Towards adaptation of the NURBS weights in shape optimization

Bézier based parametrisations in shape optimization have the drawback of using high degree polynomials to draw more complex shapes.  To overcome this drawback, Non-Uniform Rational B-Splines (NURBS) are usually used.  But, by considering the NURBS weights, in addition to the locations of the control points, as optimization variables, the dimension of the problem greatly increases and this would make the optimization process stiffer.  In this work we propose, then, an algorithm to adapt the weights of NURBS in the parametrization of shape optimization problems.  Unlike the coordinates of the control points, the weights are not considered, in this case, as variables of the optimization process.  From the knowledge of an approximate optimal shape, we consider the set of all NURBS parametrizations of the same degree that approximate the shape in the sense of least squares.  Then, we elect the parametrization associated with the most regular control polygon (least length of the  control polygon).  Numerical results show that the adaptive parametrization  improves  the performance of the optimization process.