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Fixing A Hole

The Society for Industrial and Applied Mathematics (SIAM) unites applied mathematicians worldwide, facilitating collaboration and discovery. In its recent news periodical, SIAM featured the work of Associate Dean Tom Sederberg who has developed T-Splines: a mathematical representation for manufacturable surfaces that many of his contemporaries consider unrivaled in both quality and efficiency.

Problem Meet Solution

Since their inception three decades ago, computer aided design (CAD) software programs have used a mathematical description of surfaces called NURBS. A single NURBS surface is basically a square sheet in space that designers can mold. Though individual NURBS surfaces appear smooth, they reveal their gaps and flaws when pieced together to create models, such as car or airplane bodies.

To dramatize these flaws with NURBS, Sederberg used the “Utah teapot,” a well-known NURBS model of a teapot created in 1975. He simulated putting digital tea into it and watched as the liquid dripped from a crack located between the bowl and spout. Besides frustrating designers, these defects in NURBS-based models present serious problems for analysis and manufacturing.

In 2003, Seberberg invented a novel technology called T-Splines that solves the problems created by NURBS. Since its introduction to the market, this technology has rapidly achieved international recognition. T-Splines has many advantages over NURBS, including the fact that it allows designers to create a complete surface without any gaps.

Incorporating IGA

The SIAM news article also reported on what may prove to be a landmark research conference — IGA 2011, which focused on unifying the fields of CAD and computer-aided engineering (CAE). Held January 13-15 at the University of Texas in Austin, this conference was organized by Dr. Thomas J.R. Hughes, a pioneer in the field of CAE.

Around the same time Sederberg invented T-Splines, Hughes developed a technology called isogeometric analysis (IGA). Historically, CAE splits a CAD model into small pieces called “finite elements” and performs computer simulations on these elements to determine how a physical representation of the model would actually behave.

However, this process of splitting a CAD model into finite elements has some issues. Not only is it time consuming, but it also introduces changes into the model that can significantly alter the analysis. IGA eliminates these problems by allowing a direct analysis of the entire CAD model — without splitting it into finite elements.

Hughes’ initial studies of IGA used NURBS, but he quickly discovered that the unavoidable gaps in NURBS models render them unfit for IGA. Over the past few years, Hughes’ and Sederberg’s research groups have teamed up to study the use of T-Splines in IGA. Their initial results have laid a solid mathematical foundation for using T-Splines in IGA and were a central focus of the IGA 2011 conference. Indeed, T-Splines is now the clear preferred technology for IGA and appears poised to help unify the CAD and CAE industries.

Dr. Sederberg Forum Address from CPMS Television on Vimeo.

By Natalie Rice Posted on