Keywords: structural, shape, optimisation, NURBS, reparameterisation.

Structural Shape Optimisation (SSO), determines the geometric shape of a domain in order to optimise structural performance. Typical performance measures include stress, displacement and natural frequency but the minimisation of weight as a basis for evaluating merit is probably the most obvious [1]. In most structural optimisation problems, the inherent challenges originate from the non-linearity of the objective and constraint functions which may also be implicit functions with respect to the shape design variables [2,3].

The question of how to constrain the structural finite element model to ensure it remains feasible is a major issue because constraints also need to direct the optimiser to an acceptable optimum through their gradients. To date, the mesh nodes of the FE model have been used to specify the locations of the stress constraints as the shape evolves throughout the design space. Major gaps in the generality of this methodology are exposed when mesh refinement or full remeshing is required. The robustness of the method is called into question as the constraints after a new mesh are no longer related to the previous constraints.

This issue is tackled here through the tools offered by the mathematical description of the CAD geometry. As the geometry is defined using NURBS (Non Uniform Rational B-Spline), each stress sample point can be tracked or mapped by always evaluating its Cartesian position at the same parametric location. This is done through B-spline refinement. This technique has the very important property of convergence of the curve's control polygon to the curve. As more values are added to a knot vector of the spline and the curve refined, the control polygon for the curve approaches the actual curve in the parametric area where the new knot vector values are added. By refining a curve over its entire parametric domain, the control polygon can be made arbitrarily close to the curve everywhere. Very efficient and elegant algorithms are available for the refinement of NURBS curves. Points on NURBS curves can be evaluated by using special cases of these algorithms [4].

This surface point mapping procedure enables the provision of meaningful stress constraints to be supplied to the first order optimisation algorithm. A number of examples of the surface point mapping methodology are provided together with SSO examples which demonstrate the effectiveness of the technique.

1

Vanderplaats G., "Thirty years of modern structural optimization', Advances in Engineering Software, 16, 81-88, 1993. doi:10.1016/0965-9978(93)90052-U

2

Ding Y., "Shape optimization of structures: A literature survey", Computers & Structures, 24(6), 985-1004, 1986. doi:10.1016/0045-7949(86)90307-X

3

Pourazady M., Fu Z., "An integrated approach to structural shape optimization", Computers & Structures, 60, 279-289, 1994. doi:10.1016/0045-7949(95)60363-8

4

Bloomenthal M.D., Error Bounded Approximate Reparametrization of Non-Uniform Rational B-Spline Curves, PhD thesis, University of Utah, 1999.

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