## 3.7  Special Numerical Integration Methods

Special methods are required to evaluate the singular integrals that arise in the discrete Lk and Nk operators when the collocation point lies on the element.

### 3.7.1  Two-dimensional problems

The regular integrals that arise are approximated by a standard quadrature rule such as a Gauss-Legendre rule which is specified in the parameter list to the subroutines. Tables of Gauss-Legendre rules are given in Stroud and Secrest [79] and can also generated from the NAG library [63], for example. The non-regular integrals that arise in the formulae (3.36) and (3.37) are computed via the following methods. See Jaswon and Symm [34] and Kirkup [40] for the background to these methods.

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

 { L0 ~e }DG*(p) = óõDG* G0(p,q)   dSq  ,
(3.38)
 { N0 ~e }DG*(p; np) = ¶ ¶np óõDG* ¶G0 ¶nq (p,q)  dSq   ,
(3.39)
where DG* is a straight line panel, p Î DG* (though not on an edge or corner of the panel). Let it be assumed that the panel DG* has length a+b with q = q(x) and p = q(3.32) for x Î [-a,b]. This gives the following formulae for (3.38) and (3.40):
 { L0 ~e }DG*(p) = 1 2 p [ a + b - a loga -b logb ] ,
(3.40)
 { N0 ~e }DG*(p; np) = - 1 2 p [ 1 a + 1 b ] .
(3.41)

### 3.7.2  Three-dimensional problems

The regular integrals that arise are approximated by a quadrature rule defined on a triangle. Laursen and Gellert [56] contains a selection of Gauss-Legendre quadrature rules for the standard triangle. The non-regular integrals that from discretising the Lk and Nk operators are computed by the following methods. See Jaswon and Symm [34], Terai [80], Banerjee and Butterfield [7] and Kirkup [40] for the background to these methods.

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

 { L0 ~e }DG*(p) = óõDG* G0(p,q)  dSq  ,
(3.42)
 { N0 ~e }DG*(p; np) = ¶ ¶np óõDG* ¶G0 ¶nq (p,q)  dSq   ,
(3.43)
where DG* is a planar triangular panel, p Î DG* (though not on an edge or corner of the panel). Let R(q) be the distance from p to the edge of the panel for q Î [0, 2 p], as illustrated in Figure 3.1.

Fig 3.1. Polar integration on the planar triangle panel.

The integrals (3.42) and (3.43) may be written in the form:

 { L0 ~e }DG* (p) = 1 4 p óõ 2 p 0 R(q) d q ,
 { N0 ~e }DG* (p; up) = - 1 4 p óõ 2 p 0 1 R(q) d q .
In order to evaluate the integrals, the triangular panel DG* is divided into three \triangle1, \triangle2 and \triangle3 by joining the point p to the vertices. The resulting triangles have the form of Figure 3.2.

Fig 3.2. Division of the planar triangle panel.

After some elementary analysis, we obtain

 {L0 ~e }DS* (p) = å \triangle1, \triangle2, \triangle3 1 4 p R(0) sinB( logtan( B+A 2 ) - logtan B 2 )   and
 {N0 ~e }DS* (p; up) = å \triangle1, \triangle2, \triangle3 1 4 p cos(B+A) - cosB R(0) sinB .

### 3.7.3  Axisymmetric three-dimensional problems

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) = óõDGol>G* G0(p,q)  dSq  ,
(3.44)
 { N0 ~e }DG*(p; np) = ¶ ¶np óõDG* ¶G0 ¶nq (p,q)  dSq   ,
(3.45)
where DG* is a conical shell panel, p Î DG* (though not on an edge of the panel).

The integral in (3.44) 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 in line with a method described in references [25] and [40]. The resulting regular integral on both parts is computed via the quadrature rule supplied to the routine.

The integral in (3.45) is evaluated by using the result that if the surface of integration in (3.45) is extended to enclose a three-dimensional volume then the integral vanishes (see [40]). As each panel is a truncated right circular cone shell a 45° right circular cone is added to each flat side of the panel. The integrals over the two 45° cones are regular and are computed by a composite rule based on the quadrature 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 45° cones.