Modeling and Simulation, Part II: Overview
Holistic Strategies for Designing Multistage Materials Processes
W.G. Frazier and E.A. Medina


TABLE OF CONTENTS 


New process design and control methods are needed to significantly improve productivity and reduce the costs of multistage materials processes such as hotmetal forging. Current practices for accomplishing basic design tasks, such as selecting the number of forming steps and specifying the processing conditions for each operation, produce feasible solutions that are often far from optimal. Substantial improvements in effectiveness and efficiency can be realized through holistic approaches that optimize the whole system performance instead of just individual subsystems, such as workpiece material behavior, material flow in dies, and equipment responses. To address this situation, a framework for achieving holistic design using various types of continuous and discrete dynamical models and optimizationbased design methods is proposed.
Our analytical capabilities for understanding material processes have advanced significantly during the past 20 years, and it is an opportune time to examine future research directions in the area of process design. Innovations in affordable, highperformance computers, reliable numerical algorithms, and accurate process models have enabled a general acceptance of powerful computational tools by industrial process analysts. Today, process analysis tools, such as the finiteelement method, provide greater insight into complex thermal, mechanical, and chemical phenomena associated with metalworking, casting, coating, and other materialprocessing industries. Notwithstanding these benefits, surveys by software companies indicate that these analytical tools are used by less than ten percent of all process designers. Moreover, the standard practice for using these numerical simulations still involves trialanderror determination of process design variables. This has led some experienced forging industry engineers to conclude that extensive usage of finiteelement models has not changed a single design rule. Others have pointed out that the engineering focus should be placed on solving the materialprocessing problem rather than in dealing with model building, numerical issues, and subjective interpretation of simulation results.
Process design is a challenging endeavor involving difficult decisions, such as choosing the overall sequence of manufacturing operations, selecting the number of forming steps, and specifying the processing conditions. These decisions should be driven by the real need to achieve several possibly competing objectives associated with product specification, processing cost, and productivity. Software tools need to be developed that more directly assist in making design decisions, since designers generally have limited time for performing and interpreting detailed analyses.
Perhaps the most effective and direct means for dealing with modeling, optimization, and robustness issues is through a framework based upon dynamical systems principles^{1} and trajectory optimization.^{2} In this framework, a system is viewed as an interconnection of various interacting components, each being described by a set of continuous or discretestate equations (relations describing the time and spatial evolution of internal variables) and inputoutput relationships that when viewed as a whole provides a complete system description. The system is designed via the application of formal methods of constrained optimization, which require the designer to translate the design problem into mathematical expressions corresponding to constraints (needs) and objectives (wants). The choice optimization algorithm to obtain a solution is determined in part by the type and fidelity of the models as well as the objective of the particular design task.
A second issue is the choice of model order. Experience shows that the appropriate model order for a particular process design is highly dependent on the particular task at hand, whether it is developing a cost estimate for bidding, preliminary technical design, or final detailed design; time available for the design effort; or available computing resources. In general, model order should increase as the design progresses. Although this is a wellaccepted axiom, what is less understood is that using too high an order at early design stages can be counterproductive as the increased complexity of highorder models, more often than not, introduces more design variables and more inferior local optima in the design space. Some of these optima are often associated with the numerical nature of the model and not the true problem. These local optima create havoc with the fast local search optimization algorithms that are suitable for use with highorder models, forcing the designer to use global optimization methods that require many simulations of the computationally expensive highorder models. If sufficient time and computing resources are available, this may be acceptable, but in most cases they are not. Of course, the disadvantage of using loworder models is that the resulting optimum, while global, is not likely to be the global optimum of the highorder model case; but it is hoped that it is closer to the highorder global optimum than would have been obtained using the highorder models from the outset given finite computing resources using local or globalsearch methods. Fortunately, several researchers have become aware of these problems and are developing systematic methods for simultaneously managing model order and the search for the optimum in a single framework.^{3,4}
The optimal design of a multistage processing system consists of formulating appropriate models of materials, process, and equipment behavior together with carefully chosen design objectives and constraints. In order to be consistent, the selection of models, objectives, and constraints must be considered simultaneously, not independently.

Figure 1. A schematic of a hotforging process. 

Figure 2. A sequence of dynamical processes. 
In this article, a manufacturing enterprise (i.e., global system) consists of using a sequence of thermomechanical processing steps to produce a component. Each unit process, such as forging or heat treatment, can be decomposed into subsubsystems (i.e., workpiece material, tooling, equipment, control system, etc.) that can be decomposed further.
A typical systems representation for a metalforging system is illustrated in Figure 1. Typical models for forging processes consist primarily of relations among workpiece material, tooling, and processing equipment. Two levels of coupling are present—models within models at the same time (instant) and models coupled sequentially through the workpiece (material) itself. In most cases, processes are not coupled directly, but are coupled only through the workpiece and its influence on the processes.
Manufacturing processes can be mathematically modeled as nonlinear dynamical systems using a statevariable formulation (i.e., a system of coupled, firstorder nonlinear ordinary or partial differential equations). Symbolically, this is often written as
(t) = f_{c}(x(t),u(t))
in the continuoustime case or as
x(k + 1) = f_{d}(x(k),u(k))
in discretetime, where x and u are vectors of the system's state and control variables, respectively. The functions f_{c} and f_{d} define the relationships among the current state of the system, the current control variables, and the change of the state. The important point concerning dynamical models is that they are valid for a broad range of control signals, unlike algebraic models, which are only valid for a particular class of process controls, such as constant temperature and strain rate. The use of dynamical models provides greater predictive capability over algebraic models and provides more degrees of freedom in the time domain for optimal process design. Figure 2 illustrates a sequence of dynamical processes; each process is coupled to the next via the workpiece states.


