4 The Boundary and Shell Element Method

To demonstrate the direct and indirect boundary and shell element methods, the test problem with the domain of the unit square and with a discontinuity between ([1/2],[1/2]) and ([1/2],1) is introduced. The boundary conditions are such that j(p) = 1 for 0 < p₁ < [1/2] and p₂ = 1, j(p) = -1 for [1/2] < p₁ < 1 and p₂ = 1 and v(p)=0 on the remainder of the boundary. The shell condition is such that n(p)=V(p)=0 for all points on the shell. The test problem is illustrated in figure 3.

The discrete forms of the integral operators are computed through the use of a subroutine H2LC, described in [10]. These numerical integrations are computed to sufficient accuracy so the error does not contribute significantly to the overall error in the integral equation methods.

4.1 Direct Boundary and Shell Element Method

The following linear system of equations follows from approximations (11) to (13) and (18),

é
ê
ê
ê
ê
ê
ê
ë

(M_k,SS +

I_SS )

- L_SS

0_{S G}

- M_{k,S G}

D^a_SS

D^b_SS

0_{S G}

M_k,GS

- L_k,GS

I_GG

-M_k,GG

N_k,GS

-M^t_k,GS

0_GG

-N_k,GG

ù
ú
ú
ú
ú
ú
ú
û

é
ê
ê
ê
ê
ê
ê
ê
ë

ù
ú
ú
ú
ú
ú
ú
ú
û

é
ê
ê
ê
ê
ê
ë

0_S

g_S

0_G

ù
ú
ú
ú
ú
ú
û

(19)

On solution the approximations [^(j)]_S, [^(v)]_S, [^(F)]_G, [^(d)]_S to j_S, v_S, F_G, d_S are obtained. The discrete form of equation (5) is then used to compute the solution in the domain D.

4.2 Indirect Boundary and Shell Element Method

The following linear system of equations follows from equations (14) to (17),

é
ê
ê
ê
ê
ê
ê
ê
ë

I_SS

0_SS

- L_k,SS

0_{S G}

- M_{k,S G}

0_SS

I_SS

-(M^t_k,SS +

I_SS)

0_{S G}

- N_{k,S G}

D^a_SS

D^b_SS

0_SS

0_{S G}

0_GS

- L_k,GS

I_GG

- M_k,GG

0_GS

- M^t_k,GS

0_GG

- N_k,GG

ù
ú
ú
ú
ú
ú
ú
ú
û

é
ê
ê
ê
ê
ê
ê
ê
ê
ê
ë

ù
ú
ú
ú
ú
ú
ú
ú
ú
ú
û

é
ê
ê
ê
ê
ê
ê
ë

0_S

g_S

0_G

ù
ú
ú
ú
ú
ú
ú
û

(20)

On solution the approximations [^(j)]_S, [^(v)]_S, [^(s)]_S, [^(F)]_G, [^(d)]_S to j_S, v_S, s_S, F_G, d_S are obtained. The discrete form of (10) is then used to compute the solution in the domain D.