solving these large systems, Shadid adds. On test problems with 10 million to 250 million unknowns and running on up to 8,000-plus parallel processors, the Sandia multilevel preconditioning techniques proved to be 10 to 100 times faster than one-level methods. The iterations necessary to reach a solution also stayed constant, even as the problem size grew by more than four orders of magnitude. That means the algorithms should scale well on even bigger computers.

The Sandia researchers’ nonlinear solvers are robust — unlikely to blow up and produce nonsense — and allow them to crack elaborate problems. This includes bifurcation studies, which efficiently identify changes in stability — points where a change in model parameters leads to multiple steady states or oscillatory solutions. For example, the researchers’ codes have been used to analyze chemical vapor deposition, which places films of semiconductor materials on wafers for microelectronics. Industry wants to scale up the process to get more chips out of each wafer, but reactor instabilities produce inconsistent results. “Some days you get a good uniform layer. Some days you get a really bad, nonuniform layer,” Pawlowski says. Small changes in the system — whether a valve sticks during start-up, whether a temperature setting is slightly out of adjustment, or even whether someone bumped the reactor during the run — can lead to a fundamental change in system performance.

The group’s code can determine the point at which parameter variations can cause destabilizing effects that destroy uniform, steady-state operation. That information can help improve designs for reactors and other devices. The code calculates the bifurcation point directly, compared to programs that must be run numerous times with different operating parameters. “We can set one run, let it go, and it will directly compute the parameter values of the bifurcation point,” Pawlowski says. “You don’t have to adjust the parameters by hand and try to hopefully figure out where it changes stability.” It’s “a powerful tool to design stable operation” of an experiment or process.

Similarly, the group’s unique algorithms can help scientists and engineers choose optimal designs. “I could sit at a computer all day and set different parameters and run (a program) out, change the parameters and run it out again and again” to get an optimal answer, Pawlowski says. That may work if only a few parameters are changing, but the problem is tougher when many parameters are involved.

“You don’t want to simulate for every possible combination of parameters. You want methods that will take you directly to the optimal solution,” Pawlowski says. The Sandia codes do that quickly.