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In the past ** Larry L. Schumaker** has collaborated on articles with

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AbstractWe explore the construction of parametric surfaces which interpolate prescribed 3D scattered data using spaces of parametric splines defined on a 2D triangulation. The method is based on minimizing certain natural energy expressions. Several examples involving filling holes and crowning surfaces are presented, and the role of the triangulation as a parameter is explored. The problem of creating closed surfaces is also addressed. This requires introducing spaces of splines on certain generalized triangulations.

AbstractThis paper is concerned with the problem of constructing a triangular facet surface between a pair of parallel contours in ℝ3. In particular, we discuss edge swapping algorithms designed to compute locally optimal triangulations, and explore the use of simulated annealing as a means to getting globally optimal ones. The solution of this problem has applications in the construction of a mathematical model of a three dimensional object, starting with a series of parallel curves representing cross-sections of the object as might arise, for example, in medical imaging.

AbstractA general theory of quasi-interpolants based on trigonometric splines is developed which is analogous to the polynomial spline case. The aim is to construct quasi-interpolants which are local, easy to compute, and which apply to a wide class of functions. As examples, we give a detailed treatment including error bounds for two classes which are especially useful in practice.

AbstractA construction for high order continuous finite elements on partitions consisting of tetrahedra, hexahedra and pyramids based on polynomial Bernstein-Bézier shape functions is presented along with algorithms that allow the computation of the system matrices in optimal complexity O(1) per entry.

AbstractCr macro-element spaces are constructed on polygonal domains with an arbitrary number of sides. The spaces consist of polynomial supersplines defined on triangulations which have been partially refined with Clough–Tocher splits. In addition to giving dimension formulae, minimal determining sets, and nodal bases, we derive error bounds for the corresponding Hermite interpolation operators. A number of examples are presented to show how the spaces can be used to fill n-sided holes.

AbstractA general method is given for constructing sets of sufficient linear conditions that ensure convexity of a polynomial in Bernstein–Bézier form on a triangle. Using the linear conditions, computational methods based on macro-element spline spaces are developed to construct convexity preserving splines over triangulations that interpolate or approximate given scattered data.

AbstractPolynomial spline spaces defined on T-meshes are useful tools for both surface modeling and the finite element method. Here the approximation power of such spline spaces is established. The approach uses Bernstein–Bézier methods to get precise conditions on the geometry of the meshes which lead to local and stable bases.

AbstractSpline spaces defined on spherical triangulations with hanging vertices are studied. In addition to dimension formulae, explicit basis functions are constructed, and their supports and stability are discussed. The approximation power of the spaces is also treated.

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