Keywords: sensitivity, adjoint variable, optimization, elastodynamics, thermal.

This paper is concerned with the application of a recently established methodology, for the determination of sensitivity expressions [1], to thermo-mechanical problems arising in pressure die casting. A principal objective in pressure die casting is the optimization of die designs for the rapid production of castings with enhanced surface finish. Reported in the paper is progress made in achieving this objective with the establishment of a procedure for thermal and vibration enhancement of pressure die casting dies.

Sensitivity determination for the material derivative adjoint-variable method typically involves complex mathematical manipulation despite the relative simplicity of the derived sensitivity expressions. It is demonstrated that sensitivities for thermal and mechanical systems can be determined via a two-stage process that firstly requires the determination of the governing adjoint system of equations and associated boundary conditions by means of shape-invariant variations only. The establishment of an adjoint equation permits the immediate neglect of partial derivatives, of the primary and adjoint variables with respect to the design variables, without need for any algebraic manipulation or application of integral theorems. This results in design sensitivities being obtained with minimal manipulation.

The pressure die casting process involves die designs incorporating cooling channels positioned to facilitate the controlled extraction of energy from a solidifying casting. It is known that subcooled nucleate boiling can occur in the cooling channels, which can possibly be enhanced with appropriate cooling channel designs. A particular design objective is the maximization of boiling heat transfer with some degree of constraint on spatial temperature variation over the die cavity surface. Shape sensitivity analysis is applied to a boundary element model using the material derivative adjoint variable technique. Mesh node positions on the cooling channels are used as the design parameters. The sensitivities are used in a conjugate gradient non-linear optimisation routine. It is shown that with this approach cooling channels can be designed to maximise boiling heat transfer.

One of the failings of die casting process is that variable surface quality is a common result. This can reduce the competitiveness of these processes because secondary finishing operations are required, i.e. machining, polishing, plating, etc. Dies that are producing poor surface finishes cannot, at the present time be substituted as it is not known what changes have to be made to give the necessary improvements. However, it has been recently established that die vibration can be used to polish surfaces as they form during solidification [2]. Unfortunately the vibration patterns and the thermal conditions necessary to enhance surface finish are generally not present in existing die designs. However, in principle, it is feasible with the employment of strategically located piezoelectric translators to modify vibration patterns. The positioning of translators requires optimisation and hence sensitivity determination.

Three test problems are presented in the paper to illustrate the benefits of the proposed methodology. The first problem is selected to highlight how the method can be used to obtain the sensitivity of a die subjected to variations in shape. In the second case the sensitivities are determined for a die subjected to simultaneous changes in both cooling channel shape and non-linear boiling boundary conditions. Appropriate design variables are chosen for this die and its implementation by means of the boundary element method is shown. Results of this implementation are reported, which highlight the rapid convergence of the method.

1

Davey, K. and Clark, L.D., "Sensitivity and Optimisation for Shape and Non-linear Boundary Conditions in Thermal Boundary Elements", International Journal for Numerical Methods in Engineering, Vol. 56, pp. 553-587, 2003. doi:10.1002/nme.577

2

Alonso Rasgado, M.T. and Davey, K., "An Experimental Investigation on the Effect of Vibration on Casting Surface Finish", Journal of Manufacturing Sciences and Engineering, Trans. ASME, Vol. 124, pp. 435-443, 2002. doi:10.1115/1.1459085

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