Numerical shape optimisation in blow moulding

Blow moulding is a popular manufacturing process for the production of plastic and glass containers, e.g. bottles, jars, jerrycans. In a blow moulding process a so-called preform of molten material is brought into a mould and subsequently inflated with air as to take the mould shape. Blow moulding processes typically vary in the way the preform is produced and brought into the mould. The stretch blow moulding process is a variation of the blow moulding process in which the preform is simultaneously inflated with air and stretched with a stretch rod. A two-dimensional axial-symmetrical blow moulding simulation model is developed. The numerical simulation model is based on Finite Element Methods and uses Level Set Methods to track the moving interfaces between the melt and air. Level Set Methods mark the location of the interfaces implicitly by a so-called level set function and therefore do not require re-meshing of the finite element mesh. The e??ciency of the simulation model is illustrated by applying it to the stretch blow moulding of a plastic water bottle and the blow moulding of a glass beer bottle. The model is validated by means of volume conservation and comparison with data provided by industry. Two mathematical problems are considered in blow moulding. The forward problem is to find the final container that is blow moulded from a given preform under certain operating conditions. In practice often a container with a certain wall thickness distribution is desired. Then the corresponding initial operating conditions, such as the shape of the preform and the initial temperature distribution, are sought in order to produce a container with exactly this thickness distribution. In this case the inverse problem is considered, to find the shape of the preform, given a designed container, such that the container can be blow moulded from the preform. The solvability and sensitivity of the inverse problem are analysed. It is shown that under some circumstances the melt-air interfaces can reach a force equilibrium state during blow moulding. Consequently, constraints on the mould surface and process time are necessary so that the inverse problem is solvable and not excessively sensitive to perturbations in the shape. The sensitivity of the inverse problem with respect to perturbations in the shape can be estimated by means of an approximation of the melt flow. Numerical shape optimisation is used to find a solution of the inverse problem. The optimisation method describes the unknown preform surface by a parametric curve, e.g. spline, Bezi´er curve, and computes the optimal positions of the control points of the curve as to minimise the objective function. The objective function represents the distance between the inner surface of the computed container, which is the solution of the forward problem for the approximate preform, and the inner surface of the designed container. Gradient-based optimisation algorithms are discussed to find the optimal positions of the control points. In gradient-based optimisation information about the gradient of the objective function with respect to changes in the parameters, i.e. the positions of the control points, is used to find the optimum. However, computing the gradient is extremely computationally expensive and can form the computational overhead. Therefore, finite difference approximations of the Jacobian are combined with secant updates. An error analysis is performed to choose an optimal error tolerance for the optimisation algorithm. The optimisation methods are applied to glass blow moulding and results are compared with each other. An initial guess for the iterative optimisation algorithms is constructed by an analytical approximation of the optimum. The approximation is derived by omitting the mass flow in polar direction in spherical coordinates, so that the inverse problem can be solved analytically.