Creation of interior points

where N is the number of elements in the mesh. It is hoped that

γ ei<e for ia = 1,N

in which γ ei is the estimated error of element i. Based on the aforementioned quantities, the coefficient defined as

ξ γ

i i

ea

= e

is a natural indicator for mesh refinement. Let hi be the current element size of element i; the desirable element size h_i is given by

h = h

i i

ξi

where μ takes a value of 1/p (p = order of interpolation) if there is no singularity present or where the element is far away from singularity, and μ is set equal to 1/λ for elements close to a singularity of strength λ (Lee and Lo 1997a,b). If one sets μ equal to 1/p for all the ele-ments, inadequate refinement will occur in areas close to points of singularity, and conse-quently, more refinement steps will be needed to achieve the target accuracy.

In the above procedure, element size is defined for each element of the mesh, and this information of element size requirement has to be transferred to the grid points of the con-trol space. In the case of a structural background grid, all the grid points within a particular element of size h_i will receive the new element size requirement of h_i. As for control space based on the last mesh, the element size requirement of h_i can be directly assigned to all the node points of the element under consideration, and the desirable element size at a node equals the average value of the predicted element size from all the elements connected to it.

3.5.6.5 Creation of interior points

DT provides a general rule to connect points in 2D and in a higher dimensional space. In MG by the concept of DT, usually the boundary points are first inserted to produce an initial triangulation of these points. To complete the FE mesh, more points have to be inserted at the interior of the domain in conformity of the nodal spacing requirement, and the bound-ary edges have to be recovered so that all boundbound-ary edges of the given domain exist in the triangulation. The generation of an interior point will be discussed in Sections 3.5.6.5.1–

3.5.6.5.3, and the problem of boundary recovery will be explored in Section 3.5.7. Several methods have been proposed for the creation of interior points for Delaunay meshing and other FE MG schemes, which could be grouped into different categories.

3.5.6.5.1 Refinement method

By means of the refinement method, additional nodes are created with respect to the existing triangular elements. Depending on the strategy and the locations for the creation of addi-tional points, various refinement schemes for 2D and 3D DT have been proposed.

3.5.6.5.1.1 POINT CREATED AT CIRCUMCENTRE

Nodes are created at the circumcentre of those triangles that satisfy certain conditions such as surface area, in-radius, element aspect ratio, etc., to reduce FE solution error based on complementary variation principles (Shenton and Cendes 1985; Holmes and Snyder 1988).

3.5.6.5.1.2 POINT CREATED AT CENTROID

Nodes are created at the centroid of those triangles that are considered to be too large;

sometimes, a node can also be positioned at a point that is a weighted sum of the verti-ces of the triangle (Hermeline 1980) or based on centroidal Voronoi tessellations (Du and Gunzburger 2002; Secchi and Simoni 2003; Du et al. 2010).

3.5.6.5.1.3 POINTS CREATED ON EDGES

Realising that inserting nodes at the interior of a triangle would very often produce trian-gular elements of poor quality, Borouchaki and George (1997) proposed the creation of points on the edges of the triangles. Points are generated along each edge of the triangles in turn following a geometric progression in compliance with the element size requirement, and those points that are too close to the existing points will be filtered away. Rivara and Inostroza (1997) suggested a refinement scheme of DT based on the longest-side bisection.

Remarks: The refinement methods only suggest where a node can be generated, and a back-ground grid can be employed to provide information about the desirable element size and to control whether a proposed point is acceptable or it is rejected as it is too close to some existing points or simply it is outside the problem domain.

3.5.6.5.2 Use of background grid

The grid points of a background grid are potential positions for point creation in compliance with the given node spacing requirements. Those points can be rapidly and accurately defined without ambiguity for MG by means of the Delaunay insertion kernel or alternatively con-nected up at the interior of the domain by means of standard templates. Uniform rectangular and triangular grids can be used to generate elements of regular size, and in case of a variable element size requirement, Quadtree or kd-tree recursive spatial partitions can be applied.

3.5.6.5.2.1 REGULAR GRID

A regular grid of appropriate grid spacing can be superimposed with the given domain, as shown in Figure 3.35a. Points at grid positions with a sufficient distance from the boundary edges are inserted one by one to create a mesh of uniform node spacing. The proximity of a point to an edge can be verified by checking the area of the triangle formed between the given point and the boundary edge. As grid points of a rectangular grid are at the corner of a rectangular cell, these cyclic points will cause numerical problems to the Delaunay insertion

kernel. Instead of point insertion, points at the interior of the domain can be connected up rapidly by means of standard templates. The remaining points near the domain boundary can be processed by the point insertion kernel. Alternatively, a triangular grid of equilat-eral triangles can be employed to produce better quality triangles without differentiation between internal nodes or boundary nodes in the insertion process by the Delaunay inser-tion kernel, as shown in Figure 3.35b (Lo and Liu 2002).

