An Optimization Method Based On B-spline Shape Functions & the Knot Insertion Algorithm

Anewmethodispresented todealwith shapeoptimizationproblems. Inthismethod,thege- ometry is parameterized by B-spline shape functions withthecontrolpointsoftheB-splinecurvesbecom- ing the design variables in the optimization scheme. Thecoreideaofthemethodpresentedistointroduce the knot insertion algorithm which can keep the ge- ometry unchanged whilst increasing the number of control points. Besides this core idea, the super- reduced method and mesh reflnement are also em- ployed. Thesuper-reducedmethodisusedtogetrid oftheequalityconstraints. Henceaconstrainedopti- mization problem is converted into an unconstrained optimization problem with the constraints kept un- changed. We apply the method to two applications; flrstaprobleminvolvingPoisson'sequation,andsec- ond,anapplicationofthemethodtoanairfoildesign problem based on the airfoil NACA 0012. In both of these applications, the new method is shown to be e-cient compared with 'standard' methods. In particular, in the airfoil problem we obtain a CPU saving time of about 42% compared with the EX- TREM method. Keywords: B-spline, knot insertion algorithm, mesh reflnement, optimization