Direct Differentiation

In this approach, the equations of motion are analytically differentiated with respect to each design variable bi, i=1… Nb.

At each point in time, after an analysis step is complete, Nb sets of DAE are solved numerically to yield the required values Δ y Δ b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbd9MBZ9 gBHnharuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMC G4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4 rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9 pg0FirpepeKkFr0xfr=xfr=xb9adbaGaciGadmWaamaaciGaaqqace qbcaGcbaWaaSaaaeaacqqHuoarcaWG5baabaGaeuiLdqKaamOyaaaa aaa@3FD6@ . This approach is quite robust and is used when the numbers of design variables Nb are not too large. As the number of design variables increase, the cost of the solution also increases.