4,228 research outputs found
Determinants of grids, tori, cylinders and M\"{o}bius ladders
Recently, Bie\~{n} [A. Bie\~{n}, The problem of singularity for planar grids,
Discrete Math. 311 (2011), 921--931] obtained a recursive formula for the
determinant of a grid. Also, recently, Pragel [D. Pragel, Determinants of box
products of paths, Discrete Math. 312 (2012), 1844--1847], independently,
obtained an explicit formula for this determinant. In this paper, we give a
short proof for this problem. Furthermore, applying the same technique, we get
explicit formulas for the determinant of a torus, a cylinder, and a M\"{o}bius
ladder
Grid generation about complex three-dimensional aircraft configurations
The problem of obtaining three dimensional grids with sufficient resolution to resolve all the flow or other physical features of interest is addressed. The generation of a computational grid involves a series of compromises to resolve several conflicting requirements. On one hand, one would like the grid to be fine enough and not too skewed to reduce the numerical errors and to adequately resolve the pertinent physical features of the flow field about the aircraft. On the other hand, the capabilities of present or even future supercomputers are finite and the number of mesh points must be limited to a reasonable number: one which is usually much less than desired for numerical accuracy. One technique to overcome this limitation is the 'zonal' grid approach. In this method, the overall field is subdivided into smaller zones or blocks in each of which an independent grid is generated with enough grid density to resolve the flow features in that zone. The zonal boundaries or interfaces require special boundary conditions such that the conservation properties of the governing equations are observed. Much work was done in 3-D zonal approaches with nonconservative zonal interfaces. A 3-D zonal conservative interfacing method that is efficient and easy to implement was developed during the past year. During the course of the work, it became apparent that it would be much more feasible to do the conservative interfacing with cell-centered finite volume codes instead of the originally planned finite difference codes. Accordingly, the CNS code was converted to finite volume form. This new version of the code is named CNSFV. The original multi-zonal interfacing capability of the CNS code was enhanced by generalizing the procedure to allow for completely arbitrarily shaped zones with no mesh continuity between the zones. While this zoning capability works well for most flow situations, it is, however, still nonconservative. The conservative interface algorithm was also implemented but was not completely validated
Adaptive mesh and geodesically sliced Schwarzschild spacetime in 3+1 dimensions
We present first results obtained with a 3+1 dimensional adaptive mesh code
in numerical general relativity. The adaptive mesh is used in conjunction with
a standard ADM code for the evolution of a dynamically sliced Schwarzschild
spacetime (geodesic slicing). We argue that adaptive mesh is particularly
natural in the context of general relativity, where apart from adaptive mesh
refinement for numerical efficiency one may want to use the built in
flexibility to do numerical relativity on coordinate patches.Comment: 21 pages, LaTeX, 7 figures included with eps
A numerical stabilization framework for viscoelastic fluid flow using the finite volume method on general unstructured meshes
A robust finite volume method for viscoelastic flow analysis on general
unstructured meshes is developed. It is built upon a general-purpose
stabilization framework for high Weissenberg number flows. The numerical
framework provides full combinatorial flexibility between different kinds of
rheological models on the one hand, and effective stabilization methods on the
other hand. A special emphasis is put on the velocity-stress-coupling on
co-located computational grids. Using special face interpolation techniques, a
semi-implicit stress interpolation correction is proposed to correct the
cell-face interpolation of the stress in the divergence operator of the
momentum balance. Investigating the entry-flow problem of the 4:1 contraction
benchmark, we demonstrate that the numerical methods are robust over a wide
range of Weissenberg numbers and significantly alleviate the high Weissenberg
number problem. The accuracy of the results is evaluated in a detailed mesh
convergence study
Kinematics, workspace and singularity analysis of a multi-mode parallel robot
A family of reconfigurable parallel robots can change motion modes by passing
through constraint singularities by locking and releasing some passive joints
of the robot. This paper is about the kinematics, the workspace and singularity
analysis of a 3-PRPiR parallel robot involving lockable Pi and R (revolute)
joints. Here a Pi joint may act as a 1-DOF planar parallelogram if its
lock-able P (prismatic) joint is locked or a 2-DOF RR serial chain if its
