In this second of two articles about design of experiments (DOE) and optimization, we illustrate how they apply to forging design. The first article defined the important concepts associated with these methodologies. This article addresses the implementation of DOE and optimization in simulation software and provides case studies from practical forging applications. The examples highlight how such tools can be used to reduce development time and obtain robust, optimized processes.
Before discussing case studies, it makes sense to briefly review DOE, optimization and their integration into process simulation tools like DEFORM™. DOE is a systematic method to investigate design parameters or process variations. Structured changes are made to one or more input variables of a system. The effects that these changes have on specific output variables are then assessed. Optimization is an iterative method used to automatically determine which input provides the best result within the design space. A control program interrogates a system’s response to specific inputs. It updates inputs in subsequent simulations until it converges on the “optimum” result without defects.
A multiple-operation (MO) simulation capability permits the study of DOE and optimization variables from different operations in a manufacturing sequence. For example, a DOE study may evaluate the effect of furnace heating time, blocker geometry and finisher press speed. Studies are created by defining a nominal MO simulation, creating a DOE study and defining variables, ranges, constraints, outputs and a sampling pattern. A batch process sets up and runs all of the necessary simulations. This could involve just a few or hundreds of simulations.
Data mining and display are critical since huge amounts of data can be produced in a DOE or optimization study. The batch simulation process produces a viewable database for each simulation. Additionally, DOE outputs are compiled in the form of surface response and sensitivity plots for the entire design space. This “strategic” post-processing summarizes the influence of each input variable. In the past, post-processing 100 simulations would have taken days. Now it can be done in minutes.
Various DOE and optimization case studies have been published in recent years. Two examples highlighting how DOE has been applied to forging processes are described below. The first case involves a common forging analysis, similar to those often run by die designers or process engineers. The second models a forming operation with the addition of a subsequent die stress analysis.
Industrial fittings are often forged in “platters,” single forgings that integrate multiple fitting bodies nested together in close proximity. Parts are nested to maximize production efficiency and minimize material usage. Small amounts of flash connect the individual fittings. In this case study, a 316 stainless steel platter included four fittings nested along the long axis of the die. It was known that, despite the orthogonal alignment of components, positioning a raw bar parallel to this axis would result in too much flash in some regions and underfill in others. Therefore, the ideal bar position would likely be at an angle.
A DOE was performed with the goals of finding the optimum bar orientation and diameter for producing the forging on a small press. This first operation in the process sequence was a simple upset between flat dies. It created a preform that was then passed to a finish-forging operation. The objective of the DOE was to minimize the finisher forming load, while avoiding defects such as folds and nonfill. The first DOE variable assessed bar diameters between 1.2 inches and 1.4 inches. The second DOE variable considered rotation angles ranging from 0 degrees to 16 degrees. Illustrations of these variables are shown in Figure 1. A Latin Hypercube sampling plan with an 80-simulation limit was applied to this study.
The 2D response surface shown in Figure 2 was generated from the DOE results. Its color contour revealed the forming load as a function of bar diameter and bar rotation. Had this been the only consideration, the user would have selected the sample that produced the lowest load. Yet information regarding the presence of defects was considered as a design constraint. Each sample point on the response surface indicated whether that sample failed due to nonfill, folds or both. Therefore, the ideal design was the sample that produced the lowest load and passed all constraint checks. The optimum design was a bar diameter of 1.3 inches and a bar rotation angle of approximately 6 degrees.
Further analysis of the DOE results provided even more insight into the process. Unlike optimization, which only identifies a single local optimum, DOE reveals the output response across a broad range of variables. This allowed for the identification of not only the optimum but also “safe” variable ranges. In the response plot shown, the highlighted region in the bottom of the yellow band avoided defects while achieving a minimal load. It was estimated that a process targeting the optimum conditions would allow for approximately 2.7 degrees of bar rotation and still remain nearly optimal.
Die Stress Analysis
The second case studied a steel bevel gear that was forged in two operations, as shown in Figure 3. The cylindrical stock was forged into an axisymmetric preform in the first operation. This was followed by a finish operation, which forged the teeth into the gear. Past experience had shown that the extreme forming loads needed to fill the teeth also induced very high stresses in the tooling. DOE studies were run to determine the optimum preform shape. The optimum shape needed to ensure sufficient die fill, avoid defects and minimize stress on the finish-forging die.
An MO simulation sequence was created for the process. The preform operation was simulated as a 2D axisymmetric model. A 2D-to-3D operation automatically converted the 2D model to a 3D model during the simulation runtime. The axisymmetric cross section was revolved to an 18-degree 3D section for the finish operation using rotational symmetry of a half gear tooth. The DOE input variables under consideration were die geometry alterations from the nominal design. The top die angle and bottom pin depth of the preform tooling were variables 1 and 2, as shown in Figure 4.
Statistical and simulation tools were used to study the DOE results. This began with sensitivity analyses to characterize the relative influence of the DOE variables. They showed that changes to the top die angle were more significant to finish tool stresses than were changes to the bottom pin depth. They indicated that increased angles allowed more outward material flow and lowered effective stress in the subsequent finish operation.
A range of defects and die fill were observed during incremental refinements of the DOE study. For example, increasing the top die angle above the upper limit of the final study produced an oversized preform. This shape flashed prematurely in the finisher, resulting in excessive die wear and underfill. If this preform design had been selected, it would have required an excessively large workpiece in order to fill the finish dies.
2D and 3D response surfaces quickly identified the preform design that produced the lowest stress on the tools during the final forging operation. The response plot in Figure 5 displays the finisher die effective stress as a function of the preform die geometry variations. Samples resulting in the lowest stresses are near the top center. This area is associated with designs that have a large die angle and a short pin geometry. Yet, as in the fitting example, selecting the best option for production was not as simple as choosing the sample with the optimum value.
Careful review of the DOE results revealed the potential for small folds in the red-encircled area in Figure 5. The two samples with the tallest preform die pins, but desirably low stresses, both exhibited the defect. The adjacent sample was the optimal design strictly based on minimizing die stress alone. It would be unwise to enter production using target parameters that are on the threshold of producing a defect. Therefore, the ideal option was to choose process conditions associated with the sample that produced a slightly higher (but near optimum) stress and was significantly more robust.
Process simulation is an excellent platform for DOE and optimization studies. An individual simulation is a mature and accurate tool for product and process design. Therefore, it is understandable that a group of simulations can replace shop trials as a planned experiment. When dozens or hundreds of trials are contemplated, the use of simulation becomes compelling. This is particularly true of multi-variable studies on a process chain, like the examples described.
For the foreseeable future, such studies will be managed by designers who can assess the trade-offs between a numerical optimum and a robust process. That being said, advancements in simulation will provide valuable and timely insights into design alternatives, guiding designers to increasingly more robust product and process designs.
Co-author John Walters, a frequent contributor to FORGE, is vice president of Scientific Forming Technologies Corporation, Columbus, Ohio. He may be reached at 614-451-8330 or email@example.com. Co-author Jim Miller is principal research scientist at Scientific Forming Technologies Corporation. He may be reached at 614-451-8330 or firstname.lastname@example.org.