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We explore T-splines, a generalization of NURBS enabling local refinement, as a basis for isogeometric analysis. We review T-splines as a surface design methodology and then develop it for engineering analysis applications. We test T-splines on some elementary two-dimensional and three-dimensional fluid and structural analysis problems and attain good results in all cases. We summarize the current status of T-splines, their limitations, and future possibilities.
There are a number of candidate computational geometry technologies that may be used in isogeometric analysis. The most widely used in engineering design is NURBS (non-uniform rational B-splines), the industry standard (see, e.g. [23], [24], [28], [37], [38], [40]). The major strengths of NURBS are that they are convenient for free-form surface modeling, can exactly represent all quadric surfaces, such as cylinders, spheres, ellipsoids, etc., and that there exist many efficient and numerically stable algorithms to generate NURBS objects. They also possess useful mathematical properties, such as the ability to be refined through knot insertion, Cp-1-continuity for degree p NURBS, and the variation diminishing and convex hull properties.1 NURBS are ubiquitous in CAD systems, representing billions of dollars in development investment. The major deficiencies of NURBS are that gaps and overlaps at intersections of surfaces cannot be avoided, complicating mesh generation, and that they utilize a tensor product structure making the representation of detailed local features inefficient. Furthermore, it is impossible to represent most shapes using a single, watertight NURBS surface. T-splines are a recently developed generalization of NURBS technology (see [44]). T-splines correct the deficiencies of NURBS in that they permit local refinement and coarsening, and a solution to the gap/overlap problem. Commercial bicubic T-spline surface modeling capabilities have been recently introduced in Maya [45] and Rhino [46], two NURBS-based design systems. Extensions of T-spline surfaces to arbitrary polynomial degree have been described in Finnigan [26].
A NURBS surface is defined using a set of control points, which lie, topologically, in a rectangular grid, as shown in Fig. 3a. This means that a large percentage of NURBS control points are superfluous in that they contain no significant geometric information, but merely are needed to satisfy the topological constraints. In Fig. 3c, 80% of the NURBS control points are superfluous (colored red2). By contrast, a T-spline control grid is allowed to have partial rows of control points, as shown in Fig. 3b. A partial row of control points terminates in a T-junction, hence the name T-splines. In Fig. 3c, the purple T-splines control points are T-junctions. For the head modeled by NURBS and T-splines in Fig. 3c and d, the T-spline model requires only 24% of the control points compared to the NURBS model. For a designer, fewer control points means faster modeling time. The artist that created the T-spline model of the car in Fig. 3e estimated that it took one-fourth the time it would have taken using NURBS. Refinement, the process of adding new control points to a control mesh without changing the surface, is an important basic operation used by designers. A limitation of NURBS is that refinement requires the insertion of an entire row of control points. T-junctions enable T-splines to be locally refined. As shown in Fig. 4, a single control point can be added to a T-spline control grid. Another limitation of NURBS is that because a single NURBS surface must have a rectangular topology, most objects must be modeled using several NURBS surfaces. The hand in Fig. 5 is modeled using seven NURBS patches, one for the forearm, one for the hand, and one for each finger. It is difficult to join multiple NURBS surfaces in a single watertight model, as illustrated in Fig. 5a and b, especially if corners of valence other than four are introduced. However, it is possible to merge together several NURBS surfaces into a single gap-free T-spline, as shown in Fig. 5c.
Other methods have been devised for creating watertight smooth surfaces of arbitrary topology. A notable example is subdivision surfaces [51], which are defined in terms of various refinement rules that map a control polyhedron of arbitrary topology to a smooth surface. They have been used for shell analysis by Cirak et al. [15], [16], [17]. The appeal of subdivision surfaces is that, like T-splines, they create gap-free models and there is no restriction on the topology of the control grid. Subdivision surfaces are gaining widespread adoption in the animation industry. Most of the characters in Pixar animations are modeled using subdivision surfaces [51]. The president of Walt Disney Animation Studios and Pixar Animation Studios, Catmull, was one of the inventors of subdivision surfaces in 1978 [14]. The CAD industry has not adopted subdivision surfaces very widely because they are not compatible with NURBS. With billions of dollars of infrastructure invested in NURBS, the financial cost would be prohibitive.
As a prelude to the description of T-splines, we introduce the concept of point-based splines, or simply PB-splines [44]. Though we will not actually compute with PB-splines, we feel that they have the potential to have an impact in the area of meshless methods. Here, we examine them as a generalization of the concept of NURBS, and discuss both what is to be gained and lost by their use.
T-splines combine much of the flexibility of PB-splines with the topology and structure of NURBS [44]. They allow us to build spaces that are complete up to a desired polynomial degree, as smooth as an equivalent NURBS basis, and capable of being locally refined in a manner similar to PB-splines but while keeping the original geometry and parameterization unchanged. The properties that make T-splines useful for geometrical modeling also make them useful for finite element analysis.
T-splines represents an extension of NURBS technology that permits local refinement, watertight merging of patches, and a solution to the trimmed surface problem. These features are highly desirable in a design context and T-splines have recently become available in two NURBS-based design systems, namely, Maya [45] and Rhino [46]. In this paper we have explored T-splines as a basis for isogeometric analysis. The same features that make T-splines attractive for design, make it attractive for
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