Publication Date


Document Type


Committee Members

Ramana Grandhi (Advisor)

Degree Name

Doctor of Philosophy (PhD)


Metal forming is a process that transforms a simple shape of a workpiece into a predetermined complex shape through the application of compressive/tensile forces exerted by dies. In the design of a forming process, the only factors that are known are the final component shape and the material with which it is to be made. Then the engineer has to design a process to make defect-free product, subject to limitations of shape, material properties, cost, time, and other such factors. The design cycle can be enhanced if performance sensitivity information is available that could be used with any commercially available finite element software. Hence, this research investigates the analytical continuum-based sensitivity analysis method using boundary integral and material derivative formulations. Sensitivity derivation starts by obtaining an identity integral for the non-linear deformation process. Then the adjoint problem is introduced to obtain an explicit expression for the sensitivity of the objective and constraint functions. The applicability of sensitivity analysis is demonstrated through a steady-state metal forming process. In conventional optimization all the parameters are considered as deterministic and constant. However, in practice, they are prone to various uncertainties such as variations in billet geometry, die temperature, material properties, workpiece and forming equipment positional errors, and process parameters. A combination of these uncertainties could induce heavy manufacturing losses through premature die failure, final part geometric distortion, and production risk. Identifying, quantifying, and controlling the uncertainties will reduce risk in the manufacturing environment and will minimize the overall cost of production. Hence, in this research, a robust design methodology is developed by considering the randomness in the parameters. The developed methodology is applied for die shape optimization of an axisymmetric extrusion. Die angle and spline through points are the design variables; friction factor and ram velocity are considered as random parameters. The optimization problem is formulated to minimize the exit velocity variance by placing constraints on average strain and variance. Further, the solutions of reliability-based optimization are compared with deterministic-based optimization solutions. The results herein indicate that the robust design solution gives better product quality and reduces the total exit velocity variance.

Page Count


Department or Program

Department of Mechanical and Materials Engineering

Year Degree Awarded