October 22, 2013

Over a near 65-year-long career, Ivo Babuška has seen the field of computational engineering shift from slide rule operations to supercomputers doing the mathematical heavy lifting—now operating at nearly one thousand trillion computations per second.

Of course, these tools haven't evolved by themselves. It's scientists and engineers who are ultimately responsible for amazing computational advancements. But with progress, a certain disjunction seems to have appeared between people, computational processes, and the eventual answer, says Babuška, a senior research scientist at the Institute for Computational Engineering and Sciences and professor of the department of Aerospace Engineering and Engineering Mechanics.

So, in 2008, Babuška decided to test if modern analysts could confidently judge the accuracy of advanced computing software today by asking them to apply modern finite element analysis techniques to a classic civil engineering problem.

Working with Barna Szabó, a mechanics professor emeritus at Washington University and Juhani Pitkaranta, a mathematics professor emeritus at Aalto University, Babuška challenged readers of the International Association for Computational Mechanics' January 2008 newsletter to solve the Girkmann Problem within a five percent error margin.

Named after Karl Girkmann, the author of the textbook where it first appeared more than 60 years ago, the problem concerns finding the moment and forces (Ma and Qa in the figure) acting on a concrete thin shell structure and its support ring. Problems similar to it have been applied to construct iconic thin-shell buildings like Australia's Sydney Opera House and the John F. Kennedy International Airport in New York.

The Sydney Opera House is an example of thin shell architecture. Its construction required engineers to solve problems similar to the Girkmann Problem.

But when the answers came in, it was clear that something had gone awry.

Of the 15 responses Babuška received, half were wrong, being outside of the required five percent error margin. Of those, some were very wrong with error rates higher than 50 percent.

"This is a simple problem, therefore we wanted to see the engineering community solve such a problem and their confidence in it. They gave us numbers which were completely wrong and they believed in them," Babuška, who is also a mathematics and aerospace and engineering mechanics professor, said.

"If [a structure] were to be based on such a thing it could be disastrous."

#### Only one reason

The respondents all solved the Girkmann Problem using finite element method software, opting for either the h-version or p-version. Both break complicated mathematical problems into more manageable parts, a process called discretization, which are then solved by a computer.

One method is not innately better suited to solve the Girkmann Problem than the other, said Babuška, who specifically requested finite element methods be used to mirror modern engineering techniques. Still, the p-version answers all fell within the 5 percent error range, while only two of the 11 h-version answers resulted in a correct solution

"How is it possible that this happened is a good question," Babuška said. "There could be many various reasons. Nevertheless, in this case the reason was only one. Some of the analysts did not have sufficient engineering intuition and mathematical and engineering knowledge and possibly used the software incorrectly."

The mathematics a computational engineer utilized 60 years ago are comparatively crude methods compared to technology today, said Babuška, with engineers in the past working simplified formulations with pencil and paper. Nevertheless, these methods produced results, and instilled intuition and experience, solid enough to literally construct buildings on. This is something that Babuška sees lacking in some modern-users.

#### Learning from mistakes and estimating uncertainty

Babuška is not rallying for a rejection of technology; after all he is part of ICES, an institute grounded on using computational methods to solve problems in science and engineering. It's engineers having overconfidence in computational results that "look" right in simulation, but may prove to be incorrect when examined in greater detail, that concerns him.

One does not have to search hard to find cases of computational error causing costly disasters. A relatively recent case analogous to the Girkmann Problem is the sinking of Statoil's Sleipner A offshore platform in 1991. Here, inaccurate h-version finite element analyses of the platform's concrete floating buoyancy cells led to failure of the cell wall, ultimately sinking the platform into the North Sea during test operations. The values computed with the model were 47 percent off from the correct answers.

The total loss of Sleipner A, including redesign and downtime, is estimated to have cost over $1 billion in today's currency (about $700 million in 1990 values). It also influenced the company to opt for a return to hand computation with slide rules when designing the replacement platform while its computational analysis was under review.

A key part of ICES research concerns applying computational science driven simulations toward a variety of complex problems. But, just as important is the institute's dedication to developing an awareness of uncertainties that exist within simulations and supporting software—or knowing just how much to "trust" a particular simulation.

For example, at ICES' Center for Predictive Engineering and Computational Science (PECOS) researchers and students are applying the validation process to problems surrounding spacecraft reentry into earth's atmosphere. It's a focus influenced by the explosion of the spacecraft Columbia in 2003, a tragedy that may have been prevented with a better understanding of modeling uncertainties.

Babuška himself plays an important role in the promotion of uncertainty quantification to graduate students in ICES' Computational Science, Engineering and Mathematics program, said Tinsley Oden, director of ICES.

"Ivo Babuška is famous for his advocacy of the subject of uncertainty quantification. ICES students are reminded of the fundamental importance of this subject in virtually all aspects of their research and in every weekly seminar that Babuška attends," Oden said.

#### Blueprint ready

In Babuška's words, the computational engineer needs to have enough confidence in their answers so that he or she could "have the courage to sign the blueprints."

This is something Babuška had to do in the 1950s as a civil engineer in Czechoslovakia designing the Orlik Dam, the biggest dam in the country. Babuška also applied the same "blueprint" philosophy to the Girkmann Problem. But not having any blueprints to sign, so to speak, he had to find another way to express confidence.

Babuška is a world-renowned expert in finite element method. He has achieved honorary doctorates; memberships to scientific academies in the U.S. and abroad; medals, including the 2012 Leroy Steele Prize for Lifetime Achievement from the American Mathematical Society, and the Congress Medal, the highest award of the International Association for Computational Mechanic, and even has an asteroid named in his honor. But instead of calling on his reputation to be the blueprints, Babuška let money do the talking: He offered $1,000 to the first person to prove his answer to the Girkmann problem off by 2 percent of the exact solution.

"I am putting on the table $1,000...because I am putting down my confidence," says Babuška. "I preach that one does have to express a confidence and therefore I have to. So I do it by the money."

The money is still waiting in the bank, and will likely remain there; the current estimated error for Babuška's answer is less than 2 percent.

#### Credit where it's due

ICES is developing reliable computational simulations to help understand big problems in science today, from hurricane mapping, heart disease modeling, energy extraction and carbon sequestration. While this cutting-edge research is enabled by technology, it's made possible by people. And it's the "blueprint" philosophy of researchers like Babuška, that are helping to ensure that humans are recognized as ultimately in control of simulations; This goes for when something goes wrong, but also when scientific headway is made.

"Today desktop and laptop computers have such large computational power that it is possible to solve very large mathematical problems that lead to the prediction of various physical and engineering phenomena of great complexity and importance that, 60 years ago nobody could dream about," Babuška said.

"But having confidence, or the related problem of dangerous overconfidence, should always be in the mind of the analyst."

*Written by Monica Kortsha*