lockable P joint is released. The operation modes of the robot include a 3T
operation modes to three 2T1R operation modes with two different directions of
the rotation axis of the moving platform. The inverse kinematics and forward
kinematics of the robot in each operation modes are dealt with in detail. The
workspace analysis of the robot allow us to know the regions of the workspace
that the robot can reach in each operation mode. A prototype built at
Heriot-Watt University is used to illustrate the results of this work.Comment: International Design Engineering Technical Conferences \& Computers
and Information in Engineering Conference, Aug 2017, Cleveland, United
States. 201
Planewave density interpolation methods for 3D Helmholtz boundary integral equations
This paper introduces planewave density interpolation methods for the
regularization of weakly singular, strongly singular, hypersingular and nearly
singular integral kernels present in 3D Helmholtz surface layer potentials and
associated integral operators. Relying on Green's third identity and pointwise
interpolation of density functions in the form of planewaves, these methods
allow layer potentials and integral operators to be expressed in terms of
integrand functions that remain smooth (at least bounded) regardless the
location of the target point relative to the surface sources. Common
challenging integrals that arise in both Nystr\"om and boundary element
discretization of boundary integral equation, can then be numerically evaluated
by standard quadrature rules that are irrespective of the kernel singularity.
Closed-form and purely numerical planewave density interpolation procedures are
presented in this paper, which are used in conjunction with Chebyshev-based
Nystr\"om and Galerkin boundary element methods. A variety of numerical
examples---including problems of acoustic scattering involving multiple
touching and even intersecting obstacles, demonstrate the capabilities of the
proposed technique
An adaptive octree finite element method for PDEs posed on surfaces
The paper develops a finite element method for partial differential equations
posed on hypersurfaces in , . The method uses traces of
bulk finite element functions on a surface embedded in a volumetric domain. The
bulk finite element space is defined on an octree grid which is locally refined
or coarsened depending on error indicators and estimated values of the surface
curvatures. The cartesian structure of the bulk mesh leads to easy and
efficient adaptation process, while the trace finite element method makes
fitting the mesh to the surface unnecessary. The number of degrees of freedom
involved in computations is consistent with the two-dimension nature of surface
PDEs. No parametrization of the surface is required; it can be given implicitly
by a level set function. In practice, a variant of the marching cubes method is
used to recover the surface with the second order accuracy. We prove the
optimal order of accuracy for the trace finite element method in and
surface norms for a problem with smooth solution and quasi-uniform mesh
refinement. Experiments with less regular problems demonstrate optimal
convergence with respect to the number of degrees of freedom, if grid
adaptation is based on an appropriate error indicator. The paper shows results
of numerical experiments for a variety of geometries and problems, including
advection-diffusion equations on surfaces. Analysis and numerical results of
the paper suggest that combination of cartesian adaptive meshes and the
unfitted (trace) finite elements provide simple, efficient, and reliable tool
for numerical treatment of PDEs posed on surfaces
A Two-Dimensional MagnetoHydrodynamics Scheme for General Unstructured Grids
We report a new finite-difference scheme for two-dimensional
magnetohydrodynamics (MHD) simulations, with and without rotation, in
unstructured grids with quadrilateral cells. The new scheme is implemented
within the code VULCAN/2D, which already includes radiation-hydrodynamics in
various approximations and can be used with arbitrarily moving meshes (ALE).
The MHD scheme, which consists of cell-centered magnetic field variables,
preserves the nodal finite difference representation of div(\bB) by
construction, and therefore any initially divergence-free field remains
divergence-free through the simulation. In this paper, we describe the new
scheme in detail and present comparisons of VULCAN/2D results with those of the
code ZEUS/2D for several one-dimensional and two-dimensional test problems. The
code now enables two-dimensional simulations of the collapse and explosion of
the rotating, magnetic cores of massive stars. Moreover, it can be used to
simulate the very wide variety of astrophysical problems for which multi-D
radiation-magnetohydrodynamics (RMHD) is relevant.Comment: 22 pages, including 11 figures; Accepted to the Astrophysical
Journal. Higher resolution figures available at
http://zenith.as.arizona.edu/~burrows/mhd-code
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