77 research outputs found
Meshing Deforming Spacetime for Visualization and Analysis
We introduce a novel paradigm that simplifies the visualization and analysis
of data that have a spatially/temporally varying frame of reference. The
primary application driver is tokamak fusion plasma, where science variables
(e.g., density and temperature) are interpolated in a complex magnetic
field-line-following coordinate system. We also see a similar challenge in
rotational fluid mechanics, cosmology, and Lagrangian ocean analysis; such
physics implies a deforming spacetime and induces high complexity in volume
rendering, isosurfacing, and feature tracking, among various visualization
tasks. Without loss of generality, this paper proposes an algorithm to build a
simplicial complex -- a tetrahedral mesh, for the deforming 3D spacetime
derived from two 2D triangular meshes representing consecutive timesteps.
Without introducing new nodes, the resulting mesh fills the gap between 2D
meshes with tetrahedral cells while satisfying given constraints on how nodes
connect between the two input meshes. In the algorithm we first divide the
spacetime into smaller partitions to reduce complexity based on the input
geometries and constraints. We then independently search for a feasible
tessellation of each partition taking nonconvexity into consideration. We
demonstrate multiple use cases for a simplified visualization analysis scheme
with a synthetic case and fusion plasma applications
The Complexity of Finding Small Triangulations of Convex 3-Polytopes
The problem of finding a triangulation of a convex three-dimensional polytope
with few tetrahedra is proved to be NP-hard. We discuss other related
complexity results.Comment: 37 pages. An earlier version containing the sketch of the proof
appeared at the proceedings of SODA 200
The existence of triangulations of non-convex polyhedra without new vertices
It is well known that a simple three-dimensional non-convex polyhedron may not be triangulated without using new vertices (so-called {\it Steiner points}). In this paper, we prove a condition that guarantees the existence of a triangulation of a non-convex polyhedron (of any dimension) without Steiner points. We briefly discuss algorithms for efficiently triangulating three-dimensional polyhedra
Computational Geometry Column 42
A compendium of thirty previously published open problems in computational
geometry is presented.Comment: 7 pages; 72 reference
Matrix Minor Reformulation and SOCP-based Spatial Branch-and-Cut Method for the AC Optimal Power Flow Problem
Alternating current optimal power flow (AC OPF) is one of the most
fundamental optimization problems in electrical power systems. It can be
formulated as a semidefinite program (SDP) with rank constraints. Solving AC
OPF, that is, obtaining near optimal primal solutions as well as high quality
dual bounds for this non-convex program, presents a major computational
challenge to today's power industry for the real-time operation of large-scale
power grids. In this paper, we propose a new technique for reformulation of the
rank constraints using both principal and non-principal 2-by-2 minors of the
involved Hermitian matrix variable and characterize all such minors into three
types. We show the equivalence of these minor constraints to the physical
constraints of voltage angle differences summing to zero over three- and
four-cycles in the power network. We study second-order conic programming
(SOCP) relaxations of this minor reformulation and propose strong cutting
planes, convex envelopes, and bound tightening techniques to strengthen the
resulting SOCP relaxations. We then propose an SOCP-based spatial
branch-and-cut method to obtain the global optimum of AC OPF. Extensive
computational experiments show that the proposed algorithm significantly
outperforms the state-of-the-art SDP-based OPF solver and on a simple personal
computer is able to obtain on average a 0.71% optimality gap in no more than
720 seconds for the most challenging power system instances in the literature
On Monotone Sequences of Directed Flips, Triangulations of Polyhedra, and Structural Properties of a Directed Flip Graph
This paper studied the geometric and combinatorial aspects of the classical
Lawson's flip algorithm in 1972. Let A be a finite set of points in R2, omega
be a height function which lifts the vertices of A into R3. Every flip in
triangulations of A can be associated with a direction. We first established a
relatively obvious relation between monotone sequences of directed flips
between triangulations of A and triangulations of the lifted point set of A in
R3. We then studied the structural properties of a directed flip graph (a
poset) on the set of all triangulations of A. We proved several general
properties of this poset which clearly explain when Lawson's algorithm works
and why it may fail in general. We further characterised the triangulations
which cause failure of Lawson's algorithm, and showed that they must contain
redundant interior vertices which are not removable by directed flips. A
special case if this result in 3d has been shown by B.Joe in 1989. As an
application, we described a simple algorithm to triangulate a special class of
3d non-convex polyhedra. We proved sufficient conditions for the termination of
this algorithm and show that it runs in O(n3) time.Comment: 40 pages, 35 figure
Linear Complexity Hexahedral Mesh Generation
We show that any polyhedron forming a topological ball with an even number of
quadrilateral sides can be partitioned into O(n) topological cubes, meeting
face to face. The result generalizes to non-simply-connected polyhedra
satisfying an additional bipartiteness condition. The same techniques can also
be used to reduce the geometric version of the hexahedral mesh generation
problem to a finite case analysis amenable to machine solution.Comment: 12 pages, 17 figures. A preliminary version of this paper appeared at
the 12th ACM Symp. on Computational Geometry. This is the final version, and
will appear in a special issue of Computational Geometry: Theory and
Applications for papers from SCG '9
The Computational Complexity of Integer Programming with Alternations
We prove that integer programming with three alternating quantifiers is NP-complete, even for a fixed number of variables. This complements earlier results by Lenstra and Kannan, which together say that integer programming with at most two alternating quantifiers can be done in polynomial time for a fixed number of variables. As a byproduct of the proof, we show that for two polytopes P, Q in R^4, counting the projection of integer points in QP is #P-complete. This contrasts the 2003 result by Barvinok and Woods, which allows counting in polynomial time the projection of integer points in P and Q separately
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