br Mesh refinement For practical purposes we

Mesh refinement For practical purposes, we choose the quadrilateral mesh as the exemplar to demonstrate the process of mesh automation with boundary-driven optimization. Owing to the possible complexity of boundary conditions associated with any arbitrary surfaces, the preliminary boundary-driven mesh will be potentially composed of triangles, quadrilaterals and other polygonal face elements. To ensure a quad-dominant mesh, additional mesh refinement functions are required to remove skewed triangles, and to construct quad-dominant faces from arbitrary polygonal faces. Lastly, a mesh-smoothing operator is introduced to relax the constructed mesh topology. The conditions and procedures for refining the BDMesh elements to a well-structured quad-dominant mesh are discussed below.
Example: meshing a trimmed surface The algorithm presented above is illustrated on a trimmed surface as the target surface for boundary-driven mesh optimization. The left side image in Figure 13 shows the original untrimmed surface and the right image delineates the relationships between the original surface and the trimming curves. The resulting trimmed surface, as shown in the right side image of Figure 13, has new boundary edges formed by: (i) original untrimmed edges (shown shaded in green); (ii) exterior trimming edges (shown shaded in red); and (iii) interior trimming edges (shown shaded in blue). By taking these boundary edges into consideration, a BDCurve network is first created, as shown in the left image in Figure 14. The right image in Figure 14 illustrates the preliminary result of meshing the trimmed surface with the constructed curve network (shown shaded red and blue lines in the left image of Figure 14). In Figure 15, two smoothed mesh results are given. On the left is the smoothed result of the preliminary BDMesh, and on the right is a qaudrangulated result of the smoothed BDMesh.
Discussions and conclusions At first glance, the UV-based curve network (the left image of Figure 19) is in appearance similar to the boundary-driven curve network (the right image of Figure 19). However, they phalloidin are very different in their formation. The former is interpolated solely from the underlying iso-parameters, the latter is computed from inherent boundary conditions. In a sense, the boundary-driven curve network conforms to the inherent surface boundary conditions more strictly than the UV-based curve network. This property ensures a well-configured framework for the tessellation pattern, particularly, at the boundary edges. Although our approach looks at the entire boundary condition as the whole to optimize the tessellation result, it is also possible to specify customized sources to explore varied surface tessellation pattern generations. In Figure 20, we demonstrate an example of utilizing an additional curve on the surface as the input for BDMesh construction. The image on the left side of Figure 20 shows an additional curve (shown shaded in red) on the given surface domain. The right image illustrates the constructed BDCurve network derived from this customized source. In this example, we illustrate the capability of introducing curve inputs as an additional constraint to create the underlying tessellation structure. This capability would afford designers a vehicle to control and experiment how they intend to discretize the freeform surfaces with pattern-based elements, such as quadrilaterals, or triangles, using customized input. In comparison to aforementioned algorithms, this proposed approach is yet another method with which we examine the surface tessellation problem. The major difference from this approach to others is to the adaptability for designers to intervene the optimization process and induce distinctive freeform surface designs. As discussed in both Wang (2009) and Pottmann et al. (2008), the base tessellated pattern is seminal for freeform surface applications and plays an important role in the constructive procedures. The development from a base tessellation pattern to real architectural manifestation can be treated as a remeshing process, which subdivides and reconfigures an initial discretized model as finer elements with added physical considerations. We claim that a well-structured model, which coheres to inherent boundary conditions, will guarantee success for surface development. In Section 4, we demonstrated how this approach could be applied procedurally by gradually increasing the boundary complexity for surface tessellation. In addition, we have explored other surface tessellation examples to ensure robustness of our approach. For example we have employed the Archimedean patterns (Akleman et al., 2005) as exemplars to demonstrate the potential of subdivision applications for customizable pattern-based surface tessellation (Wang, 2012).