Surface models Curve network

Closed boundaries of surfaces are automatically defined using topology and curve information from the curve network model. Surfaces are generated then trimmed by closed boundaries. Standard NURBS surfaces are applied for this purpose. The... 

More and more products are covered by complex sculptured surfaces and multi-surface shapes. One of the primary objectives in the development of modeling systems is the ability to handle any desired shape regardless of its complexity. The traditional way is by creating single surfaces then blending them into complex ones. An advanced solution is the application of a curve network as complex input information to create a set of surfaces in an associative way. Complex surfaces are often called skins. 

Complex surfaces are modeled with specified continuity maintained by the model creating procedure across boundaries of the surfaces in the network. A surface is built up by individual surfaces. Construction curves for surfaces are generated by using segments of network curves. Closed regions are fitted with individual surfaces. Four-sided regions are enclosed by a boundary... 

The surface creation process begins by laying down curves directly on the polygonal model to define the different surfaces to be created. The curves network created on model (fig. 7) can be the basis for subsequent realization of the surfaces. 

Once the curves network is created, the model is ready to generate NURBS surfaces (fig. 8). This can be done automatically or manual. Automatic surface generation doesn t need to draw a curve, while manual surface generation can completely maintain the flow line of the original polygon surface. Manual generation of surfaces is related to the network of curves. 

Imagine a network model of the diamond structure (Fig. 1.17(e)), blue lattice), constructed from rubber tubes. Now inflate the network, swelling the hollow tubes. The resulting structure is a curved continuous network, enclosing the tunnels in the diamond network. If the inflation procedure is continued, the surface closes up around a complementary diamond network. The D-surface is the "half-way point" during the procedure. 

Practically any experimental kinetic curve can be reproduced using a model with a few parallel (competitive) or consecutive surface reactions or a more complicated network of chemical reactions (Fig. 4.70) with properly fitted forward and backward rate constants. For example, Hachiya et al. used a model with two parallel reactions when they were unable to reproduce their experimental curves using a model with one reaction. In view of the discussed above results, such models are likely to represent the actual sorption mechanism on time scale of a fraction of one second (with exception of some adsorbates, e.g, Cr that exchange their ligands very slowly). Nevertheless, models based on kinetic equations of chemical reactions were also used to model slow processes. For example, the kinetic model proposed by Araacher et al. [768] for sorption of multivalent cations and anions by soils involves several types of surface sites, which differ in rate constants of forward and backward reaction. These hypothetical reactions are consecutive or concurrent, some reactions are also irreversible. Model parameters were calculated for two and three... 

The result of polygon mesh optimization stage is a fully closed model (fig. 6) ready to generate NURBS (Non-uniform Rational B-Spline) curves or surfaces network. 

Fig. 3.6 Panel (a) Stress-strain curve of a homeotropic NE oriented by a surface treatment of the sample-bearing glass slides (Urayama et al. 2007). The data have been extracted from this reference. The stress is the nominal stress and the extension ratio 2 is defined by 2 = L/Lq, where L and Lq are the lengths of the NE for the stretched and un-stretched states. Panel (b) Stress-strain curve of the homeotropic NE oriented by an E-field (Rogez et al. 2011). For both samples, the solid lines for regimes II and III are not fitted curves but theoretical curves calculated from (3.6) to (3.8) derived from the neo-classical model, using the known values of parameters 2i and (see text). The excellent agreement between the experimental and the calculated curves shows that the elasticity of the network is Gaussian. Reprinted and adapted with kind permission of Springer Science + Business Media (Rogez and Martinoty 2011)...

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