Robust Optimization and Tailoring of Scatter in Metal Forming Processes

Metal forming is the process of deforming metals into desired shapes. To obtain a specific shape, process settings must be adjusted. Decades ago, analytical approximations and trial-and-error methods were used to find appropriate process settings, which was time-consuming and costly. Availability of computers for numerical calculations opened a new horizon for searching for an optimal process setting. Computer simulations replaced the costly experiments, and optimization algorithms were programmed to find the optimal process design efficiently. In a forming process, there are many sources of disturbance such as variation in material properties, forming temperature, and thickness. Those noise variables are either out of control or costly to control and they lead to variations in the shape of the product. The challenge is to obtain an accurate shape and reduce its variation. It is performed by adjusting the parameters that can be controlled (design variables). A class of optimization techniques that is used to reduce the sensitivity of output to the input is known as robust optimization. As the simulations of metal forming processes are costly and the computational sources are usually limited, an approximate model of the process, a metamodel, is used to describe the relation between the input and responses. The simulations are then performed only on specific combinations of parameters (design of experiments). Then the optimization algorithm searches for an optimum design at which the process has the least sensitivity to disturbances. This approach is referred to as metamodel-based robust optimization. Metamodel-based robust optimization comprises many building blocks and the accuracy and efficiency of the method depends on the choices in each step. In this thesis, an analytical approach is presented to speeding up the calculation of the statistics of a response distribution determined by a Kriging metamodel or Gaussian radial basis function networks. It also includes the calculation of the gradient of the mean and the standard deviation of the response, and the uncertainty of the objective function value. This method is validated by comparing the results with that of the Monte Carlo method. Moreover, the significance of the analytical evaluation of uncertainty of the objective function value is shown during the sequential improvement of the metamodel. The results confirm that the robust optimum can be achieved accurately with less computational effort than when using Monte Carlo. A general assumption in robust optimization is that all inputs and outputs follow a normal probability distribution. An investigation is made of how non-normality of input and the propagation of a normal input via non-linear models lead to non-normal response. In this case, the objective function and constraints for robust optimization are redefined based on a reliability level. For this purpose, two metal forming processes are investigated. Stretch-bending a dual-phase steel sheet and forming an automotive component (B-pillar) were optimized considering non-normality of input and response. It is demonstrated that by accounting for non-normal input and response a higher reliability is achieved than when considering a normal distribution. Performing robust optimization allows the minimum variation of response around the target mean to be achieved. This is referred to as a forward problem and can be inverted. In a first scenario, if the minimum variation of response does not have a satisfactory level and further reduction of variation is required, the tolerance for the noise variables must be tightened. Since it is expensive to suppress all noise variables, the cheapest combination of tolerances for noise is preferred while the response is within a specified tolerance. In a second scenario, if the variation already meets the tolerances, a cheaper process is obtained by allowing greater noise. Based on these two scenarios, a new method is developed to determine the acceptable material and process scatter from the specified product tolerance by inverse robust optimization. This problem is referred to as tailoring of scatter. A gradient-based approach is used based on the analytical evaluation of the characteristics of output distribution to solve the inverse problem. As the evaluation of the robust optimum is computationally affordable using the analytical approach, the inverse analysis is also efficient. Tailoring of scatter in forming the B-pillar is performed based on the proposed approach. This leads to Pareto fronts which show the optimal adjustment of the tolerance for each noise variable such that the specified output tolerance is met. This method is used successfully to obtain the cheapest combinations of tolerances for noise variables to meet the required quality of the process.