6,476 research outputs found
Boundary integration of polynomials over an arbitrary linear hexahedron in Euclidean three-dimensional space
This paper is concerned with explicit integration formulas and algorithms for computing volume integrals of trivariate polynomials over an arbitrary linear hexahedron in Euclidean three-dimensional space. Three different approaches are discussed. The first algorithm is obtained by transforming a volume integral into a sum of surface integrals and then into convenient and computationally efficient line integrals. The second algorithm is obtained by transforming a volume integral into a sum of surface integrals over the boundary quadrilaterals. The third algorithm is obtained by transforming a volume integral into a sum of surface integrals over the triangulation of boundary. These algorithms and finite integration formulas are then followed by an application example, for which we have explained the detailed computational scheme. The symbolic finite integration formulas presented in this paper may lead to efficient and easy incorporation of integral properties of arbitrary linear polyhedra required in the engineering design process. © 1998 Elsevier Science S.A. All rights reserved
An exact general remeshing scheme applied to physically conservative voxelization
We present an exact general remeshing scheme to compute analytic integrals of
polynomial functions over the intersections between convex polyhedral cells of
old and new meshes. In physics applications this allows one to ensure global
mass, momentum, and energy conservation while applying higher-order polynomial
interpolation. We elaborate on applications of our algorithm arising in the
analysis of cosmological N-body data, computer graphics, and continuum
mechanics problems.
We focus on the particular case of remeshing tetrahedral cells onto a
Cartesian grid such that the volume integral of the polynomial density function
given on the input mesh is guaranteed to equal the corresponding integral over
the output mesh. We refer to this as "physically conservative voxelization".
At the core of our method is an algorithm for intersecting two convex
polyhedra by successively clipping one against the faces of the other. This
algorithm is an implementation of the ideas presented abstractly by Sugihara
(1994), who suggests using the planar graph representations of convex polyhedra
to ensure topological consistency of the output. This makes our implementation
robust to geometric degeneracy in the input. We employ a simplicial
decomposition to calculate moment integrals up to quadratic order over the
resulting intersection domain.
We also address practical issues arising in a software implementation,
including numerical stability in geometric calculations, management of
cancellation errors, and extension to two dimensions. In a comparison to recent
work, we show substantial performance gains. We provide a C implementation
intended to be a fast, accurate, and robust tool for geometric calculations on
polyhedral mesh elements.Comment: Code implementation available at https://github.com/devonmpowell/r3
On the equivalence between the cell-based smoothed finite element method and the virtual element method
We revisit the cell-based smoothed finite element method (SFEM) for
quadrilateral elements and extend it to arbitrary polygons and polyhedrons in
2D and 3D, respectively. We highlight the similarity between the SFEM and the
virtual element method (VEM). Based on the VEM, we propose a new stabilization
approach to the SFEM when applied to arbitrary polygons and polyhedrons. The
accuracy and the convergence properties of the SFEM are studied with a few
benchmark problems in 2D and 3D linear elasticity. Later, the SFEM is combined
with the scaled boundary finite element method to problems involving
singularity within the framework of the linear elastic fracture mechanics in
2D
Entropy in the Classical and Quantum Polymer Black Hole Models
We investigate the entropy counting for black hole horizons in loop quantum
gravity (LQG). We argue that the space of 3d closed polyhedra is the classical
counterpart of the space of SU(2) intertwiners at the quantum level. Then
computing the entropy for the boundary horizon amounts to calculating the
density of polyhedra or the number of intertwiners at fixed total area.
Following the previous work arXiv:1011.5628, we dub these the classical and
quantum polymer models for isolated horizons in LQG. We provide exact
micro-canonical calculations for both models and we show that the classical
counting of polyhedra accounts for most of the features of the intertwiner
counting (leading order entropy and log-correction), thus providing us with a
simpler model to further investigate correlations and dynamics. To illustrate
this, we also produce an exact formula for the dimension of the intertwiner
space as a density of "almost-closed polyhedra".Comment: 24 page
Picture Changing Operators in Closed Fermionic String Field Theory
We discuss appropriate arrangement of picture changing operators required to
construct gauge invariant interaction vertices involving Neveu-Schwarz states
in heterotic and closed superstring field theory. The operators required for
this purpose are shown to satisfy a set of descent equations.Comment: 15 page
A posteriori error estimates for the Electric Field Integral Equation on polyhedra
We present a residual-based a posteriori error estimate for the Electric
Field Integral Equation (EFIE) on a bounded polyhedron. The EFIE is a
variational equation formulated in a negative order Sobolev space on the
surface of the polyhedron. We express the estimate in terms of
square-integrable and thus computable quantities and derive global lower and
upper bounds (up to oscillation terms).Comment: Submitted to Mathematics of Computatio
- …