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Equation Set (equation + set)

Distribution by Scientific Domains
Engineering40%
Earth and Environmental Science40%

Selected Abstracts

A cut-cell non-conforming Cartesian mesh method for compressible and incompressible flow

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 11 2007
J. Pattinson
Abstract This paper details a multigrid-accelerated cut-cell non-conforming Cartesian mesh methodology for the modelling of inviscid compressible and incompressible flow. This is done via a single equation set that describes sub-, trans-, and supersonic flows. Cut-cell technology is developed to furnish body-fitted meshes with an overlapping mesh as starting point, and in a manner which is insensitive to surface definition inconsistencies. Spatial discretization is effected via an edge-based vertex-centred finite volume method. An alternative dual-mesh construction strategy, similar to the cell-centred method, is developed. Incompressibility is dealt with via an artificial compressibility algorithm, and stabilization achieved with artificial dissipation. In compressible flow, shocks are captured via pressure switch-activated upwinding. The solution process is accelerated with full approximation storage (FAS) multigrid where coarse meshes are generated automatically via a volume agglomeration methodology. This is the first time that the proposed discretization and solution methods are employed to solve a single compressible,incompressible equation set on cut-cell Cartesian meshes. The developed technology is validated by numerical experiments. The standard discretization and alternative methods were found equivalent in accuracy and computational cost. The multigrid implementation achieved decreases in CPU time of up to one order of magnitude. Copyright © 2007 John Wiley & Sons, Ltd. [source]

Towards a consistent numerical compressible non-hydrostatic model using generalized Hamiltonian tools

THE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 635 2008
Almut Gassmann
Abstract A set of compressible non-hydrostatic equations for a turbulence-averaged model atmosphere comprising dry air and water in three phases plus precipitating fluxes is presented, in which common approximations are introduced in such a way that no inconsistencies occur in the associated budget equations for energy, mass and Ertel's potential vorticity. These conservation properties are a prerequisite for any climate simulation or NWP model. It is shown that a Poisson bracket form for the ideal fluid part of the full-physics equation set can be found, while turbulent friction and diabatic heating are added as separate ,dissipative' terms. This Poisson bracket is represented as a sum of a two-fold antisymmetric triple bracket (a Nambu bracket represented as helicity bracket) plus two antisymmetric brackets (so-called mass and thermodynamic brackets of the Poisson type). The advantage of this approach is that the given conservation properties and the structure of the brackets provide a good strategy for the construction of their discrete analogues. It is shown how discrete brackets are constructed to retain their antisymmetric properties throughout the spatial discretisation process, and a method is demonstrated how the time scheme can also be incorporated in this philosophy. Copyright © 2008 Royal Meteorological Society [source]

Analysis of a regularized, time-staggered discretization applied to a vertical slice model,

ATMOSPHERIC SCIENCE LETTERS, Issue 4 2006
Mark Dubal
Abstract A regularized and time-staggered discretization of the two-dimensional, vertical slice Euler equation set is described and analysed. A linear normal mode analysis of the time-discrete system indicates that unconditional stability is obtained, for appropriate values of the regularization parameters, for both the hydrostatic and non-hydrostatic cases. Furthermore, when these parameters take their optimal values, the stability behaviour of the normal modes is identical to that obtained from a semi-implicit discretization of the unregularized equations. © Crown Copyright 2006. Reproduced with the permission of the Controller of HMSO. Published by John Wiley & Sons, Ltd. [source]

A fast triangle to triangle intersection test for collision detection

COMPUTER ANIMATION AND VIRTUAL WORLDS (PREV: JNL OF VISUALISATION & COMPUTER ANIMATION), Issue 5 2006
Oren Tropp
Abstract The triangle-to-triangle intersection test is a basic component of all collision detection data structures and algorithms. This paper presents a fast method for testing whether two triangles embedded in three dimensions intersect. Our technique solves the basic sets of linear equations associated with the problem and exploits the strong relations between these sets to speed up their solution. Moreover, unlike previous techniques, with very little additional cost, the exact intersection coordinates can be determined. Finally, our technique uses general principles that can be applied to similar problems such as rectangle-to-rectangle intersection tests, and generally to problems where several equation sets are strongly related. We show that our algorithm saves about 20% of the mathematical operations used by the best previous triangle-to-triangle intersection algorithm. Our experiments also show that it runs 18.9% faster than the fastest previous algorithm on average for typical scenarios of collision detection (on Pentium 4). Copyright © 2006 John Wiley & Sons, Ltd. [source]

Performance of algebraic multi-grid solvers based on unsmoothed and smoothed aggregation schemes

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 7 2001
R. WebsterArticle first published online: 31 JUL 200
Abstract A comparison is made of the performance of two algebraic multi-grid (AMG0 and AMG1) solvers for the solution of discrete, coupled, elliptic field problems. In AMG0, the basis functions for each coarse grid/level approximation (CGA) are obtained directly by unsmoothed aggregation, an appropriate scaling being applied to each CGA to improve consistency. In AMG1 they are assembled using a smoothed aggregation with a constrained energy optimization method providing the smoothing. Although more costly, smoothed basis functions provide a better (more consistent) CGA. Thus, AMG1 might be viewed as a benchmark for the assessment of the simpler AMG0. Selected test problems for D'Arcy flow in pipe networks, Fick diffusion, plane strain elasticity and Navier,Stokes flow (in a Stokes approximation) are used in making the comparison. They are discretized on the basis of both structured and unstructured finite element meshes. The range of discrete equation sets covers both symmetric positive definite systems and systems that may be non-symmetric and/or indefinite. Both global and local mesh refinements to at least one order of resolving power are examined. Some of these include anisotropic refinements involving elements of large aspect ratio; in some hydrodynamics cases, the anisotropy is extreme, with aspect ratios exceeding two orders. As expected, AMG1 delivers typical multi-grid convergence rates, which for all practical purposes are independent of mesh bandwidth. AMG0 rates are slower. They may also be more discernibly mesh-dependent. However, for the range of mesh bandwidths examined, the overall cost effectiveness of the two solvers is remarkably similar when a full convergence to machine accuracy is demanded. Thus, the shorter solution times for AMG1 do not necessarily compensate for the extra time required for its costly grid generation. This depends on the severity of the problem and the demanded level of convergence. For problems requiring few iterations, where grid generation costs represent a significant penalty, AMG0 has the advantage. For problems requiring a large investment in iterations, AMG1 has the edge. However, for the toughest problems addressed (vector and coupled vector,scalar fields discretized exclusively using finite elements of extreme aspect ratio) AMG1 is more robust: AMG0 has failed on some of these tests. However, but for this deficiency AMG0 would be the preferred linear approximation solver for Navier,Stokes solution algorithms in view of its much lower grid generation costs. Copyright © 2001 John Wiley & Sons, Ltd. [source]