Chapter 5
The Exterior Laplace Problem



This chapter addresses the problem of solving the Laplace equation exterior to a closed surface or surfaces and subject to a boundary condition and to a specified incident field. Let the domain of the Laplace's equation be the region E exterior to the closed boundary S, as illustrated in Figure 5.1.
Fig 5.1. The domain of the exterior Laplace problem

The problem is equivalent to the solution of the Laplace equation

Ñ2 f(p) = 0    (p Î E)
in the domain E. The boundary condition is assumed to take a general form

a(p) f(p) + b(p) v(p) = f(p)    (p Î S)
(70)
where a, b and f are real-valued functions defined on the boundary.

5.1  Integral Equation Formulation

In this Section we consider the integral equation fomulations of the interior Laplace equation. An incident field along with a general boundary condition (49) is included and this leads to more generalised boundary integral equations.

5.1.1  Direct Formulation

The application of Green's second theorem to the Laplace equation gives the following equations:

{ M f}S (p) - f(p) = { L v }S (p)    (p Î E) ,
(71)

{ M f}S (p) - 1
2
 f(p) = { L v }S (p)     (p Î S) ,
(72)
where the Laplace integral operators, L and M, are defined in Section 2.1 and v(p) = [(f)/( n)]. Note that the normals to the boundary are taken to be in the outward direction.

The above equations can be utilised to solve the interior Laplace equation in the manner outlined in Chapter 1; equation (71) gives (approximations to) both f and v on the boundary S, equation (72) yields an approximation to f(p) for any point p in the domain. There is only one small difficulty with this approach and that is in the case of a Dirichlet boundary condition equation (72) is a Fredholm integral equation of the first kind, which should generally be avoided if possible (as discussed in the previous chapter).

Differentiating each term of equation (71) with respect to any vector v(p) gives

up
 { M f}S (p) - f(p)
up
 =
up
 { L v }S (p)    (p Î D) ,
or

{ N f}S (p;up) - f(p)
up
 = { Mt v }S (p;up)    (p Î D) ,
using the notation of Section 2.1.

By taking the limit as the point p approaches the boundary with the vector up being the unit outward normal to the boundary at p (that is np), and taking into account the jump properties of Section 3.1 the following boundary integral equation is obtained:

{ N f}S (p; np) = { (Mt + 1
2
 I) v }S (p; np) (p)    (p Î S) ,
(73)
where v(p) = f(p)/ np.

Since the operators Mt and N are available through the subroutines in Chapter 3, then it is straightforward to base the boundary element method on equation (73). However, for the Neumann problem we need to solve over the hypersingular operator N, which can lead to some loss of accuracy, similar to that experienced with the solution of the first kind equation discussed earlier.

Since neither of equations (72) and (73) are universally acceptable for solving the interior Laplace equation for all boundary conditions of the form (49), a hybrid equation is proposed that couples the two original equations into a single equation

{ (M - 1
2
 I + N) f}S (p; np) = { (L + (Mt + 1
2
 I) ) v }(p; np)     (p Î S) .
(74)
The equation (74) provides a suitable basis of a method for the solution of interior Laplace equation for all boundary conditions and it is the equation employed in the subroutines associated with this Chapter.

5.1.2  Indirect Formulation

Following on from the ideas in subsection 1.6, the corresponding indirect integral equation formulations to (72) and (73) can be obtained by writing f as a single or double layer potential;

f(p) = {L s0 }S(p)    or    f(p) = {M s¥ }S(p)   (p Î E)
where the s0 and s¥ are source density functions defined on S. For points on the boundary the equations become boundary integral equations;

f(p) = {L s0 }S(p)    or    f(p) = {(M+ 1
2
 I) s¥ }S(p)   (p Î S)
where the jump condition of Section 2.1 has been taken into account in the second equation.

The integral equations arrived at in this way have the same difficulties as the corresponding direct equation (51); the Dirichlet problem is replaced by a first kind equation. Again the problem can be circumvented by using a hybrid formulation; writing f as a weighted sum of single and double layer potentials

f(p) = {(L + M) s1 }S(p)   (p Î E) ,
(75)
This gives rise to the following boundary integral equation:

f(p) = {(L + (M + 1
2
 I)) s1 }S(p)   (p Î S) .
(76)

The equation (76) is only suitable for solving the Dirichlet problem since it refers to f and not v on the boundary. Differentiating equation (75) with respect to np and taking the limit as the point p approaches a point on the boundary gives the following boundary integral equation:

v(p) = { (Mt - 1
2
 I + N ) s1 }S (p;np)   (p Î S) ,
(77)
which can be used for the solution of the Neumann problem.

Both equations (76) and (77) are required in the indirect solution of the Laplace equation with a Robin boundary condition. In this case the relevant integral equation is obtained through the substitution of the forms (76) and (77) into the boundary condition (49) to give

a(p) {(L + (M + 1
2
 I)) s1 }S(p)+b(p) { (Mt - 1
2
 I + N ) s1 }S (p;np) = f(p)   (p Î S) .
(78)

5.1.3  Field Modification

In this generalisation problems there may be an incident field in the domain, termed fi, which is the field that would exist if there were no boundaries, or the free-space Laplace field. Such problems can also be solved by the boundary element method, it only requires a generalisation of the integral equations and the corresponding alteration of the resulting boundary element methods.

To complete this Section the improved integral reformulations of the Laplace equation are generalised. The improved methods based on these formulations are given in the next Section. The generalisation of the elementary methods and the Schenck method in order to include the scattering term is not explicitly carried out in this text, although the development of such methods should be clear from the formulations given.

Indirect Formulation

The inclusion of the scattering term generalises the integral equations (76), (77), (78) as follows:

f(p) = fi(p) +{ ( L + M ) s1 }S(p)      (p Î E) ,
(79)

f(p) = fi(p) +{ ( L + ( M + 1
2
 I ) ) s1 }S(p)      (p Î S) ,
(80)

v(p) = vi(p) +{ ( Mt - 1
2
 I + N ) s1 }S(p;np )      (p Î S) 
(81)
where vi = fi/ np.

For a Dirichlet (Neumann) boundary condition, the solution can be found by solving (80) ((81)) to find s and then substituting the result into (79) to find f in E. Substituting the expressions (80) and (81) for f and v into the equation for the more general Robin boundary condition (70) gives

{ ( a(p) { fi(p) + (L + ( M + 1
2
 I ) ) + b(p){ vi(p)+( Mt - 1
2
 I + Nk ) s1 }S(p; np) = f(p) .
(82)

Direct Formulation

The solution of the exterior Laplace problem on the boundary S can be determined through solving the following integral equation:

{ (M - 1
2
 I + N ) f}S (p;np) = -fi(p) - vi(p) +{ ( L + ( Mt + 1
2
 I ) )v }S (p;np)
(83)
for (p Î S), a generalisation of equation (53), subject to the boundary condition (70). Once f(p) and v(p) are obtained through solving the above equation, the solution in the domain can be obtained by the integration

f(p) = fi(p) +{ M f}S (p) -{ L v }S (p)     (p Î E) .
(84)


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