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An inclusion condition is presented to guarantee the existence of the solution of the linear complementarity problem in a given domain. The condition can be tested numerically with very small computational cost. Based on the condition algorithms are designed to compute an interval to enclose the unknown solution. Numerical results are reported to support the theoretical analysis in the paper.
We study a system of coupled convection-diffusion equations. The equations have diffusion parameters of different magnitudes associated with them which give rise to boundary layers at either boundary. An upwind finite difference scheme on arbitrary meshes is used to solve the system numerically. A general error estimate is derived that allows to immediately conclude robust convergence – w.r.t. the...
We consider the non-conforming Gauss-Legendre finite element family of any even degree k≥4 and prove its inf-sup stability without assumptions on the grid. This family consists of Scott-Vogelius elements where appropriate k-th-degree non-conforming bubbles are added to the velocities – which are trianglewise polynomials of degree k.
In this paper, we introduce construction algorithms for Korobov rules for numerical integration which work well for a given set of dimensions simultaneously. The existence of such rules was recently shown by Niederreiter. Here we provide a feasible construction algorithm and an upper bound on the worst-case error in certain reproducing kernel Hilbert spaces for such quadrature rules. The proof is...
A new backward stable algorithm (Algorithm 2) for polynomial interpolation based on the Lagrange and the Newton interpolation forms is proposed. It is shown that the Aitken algorithm and the scheme of the divided differences can be significantly less accurate than the proposed unconditionally stable Algorithm 2. Numerical examples that illustrate the advantage of a new algorithm are also given.
The Krawczyk and the Hansen-Sengupta interval operators are closely related to the interval Newton operator. These interval operators can be used as existence tests to prove existence of solutions for systems of equations. It is well known that the Krawczyk operator existence test is less powerful that the Hansen-Sengupta operator existence test, the latter being less powerful than the interval Newton...
Any two objects A and B can be viewed as two different projections of their Cartesian product A×B. Rotating and projecting A×B results in a continuous transformation of A into B. During certain rotations, the contour of the Cartesian product remains the same although its projection changes. Based on these properties, we derive a fast and simple morphing algorithm without topological constraints on...
We describe the structure and general properties of surfaces with polar layout. Polar layout is particularly suitable for high valences and is, for example, generated by a new class of subdivision schemes. This note gives an high level view of surfaces with polar structure and does not analyze particular schemes.
Point and splat-based representations have become a suitable technique both for modeling and rendering complex 3D shapes. Converting other kinds of models as parametric surfaces to splat-based representations will allow to mix surface and splat-based models and to take advantage of the existing point-based rendering methods. In this work, we present an approach to convert a parametric surface into...
It is well known that canal surfaces defined by a rational spine curve and a rational radius function are rational. The aim of the present paper is to construct a rational parameterization of low degree. The author uses the generalized stereographic projection in order to transform the problem to a parameterization problem for ruled surfaces. Two problems are discussed: parameterization with boundary...
Non-self-intersection is both a topological and a geometric property. It is known that non-self-intersecting regular Bézier curves have non-self-intersecting control polygons, after sufficiently many uniform subdivisions. Here a sufficient condition is given within ℝ3 for a non-self-intersecting, regular C2 cubic Bézier curve to be ambient isotopic to its control polygon formed after sufficiently...
Reverse engineering is concerned with the reconstruction of surfaces from three-dimensional point clouds originating from laser-scanned objects. We present an adaptive surface reconstruction method providing a hierarchy of quadrilateral meshes adapting surface topology when a mesh is refined. This way, a user can choose a model with proper resolution and topology from the hierarchy without having...
We present a multiresolution morphing algorithm using ``as-rigid-as-possible'' shape interpolation combined with an angle-length based multiresolution decomposition of simple 2D piecewise curves. This novel multiresolution representation is defined intrinsically and has the advantage that the details' orientation follows any deformation naturally. The multiresolution morphing algorithm consists of...
The standard proof of the Lane-Riesenfeld algorithm for inserting knots into uniform B-spline curves is based on the continuous convolution formula for the uniform B-spline basis functions. Here we provide two new, elementary, blossoming proofs of the Lane-Riesenfeld algorithm for uniform B-spline curves of arbitrary degree.
Filleting and rounding operations are very useful in solid modeling. The most popular means of producing a fillet surface is by means of the so-called ``rolling ball blend.'' A new method, inspired by the rolling ball blend, which overcomes both of its serious drawbacks is presented in this paper.
For a surface with non vanishing Gaussian curvature the Gauss map is regular and can be inverted. This makes it possible to use the normal as the parameter, and then it is trivial to calculate the normal and the Gauss map. This in turns makes it easy to calculate offsets, the principal curvatures, the principal directions, etc. Such a parametrization is not only a theoretical possibility but can be...
Robust Product Lifecycle Management (PLM) technology requires availability of informationally- complete models for all parts of a design-project including spatial constraints. This is the subject of the present investigation, leading to a new model for spatial constraints, the ``virtual solid'', which generalizes a similar concept used by Sapidis and Theodosiou to model ``required free-spaces'' in...
We consider a parameterized family of closed planar curves and introduce an evolution process for identifying a member of the family that approximates a given unorganized point cloud {pi}i=1,...,N. The evolution is driven by the normal velocities at the closest (or foot) points (fi) to the data points, which are found by approximating the corresponding difference vectors pi-fi in...
We describe a new technique for fitting noisy scattered point cloud data. The fitting surface is determined as zero level isosurface of a trivariate model which is an implicit least squares fit of the data based upon Radial Hermite Operators (RHO). We illustrate the value of these new techniques with several diverse applications.
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