Showing papers by "Jeff Erickson published in 2004"

Optimally Cutting a Surface into a Disk

Abstract: We consider the problem of cutting a subset of the edges of a polyhedral manifold surface, possibly with boundary, to obtain a single topological disk, minimizing either the total number of cut edges or their total length. We show that this problem is NP-hard in general, even for manifolds without boundary and for punctured spheres. We also describe an algorithm with running time n O(g+k), where n is the combinatorial complexity, g is the genus, and k is the number of boundary components of the input surface. Finally, we describe a greedy algorithm that outputs a O(log2 g)-approximation of the minimum cut graph in O(g 2 n log n) time.

Spacetime meshing with adaptive refinement and coarsening

Abstract: We propose a new algorithm for constructing finite-element meshes suitable for spacetime discontinuous Galerkin solutions of linear hyperbolic PDEs. Given a triangular mesh of some planar domain Ω and a target time value T, our method constructs a tetrahedral mesh of the spacetime domain Ω X [0,T] in constant running time per tetrahedron in R3 using an advancing front method. Elements are added to the evolving mesh in small patches by moving a vertex of the front forward in time. Spacetime discontinuous Galerkin methods allow the numerical solution within each patch to be computed as soon as the patch is created. Our algorithm employs new mechanisms for adaptively coarsening and refining the front in response to a posteriori error estimates returned by the numerical code. A change in the front induces a corresponding refinement or coarsening of future elements in the spacetime mesh. Our algorithm adapts the duration of each element to the local quality, feature size, and degree of refinement of the underlying space mesh. We directly exploit the ability of discontinuous Galerkin methods to accommodate discontinuities in the solution fields across element boundaries.

Kinetic collision detection between two simple polygons

Abstract: We design a kinetic data structure for detecting collisions between two simple polygons in motion. In order to do so, we create a planar subdivision of the free space between the two polygons, called the external relative geodesic triangulation, which certifies their disjointness. We show how this subdivision can be maintained as a kinetic data structure when the polygons are moving, and analyze its performance in the kinetic setting.

On the least median square problem

Abstract: We consider the exact and approximate computational complexity of the multivariate LMS linear regression estimator. The LMS estimator is among the most widely used robust linear statistical estimators. Given a set of n points in Rd and a parameter k, the problem is equivalent to computing the narrowest slab bounded by two parallel hyperplanes that contains k of the points. We present algorithms for the exact and approximate versions of the multivariate LMS problem. We also provide nearly matching lowerbounds for these problems, under the assumption that deciding whether n given points in Rd are affinely nondegenerate requires Ω(nd) time.

Automatic Blocking Scheme for Structured Meshing in 2d Multiphase Flow Simulation.

Abstract: We present an automatic algorithm to construct blocking scheme for multiblock structured meshes in 2D multiphase flow problems. Our solution is based on the concepts of medial axis and Delaunay triangulation. We show that the quality of the blocking scheme strongly depends on the quality of Delaunay triangulation. Therefore, well-known techniques and issues like Delaunay refinement and geometric degeneracy resurge again in multiblock structured meshes.

Separating point sets in polygonal environments

Abstract: We consider the separability of two point sets inside a polygon by means of chords or geodesic lines. Specifically, given a set of red points and a set of blue points in the interior of a polygon, we provide necessary and sufficient conditions for the existence of a chord and for the existence of a geodesic path which separate the two sets when they exist we also derive efficient algorithms for their obtention. We study as well the separation of the two sets using a minimum number of pairwise non-crossing chords.

Efficient Tradeoff Schemes in Data Structures for Querying Moving Objects

Abstract: The ability to represent and query continuously moving objects is important in many applications of spatio-temporal database systems In this paper we develop data structures for answering various queries on moving objects, including range and proximity queries, and study tradeoffs between various performance measures—query time, data structure size, and accuracy of results

Adaptive discontinuous galerkin method for elastodynamic s on unstructured spacetime grids

Abstract: We present an adaptive spacetime discontinuous Galerkin (S DG) method for linearized elastodynamics. The SDG method uses a simple Bubnov-Galerkin projection that delivers sta ble nd oscillation–free solutions, with O (N) complexity and exact momentum balance on every spacetime element. An extended version of the Tent Pitcher algorithm generates unstructured space time grids that support simultaneous grading in space and time. We pres nt results in 1D and 2D× time, emphasizing problems with shocks. 1