Boundary Element Methods–Numerical Implementation - The Boundary Element Method with Programming - page 142

whereas the second part is integrated with normal Gauss Quadrature. The Jacobian
[ [ w can be easily obtained by differentiation of equations (6.36) and (6.37). The

second integral in (6.38) can be evaluated using normal Gauss Quadrature. For quadratic
elements, the substitution for r in terms of [ is more complicated. One may
approximately substitute r= a [ where a is the length of a straight line between the end
nodes of the element. This should give a small error for elements which are nearly
straight. A more accurate computation r as a function of [ is presented by Eberwien 7 .
A SUBROUTINE which provides the coordinates and weights for a Gauss Laguerre
integration is given below.
SUBROUTINE Gauss_Laguerre_coor(Cor,Wi,Intord)
!------------------------------------
! Returns Gauss_Laguerre coordinates and Weights
! for 1 to 4 Gauss points
!------------------------------------
IMPLICIT NONE
REAL, INTENT(OUT) :: Cor(8) ! Gauss point coordinate
REAL, INTENT(OUT) :: Wi(8) ! weigths
INTEGER,INTENT(IN) :: Intord ! integration order
SELECT CASE (Intord)
CASE (1)
Cor(1)= 0.5 ; Wi(1) = 1.0
CASE (2)
Cor(1)= .112008806 ; Cor(2)=.602276908
Wi(1) = .718539319 ; Wi(2) =.281460680
CASE (3)
Cor(1)= .063890793 ; Cor(2)= .368997063 ; Cor(3)= .766880303
Wi(1) = .513404552 ; Wi(2) = .391980041 ; Wi(3) =.0946154065
CASE (4)
Cor(1)= .0414484801 ; Cor(2)=.245274914 ; Cor(3)=.556165453
Cor(4)= .848982394
Wi(1) = .383464068 ; Wi(2) =.386875317 ; Wi(3) =.190435126
Wi(4) = .0392254871
CASE DEFAULT
CALL Error_Message('Gauss points not in range 1-8')
END SELECT
END SUBROUTINE

6.3.5 Subdivision of region of integration

In some cases, when point P i is near the element, the number of Gauss points required
will exceed 4 in table 6.1. In this case it is necessary to subdivide the element into sub
regions of integration. A simple approach is to subdivide the element into equal
subdivisions depending on the value of R/L. If according to the R/L value the maximum
number of Gauss points available is exceeded, the element is subdivided into K regions
where