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Electromagnetic Scattering 99 purpose, we choose to employ the Delaunay–Voronoï dual diagram, with the integrals taken over the edges of the Delaunay and Voronoï cells. To illustrate the process, consider a triangular element m of the Delaunay mesh. This element will share an edge with N m elements, with numbers m i ,1≤ i ≤ N m , where N m = 3, unless the element has an edge representing the boundary of the domain. Suppose the Delaunay edge mm i is the common edge between elements m and m i and let the length of this edge be denoted by l mmi . Similarly, suppose that the Voronoï edgemm i is the line segment connecting the circumcentres of element m and element m i . The length of this Voronoï edge will be denoted by h mmi . As basic unknowns in the solution algorithm, we consider the value of the z-component of the magnetic field at the Voronoï vertices, and denote this by H m , and the projection of the electric field at the midpoint of the Delaunay edge mm i , in the direction of the edge, and denote this by E mmi . In this case, the laws of Ampère and Faraday can be approximated, using central differencing, as H (n+1/2) m E (n+1) mm i = H (n−1/2) m − ∆t ∑N m S m i=1 E (n) mm i l mmi , (11) = E mm (n) i + ∆t [ ] H m (n+1/2) − H m (n+1/2) h i , (12) mmi where S m is the area of element m. This is a staggered explicit scheme, where the time step size for a stable implementation may be determined from the requirement [TH00] ∆t < C min {l min ,h min } . (13) Here l min and h min are the minimum Delaunay and Voronoï edge lengths respectively and C is a safety factor. This implies the use of meshes which do not include either very short Delaunay, or very short Voronoï, edges. However, Voronoï edge lengths may vanish completely, on a general unstructured mesh, when two adjacent triangles have a common circumcentre. When this happens, the simple remedy is to merge these two triangles to form a single quadrilateral element. The discrete formulae of the equations (11) and (12) may be applied directly to this quadrilateral, with appropriate redefinition of N m . Moreover, the same merging procedure can be adopted when more than two triangles share a common circumcentre and the discrete equations applied again to the polygonal cell that is created by merging the triangles in this manner. This merging process is illustrated in Figure 1. If the mesh contains short non-zero Voronoï sides, the merging process may still be carried out, to overcome the severe restriction on the time step. However, this will reduce the accuracy of the scheme, due to the slight local non-orthogonality introduced by the merging. The boundary condition on the tangential component of the electric field can be directly imposed at the surface of the PEC. The far field boundary condition is again approximated by the addition of an artificial PML, with the external boundary of the truncated domain taken to be rectangular in shape.
100 I. Sazonov et al. Fig. 1. An example of a Delaunay–Voronoï dual diagram showing two mutually orthogonal meshes suitable for use with a co-volume solution scheme. The dotted lines indicate Voronoï edges and the dots represent Voronoï vertices. Quadrilateral and pentagonal elements, formed by the merging of triangles, are indicated by bold lines. 5 Mesh Generation With algorithms of the form considered here, wave propagation problems are normally simulated on a mesh, which is as uniform as possible, with a prescribed element size δ which is related to the wavelength. For two-dimensional simulations, in the absence of boundaries, the ideal mesh for the co-volume method is simply a mesh of equilateral triangles, with the Delaunay edge length l = δ. In this case, the Voronoï elements are perfect hexagons, with edge length h = δ/ √ 3 ≈ 0.577 δ. This ideal mesh has the highest quality but, for general scattering simulations, it almost certainly will not be able to represent the geometry of the scatterer. To overcome this problem, a method based on stitching the ideal mesh to a near-boundary unstructured mesh has been developed [SWH + 06]. In the vicinity of each boundary, a body fitted local mesh is constructed, with the properties close to those of the ideal mesh. Near-boundary elements are generated by a modified form of the advancing front method. The ideal mesh is employed, away from boundaries, in the major portion of the domain. An additional temporary layer of near-boundary elements is generated to assist the process of connecting the near-boundary mesh to the ideal mesh. The new nodes of this extra layer are marked as potential nodes for connection. For each of these potential nodes, the closest node in the ideal mesh is identified. Joining, consecutively, these identified nodes of the ideal mesh, we obtain a closed polygon, or set of polygons. The gap between the near-boundary elements and the ideal mesh element is triangulated using the Delaunay method. Here, points of the ideal mesh which lie in the gap will also be used during the triangulation. Standard mesh enhancement procedures, such as edge swapping and Laplace smoothing, are used at the end to improve the quality of the generated elements.
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Partial Differential Equations
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Partial Differential Equations Mode
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Dedicated to Olivier Pironneau
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VIII Preface computers has been at
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Contents List of Contributors .....
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List of Contributors Yves Achdou UF
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List of Contributors XV Claude Le B
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Discontinuous Galerkin Methods Vive
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Discontinuous Galerkin Methods 5 2
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Discontinuous Galerkin Methods 7 g
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Discontinuous Galerkin Methods 9 (
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Discontinuous Galerkin Methods 15 W
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Let a h and b h denote the bilinear
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Discontinuous Galerkin Methods 19 t
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Discontinuous Galerkin Methods 21 l
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Table 1. Primal DG for transport Di
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Discontinuous Galerkin Methods 25 [
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Mixed Finite Element Methods on Pol
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Mixed FE Methods on Polyhedral Mesh
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4 Hybridization and Condensation Mi
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is symmetric and positive definite,
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with some coefficient α ∈ R wher
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Domain Decomposition Approach for C
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Numerical Analysis of a Finite Elem
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so that |u| 2 1,ω h ≤ 2 Numerica
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Computing the Eigenvalues of the La
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Eigenvalues of the Laplace-Beltrami
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A Fixed Domain Approach in Shape Op
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