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CivilComp Proceedings
ISSN 17593433 CCP: 84
PROCEEDINGS OF THE FIFTH INTERNATIONAL CONFERENCE ON ENGINEERING COMPUTATIONAL TECHNOLOGY Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Paper 3
Unstructured Dual Meshes for CoVolume Integration Schemes I. Sazonov, O. Hassan, K. Morgan and N.P. Weatherill
Civil & Computational Engineering Centre, University of Wales Swansea, United Kingdom I. Sazonov, O. Hassan, K. Morgan, N.P. Weatherill, "Unstructured Dual Meshes for CoVolume Integration Schemes", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Fifth International Conference on Engineering Computational Technology", CivilComp Press, Stirlingshire, UK, Paper 3, 2006. doi:10.4203/ccp.84.3
Keywords: Delaunay triangulation, Voronoï diagram, covolume integration schemes.
Summary
Yee's scheme, for the solution of the Maxwell equations [1], is an example of a
covolume solution technique that is staggered in both space and time. It exhibits a high
degree of computationally efficiency, in terms of both CPU and memory requirements compared
to, for example, a finite element time domain method (FETD). Initially proposed for
structured grids, Yee's scheme can be generalized for unstructured meshes and this will
enable its application to industrially complex geometries [2]. However, despite
the fact that real progress has been achieved in unstructured mesh generation methods over
the last two decades, such schemes have not generally proved to be effective for simulations
involving domains of complex shape. This is a result of the difficulties encountered when
attempting to generate high quality meshes, satisfying the requirements necessary for use of
covolume schemes.
As it is staggered in space, Yee's scheme needs two mutually orthogonal meshes for the electric and magnetic fields and the dual DelaunayVoronoï diagram is the obvious choice. The time step limitation of Yee's algorithm is proportional to the shortest edge in both meshes, so it is critical that the Voronoï diagram is as well behaved as the Delaunay triangulation. In addition, from view point of accuracy, the Delaunay mesh should not contain bad elements, where an element is defined as bad in this context if its circumcentre is located outside the element, or, at least, the number of bad elements should be minimized. Using present meshing methods, it is possible to build a smoothly varying Delaunay triangulation, with a high quality index. However, the corresponding Voronoï diagram is often highly irregular, with some very short Voronoï edges. This increases the total number of required time steps and, hence, reduces the practicality of the method. In the three dimensional case, standard mesh generation methods, like the advancing front technique can produce meshes, with about 30% bad elements, which are unsuitable for use with a covolume method. In this paper, this difficulty will be addressed and nonstandard mesh generation procedures studied. In the two dimensional case, the problem can be solved by use of a stitching method [3] or by the centroidal Voronoï tesselation (CVT) method [4], supplemented by constrained smoothing [5]. In the stitching approach, the problem of triangulation is split into a set of relatively simple problems of local triangulation. Firstly, in the vicinity of boundaries, body fitted local meshes are built with properties close to those regarded as being ideal. An ideal mesh is employed, away from boundaries, in the major portion of the domain. The mesh fragments are then combined, to form a consistent mesh, with the outer layer of the near boundary elements stitched to a region of ideal mesh by a special procedure, in which the high compliance of mesh fragments is used. The work of generalising the stitching method to the three dimensional case is currently in progress. The CVT method is easier to generalise to three dimensions. It produces meshes with around 35% bad elements and, as these elements are located near boundaries, the essential part of the work in both methods is the near boundary triangulation. References
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