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71
Efficient, Fair Interpolation using CatmullClark Surfaces
, 1993
"... We describe an efficient method for constructing a smooth surface that interpolates the vertices of a mesh of arbitrary topological type. Normal vectors can also be interpolated at an arbitrary subset of the vertices. The method improves on existing interpolation techniques in that it is fast, robus ..."
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Cited by 204 (9 self)
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We describe an efficient method for constructing a smooth surface that interpolates the vertices of a mesh of arbitrary topological type. Normal vectors can also be interpolated at an arbitrary subset of the vertices. The method improves on existing interpolation techniques in that it is fast, robust and general. Our approach is to compute a control mesh whose CatmullClark subdivision surface interpolates the given data and minimizes a smoothness or "fairness" measure of the surface. Following Celniker and Gossard, the norm we use is based on a linear combination of thinplate and membrane energies. Even though CatmullClark surfaces do not possess closedform parametrizations, we show that the relevant properties of the surfaces can be computed efficiently and without approximation. In particular, we show that (1) simple, exact interpolation conditions can be derived, and (2) the fairness norm and its derivatives can be computed exactly, without resort to numerical integration.
Scattered Data Interpolation with Multilevel Splines
 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS
, 1997
"... This paper describes a fast algorithm for scattered data interpolation and approximation. Multilevel Bsplines are introduced to compute a C²continuous surface through a set of irregularly spaced points. The algorithm makes use of a coarsetofine hierarchy of control lattices to generate a sequen ..."
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Cited by 158 (10 self)
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This paper describes a fast algorithm for scattered data interpolation and approximation. Multilevel Bsplines are introduced to compute a C²continuous surface through a set of irregularly spaced points. The algorithm makes use of a coarsetofine hierarchy of control lattices to generate a sequence of bicubic Bspline functions whose sum approaches the desired interpolation function. Large performance gains are realized by using Bspline refinement to reduce the sum of these functions into one equivalent Bspline function. Experimental results demonstrate that highfidelity reconstruction is possible from a selected set of sparse and irregular samples.
Curves and Surfaces for CAGD
, 1993
"... This article provides a historical account of the major developments in the area of curves and surfaces as they entered the area of CAGD – Computer Aided Geometric Design – until the middle 1980s. We adopt the definition that CAGD deals with the construction and representation of freeform curves, s ..."
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Cited by 79 (1 self)
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This article provides a historical account of the major developments in the area of curves and surfaces as they entered the area of CAGD – Computer Aided Geometric Design – until the middle 1980s. We adopt the definition that CAGD deals with the construction and representation of freeform curves, surfaces, or volumes. 1.
Curved PN Triangles
 In Symposium on Interactive 3D Graphics
, 2001
"... quadratically varying normals). To improve the visual quality of existing trianglebased art in realtime entertainment, such as computer games, we propose replacing flat triangles with curved patches and higherorder normal variation. At the hardware level, based only on the three vertices and three ..."
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Cited by 72 (3 self)
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quadratically varying normals). To improve the visual quality of existing trianglebased art in realtime entertainment, such as computer games, we propose replacing flat triangles with curved patches and higherorder normal variation. At the hardware level, based only on the three vertices and three vertex normals of a given flat triangle, we substitute the geometry of a threesided cubic Bézier patch for the triangle’s flat geometry, and a quadratically varying normal for Gouraud shading. These curved pointnormal triangles, or PN triangles, require minimal or no change to existing authoring tools and hardware designs while providing a smoother, though not necessarily everywhere tangent continuous, silhouette and more organic shapes. CR Categories: I.3.5 [surface representation, splines]: I.3.6— graphics data structures
Modeling with Cubic APatches
, 1995
"... We present a sufficient criterion for the Bernstein Bezier (BB) form of a trivariate polynomial within a tetrahedron, such that the real zero contour of the polynomial defines a smoothand singlesheeted algebraic surface patch, We call this an Apatch. We present algorithms to build a mesh of cubic ..."
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Cited by 68 (36 self)
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We present a sufficient criterion for the Bernstein Bezier (BB) form of a trivariate polynomial within a tetrahedron, such that the real zero contour of the polynomial defines a smoothand singlesheeted algebraic surface patch, We call this an Apatch. We present algorithms to build a mesh of cubic Apatches to interpolate a given set of scattered point data in three dimensions, respecting tbe topology of any surface triangulation T of the given point set. In these algorithms we first specify “normals” an the data points, then build a simplicial hull consisting of tetrahedral surrounding the surface triangulation 2’, and finally construct cubic Apatches within each tetrahedron. The resulting surface constructed is C¹ (tangent plane) continuous and single sheeted in each of the tetrahedral. We also show how to adjust the free parameters of the Apatches to achieve both local and global shape control.
A twolevel additive Schwarz preconditioner for nonconforming plate elements
 Numer. Math
, 1994
"... Abstract. Twolevel additive Schwarz preconditioners are developed for the nonconforming P1 finite element approximation of scalar secondorder symmetric positive definite elliptic boundary value problems, the Morley finite element approximation of the biharmonic equation, and the divergencefree no ..."
