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For many classic mathematical physics problems, such as heat conduction, particle transport/diffusion etc., physicists have established a system of ordinary or partial differential equations that are valid in a certain region (or domain), under continuum approximation, and certain types of boundary and initial conditions. In most of these cases, however, there are no analytical solutions that can be obtained. Therefore it is necessary to recast the problem in pure algebraic form, involving only basic arithmetic operations by discretising the continuum problem through which the infinite set of numbers representing the unknown function or functions is replaced by a finite number of unknown parameters. One of the simplest discretization scheme is

the finite different process which can be regarded as a subclass of finite element

methods. Various trial function approximations fall under the general classification of finite element methods. The essential idea of the finite element process lies in that if

the sub-domains are of a relatively simple shape and if the definition of the trial functions over these sub-domains can be made in a repeatable manner, then it is possible to deal in this fashion with assembled regions of complex shapes quite readily. Particularly, if the trial functions are to be defined in a piece-wise manner, it is advantageous to assign to them a narrow “base” and make their value zero everywhere except in the element in question and in the sub-domains immediately adjacent to this element. This will give sparse matrices in the final approximation equations, resulting in yet another advantage of the finite element process.

(i) Construction of mesh

In order for the numerical calculation to be performed, a physical area (domain) of a pixel in study with dimension of L xx L y X L z must be subdivided into a number of small

cells by dividing L x into a number of divisions N & x in the x-direction, L y into A^y, and L zinto NAz. The total number of cells is the product of NAxxN AyxN Az. A vertex of a cell

is called a node or in physics term a lattice point. The total independent number of nodes for a given pixel is given by (A^Ax+l)x(AAy+l)x(AAz+l)- The node is numbered serially by a single index integer, say, i, rather than assigned a set of integer indices like (i,j,k) in order to use finite element technique. Each node has a unique integer number (index) to be referred to and also has a unique x-y-z co-ordinates. Director and potential fields are calculated by referring to the node as a requirement of discretisation. In the case of 2D model where x axis denotes pixel width, z denotes the height (cell gap), and y is the depth, assuming extended to infinity so that we do not have to consider. The dividing interval Ax is not necessarily required being equal to Az, depending on the problem in study although Ax being equal to Az gives an ideal aspect ratio, therefore, more accurate numerical approximation, but not necessarily the most efficient algorithm. In the case of liquid crystal displays, normally, a pixel width Lx is in the range from 30 (Ltm to 90 |LLm while the height L z goes only from 4 J i m to 5 jam. L x is much greater than Lz. However the light propagates and the voltage are

applied across z, therefore we have to pay close attention to variation across z. Hence, in this work, Ax is in the most cases chosen to be in the order of six times Az unless stated otherwise. The issue of the influences of different aspect ratios on the computational results is addressed and discussed in the Section 6.16. Figure 11 shows a sample mesh of a pixel whose dimension is Lx=30 |im and L z= 5 jLim. The choice of

discretisation intervals are 1.5pm in x and 0.25jLim in z respectively (which gives a

aspect ratio of 6). The number of divisions N^=20 and N&z=20, which makes

20x20=400 cells, or 800 triangle elements, and 441 (=21x21) nodes. Two shortest thick horizontal solid lines at the top of the mesh represent the two bus electrodes, the line at the top centre represents the pixel electrode, and the long line at the bottom is the ground electrode. The pixel area is divided into appropriate blocks (boxes) according to the boundary conditions. The co-ordinates of each box’s boundary lines are given as well as numbers of divisions N b^ N^ of each box in x and in z . In the

sample mesh shown in Figure 11, the pixel is divided into 7 boxes, co-ordinates of each box and division numbers are shown in the Table 1.

1 2 3 4 5 6 7 (*1,*2) (-15,-12) (-12,-9) (-9,-6) (-6,6) (6,9) (9,12) (12,15) N b iy Ax 2 2 2 8 2 2 2 (-2.5 2.5) N b ly Az 2 0

Table 1 Parameters used in generating sample mesh as shown in Figure 11

.— .--- - —-— —-— - — —— — — __ -— ■— — ^ " — ^ ^ —— ^ ^ — ___ __-— — — — ■ _-—1 ___ ^ ^ —■— — —■— —- — — -— " -— ■— ____ ' ,— ' ,---- ' — —■-— — " — —— —■— —-— —-— --- — ^ — •—

Figure 11 A sample mesh of a pixel with two bus electrodes, one pixel electrode and a ground electrode