a 
b 
Figure 3. A (a) strainrate change test and (b) constant strainrate test showing that different dynamical processing conditions can yield similar material microstructures. 
Dynamical models of material behavior are especially important because of the timevarying behavior of quantities such as microstructure, flow stress, and defect formation. Figure 3 illustrates how the microstructure and flow stress of a TiAl alloy responds to different transient processing conditions. In this example, the material responses under constant strainrate conditions versus increasing step changes in strain rate are compared. Because several different processing histories (trajectories) can lead to nearly the same end result, it is important to be able to find the best trajectory to realize the desired objectives. Consequently, models and design techniques for controlling microstructure during thermomechanical processing have been developed to address critical issues, such as stability, transient and steadystate response, and robustness of material trajectories, which are defined here as the time evolution of material attributes (phase, grain size, dislocation density, etc.).
Dynamical models of equipment systems are also useful, if not essential, for determining the desired adjustable parameter settings for coincident tracking of the equipment response with optimized commands. In general, furnaces, vacuum chambers, and forming machinery possess a range of timevarying performance capabilities that can be tuned to the needs of a given process. An example using a highfidelity dynamical model of a 907 tonne forge press is described in Reference 5.
For processes modeled by dynamical relationships, the optimization of timevarying rather than stationary quantities is required. This leads to the notion of trajectory optimization (i.e., the calculation of the best path for some quantity to follow over time). A mathematical description of this type of problem in the continuoustime case is to minimize
subject to the constraints
c_{i}(x(t_{i}),u(t_{i})) 0, i = 1 to N_{c}
d_{i}(x(t),u(t)) 0, t [t_{l},t_{u}]_{i}, i = 1 to N_{d} 
and the material and process models

^{w}

= f 
^{w} _{j} 
(x 
^{p} _{j} 
,x^{w},u_{j}), j = 1 to N_{f}


^{p} ^{j} 
= f 
^{p} _{j} 
(x 
^{p} _{j} 
,x^{w},u_{j}), j = 1 to N_{f} 
where

x^{wf} 



x^{p}_{1} 

u_{1} 
x= 

and = 


x^{p}_{Nf} 

u_{Nf} 

Figure 4. An illustration of a cogging operation. 
A block diagram illustrating the model for the case of three sequential processes is shown in Figure 2, in which J is the overall design objective (a targeted goal); h_{j} are objective terms defined at a point in time and often used for final workpiece objectives, such as microstructure; g_{i} is the integrand of objective terms defined on a time interval (e.g., energy required, processing cost, and processing time); c_{i} are the constraints defined at a point in time (often final value constraints, such as strain, cross head travel limit, etc.); d_{i} are constraints defined over a time interval, such as temperature limits, strainrate limits, load limits of equipment, and maximum temperature and pressure; x^{w} is the vector of workpiece states (e.g., dislocation density, average grain size, phase fractions, temperature distribution, etc.); x^{p}_{j} are vectors of process states (e.g., ram speed, pressure, and ambient temperature.); u_{j} are vectors of process controls, desired ram speed, desired pressure, and desired temperature; and N_{f}, N_{c}, N_{d}, N_{g}, and N_{h} are the number of discrete processes, pointwise constraints, interval constraints, pointwise objective terms, and interval objective terms, respectively. The subscript j in the quantities x^{p}_{j} and u_{j} refers to a particular thermomechanical process, such as forging, extrusion, or heat treatment.

