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

Interpolatory Catmull-Clark volumetric subdivision over unstructured hexahedral meshes for modeling and simulation applications

International audienceVolumetric modeling is an important topic for material modeling and isogeometric simulation. In this paper, two kinds of interpolatory Catmull-Clark volumetric subdivision approaches over unstructured hexahedral meshes are proposed based on the limit point formula of Catmull-Clark subdivision volume. The basic idea of the first method is to construct a new control lattice, whose limit volume by the CatmullClark subdivision scheme interpolates vertices of the original hexahedral mesh. The new control lattice is derived by the local push-back operation from one CatmullClark subdivision step with modified geometric rules. This interpolating method is simple and efficient, and several shape parameters are involved in adjusting the shape of the limit volume. The second method is based on progressive-iterative approximation using limit point formula. At each iteration step, we progressively modify vertices of an original hexahedral mesh to generate a new control lattice whose limit volume interpolates all vertices in the original hexahedral mesh. The convergence proof of the iterative process is also given. The interpolatory subdivision volume has C 2-smoothness at the regular region except around extraordinary vertices and edges. Furthermore, the proposed interpolatory volumetric subdivision methods can be used not only for geometry interpolation, but also for material attribute interpolation in the field of volumetric material modeling. The application of the proposed volumetric subdivision approaches on isogeometric analysis is also given with several examples

Composite primal/dual √3-subdivision schemes

We present new families of primal and dual subdivision schemes for triangle meshes and 3-refinement. The proposed schemes use two simple local rules which cycle between primal and dual meshes a number of times. The resulting surfaces become very smooth at regular vertices if the number of cycles is ⩾2. The C^1-property is violated only at low-valence irregular vertices, and can be restored by slight modifications of the local rules used. As a generalization, we introduce a wide class of composite subdivision schemes suitable for arbitrary topologies and refinement rules. A composite scheme is defined by a simple upsampling from the coarse to a refined topology, embedded into a cascade of geometric averaging operators acting on coarse and/or refined topologies. We propose a small set of such averaging rules (and some of their parametric extensions) which allow for the switching between control nets associated with the same or different topologic elements (vertices, edges, faces), and show a number of examples, based on triangles, that the resulting class of composite subdivision schemes contains new and old, primal and dual schemes for 3-refinement as well as for quadrisection. As a common observation from the examples considered, we found that irregular vertex treatment is necessary only at vertices of low valence, and can easily be implemented by using generic modifications of some elementary averaging rules

Point-Normal Subdivision Curves and Surfaces

This paper proposes to generalize linear subdivision schemes to nonlinear subdivision schemes for curve and surface modeling by refining vertex positions together with refinement of unit control normals at the vertices. For each round of subdivision, new control normals are obtained by projections of linearly subdivided normals onto unit circle or sphere while new vertex positions are obtained by updating linearly subdivided vertices along the directions of the newly subdivided normals. Particularly, the new position of each linearly subdivided vertex is computed by weighted averages of end points of circular or helical arcs that interpolate the positions and normals at the old vertices at one ends and the newly subdivided normal at the other ends. The main features of the proposed subdivision schemes are three folds: (1) The point-normal (PN) subdivision schemes can reproduce circles, circular cylinders and spheres using control points and control normals; (2) PN subdivision schemes generalized from convergent linear subdivision schemes converge and can have the same smoothness orders as the linear schemes; (3) PN $C^2$ subdivision schemes generalizing linear subdivision schemes that generate $C^2$ subdivision surfaces with flat extraordinary points can generate visually $C^2$ subdivision surfaces with non-flat extraordinary points. Experimental examples have been given to show the effectiveness of the proposed techniques for curve and surface modeling.Comment: 30 pages, 17 figures, 22.5M

Ισογεωμετρική Στατική Ανάλυση με T-SPLines

Σκοπός αυτής της διπλωματικής είναι η διερεύνηση της ισογεωμετρικής στατικής ανάλυσης χρησιμοποιώντας ΄ενα νέο έιδος συναρτήσεων σχήματος , τις T-SPLines. Τόσο οι T-SPLines όσο και η ανάλυση πεπερασμένων στοιχείων εετάστηκαν ξεχωριστά αφού αποτελούν τις δύο συνιστώσες της ισογεωμετρικής μεθόδου. Τα θέματα που εξετάστηκαν είναι οι T-SPLines και οι ιδιότητές τους, οι τεχνικές πύκνωσης του δικτύου , η μόρφωση του μητρώου στιβαρότητας, η επεξεργασία των αποτελεσμάτων της ανάλυσης (πεδίο μετατοπίσεων, τάσεων και παραμορφώσεων) και εφαρμογές 2Δ για τη διερεύνηση διαφόρων φορέων.The scope of this thesis if the investigation of static isogeometric analysis unsing a new type of shape functions T-SPLines. T-SPLines and finite elements have been examined separately, as the two components of the isogeometric method. The topics considered are T-SPLine formulation and properties, refinement techniques, stiffness matrix formulation , result post-processing (displacement, stress and strain field) and linear 2D applications investigating models of various representations.Δημήτριος Γ. Τσαπέτη

Drape simulation using solid-shell elements and adaptive mesh subdivision

In this paper, 4-node quadrilateral and 3-node triangular solid-shell elements are applied to drape simulations. With locking issues alleviated by the assumed natural strain method and plane-stress enforcement, static and dynamic drape problems are attempted by the quadrilateral element. If the drape is deep and the mesh density is inadequate, non-realistic sharp folds are predicted due to the non-physical interpenetration of top and bottom element surfaces. To avoid the interpenetration, a reversible adaptive subdivision based on the 1–4 splitting method is developed. To ensure displacement compatibility among elements at different subdivision levels, macro-transition elements are formed by quadrilateral and triangular solid-shell elements. To reduce the dynamic oscillation induced by newly inserted nodes, the discrete Kirchhoff condition is employed to determine the related nodal variables. Dynamic drape examples using adaptive meshing are presented. It can be seen that the predictions look realistic and deep drapes can be predicted with the interpenetration avoided yet the required number of nodes can be kept relatively small.postprin