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16
Subdivision Surfaces: A New Paradigm For ThinShell FiniteElement Analysis
 INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING
, 2000
"... We develop a new paradigm for thinshell finiteelement analysis based on the use of subdivision surfaces for: i) describing the geometry of the shell in its undeformed configuration, and ii) generating smooth interpolated displacement fields possessing bounded energy within the strict framework ..."
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Cited by 77 (20 self)
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We develop a new paradigm for thinshell finiteelement analysis based on the use of subdivision surfaces for: i) describing the geometry of the shell in its undeformed configuration, and ii) generating smooth interpolated displacement fields possessing bounded energy within the strict framework of the KirchhoffLove theory of thin shells. The particular subdivision strategy adopted here is Loop's scheme, with extensions such as required to account for creases and displacement boundary conditions. The displacement fields obtained by subdivision are H 2 and, consequently, have a finite KirchhoffLove energy. The resulting finite elements contain three nodes and element integrals are computed by a onepoint quadrature. The displacement field of the shell is interpolated from nodal displacements only. In particular, no nodal rotations are used in the interpolation. The interpolation scheme induced by subdivision is nonlocal, i. e., the displacement field over one element depend on the nodal displacements of the element nodes and all nodes of immediately neighboring elements. However, the use of subdivision surfaces ensures that all the local displacement fields thus constructed combine conformingly to define one single limit surface.
4–8 Subdivision
, 2001
"... In this paper we introduce 4–8 subdivision, a new scheme that generalizes the fourdirectional box spline of class C4 to surfaces of arbitrary topological type. The crucial advantage of the proposed scheme is that it uses bisection refinement as an elementary refinement operation, rather than more co ..."
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Cited by 51 (5 self)
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In this paper we introduce 4–8 subdivision, a new scheme that generalizes the fourdirectional box spline of class C4 to surfaces of arbitrary topological type. The crucial advantage of the proposed scheme is that it uses bisection refinement as an elementary refinement operation, rather than more commonly used face or vertex splits. In the uniform case, bisection refinement results in doubling, rather than quadrupling of the number of faces in a mesh. Adaptive bisection refinement automatically generates conforming variableresolution meshes in contrast to face and vertex split methods which require a postprocessing step to make an adaptively refined mesh conforming. The fact that the size of faces decreases more gradually with refinement allows one to have greater control over the resolution of a refined mesh. It also makes it possible to achieve higher smoothness while using small stencils (the size of the stencils used by our scheme is similar to Loop subdivision). We show that the subdivision surfaces produced by the 4–8 scheme are C^4 continuous almost everywhere, except at extraordinary vertices where they are is C¹continuous.
A Unified Framework for Primal/Dual Quadrilateral Subdivision Schemes
 CAGD
, 2001
"... Quadrilateral subdivision schemes come in primal and dual varieties, splitting faces or respectively vertices. The scheme of CatmullClark is an example of the former, while the DooSabin scheme exemplifies the latter. In this paper we consider the construction of an increasing sequence of alternati ..."
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Cited by 34 (5 self)
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Quadrilateral subdivision schemes come in primal and dual varieties, splitting faces or respectively vertices. The scheme of CatmullClark is an example of the former, while the DooSabin scheme exemplifies the latter. In this paper we consider the construction of an increasing sequence of alternating primal/dual quadrilateral subdivision schemes based on a simple averaging approach. Beginning with a vertex split step we successively construct variants of DooSabin and CatmullClark schemes followed by novel schemes generalizing Bsplines of bidegree up to nine. We prove the schemes to be C¹ at irregular surface points, and analyze the behavior of the schemes as the number of averaging steps increases. We discuss a number of implementation issues common to all quadrilateral schemes. In particular we show how both primal and dual quadrilateral schemes can be implemented in the same code, opening up new possibilities for more flexible geometric modeling applications and pversions of the Subdivision Element Method. Additionally we describe a simple algorithm for adaptive subdivision of dual schemes.
