A new method using b-splines as shape functions and the knot insertion algorithm for shape optimization - Xu, Bo (Cranfield University, 2007)
A new method is developed to deal with shape optimization problems. The core idea of the method is to introduce the knot insertion algorithm which keeps the geometry unchanged while increasing the number of control points. In addition to this idea, the super-reduced idea and the mesh refinement are also employed to deal with the equality constrained optimization problem. The developed method has been tested on several applications. The first application is a Poisson equation problem. The result produced by the new method is compared with the result produced by the BFGS method because the BFGS method is considered to be one of the best methods in optimization. The result shows that the new method is more efficient than the BFGS method. The second application of the method is an airfoil design problem. The performance of using the new method is compared with the performance of using the EXTREM method in two design cases, RAE 2822 and NACA 0012. In both of these cases, the new method is much more efficient than the EXTREM method. The 3-dimensional tested case used is a mathematical problem. In this case, one finds that the discretization error is much bigger when compared with the 2-dimensional case. However the method still converges quickly to the optimum solution. As well as the above applications, the use of high order B-splines and multi-objective optimization have also been investigated. In summary, a new method is developed for shape optimization problems and validation has been carried out on several numerical examples. With the idea of the new method, it is possible to improve the efficiency of the method currently used as long as B-splines are used to describe the geometry.