Figure 5. A plot of the maximum and minimum billet temperatures throughout the process. The initial heatup is not shown. 
It is important to recognize that optimization is essentially just one method for performing a design, and whether the resulting design is the best one depends entirely on the criteria specified by the designer. As illustrated, optimization techniques require the specification of two types of criteria: objectives (wants) and constraints (needs). To achieve the desired goals, the designer must specify all relevant criteria and must carefully determine the criteria that are objectives and the criteria that are constraints. As an example, it may be desired to minimize the production costs (objective), while maintaining specified productquality standards (constraint). On the other hand, the opposite scenario may be desired (i.e., maximize the product quality [objective] while not exceeding a specified cost [constraint]). Effective optimization strategies consider the entire processingdesign problem, not just some parts, thereby avoiding the overoptimization of parts of the process at the expense of the whole manufacturing enterprise. Although not described in detail, the same formulation and comments apply to discretetype models and optimization methods.
This example focuses on how to determine the optimal heating times for controlling the temperature distribution in a large billet of a titanium alloy during multistage cogging operations. The cogging process is primarily used as a method for modifying the dimensions of the cross section of a workpiece without significantly affecting the shape. An illustration of the cogging process is given in Figure 4. To avoid the formation of microstructural defects for this alloy, the temperature of the billet must be maintained within a narrow range (830–950°C) during plastic deformation. Because of rapid cooling while the workpiece is out of the furnace, the entire operation cannot be completed within a single cogging step. Therefore, the workpiece is repeatedly returned to the furnace to be reheated. The current industrial process requires six cycles (12 steps) of heating and deforming (for a total of 12.5 hours) to be completed and, according to simulation studies, does not maintain the temperature in the desired range. In this application, a continuoustime trajectory optimization algorithm was used to meet the temperature constraints and minimize the total production time. Details of an optimization procedure similar to the one used to solve this problem are discussed elsewhere.^{6} This particular algorithm uses analyticalgradient (Frechet derivative) information and is primarily suited for seeking local optimum (final design).

Figure 6. A video clip (~400 kb) of the temperature distribution at the end of the initial heat up. 
Application of the design algorithm to a nonlinear, finitedifference, heattransfer model of the process yielded furnace times that maintained the temperature within the desired constraints throughout the entire workpiece and yielded a total processing time of approximately 7.5 hours—a significant reduction. The characteristics of the solution include a significant amount of differential heating with a rather steep gradient from the surface to the interior. It is also interesting to note that, except for the initial heatup, the furnace times are monotonically increasing. Figure 5 shows the maximum and minimum temperatures within the billet as a function of time. The blue vertical bands represent the periods when the workpiece is out of the furnace. The middle third of each band corresponds to the period when the billet is being deformed. The figure illustrates the transient nature of the process in that the maximum temperature continues to fall even after the billet reenters the furnace because of the internal thermal gradients. A similar phenomenon occurs with the minimum temperature. Figure 6 comprises a movie (~400 kb) illustrating the temperature distribution during the first five steps (heatcoolheatcoolheat).
In this example, a generalized hillclimbing global optimization algorithm^{7} was applied to the design of a sequence of manufacturing processes using loworder process and material models. The models are inherently nonsmooth functions of many of the design parameters. The design objective is to minimize the cost of production given a fixed lot size of turbineengine disk components. A schematic description of the design system is given in Figure 7.
A diagram showing the processes and the possible physical design variables is given in Figure 8. Application of the optimization algorithm yielded an initial cost of $5,305 and a final cost of $1,191 for the first run and initial and final costs of $11,614 and $2,245, respectively, for the second run. The resulting process design is intended to determine the manufacturing sequence and process parameters to the first order. For example, in the forging processes, the volume of material required and the nominal shape of the forging dies are determined, but the exact shape of the dies is not. Detailed design, which should only impact cost at the second order, is still required to complete the process design.


Figure 7. The organizational structure of the design system. 
Figure 8. The process routes and design parameters. 
The importance of considering dynamical effects in the design of thermomechanical materials processes cannot be overstated since these processes are inherently transient (timevarying) in nature. This design philosophy has been used very successfully by aerospace engineers for more than two decades to design flight trajectories and control parameters of multistage missiles, launch vehicles, and satellites. The design has also shown that the mathematical structure of the problem is precisely the same as multistage materials processes. Certainly the complexities of modeling the behavior of materials during processing, combined with an emphasis on the equilibrium behavior of material systems, have been a bane to the widespread use of dynamical systems engineering principles in materials and manufacturing processes design. However, the evidence of more materials and manufacturing researchers addressing these issues indicates that things are changing.
References
1. R.E. Skelton, Dynamic Systems Control (New York: John Wiley and Sons, 1988).
2. R.E. Stengel, Optimal Control and Estimation (New York: Dover, 1994).
3. N.M. Alexandrov et al., "A Trust Region Framework for Managing Use of Approximation Models in Optimization" (1998), pp. 16–23.
4. N.M. Alexandrov, "On Managing the Use of Surrogates in General Nonlinear Optimization and MDO," Proceedings of the 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, AIAA paper AIAA984798 (1998).
5. W.G. Frazier et. al., "Model and Simulation of Metal Forming Equipment." J. of Mat. Engineering and Performance, 6, pp. 153–160.
6. W.G. Frazier et al., "Application of Control Theory Principles to Optimisation of Grain Size during Hot Extrusion," Materials Science and Technology, 14 (1) (1998), pp. 25–31.
7. S. Jacobson (private communication).
W.G. Frazier is with the Air Force Research Laboratory, WrightPatterson Air Force Base. E.A. Medina is with Austral Engineering and Software.
For more information, contact W.G. Frazier, Air Force Research Laboratory, WPAFB, Ohio; email william.frazier@afrl.af.mil.
Copyright held by The Minerals, Metals & Materials Society, 1999
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