Multiresolution Mesh Representation: Models and Data Structures
 Tutorials on Multiresolution in Geometric Modelling
, 2002
"... Multiresolution meshes are a common basis for building representations of a geometric shape at dierent levels of detail. The use of the term multiresolution depends on the remark that the accuracy (or, level of detail) of a mesh in approximating a shape is related to the mesh resolution, i.e., to t ..."
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Cited by 24 (17 self)
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Multiresolution meshes are a common basis for building representations of a geometric shape at dierent levels of detail. The use of the term multiresolution depends on the remark that the accuracy (or, level of detail) of a mesh in approximating a shape is related to the mesh resolution, i.e., to the density (size and number) of its cells. A multiresolution mesh provides several alternative meshbased approximations of a spatial object (e.g., a surface describing the boundary of a solid object, or the graph of a scalar eld).
Smoothness of stationary subdivision on irregular meshes
 Constructive Approximation
, 1998
"... We derive necessary and sufficient conditions for tangent plane and C kcontinuity of stationary subdivision schemes near extraordinary vertices. Our criteria generalize most previously known conditions. We introduce a new approach to analysis of subdivision surfaces based on the idea of the univers ..."
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Cited by 22 (1 self)
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We derive necessary and sufficient conditions for tangent plane and C kcontinuity of stationary subdivision schemes near extraordinary vertices. Our criteria generalize most previously known conditions. We introduce a new approach to analysis of subdivision surfaces based on the idea of the universal surface. Any subdivision surface can be locally represented as a projection of the universal surface, which is uniquely defined by the subdivision scheme. This approach provides us with a more intuitive geometric understanding of subdivision near extraordinary vertices. AMS MOS classification: 65D10, 65D17, 68U05
Subdivision Scheme Tuning around Extraordinary Vertices
, 2004
"... In this paper we extend the standard method to derive and optimize subdivision rules in the vicinity of extraordinary vertices (EV). Starting from a given set of rules for regular control meshes, we tune the extraordinary rules (ER) such that the necessary conditions for C continuity are satisfie ..."
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Cited by 15 (1 self)
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In this paper we extend the standard method to derive and optimize subdivision rules in the vicinity of extraordinary vertices (EV). Starting from a given set of rules for regular control meshes, we tune the extraordinary rules (ER) such that the necessary conditions for C continuity are satisfied along with as many necessary C conditions as possible. As usually done, our approach sets up the general configuration around an EV by exploiting rotational symmetry and reformulating the subdivision rules in terms of the subdivision matrix' eigencomponents. The degrees of freedom are then successively eliminated by imposing new constraints which allows us, e.g., to improve the curvature behavior around EVs. The method is flexible enough to simultaneously optimize several subdivision rules, i.e. not only the one for the EV itself but also the rules for its direct neighbors. Moreover it allows us to prescribe the stencils for the ERs and naturally blends them with the regular rules that are applied away from the EV. All the constraints are combined in an optimization scheme that searches in the space of feasible subdivision schemes for a candidate which satisfies some necessary conditions exactly and other conditions approximately. The relative weighting of the constraints allows us to tune the properties of the subdivision scheme according to application specific requirements. We demonstrate our method by tuning the ERs for the wellknown Loop scheme and by deriving ERs for a # 3type scheme based on a 6direction Boxspline.
Integrated Modeling, FiniteElement Analysis, and Engineering Design for ThinShell Structures using Subdivision
 ComputerAided Design
, 2002
"... Many engineering des98 applications require geometric modeling and mechanical s imulation of thin flexibles tructures ,s uchas thos e found in the automotive and aerosH ce indus ries Traditionally, geometric modeling, mechanical s mulation, and engineering des ign are treated as s eparate modules re ..."
