Chapter 4
The Interior Laplace Problem



The underlying problem addressed in this Chapter is that of computing the solution of Laplace's equation within a bounded domain subject to a specified boundary condition. Let the Laplace equation govern an arbitrary closed region D with boundary S, as illustrated in Figure 4.1.

Fig 4.1. The domain of the interior Laplace problem.

The aim is to solve the Laplace equation

Ñ2 f(p) = 0    (p Î D) .
The boundary condition is assumed to take a general form

a(p) f(p) + b(p) v(p) = f(p)
(49)
where a, b and f are real-valued functions defined on the boundary.

In this Chapter the application of the BEM to the interior Laplace problem is developed further so that the solution with a general boundary, boundary condition and incident field can be obtained. The subroutines LIBEM2, LIBEM3 and LIBEMA for solving the two-, three- and axisymmetric three-dimensional problems are described and demonstrated.


4.1  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.

4.1.1  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 D) ,
(50)

{ M f}S (p) + 1
2
 f(p) = { L v }S (p)     (p Î S) ,
(51)
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 (50) gives (approximations to) both f and v on the boundary S, equation (51) 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 (51) is a Fredholm integral equation of the first kind.

In general first kind equations are found to be difficult to solve and the matrices that arise in their equivalent linear systems are ill-conditioned (see [8], for example). Even though first kind equations like (50), having singular kernels, do not present the severe numerical problems that those with smooth kernels do, it is found that their solution can lead to a marginal loss of accuracy since the matrices that arise have higher condition numbers and hence magnify any numerical error. Boundary integral equation reformulations of the interior Laplace equation provide us with a selection from which the possibility of having to solve a first kind equations can be avoided.

Differentiating each term of equation (50) 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) ,
(52)
where v(p) = f(p)/ np.

Since the operators Mt and N are available through the subroutines in Section 3, then it is straightforward to base the boundary element method on equation (52). 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 (51) and (52) 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) .
(53)
The equation (53) 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.

4.1.2  Indirect Formulation

Following on from the ideas in subsection 1.6, the corresponding indirect integral equation formulations to (51) and (52) 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 D)
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 D) ,
(54)
This gives rise to the following boundary integral equation:

f(p) = {(L + (M - 1
2
 I)) s1 }S(p)   (p Î S) .
(55)

The equation (55) is only suitable for solving the Dirichlet problem since it refers to f and not v on the boundary. Differentiating equation (54) 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) ,
(56)
which can be used for the solution of the Neumann problem.

Both equations (55) and (56) 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 (55) and (56) 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) .
(57)

4.1.3  Field Modification

The potential field need not be a result of the boundary and boundary condition alone; the surface may simply act to modify an existing field. In such cases there is an incident field in the domain, termed fi, which is the field that would exist if there were no boundaries. 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 boundary element methods.

Direct formulation

In the simplest case, the equation (50) may be generalised as follows:

f(p) = fi(p)-{ M f}S (p) + { L v }S (p)    (p Î D) ;
(58)
the solution f(p) is equated to the incident field fi(p) and modified by the other terms. The boundary integral equation that arises from the formulation (58) is as follows:

{ M f}S (p) + 1
2
 f(p) = fi(p) + { L v }S (p)     (p Î S) .
The corresponding generalisation of (52) is

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

The formulation employed in subroutines LIBEM2, LIBEM3 and LIBEMA is a hybrid of these equations

{ (M + 1
2
 I + N) f}S (p; np) = fi(p) + vi(p) + { (L + (Mt - 1
2
 I) ) v }S(p; np)     (p Î S) .
(59)
The equation (59) is the generalisation of (53) and the equations are equivalent when there is no incident field (fi(p) = 0, vi(p) = 0 for all p Î D ÈS).

Indirect formulation

Generalising equations (54)-(56) to include the incident field gives rise to the following integral equations:

f(p) = fi(p) + {(L + M ) s}S(p)   (p Î D) ,
(60)

f(p) = fi(p) + {(L + (M - 1
2
 I)) s}S(p)   (p Î S) ,
(61)

v(p) = vi(p) + { (Mt + 1
2
 I + N) s1 }S (p;np)   (p Î S) .
(62)

The indirect boundary integral equation for the solution of the interior Laplace equation with the general Robin boundary condition (49) and with an incident field is as follows:

a(p) { fi(p) + (L + (M - 1
2
 I)) s1 }(p)+b(p) { vi(p)+ (Mt + 1
2
 I + N ) s1 }S (p;np) = f(p)
(63)
for (p Î S).

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