Airplane travel is different for Krzysztof “Chris” Fidkowski than for most people. As an assistant aerospace engineering professor at the University of Michigan, he uses high-performance computers to simulate airfoils and aircraft, and to seek better ways to calculate bigger fluid dynamics problems.
“I fly a lot, and I know planes are safe, so I try to shelve the engineering side of things because every little thing you see outside might make you wonder,” he chuckles. Yet Fidkowski, a Department of Energy Computational Science Graduate Fellowship (DOE CSGF) recipient from 2003 to 2007, likes watching ailerons and flaps deploy to control the aircraft. “Sometimes if you look very closely and the conditions are right, you might see a shock in the air, right over the wing.”
Shocks and other flow phenomena are just what Fidkowski simulates. Yet it’s impossible to capture each air molecule’s movements, “so we’re always making approximations to the fluid dynamics and that gives us errors” – the plus-minus that defines the output’s accuracy. Fidkowski, a native of Poland who dreamt of becoming an astronaut, focuses on cutting error by adapting the computational meshes researchers use to analyze things like airflow over wings. “We have the capability to reduce that error bar – that plus-minus – in fairly straightforward ways,” usually by putting more mesh points in areas important to accurately predict an output.
This spring, Fidkowski’s work on error estimation and mesh resolution earned him an Early Career Research Program award from the Department of Energy’s Office of Science. The grant supports scientists in their early years, when they do their most formative work.
In a paper with graduate student Marco Ceze, Fidkowski describes an anisotropic hp-adaptation framework to cut output error. Like many fluid dynamics simulations, it focuses on the boundary layer – the critical, super-thin blanket of air nearest to the aircraft wing or body. Simulating airflow over the entire aircraft with an isotropic mesh small enough to resolve the boundary layer is far too demanding for even the best supercomputer. Instead, the algorithm makes boundary layer mesh elements anisotropic – flattened in one direction – with the hp-adaptation method flagging and refining those generating the most error.
As a rule, areas of nearly discontinuous airflow, like shocks or shears, are best represented with h refinement, so named because h denotes the length of a mesh element’s side. It bisects each error-prone element to provide greater precision. In areas with smoother flows, error-causing elements are mathematically refined with higher-degree polynomial (p) approximations. “Increasing polynomial order is usually the best way to resolve smooth regions of the flow, where stuff happens relatively slowly,” Fidkowski says.
Researchers can hand-design meshes to apply the appropriate method based on expected flow characteristics. But “we don’t leave it up to the user to decide that,” Fidkowski says. The algorithm chooses the best option automatically based on output error. Users specify a single output, such as drag. The algorithm picks the best refinement option, h or p, that most reduces drag error with the least computational effort. Sometimes the algorithm detects flow discontinuities where none were suspected and automatically bisects mesh elements. “That’s not something we would have designed by hand,” Fidkowski adds.
Although Fidkowski focuses on aerospace, his methods could model other flows. He’s part of a Michigan group participating in the Consortium for Advanced Simulation of Light Water Reactors, one of five DOE Energy Innovation Hubs. “There might be a few outputs you really care about – critical outputs for the safety of the reactor that you want to get just right. You don’t want numerical errors and discretization errors polluting your results.”
Regardless of the application, Fidkowski wants to design more efficient algorithms that solve bigger problems on the exotic computer architectures to come. “We want to do real-world problems. We’re just skimming the surface of that.” Fidkowski also wants to make error estimation a routine part of uncertainty quantification, which puts a number on the degree to which a simulation’s results can be trusted, and of optimization to improve designs.
“The most exciting part is getting an algorithm to work,” he says – something that rarely happens on the first try. It’s just as important to disseminate what he learns to students. The flexibility to research and to teach thrills him and fills his days. “That’s when you know you’re doing things you like.”