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Cited by 15 (2 self)
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Many engineering des98 applications require geometric modeling and mechanical s imulation of thin flexibles tructures ,s uchas thos e found in the automotive and aerosH ce indus ries Traditionally, geometric modeling, mechanical s mulation, and engineering des ign are treated as s eparate modules requiring di#erent methods and represR tations Due to the incompatibility of the involved repres9 tations the trans ition from geometric modeling to mechanicals imulation,as wellas in the oppos ite direction, requires s ubs tantial e#ort. However, for engineering des ign purpos es e# cient trans ition between geometric modeling and mechanicalsa ulationis esio tial. We propos e the us ofs ubdivis ons5AP ces as a common foundation for modeling, s mulation, and des gn in a unified framework. Subdivis ons686 ces provide a flexible and e#cient tool for arbitrary topology freeforms urface modeling, avoiding many of the problems inherent in traditionals pline patch bas d approaches The underlying bass functions are als ideallys5 ted for a finiteelement treatment of thes ocalled thins hell equations , which des cribe the mechanical behavior of the modeleds tructures The res ulting s lvers are highly s alable, providing an e#cient computational foundation for des ign exploration and optimization. We demons rate our claims withs everal des5R examples s howing the versH6P64 y and high accuracy of the propos d method. Key words: Subdivis on Surfaces FiniteElements Shells ; 1
Approximation Properties of Subdivision Surfaces
"... Splines are piecewise polynomial functions defined on a domain in Euclidean space. Because they are easily computed and have highorder approximation power, they are useful for modeling surfaces. Modeling a complex surface with splines typically requires a number of spline patches, which must be smo ..."
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Cited by 10 (0 self)
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Splines are piecewise polynomial functions defined on a domain in Euclidean space. Because they are easily computed and have highorder approximation power, they are useful for modeling surfaces. Modeling a complex surface with splines typically requires a number of spline patches, which must be smoothly joined, making splines cumbersome to use. Subdivision schemes generalize splines to domains of arbitrary topology. Thus, subdivision functions can be used to model complex surfaces without the need to join patches. Like splines, subdivision schemes have a multiresolution structure (i.e, a nested sequence of function spaces) associated to subdivisions of the domain. This thesis shows that a particular class of subdivision functions also have highorder approximation power. Although only one subdivision scheme, Loop's, is analyzed, the approach appears to be more general. The main result is an approximation theorem in Sobolev spaces H s of functions with square integrable derivatives up to order s. It is shown that each function f in H r , r # 3 can be approximated in H s , s 0 is arbitrary. This approximation theorem provides a theoretical foundation for various applications of subdivision schemes, such as the solution of thin shell problems in elasticity.
Matrixvalued subdivision schemes for generating surfaces with extraordinary vertices
 Comput. Aided Geom. Design
"... Subdivision templates of numerical values are replaced by templates of matrices in this paper to allow the introduction of shape control parameters for the feasibility of achieving desirable geometric shapes at those points on the subdivision surfaces that correspond to extraordinary control vertice ..."
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Cited by 9 (6 self)
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Subdivision templates of numerical values are replaced by templates of matrices in this paper to allow the introduction of shape control parameters for the feasibility of achieving desirable geometric shapes at those points on the subdivision surfaces that correspond to extraordinary control vertices. Formulation of the matrixvalued subdivision surface is derived. Based on refinable bivariate spline function vectors for matrixvalued subdivisions, the notion of characteristic map introduced by Reif is extended from (scalar) surface subdivisions to matrixvalued subdivisions. The C 1 and C kcontinuity of Reif and Prautzsch for matrixvalued subdivisions are discussed. To illustrate the general theory, the smoothness of matrixvalued triangular subdivision schemes for extraordinary vertices with valences 3 and 4 is analyzed. The issue of effective choices of the shape control parameters will also be discussed in this paper. Keywords: Matrixvalued surface subdivision, matrixvalued templates, surface shape control, extraordinary vertices, characteristic map, C kcontinuity 1