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Computational Geometry Column 42
A compendium of thirty previously published open problems in computational
geometry is presented.Comment: 7 pages; 72 reference
Finite element simulation of three-dimensional free-surface flow problems
An adaptive finite element algorithm is described for the stable solution of three-dimensional free-surface-flow problems based primarily on the use of node movement. The algorithm also includes a discrete remeshing procedure which enhances its accuracy and robustness. The spatial discretisation allows an isoparametric piecewise-quadratic approximation of the domain geometry for accurate resolution of the curved free surface.
The technique is illustrated through an implementation for surface-tension-dominated viscous flows modelled in terms of the Stokes equations with suitable boundary conditions on the deforming free surface. Two three-dimensional test problems are used to demonstrate the performance of the method: a liquid bridge problem and the formation of a fluid droplet
The monotonicity of f-vectors of random polytopes
Let K be a compact convex body in Rd, let Kn be the convex hull of n points
chosen uniformly and independently in K, and let fi(Kn) denote the number of
i-dimensional faces of Kn. We show that for planar convex sets, E(f0(Kn)) is
increasing in n. In dimension d>=3 we prove that if lim(E((f[d
-1](Kn))/(An^c)->1 when n->infinity for some constants A and c > 0 then the
function E(f[d-1](Kn)) is increasing for n large enough. In particular, the
number of facets of the convex hull of n random points distributed uniformly
and independently in a smooth compact convex body is asymptotically increasing.
Our proof relies on a random sampling argument
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