3.5.6.5.2.2 QUADTREE PARTITION

The idea of Quadtree partition of space has been introduced in Section 3.4. A rectangular box that is large enough to contain the given domain is created. A Quadtree partition of the rectangular box is carried out in compliance with the nodal spacing specification, as shown in Figure 3.36a. Corner points and mid-side points of the cells can be triangulated by standard templates or inserted by the Delaunay insertion kernel. As corner points of a cell are cyclic, in the insertion process, the centroids of the cells are also inserted to avoid numerical problems.

However, similar to the regular grid, a recursively refined triangular grid analogous to the

(a) (b)

Figure 3.35 Creation of interior points by means of background grid: (a) rectangular grid; (b) triangular grid.

Solid line = domain boundary, broken line = triangulation of boundary points.

(a) (b)

Figure 3.36 Creation of interior points by recursive spatial partition: (a) quadtree decomposition; (b) recur­

sive triangular grid.

Quadtree subdivision can be used to produce triangles with better shape quality without caus-ing numerical difficulties in the Delaunay insertion, as shown in Figure 3.36b.

3.5.6.5.2.3 RANDOM POINT GENERATION

Elements of uniform size will be generated using grid points of a regular grid, and elements of variable size can be created by a recursive subdivision of a rectangular or a triangular grid, as discussed in Section 3.5.6.5.2.2. MG by means of a random point was proposed many years ago by Fukuda and Suhara (1972) and Cavendish (1974) to generate triangles of roughly equal size but of rather arbitrary shapes. However, to generate elements of variable size using a regu-lar grid without recursive spatial partition, the technique of random point generation can also be employed. Points are generated by a random process within a cell, which will be rejected if they are too close to the existing points within the cell or with respect to points of the adjacent cells. Each cell of the grid will be considered in turn, and the point insertion process will pass on to the next cell when an expected number of points have been generated. When all the cells of the grid have been considered, the domain will be filled up with randomly generated points in compliance with the element size requirement in a statistical sense. A grid with recursive spatial partition can also be used in conjunction with the random generation process to pro-duce elements with size in better compliance with the specified node spacing.

3.5.6.5.2.4 ‘VARIOGRAM’ – DISTANCE FROM BOUNDARY

For a planar domain bounded by line segments, points are inserted one by one within the given domain. An inserted point has to be located at a position that is farthest away from all the existing points, including the boundary points and the previously inserted points (Tacher and Parriaux 1996). In the actual implementation, a background grid with grid-point spacing compatible with the smallest element is prepared. The distance of each grid point from the boundary is computed by means of an induction formula based on the pre-vious grid point. A variogram is formed by listing the distance of all the grid points in an ascending order of magnitude. Points are accepted as long as the distance to the boundary is greater than some prescribed threshold value.

3.5.6.5.3 Other techniques

Apart from the refinement methods and those based on a background grid, many other tech-niques have been proposed to generate interior points within a given domain. An exhaustive account on all these methods is quite impossible and impractical; apart from the packing of circles, ellipses and spheres (Lo and Wang 2005b,c,d), two more methods that make use of entirely different concepts in point creation are included for a detailed discussion in Sections 3.5.6.5.3.1 and 3.5.6.5.3.2.

3.5.6.5.3.1 CONTOUR LINE

Right from the beginning in the development of FE MG, two major trends were the struc-tured and unstrucstruc-tured meshes. While various mapping methods were available for the generation of structured mesh, the generation of unstructured mesh was still at the infant stage. Apart from the mesh subdivision techniques (Thacker 1980), an early attempt for the automatic generation of unstructured mesh is by means of random point generation (Fukuda and Suhara 1972; Cavendish 1974), which was later modified by Shaw and Pitchen (1978) to generate nodes in a more systematic manner over rectangular cells. When the AFT

was proposed in the classical paper by Lo (1985), interior points were generated with the aid of contour lines cutting across the planar regions. This method was also later extended to 3D to generate interior nodes for ADF MG of 3D objects.

Direct deterministic approach and speed are the major advantages of MG by means of contours to produce meshes of uniform element size. Although well-shaped triangles can be produced by using contour, with the emergence of more powerful computers for solutions of ever more challenging physical problems, meshes of non-uniform element size or even anisotropic meshes of directional size specifications are required. Nevertheless, high-quality meshes of uniform element size can be rapidly generated by this simple approach as a first mesh in an adaptive refinement analysis or for further processing.

Let Ω be a planar domain with a boundary composed of Nb line segments Γ = {AiB_i, i = 1,N_b}. The following are the steps of the contour line method for the generation of interior nodes within Ω.

