Postprocessing is only done for geometrical design variables. The postprocessing procedure is coded in sensitivity.c and consists of the following steps:
Now the steps are treated in more detail:
N^T (b-p)=0. | (870) |
Since p belongs to the subspace it can be written as a linear combination of the basis vectors p=Nx, where x is a mx1 vector of coefficients. Consequently:
N^T N x = N^T b, | (871) |
from which x can be solved yielding:
p=N(N^T N)^{-1} N^T b. | (872) |
The complement of the projection vector is I-N(N^T N)^{-1} N^T. Denoting A=(N^T N)^{-1}, the constrained sensitivies c are obtained from the unconstrained sensitivities b by:
c=(I-NAN^T) b, | (873) |
or, in component notation:
c_i=b_i-\sum_k w_{ik}, | (874) |
where
w_{ik}=\left( \sum_j N_{ij} A_{j \underline{k}} \right) \left( \sum_l (N^T)_{\underline{k}l} b_l\right) | (875) |
(no summation over k in the last equation).
Active constraints are constraints which
To this end the algorithm starts with all constraints which are fulfilled an removes the constraints one-by-one for which the Lagrange multiplier points to the feasible part of the space.