Civil Engineering Reference
In-Depth Information
UP= U(r,ko,Cdim) ; TP= T(r,dxr,Vnorm,Cdim) ! Potential
ELSE
UP= UK(dxr,r,E,ny,Cdim) ; TP= TK(dxr,r,Vnorm,ny,Cdim)
END IF
Direction_P1: DO
ii=1,Ndof
IF
(Isym == 0)
THEN
iD= Ndof*(i-1) + ii ! line number in array
ELSE
iD= Ndest(i,ii) ! line number in array
END IF
IF
(id == 0)
CYCLE
Direction_Q1: DO
jj=1,Ndof
Node_points1: DO
n=1,Nodel
nD= Ndof*(n-1) + jj ! column number in array
dUe(iD,nD)= dUe(iD,nD) + Ni(n)*UP(ii,jj)*Jac*Weit
IF
(Inci(n) /= i)
THEN
! diagonal elements of dTe
dTe(iD,nD)= dTe(iD,nD) + Ni(n)*TP(ii,jj)*Jac*Weit
END IF
END DO Node_points1
END DO Direction_Q1
END DO Direction_P1
END DO Gauss_points_eta1
END DO Gauss_points_xsi1
END DO Triangles
END DO Colloc_points1
RETURN
END SUBROUTINE Integ3
6.4
CONCLUSIONS
In this chapter we have discussed in some detail, numerical methods which can be used
to perform the integration of Kernel-shape function products over boundary elements.
Because of the nature of these functions, special integration schemes had to be devised,
so that the precision of integration is similar for all locations of
P
i
relative to the
boundary element over which the integration is carried out. If this is not taken into
consideration, results obtained from a BEM analysis will be in error and, in extreme
cases, meaningless.
The number of integration points which has to be used to obtain a given precision of
integration is not easy to determine. Whereas error estimates have been worked out by
several researchers based on mathematical theory, so far they are only applicable to
regular meshes and not to isoparametric elements of arbitrary curved shape. The scheme
proposed here for working out the number of integration points has been developed on a
semi-empirical basis, but has been found to work well.
We have now developed a library of subroutines which we will need for the writing
of a general purpose computer program. All that is needed is the assembly of coefficient
matrices from element contributions, to specify the boundary conditions and to solve the
system of equations.
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