1. The minimum (y_min) and the maximum (y_max) of the y-value of the nodal points are determined.

2. Imaginary horizontal lines at different levels are drawn between y_min and y_max across the domain, as shown in Figure 3.37.

3. The spacing between any imaginary line is exactly equal to the average element size of the region.

4. Determination of the intersection of a horizontal line with the domain boundary is shown in Figure 3.37. The intersections between a horizontal line (y = h) and the domain boundary Γ are determined by considering the line segments of Γ one by one.

Consider the intersection of horizontal line y = h with line segment A_iB_i as an example.

Write x₁ = x(A_i), y₁ = y(A_i), x₂ = x(B_i) and y₂ = y(B_i). There is intersection if i. (y₁ – h)(y₂ – h) < 0

ii. (y₁ – h)(y₂ – h) = 0 and (h > y₁ or h > y₂)

It is considered as no intersection in other cases. The point of intersection is given by

x h y

y y x x h

1 1

2 1 2 1

+ −

− ( − ),

As the boundary of a region is composed of closed loops, a horizontal line must cut the domain boundary at an even number of points, and the intersection points are arranged in an ascending magnitude of x, i.e. from left to right.

y = h

1 2 3 4 5 6

(x₁, y₁) (x₂, y₂)

Proposed nodal positions

Figure 3.37 Intersection between horizontal lines and the domain boundary.

5. Assume that there are 2n cuts between the horizontal line y = h and the boundary Γ;

the cuts are considered two by two, beginning with the first and second cuts. Nodes are generated on this horizontal line segment between the cuts according to the pre-scribed spacing. This only suggests a series of potential positions where points are to be generated; however, whether a node is finally accepted depends also on whether it is too close to the domain boundary. The distance from a point to a line segment is discussed in Section 2.4.1, and the result could be applied here to check if a point is too close to a boundary edge. In case a point P is too close to a line segment AB such that triangle PAB is likely to form in the mesh, one has to ensure that the quality of triangle PAB is higher than the expected threshold value. The quality measure of triangular and tetrahedral elements will be discussed in Section 6.2.1.3.

6. After the first two cuts, the process is continued with the third and fourth cuts in a similar manner until it terminates with the (2n − 1)th and the 2nth cuts, then one pro-ceeds with the next horizontal line.

Remarks: The method of intersection by using contour lines is simpler and faster than the other methods, as previously generated nodes need not be taken into account in the genera-tion of a new node since its spacing is guaranteed by the generagenera-tion procedure. A vigorous check on domain boundaries is not necessary, and very often, a uniform distribution of nodes is resulted, which can be directly linked up layer by layer to form well-shaped trian-gles (Lo et al. 1982). In case the number of cuts recorded on a horizontal line is not an even number, a more robust method can be adopted to ensure that intersection points always come in pairs (Lo 1988a,b). The idea is to assign a value of +1 or −1 to each boundary point depending on whether it is above or below the horizontal line. If a boundary point is on the horizontal line, it doesn’t matter whether it is given a positive value or a negative value. The intersection between a boundary edge and a horizontal line is reduced to checking the pari-ties of the two nodal points of the boundary edge. The intersection is recorded if the nodal values are different, i.e. one node is +1, and the other node is −1; otherwise, if nodal values are equal, it is considered as no intersection. No matter how +1 and −1 values are assigned to boundary nodes, by going through segments of a closed loop, the change of parity (+1 and

−1 values) is always an even number. However, in practice, a simpler perturbation method can also be adopted, i.e. the level of the horizontal line is slightly adjusted from y = h to y = h +Δh with a typical value of Δh = 1% of the interval between horizontal lines. Instead of cutting the domain with horizontal lines, an interior point can also be generated by cutting the domain with vertical lines. Usually, one can have better results by cutting the domain along the longer dimension, i.e. if (y_max − y_min) > (x_max − x_min), use horizontal cut lines; on the contrary, if (y_max − y_min) < (x_max − x_min), use vertical cut lines.

3.5.6.5.3.2 GENERATION OF INTERIOR NODES BY AFT

In terms of quality and speed, the AFT is perhaps one of the most efficient methods to cre-ate points in compliance with a given element size specification over 2D or 3D domains. As each point is optimised as far as possible in the ADF meshing, the quality of the elements is verified and guaranteed in the element construction process. The method is also very effective for the generation of isotropic mesh of variable element size (Lo 2013b) as well as meshes with severe anisotropic characteristics (Borouchaki et al. 1997a,b). Moreover, the DT and AFT can be merged harmoniously into one general scheme in which the merits of both methods can be fully exploited, namely, the point generation by the AFT and the point connection following the Delaunay criterion. The details of the ADF–Delaunay method and

the Delaunay–ADF method will be described in detail in Section 3.7 after the introduction of DT in this section and AFT in Section 3.6.