(ii) Triangular elements

The elements used in the finite element method can have any shape so far as they constitute a tesselation of the problem domain. In this respect, the most common shapes are triangular and quadrilateral. Triangular elements are a popular choice for

the analysis of problem involving two space dimensions because of the ease with which arbitrarily shaped domains may be closely approximated by an assembly of triangles plus the characteristic that a sufficient number of nodes may be generated at all time to construct a family of complete polynomials [88],[89]. Therefore in this work, the triangle element is used. The triangle element is obtained by dividing each rectangular cell into two triangles as shown in the Figure 11. The triangle element is numbered serially. Three vertices (nodes) of each triangular element are numbered locally. The node number (i.e., an integer index) to which each vertex corresponds is identified and stored. The co-ordinates of each vertex is given by its corresponding nodal co-ordinates. In this way, the position of each triangular element is solely determined by its three vertices’ co-ordinates, hence three corresponding nodal co­ ordinates. The nearest neighbours elements to each given node are found by the mesh generator program and stored as an output which is used as an input to the main program.

(iii) Construction of shape function

(a) Triangular element family and Pascal triangle

The following forms of triangular elements constitute a family. They are (1) linear [ Figure 12(a) ], (2) quadratic [Figure 12(b)], and (3) cubic [Figure 12(c)] which

4 2

1 1 4 5 2

Figure 12 Triangular element family: (a) linear, (b) quadratic, and (c) cubic correspond, respectively, to first order, second order, and third order polynomials expansions as shown in Pascal triangle (see Figure 13). Among them, only linear element interests us. Therefore in what follows, the focus will only be put on the first order polynomial.

Consider a series of triangles generated on a pattern indicated in Figure 12. The number of nodes in each member of the family is now such that a complete

polynomial expansion, of the order needed for inter-element compatibility, is ensured. This follows by comparison with the Pascal triangle of Figure 13 in which the number of nodes exactly match the number of polynomial terms required. This particular feature puts the triangle family in a special, privileged position.

order 0 order 1 order 2 . 3 order 3 JC

Figure 13 The Pascal triangle (Linear expansion shaded - 3 terms)

However, once again, a direct generation of shape functions will be preferred - and indeed will be shown to be particular easy.

Before proceeding further it is convenient to define a special set of normalised coordinates for a triangle.

(b) Area coordinates

Whilst Cartesian directions parallel to the sides of a rectangle were a natural choice for that shape, in the triangle these are not convenient.

L i =0.25

Figure 14 Area co-ordinates

A convenient set of co-ordinates, L\, La, and L3 for a triangle 1,2,3 (Figure 14), can be

defined by the following linear relation between these and the Cartesian system:

x = Ljjq + L 2x2 + L3x 3 = Lix i

y = Llyi + L2y 2 + L 3y 3 = L iy i (6 6)

Notice that among L\, L2, and L 3 , only two of them are independent according to the third equation, and to every set L\, L2, L 3 , corresponds a unique set of Cartesian co­ ordinates. At point 1, x must be equal to jci, and y must be equal to y2, this requires

Lj=l and L2=L^=0, etc.. A linear relation between the new and cartesian coordinates implies that contours of L\ are equally placed straight lines parallel to side 2-3 on which Li=0, etc. as shown in Figure 14.

It is easy to see that there exists an equivalent definition of the coordinate L\, L2, P3 of

a point P by ratios of the areas of the triangles AP23, A PI3, and A PI2 to that of the total triangle A123:

areaAP23 _ area API 3 _ area API 2

U = ---, L2 = , L3 =--- (67)

area A123 area A123 area A123

That is why they are named area co-ordinates.

Solving Eq. (6 6) for x and y gives

a] +b]x + c] y a2 +b2x + c2y (68) a3 +b2x + c3y in which d = ~ d e tl x 2 y 2 =area A123 1 x3 y 3 (69) and ^ l [ 2 , 3 ] — -X' 2 [ 3 , l ] 3 ; 3 [ l , 2 ] X 3 [ l , 2 ] 3 ; 2 [ 3 , l ] ^ l [ 2 , 3 ] — 3 ^ 2 [ 3 , 1 ] “ 3 ^ 3 [ l , 2 ] (70) C ] [ 2 , 3 ] — ^ [ l ^ ] X 2 [ 3 , l ]

with cyclic rotation of indices 1,2, and 3 shown in the square brackets.

In the case of linear triangular element, the shape functions are simply the area coordinates:

N\(x,y) = Li N2(x,y) = L2 N3(x,y)= L3 (71)

This is obvious as each individually gives unity at one node, zero at others, i.e.,

AT,. (/>J=L,(/>y) = 8 i,,/.y = 1,2,3 (72)

and varies linearly everywhere. Here Pj=(jCj,yj).

it should be emphasised that for every triangle the formulation of the above shape functions (71) apply - that is the essence in the finite elements methods.