In this first of two articles on how design of experiments (DOE) and optimization can be applied to forging, we will define the terms and illustrate how they fit forging design and process development. To maximize the return for the time and capital invested, all experiments will be performed using process simulation. The second article will show case studies of forging applications and illustrate how these tools can speed up the development process and provide insight into the required process controls for a cost-effective, robust process.
As companies continue to search for a lower cost with more robust processes, design of experiments (DOE) and optimization have continued to play a role in product and process development. These methods can be applied in a manufacturing environment in the form of shop trials, in an R&D lab with subscale experiments or on paper with any form of valid model. The methods can be used for a range of objectives, including the development of the optimum flap system for a new aircraft, the lowest drag race car or the best preform for a forging.
As forgers are sometimes all too aware, manufacturing processes are filled with challenges associated with process variables. Success with a new forge tool or production procedure depends on one’s ability to choose effective designs or processing conditions. Even once generally acceptable input targets have been chosen, the risk of quality issues increases when process conditions are hard to maintain.
This raises many questions regarding the effect of variables on a process. How sensitive is product quality to process variability? What happens if a process anomaly (outlier) occurs? In what ranges should process variables be kept in order to have success? What is the best tooling design for producing a given part? What process settings lead to defects or other production issues?
State-of-the-art process simulation systems, such as DEFORM, are bringing to market tools that automate and manage systematic studies of process variables. New developments include DOE, optimization, sensitivity studies, probabilistic modeling and statistical sampling tools. DOE and optimization methods are summarized to explain how they are used to automate systematic studies of process variables.
Design of Experiments
DOE is a systematic approach to investigate a system or process. A series of structured tests are designed, in which planned changes are made to the input variables of a process or system. The effects of these changes on a predefined output are then assessed. A powerful tool emerges when DOE is applied in a multi-operation simulation environment capable of modeling a process chain. Trade-offs can be observed between different stages of the process. Global optimums can be studied. More importantly, the response of each input can be evaluated. Finally, a subsequent optimization study can be conducted in a more well-defined range.
Sampling in a DOE is commonly performed as a full factorial study, which is one that requires results from all combinations of variables. For example, a DOE with three variables at five levels each requires 125 tests (5 x 5 x 5). If the number of full-factorial iterations becomes cumbersome, then statistical sampling methods can be used to study the response to changes with fewer tests. The key in deciding on the sampling density is the stability of the process. Stable processes can be tested with fewer sampling points than highly unstable ones. A range of sampling methods is shown in Figure 1.
A critical element of a simulation, DOE is the ability to set up and run the models efficiently. The time to set up a model with 50 or 100 simulations should be similar to setting up a single model. This includes scanning the simulation to organize and display results using statistical post-processing tools.
Optimization is an iterative method used to analyze design or process variations within a design space to determine the conditions that best satisfy an objective. A control program interrogates the response to predesigned inputs and updates them for subsequent simulations to find the “optimum” result. Optimization simulations can be run as a series of parallel simulations. Preprocessing, simulation and post-processing are common between DOE and optimization.
In a DOE study, the user specifies the DOE variables and sampling. They also specify an output and constraints. Then the user reviews the outputs to decide which sample or region best meets the requirements. In an optimization run, an objective is specified. This numerical target could be minimum volume, highest minimum strain, lowest load or energy, among others. Constraints are used to identify samples that may approach the objective but lead to quality problems. These problems might be an underfill, fold, shear band (high strain) or other behavior.
Sampling is not predefined prior to starting an optimization run. Initial runs are used to scan the range of input variables to assess initial surface response. The “next” sample is determined based on the response to the last change and current understanding of the trend. Optimization runs continue to look for improvement until the limit on the number of experiments has been reached or the optimum has been reached within defined convergence criteria. Figure 1 includes an optimization sampling plot, where the circled region was determined to be the local optimum.
There are endless ways for forgers to apply DOE and optimization to simulation. They can design optimized preform shapes and tooling progressions. Process conditions can be tailored to produce desired-state variables and microstructures. Tool designs can be refined to minimize die stress and attain maximum tool life. Process changes can be developed to minimize forming load requirements. The impact of part position on flash, fill and defects can also be evaluated. Various stock sizes can be assessed to optimize material use without affecting the production of a good part. These are a few of many realistic applications.
The following case studies illustrate how DOE has been applied to forging processes. The first case involved a basic forming analysis similar to those commonly run by designers or engineers. The second case modeled a forming operation and subsequent die stress analysis. An optimization case study is not covered here, but examples have appeared in literature for a number of years.
A simple DOE case study will illustrate the application of DOE to metal forming. Numerous examples of using simulation to study drawing processes have been reported. Objectives can include maximum reduction, minimum pull force or maximum die life. The most common failure mode is a tensile fracture, with the Cockroft and Latham damage model correlating well with necking-type tensile failure. In this case, a workpiece is drawn through two stationary dies with a fixed initial workpiece and final diameter.
In this DOE, two variables are studied. The first is the diameter of the first-draw die. A wide range was established to illustrate known failures. A second variable is the included angle of the dies for both reductions. These included die angles that were coupled. In other words, the die angle of the first-draw die and second-draw die remains identical. The half angles ranged from 4 degrees to 15 degrees. A full factorial DOE with nine samples of each variable was simulated.
After the simulations were run, the DOE post-processor was used to study the response to the various changes. To capture areas with potential chevron cracks and necking, a damage value of 2.0 was set as a constraint. In these cases, the DOE sample was deemed to have failed as is shown in the surface-response plots in the following figures.
One output from a DOE study is a Tornado Chart, a bar chart that shows which of the DOE variables best correlates to changes in the objective. In this case, the first-draw die reduction (longer bar length) was dominant. Additionally, the Tornado Chart shows a direct or inverse relationship to the objective. For example, a larger billet length of any diameter results in an increase in weight (direct relationship). Forging at a higher temperature, which requires a lower forging load (or fewer hammer blows), is an inverse relationship.
DOE is very powerful in that a wide range of design space or process controls can be studied. Subsequently, a refined (zoomed-in) region can be further studied at higher resolution or using optimization. When an optimization is used in the same space as a DOE study, it uses the surface response of the objective to start the search for the optimum solution. When an optimization study to minimize damage was run, an interesting observation occurred. While the numerically best solution was found, it was very close to increasing damage. This would have required very tight process control to avoid the defects from the undesirable region. While the numerical optimum was technically correct, a slightly less optimum region would result in a more robust process as the inevitable variations come into play.
The focus of this first article was to summarize the terminology and tools related to design of experiments and optimization. This provides a better understanding for how the methods compare, contrast and even complement each other. The next article will review the application of these tools to forging, die stress analysis and material modeling.