3.8  Subroutine L3ALC

In this section the subroutine L3ALC is described. The subroutine computes the discrete form of the axisymmetric three-dimensional Laplace integral operators. Details of the methods employed by the subroutine are given. The subroutine is used to compute the discrete operators and the method outlined in Subsection 3.5.2 is applied. In the test problem the boundary is a sphere with a Neumann boundary condition and results are given.

3.8.1  Background Methods

The regular integrals that arise are approximated by a two-dimensional quadrature rule defined on a rectangle which is specified in the parameter list to the subroutine. These integrals can be approximated using a Gauss-Legendre rule in the generator and q directions. The non-regular integrals that arise in the formula are computed by the following methods.

The M0 and M0t operators have regular kernels, hence the aim is to find expressions for the following:

{ L0
e'
 
}DG'(p) =


DG' 
 G0(p,q)  dSq  ,
(47)

{ N0
e'
 
}DG'(p; vp) =
vp



DG' 
  G0
nq
(p,q)  dSq   ,
(48)
where DG' is a conical element, p DG' (though not on an edge of the element).

The integral in (47) is evaluated through dividing the integral with respect to the generator direction into two parts at p and transforming the integral through changing the power of the variable, as introduced in Duffy [9]. The resulting regular integral on both parts is computed via the quadrature rule supplied to the routine.

The integral in (48) is evaluated by using the result that if the surface of integration in (48) is extended to enclose a three-dimensional volume then the integral vanishes. As each element is a truncated right circular cone, as illustrated in figure 3.1, a 45o right circular cone is added to each flat side of the element. The integrals over the two 45o cones are regular and are computed by a composite rule based on the quadrature rule supplied to the subroutine. The solution is thus equal to minus the sum of the integrals over the two 45o cones.

3.8.2  Subroutine L3ALC

In this section the subroutine L3ALC is introduced and demonstrated through the test problem L3ALC_T .

3.8.3  Test Problem and Results

This program in file L3ALC.FOR is a test for the subroutine L3ALC. The program computes the solution of the Laplace problem exterior to a sphere centred at the origin and with an axisymmetric solution via the integral equation (M+I/2)f = L v. The boundary condition and solution are assumed to be independent of theta (ie axisymmetric).

In order to use L3ALC , the sphere is approximated by a set of conical elements. Each element is decribed by a straight line on the generator (R-z plane) swept through 2 p (in the theta direction). The boundary functions are approximated by a constant at the centre of each element.

During execution the program gives the solution at the collocation points (the points at the centre of each element) and the solution at the selected interior points. The program also give the exact solution at the same points so that computed and exact solutions may be compared.

In L3ALC_T.FOR the surface is a sphere of unit radius, centred at the origin. The sphere is approximated by 16 elements. The analytic problem that is considered is f(p) = p3. The computed and exact results are compared at the collocation points in the output to the program. The solutions are also compared at (0,0,0.5); the exact solution is 0.5, the computed solution is 0.501790 to six decimal places.


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