Unstructured meshing techniques

Discretizing the problem space into finer meshes minimizes the time step and consequently increase the total run time. Therefore, it is an important problem on how to achieve high spatial resolution problems where fine details of the problem are present while keeping reasonable computer resources. Such problems are thin wire embedded in a large region [118-123], antennas [124, 125] and optoelectronics [27, 28].

Three different types of conformable surface models have been suggested; namely, locally distorted grid models [13, 17-19, 126, 127], globally distorted grid models (body fitted) [14] and unstructured models [23, 24, 29, 128].

Unstructured grid methods initially developed as an alternative to discretize complex geometries. The mesh involves elements of different size and shapes with irregular connectivity. These methods make use of triangular elements for 2D problems and tetrahedral for 3D problems [11, 30, 129].

In FDTD method, a generalized Yee’s algorithm that is based on unstructured and irregular grids is presented [29, 128]. The unstructured grids are based on triangular pyramids, which provide second-order accuracy in condition that the interior angles of the triangular faces are less than 60o. The general Yee’s algorithm is based on solving the integral form of Maxwell’s equations with a closed 3D volume. For stability, the time step is chosen to be,

(2.104)

where c is the speed of light, li is the three edges in each cell sharing a common vertex.

There are different techniques for generating unstructured meshes, two of the most famous ones are Advancing-Front technique and Delaunay based approaches.

In unstructured mesh generation techniques, nodes are distributed in the domain prior the triangulation is carried on. Advancing front technique (AFT) create nodes wherever needed in a local procedure in the same time elements are generated. The elements created are triangles and the creation starts from the domain boundary. AFT involves propagating a layer from boundaries into domain. The method starts by discretizing the boundaries of the geometry (known as front) into edges. These edges form the initial front that will advance into the field. Starting with an edge the first triangle is created by joining the two ends of the edge to an existing point or to a point on the front. Then the current edge is removed from the front and the remaining two edges of the new triangle are assigned to the new front as shown in Figure 2.17.

Figure 2.17: Advancing front technique (AFT) [130]

The procedure is iterative as it is repeated till all edges of the initial front are removed as shown in Figure 2.17. New points placed in the domain are chosen to form triangle of good shape and size, the position is determined by a prescribed field function. Advancing front method result smooth high quality triangulation in most regions of the domain but difficulties are encountered at the places where fronts collide, usually in the centre of the domain [129, 131].

Delaunay-based approaches start with random number of points, where triangles of Delaunay types are initially created. For arbitrary number of points there are many possible triangulations. Figure 2.18 shows two different Delaunay triangulations; the first one contains long triangles (bad triangles) and the second is for good quality (uniform) Delaunay triangles. The initial triangulation is formed by forming the maximum pairs of nodes without crossing any line. The first node is then added and connected to each vertex of the enclosing triangle, add the second node and if it falls into one of the triangles you already created then subdivide that triangle as shown in Figure 2.19. The process is then repeated until the last node is added and the whole domain is triangulated [32, 130] as shown in Figure 2.20. All triangles should obey the Delaunay condition that the circumcircle containing the triangle vertices should not

1

2

contain other points as shown in Figure 2.20. Delaunay triangles are mostly of good shape and it has the advantage of maximizing the minimum angle of all triangles, it does a better job of connecting nearby nodes rather than far points which avoid the formation of long triangles and produce more uniform shapes [130].

(a) (b)

Figure 2.18: (a) Delaunay triangulation with long triangles, (b) Uniform Delaunay triangulation [132].

Figure 2.19: Transforming non-Delaunay triangle (left) into Delaunay type triangle (right) [12, 130].

Delaunay mesh can be expressed by its dual mesh known as the voronoi mesh which is formed by connecting the triangles circumcentres as shown in Figure 2.20. More details on Delaunay meshes are discussed in Chapter 3.

Figure 2.20: Delaunay mesh (light) and its dual mesh Voronoi tessellation (dark) [12, 130].

Unstructured mesh in the TLM is a meshing criteria that is adopted in [23, 24] to discretize problems with fine features and that of curved boundaries. Unstructured mesh is used to discretise the problem space into triangular shapes in 2D problem and tetrahedral in 3D problems that fulfil Delaunay condition. TLM node must coincide with the circumcentre of the Delaunay triangle. In 2D problems, each node combines three transmission lines that connect the TLM node to its neighbouring nodes. Since the triangles in the mesh are of different sizes, the maximum useable time step is proportional to the length of shortest transmission line in the entire problem [23], (2.105) where  and µ are the medium permittivity and permeability, respectively.

Stubs are inserted at the end of each transmission line to compensate for the discrepancy in transmission lines lengths, and to allow signal to travel with the same velocity between nodes. In Chapter 3, 2D unstructured TLM (2D UTLM) will be discussed in detail.

The most critical point that faces the UTLM is the optimization between the mesh size and the time step. In some Delaunay triangles, the circumcentre can lay outside the triangle which affects the distance between the two adjacent circumcentres. This distance can be very small or even equal zero, which lead to very small time step. A small time step has big impact on the computational resources, so a solution must be applied to maximise the time step. This point will be addressed and solved in Chapter

4. Unstructured mesh has better dispersion characteristics as compared to the structured mesh but has the disadvantage of intermodal coupling, in which any excited is coupled to other spatial modes. A complete dispersion and modal coupling study is discussed in Chapter 5.