The linear and the second order saw-tooth correctors are shown schematically in Figures 8 and 9. The decomposition of the second order corrector into a linear and a quadratic term is also shown in Figure 10.
Figure 8: General representation of linear saw-tooth correctors.
Figure 10: Decomposition
of a second order saw-tooth corrector into a
linear and a quadratic term.
Figure 9: General representation of second order
saw-tooth correctors.
In order to apply the saw-tooth correction, the matrix equation (20) is modified as follows:
where 
and 
are the matrices
of influence coefficients for the linear saw-tooth (with unit slope)
on each panel, and 
and 
are the matrices of influence coefficients for the quadratic terms
(with unit strength)
on each panel. The self-influence coefficients for the linear and the quadratic
terms are evaluated analytically, and their values are given in Appendices A.2
and A.3, respectively. The other coefficients are evaluated via 8-point Gauss
quadrature.
and 
are
vectors with 2N components the
derivatives (evaluated at the panel mid-points) with respect to arc-length
for [u] or [p], respectively.
and 
are
vectors with 2N components the
coefficients of the quadratic terms with respect to arclength
(evaluated at the panel mid-points)
of the functions [u] or [p], respectively.
Using a Taylor expansion around the
mid-point of each panel, and correspond to the values of
the second derivatives with respect to arc-length for [u] or [p],
respectively, divided by two.
In case only the linear correctors need to be applied, the two last terms in equation (21) are not included. Also, in the event third or higher order terms need to be included, the corresponding correctors need to be added to the right-hand side of equation (21).
Equation (21) is solved in an iterative sense.
At each iteration the matrix of the unknown displacements and tractions
(2N in total) is identical to that in the case of the low-order
method. Thus, LU decomposition and backward substitution is used
in order to determine the unknowns at each iteration.
For the first iteration (i=1)
the saw-tooth correctors are taken equal to zero, i.e. the pure low-order
BEM problem is solved. In subsequent iterations the values of
, , , and are updated by using the [u]
and [p] from the previous iteration (i-1). This is accomplished by
employing second-order accurate finite difference schemes on the values
of 
and 
(corresponding to the panel
mid-points) with respect to
arc-length. Centered, backward, or forward differencing is used, depending
on the location of the panel mid-points with respect to the corners.
In the case [u] or [p] are known, the corresponding , ,
, and , as well as the related correctors are evaluated only
once and kept the same throughout the iterative process. A convergence criterion
of 
relative error in the evaluated maximum deflection, is usually
imposed. Unlike the case of potential flow BEM around a hydrofoil, described
earlier, where even the linear saw-tooth correction method seemed to converge
quickly with number of panels (2-3 iterations), in the case of structural BEM
the convergence is achieved in sometimes more than 100 iterations.
This is primarily due to the fact that our first iteration in the
iterative solution is often a ``very bad initial guess'', especially
in the case of very thin hydrofoils. A discussion on the improvement of
the convergence is given in a later section.
