Coupled Boundary Element/Finite Element Analysis - The Boundary Element Method with Programming - page 433

An example of a partially coupled analysis is shown in Figure 16.4 (b). Here we
consider the additional effect of a ground surface and another (existing) excavation. In a
partially coupled analysis we first solve the problem with the interface nodes fixed and
obtain an interface traction vector ^` 0

c t . Then we compute the pseudo stiffness matrix of
the region. Before we assemble our finite element system, both ^` 0

c t and K BE have to be
pre-multiplied with N, yielding a nodal point force vector as well as a stiffness matrix.
The only additional programming required for the implementation of a coupled
analysis capability is the assembly of transformation matrix N and the pre-multiplication
of the stiffness matrix K BE and, in the case of a partially coupled analysis, the traction
vector t BE with this matrix. If required, a “symmetrisation” procedure may be applied as
explained above.
We develop a function Mtrans which returns the transformation matrix N , an array of
dimension Ndofsc x Ndofsc, where Ndofsc is the number of interface degrees of
freedom. The input parameters of this function are number of interface elements, number
of interface nodes, incidence vector for each element and coordinates of interface nodes.

FUNCTION MTrans(Nelc,Ndofsc,xPc,Incic)
!-----------------------------------------
! Function returns the assembled matrix N
! for the conversion of a pseudo stiffnes matrix
! into a true stiffness matrix
!-----------------------------------------
INTEGER, INTENT (IN):: Nelc ! No. of interface elements
INTEGER, INTENT (IN):: Ndofsc ! No. of interface nodes
REAL, INTENT (IN) :: xPc(:,:) ! Coords of interface nodes
INTEGER, INTENT (IN):: Incic(:,:) ! Incidences of interface elem
REAL :: Mtrans(Ndofsc,Ndofsc) ! Function returns array
REAL :: MMjn(Ndof,Ndof)
REAL :: Glcor(2),Wi(2),Wie(2),Ni(Nodel),Elcor(Cdim,Nodel)
REAL :: xsi,eta,Jac,Weit,Mjn
INTEGER :: Inci(nodel)
Mtrans= 0.
ldim= Cdim - 1
Mi= 2 ; Ki= 1 ; Wie=1.0
CALL Gauss_coor(Glcor,Wi,2) ! 2x2 integration

IF (Cdim == 3) THEN
Ki=2
Wie= Wi
END IF
Interface_elements: &
DO Nel= 1,Nelc
Inci(:)= Incic(nel,:)
Elcor(:,:)= xPc(:,Inci(:))
Nodes_of_elem1: &
DO j=1,nodel
Nodes_of_elem2: &
DO n=1,nodel
Mjn= 0.