Review of Subdivision Schemes and their Applications

Background: Subdivision surfaces modeling method and related technology research gradually become a hot spot in the field of computer-aided design(CAD) and computer graphics (CG). In the early stage, research on subdivision curves and surfaces mainly focused on the relationship between the points, thereby failing to satisfy the requirements of all geometric modeling. Considering many geometric constraints is necessary to construct subdivision curves and surfaces for achieving high-quality geometric modeling. Objective: This paper aims to summarize various subdivision schemes of subdivision curves and surfaces, particularly in geometric constraints, such as points and normals. The findings help scholars to grasp the current research status of subdivision curves and surfaces better and to explore their applications in geometric modeling. Methods: This paper reviews the theory and applications of subdivision schemes from four aspects. We first discuss the background and key concept of subdivision schemes. We then summarize the classification of classical subdivision schemes. Next, we show the subdivision surfaces fitting and summarize new subdivision schemes under geometric constraints. Applications of subdivision surfaces are also discussed. Finally, this paper gives a brief summary and future application prospects. Results: Many research papers and patents of subdivision schemes are classified in this review paper. Remarkable developments and improvements have been achieved in analytical computations and practical applications. Conclusion: Our review shows that subdivision curves and surfaces are widely used in geometric modeling. However, some topics need to be further studied. New subdivision schemes need to be presented to meet the requirements of new practical applications.

SUBDIVISION SURFACE MID-SURFACE RECONSTRUCTION OF TOPOLOGY OPTIMIZATION RESULTS AND THIN-WALLED SHAPES USING SURFACE SKELETONS

Polygon meshes and particularly triangulated meshes can be used to describe the shape of different types of geometry such as bicycles, bridges, or runways. In engineering, such polygon meshes can occur as finite element meshes, resulting from topology optimization or laser scanning. This article presents an automated parameterization of polygon meshes into a parametric representation using subdivision surfaces, especially in topology optimization. Therefore, we perform surface skeletonization on a volumetric grid supported by the Euclidian distance transformation and topology preserving and shape-preserving criterion. Based on that surface skeleton, an automated conversation into a Subdivision Surface Control grid is established. The final mid-surface-like parametrization is quite flexible and can be changed by variating the control gird or the local thickness.

Grid-Free Surface-Based Geological Modelling using Subdivisions Surfaces and NURBS – Advantages for Geothermal Applications

<p>Pragmatic and cost-effective representations of geological structures and features (e.g., heterogeneities, faults and folds) in full 3-D geological models are challenging. Implementations are highly dependent on the flexibility of the representation method. We investigate the use of parametric surface-based geological modelling methods for the purpose of low-dimensional model representations. Specifically, we focus on two grid-free and controllable parametric surfaced-based modelling methods: NURBS and subdivision surfaces. NURBS are the standard method in Computer-Aided Design (CAD) and have been used in geological reservoir modelling before. Subdivision surfaces are a common representation in the gaming and animation industry. They are very interesting as they can support watertight modelling and arbitrary topology (preserving the relationship between different parts of the model). However, this method is, to date, rarely used in geological modelling.</p><p>Unlike implicit modelling, parametric surfaced-based modelling is a grid-free representation and exploits the boundary surfaces of the model. Also, the geological features (e.g., heterogeneities, faults, folds) can be represented by there bounding surfaces instead of grid-cells. Therefore, they do not suffer from the limitation of grid cells (e.g., Stair-stepping), which are often present in implicit representations.</p><p>We discuss the advantages and shortcomings of both NURBS and subdivision surfaces for geological modelling. Furthermore, we investigate the approximation of geological structures by subdivision surfaces in this presentation. The approximated models are watertight (closed), controllable with few control points, smooth, and have less than 5% of the number of the vertices of the original model. Reducing the number of vertices of the model while preserving the topology can decrease the cost of both modelling and simulations. As the final step, we present the advantages of grid-free surface-based geological modelling for thermal finite element analyses by using a state-of-the-art finite-element solver, namely the MOOSE framework</p>

Construction and analysis of unified 4-point interpolating nonstationary subdivision surfaces

Subdivision schemes (SSs) have been the heart of computer-aided geometric design almost from its origin, and several unifications of SSs have been established. SSs are commonly used in computer graphics, and several ways were discovered to connect smooth curves/surfaces generated by SSs to applied geometry. To construct the link between nonstationary SSs and applied geometry, in this paper, we unify the interpolating nonstationary subdivision scheme (INSS) with a tension control parameter, which is considered as a generalization of 4-point binary nonstationary SSs. The proposed scheme produces a limit surface having $C^{1}$ C 1 smoothness. It generates circular images, spirals, or parts of conics, which are important requirements for practical applications in computer graphics and geometric modeling. We also establish the rules for arbitrary topology for extraordinary vertices (valence ≥3). The well-known subdivision Kobbelt scheme (Kobbelt in Comput. Graph. Forum 15(3):409–420, 1996) is a particular case. We can visualize the performance of the unified scheme by taking different values of the tension parameter. It provides an exact reproduction of parametric surfaces and is used in the processing of free-form surfaces in engineering.