5.3  Improved Formulations

The term improved formulations is used to include the integral equation representations that were introduced in the 1960s and early 1970s for which the integral operator over which a solution was sought was always non-singular. Hence the formulations were intended to form a firm foundation for the boundary element method solution of the exterior Helmholtz equation. The general drawback with these methods is the inclusion of the Nk operator, which is very difficult to discretise. However, for the elements considered in this manual, the discrete Nk operator is available through the subroutines outlined in Chapter 3. Hence it is feasible to implement the improved methods and, because of their robustness in comparison with the methods outlined earlier, boundary element methods that result from them are implemented in the accompanying software and considered in the next Sections. In this Chapter a particular implementation of an improved method is described and this forms the basis of the subroutines HEBEM2, HEBEM3 and HEBEMA that are introduced in Section 5.5.

5.3.1  Improved Indirect Formulation

The equations for the improved indirect formulation that are directly applicable to the solution of the Dirichlet problem were apparently introduced independently by Brakhage and Werner [12], Leis [57] and Panich [66]. The formulation involves writing f(p) for p Î E as a linear sum of single- and double-layer potentials;
f(p) = { ( Lk + nMk ) sn }S(p)      (p Î E) .
(5.13)
For points on the boundary the equation becomes
f(p) = { ( Lk + n( Mk - 1
2
 I ) ) sn }S(p)      (p Î S) .
(5.14)
For a given Dirichlet boundary condition the boundary integral equation (5.14) has a unique solution provided the parameter n is such that Im(n) ¹ 0. Having computed sn(p) for p Î S, the solution in the domain can be computed through substituting it into an approximation based on (5.13).

An integral equation that is suitable for solving the exterior Neumann problem can be derived through differentiating (5.13) with respect to the outward normal to the boundary and taking the limit as p approaches the boundary at the base of the normal. This gives the following boundary integral equation:

v(p) = { ( Mkt + 1
2
 I + nNk ) sn }S(p;np)      (p Î S) ,
(5.15)
which is attributed to Kussmaul [54]. In correspondence with the earlier formulation (5.14), the equation (5.15) has a unique solution provided Im(n) ¹ 0. The solution of the Neumann problem can be obtained from finding an approximation to sn by solving the boundary integral equation (5.15). The solution in the domain can then be obtained by approximating (5.13).

For the general Robin problem the boundary condition takes the form (5.1). Substituting the expressions for f(p) and v(p) obtained earlier into the boundary condition gives the following integral equation:

{ (a(p) ( Lk + n( Mk - 1
2
 I ) ) + b(p) ( Mkt + 1
2
 I + nNk ) )sn }S(p;np) = f(p) .
(5.16)
Given a suitable Robin condition, the functions a(p), b(p) and f(p), an approximation to sn(p) for p Î S can be obtained from the numerical solution of the integral equation (5.16). The solution at points of the domain E can then be found by the substitution of the result into (5.13).

5.3.2  Improved Direct Formulation

The improved direct formulation originates in the paper by Burton and Miller [15]. The formulation is a hybrid of the elementary direct formulation (5.3) and equation that arises through differentiating that equation with respect to the normal to the boundary;
{ Nk f}S (p;np) = { ( Mkt + 1
2
 I ) v }S (p;np)     (p Î S) .

The improved direct integral equation formulation is simply a linear combination of (5.3) with this equation, giving the following:

{ (Mk - 1
2
 I + mNk ) f}S (p;np) = { ( Lk + m( Mkt + 1
2
 I ) )v }S (p;np)     (p Î S) .
(5.17)

The boundary integral equation can be used to solve both the Neumann and Dirichlet problems for the exterior Helmholtz equation; the numerical solution of (5.17) gives both functions f and v on S and approximation of (5.2) gives the solution at any point in the exterior. The integral equation (5.17) has a unique solution provided Im(m) ¹ 0 [15]. For the general Robin problem the integral equation (5.17) must be solved alongside the specified Robin condition (5.1). A method for carrying this out will be given in the next Section.

5.3.3  Scattering

When an incident Helmholtz field is modified by an obstacle the result is termed the scattered field. The concept of scattering is analagous to that of field modification considered in the previous Chapter, and the inclusion of the incident field term in the integral equation formulations is entirely similar. In scattering problems there is an incident field in the domain, termed fi, which is the field that would exist if there were no boundaries, or the free-space Helmholtz 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 Helmholtz 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 (5.13), (5.14), (5.15) as follows:

f(p) = fi(p) +{ ( Lk + nMk ) sn }S(p)      (p Î E) ,
(5.18)
f(p) = fi(p) +{ ( Lk + n( Mk + 1
2
 I ) ) sn }S(p)      (p Î S) ,
(5.19)
v(p) = vi(p) +{ ( Mkt - 1
2
 I + nNk ) sn }S(p;np )      (p Î S) 
(5.20)
where vi = [(fi)/( np)].

For a Dirichlet (Neumann) boundary condition, the solution can be found by solving (5.19) ((5.20)) to find sn and then substituting the result into (5.18) to find f in E. Substituting the expressions (5.19) and (5.20) for f and v into the equation for the more general Robin boundary condition (5.1) gives

{ ( a(p) { fi(p) + (Lk + n( Mk + 1
2
 I ) ) + b(p){ vi(p)+( Mkt - 1
2
 I + nNk ) sn }S(p; np) = f(p) .
(5.21)

Direct formulation

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

{ (Mk - 1
2
 I + mNk ) f}S (p;np) = -fi(p) - mvi(p) +{ ( Lk + m( Mkt + 1
2
 I ) )v }S (p;np)
(5.22)
for (p Î S), a generalisation of equation (5.17), subject to the boundary condition (5.1). 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) +{ Mk f}S (p) -{ Lk v }S (p)     (p Î E) .
(5.23)
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