Partial Differential Equations - Modelling and ... - ResearchGate

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.