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

A new method is presented to deal with shape optimization problems. In this method, the geometry is parameterized by B-spline shape functions with the control points of the B-spline curves becoming the design variables in the optimization scheme. The core idea of the method presented is to introduce the knot insertion algorithm which can keep the geometry unchanged whilst increasing the number of control points. Besides this core idea, the superreduced method and mesh refinement are also employed. The super-reduced method is used to get rid of the equality constraints. Hence a constrained optimization problem is converted into an unconstrained optimization problem with the constraints kept unchanged. We apply the method to two applications; first a problem involving Poisson’s equation, and second, an application of the method to an airfoil design problem based on the airfoil NACA 0012. In both of these applications, the new method is shown to be efficient compared with ‘standard’ methods. In particular, in the airfoil problem we obtain a CPU saving time of about 42% compared with the EXTREM method.