Smooth Subdivision Surfaces: Mesh Blending and Local Interpolation

Subdivision surfaces are widely used in computer graphics and animation. Catmull-Clark subdivision (CCS) is one of the most popular subdivision schemes. It is capable of modeling and representing complex shape of arbitrary topology. Polar surface, working on a triangle-quad mixed mesh structure, is proposed to solve the inherent ripple problem of Catmull-Clark subdivision surface (CCSS). CCSS is known to be C1 continuous at extraordinary points. In this work, we present a G2 scheme at CCS extraordinary points. The work is done by revising CCS subdivision step with Extraordinary-Points-Avoidance model together with mesh blending technique which selects guiding control points from a set of regular sub-meshes (named dominative control meshes) iteratively at each subdivision level. A similar mesh blending technique is applied to Polar extraordinary faces of Polar surface as well. Both CCS and Polar subdivision schemes are approximating. Traditionally, one can obtain a CCS limit surface to interpolate given data mesh by iteratively solving a global linear system. In this work, we present a universal interpolating scheme for all quad subdivision surfaces, called Bezier Crust. Bezier Crust is a specially selected bi-quintic Bezier surface patch. With Bezier Crust, one can obtain a high quality interpolating surface on CCSS by parametrically adding CCSS and Bezier Crust. We also show that with a triangle/quad conversion process one can apply Bezier Crust on Polar surfaces as well. We further show that Bezier Crust can be used to generate hollowed 3D objects for applications in rapid prototyping. An alternative interpolating approach specifically designed for CCSS is developed. This new scheme, called One-Step Bi-cubic Interpolation, uses bicubic patches only. With lower degree polynomial, this scheme is appropriate for interpolating large-scale data sets. In sum, this work presents our research on improving surface smoothness at extraordinary points of both CCS and Polar surfaces and present two local interpolating approaches on approximating subdivision schemes. All examples included in this work show that the results of our research works on subdivision surfaces are of high quality and appropriate for high precision engineering and graphics usage

Adaptive mesh refinement techniques for high-order finite-volume WENO schemes

This paper demonstrates the capabilities of Adaptive Mesh Refinement Techniques (AMR) on 2D hybrid unstructured meshes, for high order finite volume WENO methods. The AMR technique developed is a conformal adapting unstructured hybrid quadrilaterals and triangles (quads & tris) technique for resolving sharp flow features in accurate manner for steady-state and time dependent flow problems. In this method, the mesh can be refined or coarsened which depends on an error estimator, making decision at the parent level whilst maintaining a conformal mesh, the unstructured hybrid mesh refinement is done hierarchically.When a numerical method can work on a fixed conformal mesh this can be applied to do dynamic mesh adaptation. Two Refinement strategies have been devised both following a H-P refinement technique, which can be applied for providing better resolution to strong gradient dominated problems. The AMR algorithm has been tested on cylindrical explosion test and forward facing step problems

Feature Adaptive Ray Tracing of Subdivision Surfaces

abstract: Subdivision surfaces have gained more and more traction since it became the standard surface representation in the movie industry for many years. And Catmull-Clark subdivision scheme is the most popular one for handling polygonal meshes. After its introduction, Catmull-Clark surfaces have been extended to several eminent ways, including the handling of boundaries, infinitely sharp creases, semi-sharp creases, and hierarchically defined detail. For ray tracing of subdivision surfaces, a common way is to construct spatial bounding volume hierarchies on top of input control mesh. However, a high-level refined subdivision surface not only requires a substantial amount of memory storage, but also causes slow and inefficient ray tracing. In this thesis, it presents a new way to improve the efficiency of ray tracing of subdivision surfaces, while the quality is not as good as general methods.Dissertation/ThesisMasters Thesis Computer Science 201

Flexible G1 Interpolation of Quad Meshes

International audienceTransforming an arbitrary mesh into a smooth G1 surface has been the subject of intensive research works. To get a visual pleasing shape without any imperfection even in the presence of extraordinary mesh vertices is still a challenging problem in particular when interpolation of the mesh vertices is required. We present a new local method, which produces visually smooth shapes while solving the interpolation problem. It consists of combining low degree biquartic Bézier patches with minimum number of pieces per mesh face, assembled together with G1-continuity. All surface control points are given explicitly. The construction is local and free of zero-twists. We further show that within this economical class of surfaces it is however possible to derive a sufficient number of meaningful degrees of freedom so that standard optimization techniques result in high quality surfaces

Addressing Integration Error for Polygonal Finite Elements Through Polynomial Projections: A Patch Test Connection

Polygonal finite elements generally do not pass the patch test as a result of quadrature error in the evaluation of weak form integrals. In this work, we examine the consequences of lack of polynomial consistency and show that it can lead to a deterioration of convergence of the finite element solutions. We propose a general remedy, inspired by techniques in the recent literature of mimetic finite differences, for restoring consistency and thereby ensuring the satisfaction of the patch test and recovering optimal rates of convergence. The proposed approach, based on polynomial projections of the basis functions, allows for the use of moderate number of integration points and brings the computational cost of polygonal finite elements closer to that of the commonly used linear triangles and bilinear quadrilaterals. Numerical studies of a two-dimensional scalar diffusion problem accompany the theoretical considerations

Geometric Structures on Matrix-valued Subdivision Schemes

Surface subdivision schemes are used in computer graphics to generate visually smooth surfaces of arbitrary topology. Applications in computer graphics utilize surface normals and curvature. In this paper, formulas are obtained for the first and second partial derivatives of limit surfaces formed using 1-ring subdivision schemes that have 2 by 2 matrix-valued masks. Consequently, surface normals, and Gaussian and mean curvatures can be derived. Both quadrilateral and triangular schemes are considered and for each scheme both interpolatory and approximating schemes are examined. In each case, we look at both extraordinary and regular vertices. Every 3-D vertex of the refinement polyhedrons also has what is called a corresponding “shape vertex.” The partial derivative formulas consist of linear combinations of surrounding polyhedron vertices as well as their corresponding shape vertices. We are able to derive detailed information on the matrix-valued masks and about the left eigenvectors of the (regular) subdivision matrix. Local parameterizations are done using these left eigenvectors and final formulas for partial derivatives are obtained after we secure detailed information about right eigenvectors of the subdivision matrix. Using specific subdivision schemes, unit normals so obtained are displayed. Also, formulas for initial shape vertices are postulated using discrete unit normals to our original polyhedron. These formulas are tested for reasonableness on surfaces using specific subdivision schemes. Obtaining a specified unit normal at a surface point is examined by changing only these shape vertices. We then describe two applications involving surface normals in the field of computer graphics that can use our results

Efficient implementation of characteristic-based schemes on unstructured triangular grids

Using characteristics to treat advection terms in time-dependent PDEs leads to a class of schemes, e.g., semi-Lagrangian and Lagrange–Galerkin schemes, which preserve stability under large Courant numbers, and may therefore be appealing in many practical situations. Unfortunately, the need of locating the feet of characteristics may cause a serious drop of efficiency in the case of unstructured space grids, and thus prevent the use of large time-step schemes on complex geometries. In this paper, we perform an in-depth analysis of the main recipes available for characteristic location, and propose a technique to improve the efficiency of this phase, using additional information related to the advecting vector field. This results in a clear improvement of execution times in the unstructured case, thus extending the range of applicability of large time-step schemes