68 research outputs found
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Compatable smooth interpolation in triangles
Boolean sum smooth interpolation to boundary data on a triangle is described. Sufficient conditions are given so that the functions when pieced together form a CN-1(Ω) function over a triangular subdivision of a polygonal region Ω and the precision sets of the interpolation functions are derived. The interpolants are modified so that the compatability conditions on the function which is interpolated can be removed and a C1 interpolant is used to illustrate the theory. The generation of interpolation schemes for discrete boundary data is also discussed
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Error analysis of galerkin methods for dirichlet problems containing boundary singularities
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Interpolation remainder theory from taylor expansions with non-rectangular domains of influence
Sobolev norm error bounds are derived for interpolation remainders on triangles using two types of Taylor expansion. These bounds are applied to the finite element analysis of Poisson's equation on a triangulation of a polygonal region
Smooth polynomial interpolation to boundary data on triangles
Boolean sum interpolation theory is used to derive a polynomial interpolant which interpolates a function F ∈ CN(∂T), and its derivatives of order N and less, on the boundary 3T of a triangle T. A triangle with one curved side is also considered
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Sard kernel theorems on triangular and rectangular domains with extensions and applications to finite element error bounds
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Exact and approximate boundary data interpolation in the finite element method
Matching boundary data exactly in an elliptic problem avoids one of Strang's "variational crimes". (Strang and Fix (1973)). Supporting numerical evidence for this procedure is given by Marshall and Mitchell (1973), who considered the solution of Laplace's equation with Dirichlet boundary data by bilinear elements over squares and measured the errors in the L2 norm. Then Marshall and Mitchell (1978) obtained some surprising results: for certain triangular elements, matching the boundary data exactly produced worse results than the usual procedure of interpolating the boundary data
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Sard kernels for certain bivariate cubatures
Sard's kernel theorems [5] concern the result of applying a bounded linear functional to an appropriate Taylor expansion. The smoothness assumed for the functions determines the Taylor expansion, which in turn determines a norm on the function space. This norm of course defines which linear functionals are bounded
Blending using ODE swept surfaces with shape control and C1 continuity
Surface blending with tangential continuity is most widely applied in computer aided design, manufacturing systems, and geometric modeling. In this paper, we propose a new blending method to effectively control the shape of blending surfaces, which can also satisfy the blending constraints of tangent continuity exactly. This new blending method is based on the concept of swept surfaces controlled by a vector-valued fourth order ordinary differential equation (ODE). It creates blending surfaces by sweeping a generator along two trimlines and making the generator exactly satisfy the tangential constraints at the trimlines. The shape of blending surfaces is controlled by manipulating the generator with the solution to a vector-valued fourth order ODE. This new blending methods have the following advantages: 1). exact satisfaction of 1C continuous blending boundary constraints, 2). effective shape control of blending surfaces, 3). high computing efficiency due to explicit mathematical representation of blending surfaces, and 4). ability to blend multiple (more than two) primary surfaces
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