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Cited by 58 (5 self)
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Abstract. Twolevel additive Schwarz preconditioners are developed for the nonconforming P1 finite element approximation of scalar secondorder symmetric positive definite elliptic boundary value problems, the Morley finite element approximation of the biharmonic equation, and the divergencefree nonconforming P1 finite element approximation of the stationary Stokes equations. The condition numbers of the preconditioned systems are shown to be bounded independent of mesh sizes and the number of subdomains in the case of generous overlap. 1.
A Scalable Substructuring Method By Lagrange Multipliers For Plate Bending Problems
 SIAM J. Numer. Anal
, 1997
"... . We present a new Lagrange multiplier based domain decomposition method for solving iteratively systems of equations arising from the finite element discretization of plate bending problems. The proposed method is essentially an extension of the FETI substructuring algorithm to the biharmonic equat ..."
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Cited by 27 (13 self)
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. We present a new Lagrange multiplier based domain decomposition method for solving iteratively systems of equations arising from the finite element discretization of plate bending problems. The proposed method is essentially an extension of the FETI substructuring algorithm to the biharmonic equation. The main idea is to enforce continuity of the transversal displacement field at the subdomain crosspoints throughout the preconditioned conjugate gradient iterations. The resulting method is proved to have a condition number that does not grow with the number of subdomains, and grows at most polylogarithmically with the number of elements per subdomain. These optimal properties hold for numerous plate bending elements that are used in practice including the HCT, DKT, and a class of nonlocking elements for the ReissnerMindlin plate models. Computational experiments are reported and shown to confirm the theoretical optimal convergence properties of the new domain decomposition method. C...
Convergence of nonconforming multigrid methods without full elliptic regularity
 Math. Comp
, 1995
"... Abstract. We consider nonconforming multigrid methods for symmetric positive definite second and fourth order elliptic boundary value problems which do not have full elliptic regularity. We prove that there is a bound (< 1) for the contraction number of the Wcycle algorithm which is independent ..."
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Cited by 23 (0 self)
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Abstract. We consider nonconforming multigrid methods for symmetric positive definite second and fourth order elliptic boundary value problems which do not have full elliptic regularity. We prove that there is a bound (< 1) for the contraction number of the Wcycle algorithm which is independent of mesh level, provided that the number of smoothing steps is sufficiently large. We also show that the symmetric variable Vcycle algorithm is an optimal preconditioner. 1.
Analysis Of Lagrange Multiplier Based Domain Decomposition
, 1998
"... The convergence of a substructuring iterative method with Lagrange multipliers known as Finite Element Tearing and Interconnecting (FETI) method is analyzed in this thesis. This method, originally proposed by Farhat and Roux, decomposes finite element discretization of an elliptic boundary value pro ..."
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Cited by 20 (5 self)
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The convergence of a substructuring iterative method with Lagrange multipliers known as Finite Element Tearing and Interconnecting (FETI) method is analyzed in this thesis. This method, originally proposed by Farhat and Roux, decomposes finite element discretization of an elliptic boundary value problem into Neumann problems on the subdomains, plus a coarse problem for the subdomain null space components. For linear conforming elements and preconditioning by Dirichlet problems on the subdomains, the asymptotic bound on the condition number C(1 log(H=h)) fl , where fl = 2 or 3, is proved for a second order problem, h denoting the characteristic element size and H the size of subdomains. A similar method proposed by Park is shown to be equivalent to FETI with a special choice of some components and the bound C(1 log(H=h)) 2 on the condition number is established. Next, the original FETI method is generalized to fourth order plate bending problems. The main idea there is to enfor...
Overview and Recent Advances in Natural Neighbour Galerkin Methods
 ARCHIVES OF COMPUTATIONAL METHODS IN ENGINEERING  STATE OF THE ART REVIEWS
, 2003
"... In this paper, a survey of the most relevant advances in natural neighbour Galerkin methods is presented. In these methods (also known as natural element methods, NEM), the Sibson and the Laplace (nonSibsonian) interpolation schemes are used as trial and test functions in a Galerkin procedure. Natu ..."
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Cited by 16 (5 self)
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In this paper, a survey of the most relevant advances in natural neighbour Galerkin methods is presented. In these methods (also known as natural element methods, NEM), the Sibson and the Laplace (nonSibsonian) interpolation schemes are used as trial and test functions in a Galerkin procedure. Natural neighbourbased methods have certain unique features among the wide family of socalled meshless methods: a welldefined and robust approximation with no userdefined parameters on nonuniform grids, and the ability to exactly impose essential (Dirichlet) boundary conditions are particularly noteworthy. A comprehensive review of the method is conducted, including a description of the Sibson and the Laplace interpolants in two and threedimensions. Application of the NEM to linear and nonlinear problems in solid as well as fluid mechanics is studied. Other issues that are pertinent to the vast majority of meshless methods, such as numerical quadrature, imposing essential boundary conditions, and the handling of secondary variables are also addressed. The paper is concluded with some benchmark computations that demonstrate the accuracy and the key advantages of this numerical method.