Zero Is Expensive!

May 18, 2012

Ewald Quak

Among the topics that triggered lively discussions at the "Forward Looking Session" at the SIAM/ACM Conference on Geometric Design and Physical Modeling (Orlando, Florida, October 2011; was the question of how much precision it is reasonable to demand in geometric computing. With her comment "Zero is expensive!" Sara McMains of UC Berkeley was referring of course to the fact that demands for ever-increasing precision of results cannot be met for free.

In his talk at the session, George Allen of Siemens PLM Software in Shanghai, a veteran developer of commercial CAD/CAM systems, elaborated on the question "how good is good enough?" In his view, a customer's expectations of the precision of results can be too high simply because of the low quality of the input data: Even the cleverest of algorithms will observe the rule of "garbage in, garbage out."

Many in the audience seemed to disagree with Allen's observations. Still, is the demand for strictly watertight geometric 3D models realistic? After all, the Vikings reached America in ships that were certainly not watertight, yet had been skillfully built to allow only as much water to leak in as could be safely bailed out. On the other hand, the demand for user-adjustable accuracy is clearly a reasonable one.

So where is the borderline between insignificant errors and meaningful features? This has been a question for 50 years, since the early days of CAD/CAM, and its treatment is a never-ending story. Participants generally agreed that a useful algorithm is more than just a description of the computational steps; it should be accompanied by information on the quality of input data necessary for the algorithm to work satisfactorily, and on the quality of the output that can then be expected. Practical ways of formalizing the level of input and output data quality are needed, and for designing algorithms that can cope with imperfect data by, for example, making information from neighboring regions available at identified gaps.

The relation between output quality of CAD systems and input quality requirements for finite element analysis or other numerical simulation methods is also at the heart of another topic discussed both in the "Forward Looking Session" and during the whole meeting: isogeometry---the use of NURBS (Non-Uniform Rational B-Splines) as basis functions not only in design but also in subsequent simulations. Tom Hughes of the Institute of Computational Engineering and Sciences at UT Austin, an invited speaker at the conference, introduced isogeometric analysis in 2005 to overcome the bottleneck between product design and PDE-based simulation for functional free-form surfaces and to open the possibility of integrated analysis-based shape optimization. (See Dana Mackenzie's article "Curing Ill Surfaces," SIAM News, April 2011, page 1.)

Since its inception, the isogeometric approach has attracted great interest, but many issues, especially concerning its use in industrial practice, still need to be addressed. While isogeometry was discussed at previous meetings, this time---in addition to the invited lecture by Hughes---five minisymposia and two contributed paper sessions were devoted to related topics.

One of the minisymposium organizers, Tor Dokken of the Norwegian research institute SINTEF, and the project leader of a European research project called "TERRIFIC---Enhancing Interoperability" in the European Factory of the Future programme, discussed isogeometry in the Forward Looking Session and the subsequent panel discussion. In fact, after the conference, Dokken and Hughes got together to start formulating a research roadmap for isogeometry; we look forward to its release. Figure 1 gives a glimpse of the differences between the established approach and the ideas behind the isogeometric paradigm.

Figure 1. Left, the current one-way flow of models from CAD to finite element analysis. Right, the isogeometric model provides interoperability between CAD and analysis. Courtesy of Tor Dokken.

Another high-potential evolving area addressed in both minisymposia and contributed talks was the use of geometric design-related methods in bio-nanotechnology. In an overview of the relevant challenges, Chandrajit Bajaj, also of ICES at UT Austin, urged interested members of the audience to get involved in bio-nanotechnology research as this field will benefit greatly from the collective knowledge represented by the SIAM Activity Group on Geometric Design.

SIAG/GD is one of the smaller SIAM activity groups but has a large percentage of members from industry. For the past 25 years, the biennial conferences organized by the SIAG have counted among the main general international conferences in geometric modeling and related areas, and they have been well attended by mathematicians and engineers from academia, industry, and government. To increase their impact, the last two events, in San Francisco in 2009 and now in Orlando in 2011, were held jointly with ACM SIGGRAPH, incorporating the annual ACM Symposia on Solid and Physical Modeling in those years.

The Forward Looking Session, introduced at the 2007 meeting, organized by the author of this report, features leading speakers from industry and academia whose task is to challenge conference participants with academically interesting open problems of substantial practical industrial relevance. By 2011, one of the original intentions for such sessions had been realized, as demonstrated in the account of progress made on previously introduced challenges---in this case aircraft design, by Tom Grandine and Matthew Patterson of The Boeing Company. We are happy to report that this does not indicate a shortage of new problems in the area, as the speakers were easily able to convince the audience.

As on previous occasions, the session presentations were followed by a panel discussion, with the speakers fielding questions from the audience as to where things are heading in our field and taking the opportunity to speculate as to where they should or should not be heading. These lively discussions are always greatly appreciated by the audience.

Apart from issues already mentioned, like error handling and the future of the isogeometry paradigm, the impact of commercial and financial pressures and of intellectual property rights was addressed in this discussion. Here, speakers were quick to point out that besides the necessary protection of intellectual and financial assets, access for and inclusion of as many interested researchers as possible are indeed very important. Even commercial CAD systems offer favourable academic licences, and open toolboxes, in, for example, isogeometry, are being developed. As current SIAG chair Konrad Polthier of the German research center MATHEON in Berlin pointed out, in the future SIAG/GD will promote industrial visits for students as part of its activities.

Ewald Quak divides his time between Estonia (Center for Nonlinear Studies at Tallinn University of Technology) and Norway (Center of Mathematics for Applications, University of Oslo, and the SINTEF Department of Applied Mathematics).

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