3 Application of Collocation

In this section it is shown how collocation is applied to derive the discrete form of the integral equations. The boundary and shell are divided into uniform elements. The boundary S is divided into n_S elements DS₁, DS₂, ..., DS_{n_S}, the shell G is divided into n_G elements DG₁, DG₂, ... DG_{n_G} and the boundary functions and shell functions are approximated by a constant on each element. Let p₁, p₂, ..., p_{n_S} and q₁, q₂, ..., q_{n_G} be the collocation point with p_i Î DS_i for i=1, 2,..., n_S and q_i Î DG_i for i=1, 2, ..., n_G and each lying at the centre of the respective element. Thus we have c(p_i) = [1/2] (i = 1, 2, ..., n_S ) and c(q_i) = [1/2] (i = 1, 2, ..., n_G).

3.1 Notation

It is helpful to introduce the following notation. Define the vectors of the function values at the collocation points as follows:

j_S = [ j(q₁), j(p₂), ..., j(p_{n_S})]^T ,

d_G = [ d(p₁), d(q₂), ..., d(q_{n_G})]^T ,

vectors v_S, s_S, a_S, b_S, g_S, n_G, F_G, V_G are defined similarly.

Let the matrices L_SS, L_SG, L_GS, and L_GG be defined as follows:

[L_SS]_ij = { L e }_{DS_j} (p_i) (i=1,2,...,n_S),(j=1,2,...,n_S) ,

[L_SG]_ij = { L e }_{DG_j} (p_i) (i=1,2,...,n_S), (j=1,2,...,n_G) ,

[L_GS]_ij = { L e }_{DS_j} (q_i) (i=1,2,...,n_G), (j=1,2,...,n_S) ,

[L_GG]_ij = { L e }_{DG_j} (q_i) (i=1,2,...,n_G), (j=1,2,...,n_G)

where e is the unit function. Notation for the other integral operators (M, M^t, and N) is developed in a similar way.

3.2 Discrete form of the Integral Equations

The adoption of the notation above allows us to construct the following linear systems of approximations which are the discrete analogues the direct integral equation formulation (1-3):

[M_SS+

I_SS]j_S ť L_SSv_S+ M_SG d_G- L_SG n_G,

(10)

F_G ť -M_GSj_S+ L_GSv_S+ M_GG d_G- L_GG n_G,

(11)

V_G ť -N_GSj_S+ M^t_GSv_S+ N_GG d_G- M^t_GG n_G.

(12)

Similarly, the discrete form of the indirect integral equation formulation (5-8) is as follows:

j_S ť L_SS s_S+ M_{S G} d_G- L_{S G} n_G,

(13)

v_S ť [M^t_SS +

I_SS ] s_S+ N_{S G} d_G- M^t_{S G} n_G,

(14)

F_G ť L_GSs_S+ M_GG d_G- L_GG n_G,

(15)

V_G ť M^t_GSs_S+ N_GG d_G- M^t_GG n_G.

(16)

The boundary condition can be written in the following form

D^a_SS j_S +D^b_SS v_S = g_S,

(17)

where D^a_SS = diag(a₁, a₂, ..., a_{n_S}) and D^b_SS = diag(b₁, b₂, ..., b_{n_S}). Similar equations can be obtained for the shell condition.